Research Directions in Distributed Parameter Systems (Frontiers in Applied Mathematics)

where the mean-value -My(^) is defined by

and |y| is the measure of the set Y. Hence, for i, j = (1,... , n),

Obviously, due to the oscillatory character of coefficients a,ij, this convergence is not strong. So, in the left-hand side integral in (3.9), one has products of sequences only converging weakly. In general, the limit of such products is not the product of limits so that in our model example, one clearly has

The only information we have derived above is that

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As before, we have to give an expression relating £° to UQ. As a matter of fact, it is now classical in periodic homogenization theory that hypotheses made on the matrix A imply the existence of a matrix AO with constant coefficients a f,-i (*> j' = 1> • • • > n), such that £° = ,4oVu° so that u° is the unique solution of

This implies that the whole sequence u£ converges to u°, so (3.12) is the homogenized problem, AO is the homogenized (or effective) matrix, a^ are the homogenized coefficients and u° the homogenized solution. The following result holds true (see for instance, Bensoussan, Lions and Papanicolaou [6] and Cioranescu and Donate [14]). Theorem 3.1. The limit function u° 6 HQ($I) from convergence (3.4) is the unique solution of the homogenized problem (3.12). The constant matrix A0 = (a?fc)i
where %> (j = 1,... , n) is the solution of the system

One also has

where x* (i = 1,... , n) is the solution of the system (adjoint to (3.14)j

Formula (3.13) (or (3.15)) shows that the homogenized coefficients are not obtained as a simple average of the original coefficients of the composite. They

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contain the usual averages .My(ay), the mixture law, to which one has to add correcting terms expressed by means of the "corrector" functions x* • To summarize, the homogenization process of problem (3.2) consists of replacing the initial partial differential equation with rapidly oscillating coefficients that describe the composite material by a partial differential equation with constant coefficients. To compute these coefficients (characterizing the fictitious homogenized material), one has to solve partial differential equations with no rapid oscillating coefficients on the fixed reference cell Y. Here, one can immediately see the interest of homogenization methods for the numerical computation. A direct numerical computation of the solution of problem (3.2) needs a very fine discretization mesh if one has a large number of heterogeneities. This means a long computational time and many sources of errors due to the high oscillatory character of the coefficients. As mentioned above, none of these difficulties appear when computing the homogenized solution since to do it, we have to solve problems without rapid oscillations. Moreover, under some additional assumptions, an error estimate is obtained, showing that for e small enough, the homogenized solution u° is a good approximation of the physical one ue. To prove this estimate, one first shows that UE can be written as an asymptotic expansion with respect to e. One has the following theorem. Theorem 3.2. The solution u£ of (3.2) admits the following asymptotic expansion:

where xk is defined by (3.14) and Oke by

Moreover, if f € C°°(Sl), dtl is of class C°° and %k, &kl € W1,00(y) VK,t = 1,... ,n, then there exists a constant C independent of e, such that

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Figure 3.4. Domain with two micro-scales e and 6. Theorem 3.3 (corrector result). Suppose that VyXi e (Lr(Y))n, i = 1,... ,n, and Vu° e (L8(fl))n with 1 < r, s < oo and such that l/r + 1/s = 1/2. Then

In Section 3.2, we will briefly show how one can prove these three main results by classical homogenization methods for periodic materials, namely the multiplescale method, Tartar's oscillating test functions method and the two-scale convergence. In Section 3.3, we present a new method, the periodic unfolding method, which simplifies the proofs of the theorems significantly and completely justifies the asymptotic expansion (3.17). The main advantage of this method is that it is well adapted to the multi-scale case. We describe this case in Section 3.6. Before this, however, we introduce another kind of material with a highly oscillating character, the periodically perforated domains. This is the subject of Section 3.4. As for the model problem (3.2), under appropriate hypotheses on the boundary of perforated domains, one has general homogenization results which are similar to Theorems 3.1-3.3. A special case of perforated materials are truss-like structures (in which one has very "big holes" and little material). Again, the homogenization methods are well adapted to describing the overall behavior of these structures. In Section 3.5, we recall some of the main results in this field, in particular those related to grids or to truss-like beams. As already mentioned, we will apply the periodic unfolding method to the case where one has more than one micro-scale. Such a situation is depicted in Figure 3.4, where two micro-scales appear, e and 6. The domain fi is covered by an el^-paving; in turn, a zone Yi inside the cell Y is covered by a SZ-psving. In the general case, one can have a chain of micro-scales, and any kind of situation is allowed such as heterogeneities, holes or trusses on different scales. A general homogenization result is given in Theorem 3.17.

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63

The Classical Methods in Homogenization

The aim of this section is to present the three, by now classical, methods that are used to prove Theorems 3.1, 3.2 and 3.3: the multiple scale method, Tartar's oscillating test functions method and the two-scale convergence method. All of these methods, as well as the unfolding periodic method of Section 3.3, are based on the fact that, as noticed above, we are in the presence of two different scales, x and x/e.

3.2.1

The Multiple-Scale Method

Based on the the two-scale aspect of problem (3.2), we look for its solution u£ as a formal asymptotic expansion of the form

where the function HJ = Uj(x, y) for j = 1,2,... is such that

This means that the two scales of the problem are now separated. Obviously, for any function w(x) =
where y = x/e. Using this rule and expansion (3.18) in (3.2), and equating the power-like terms of s, we obtain an infinite chain of system of equations defining the functions Uj, j = 1,2,.... To solve these systems, we apply a variant of the Lax-Milgram Theorem for the case of spaces of periodic functions. To do so, let us introduce the bilinear form

and the functional spaces

The last space is the quotient of Hper(Y) by the following relation of equivalence: ipi ~ ^2 ^=^ ipi — V'a = constant in y. Then, by the Lax-Milgram Theorem, one has the following lemma. Lemma 3.4. (i) Let L be a continuous linear form on V. problem has a unique solution u e V:

Then the following

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(ii) Let C be a continuous linear form on the space V. Then the following problem has a unique solution U € V:

The first system of the chain, defining UQ, is

Then, Lemma 3.4 implies immediately that UQ is independent of y. so that

Consequently, the second system of the chain, defining u\ (which exists, due, again to Lemma 3.4), can be written in the form

Since ^L is independent of y, it is straightforward that u\ is of the following oarticular form:

where %* for i = 1,... , n is defined by (3.14). Using information from (3.22) and (3.24) leads to the next system

where

Notice that u% is a solution of (3.20) with JC(tp) = fy F(f,uo,ui) dy, which is a linear and continuous form on Vy- To apply Lemma 3.4, this form must have the

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same value for two elements tf^i, fa, in the same class of equivalence in V (i.e., ifci — ^2 = constant in y ). To have £(V>i) — £(^2), one needs that

or, equivalently,

This is precisely the homogenized equation from (3.12) if one takes into account (3.22) and (3.24). Continuing the same procedure as above, one obtains an expression of U2 by means of UQ, namely the last term in expansion (3.17). The proof of the error estimate in Theorem 3.2 makes use of this expansion and of elliptic estimates for systems (3.12) and (3.14).

3.2.2

Tartar's Oscillating Test Functions Method

This method, introduced by Tartar [17, 18], is based on the explicit construction of suitable oscillating test functions allowing us to pass to the limit in problem (3.2). The spirit of the method is the use of the adjoint of system (3.2) to eliminate the terms containing products of only weakly converging functions and where the passage to the limit is not possible. To be more precise, let (f> € 'D(fJ) and let z£ be a sequence of functions such that z^(p €E .Hg(fi) so that they can be used in (3.9) to get

Suppose furthermore that ze satisfies the adjoint problem

whence (after multiplying by (pu€ and integrating by parts)

Adding now (3.26) and (3.27) yields

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Set (compare with (3.5))

Recalling notation (3.5), (3.28) can be written in the form

By Tartar's method, we construct a set of test functions such that

so that one can pass to the limit in (3.30) to get

Integrating by parts and recalling equation (3.8), we finally obtain

that is to say,

As a matter of fact, we will obtain precise values of z° and n° in order to get a formula expressing E° in terms of u°. Another feature of Tartar's method is that the family of test functions is constructed on the reference cell (i.e., depending on y) and then passes to fl by the usual change of variables x = ey. These functions are built up by using the set of functions xi (i = 1,... , n) defined by (3.16). We set

Consequently, the family of vector functions n = (n1Ei,... ,nEin) defined by (3.29) satisfies

Observe that

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Prom definition (3.33), one checks that

Moreover, according to convergence (3.10),

Therefore, for any i = 1,... , n, (3.32) reads

which is precisely the relationship we were looking for and, when used in equation (3.8), yields (3.12). Afterwards, one shows that the homogenized matrix AO is elliptic. This implies the uniqueness of u° and ends the proof of Theorem 3.1. D To end this subsection, let us mention that one can prove the convergence of energy related to system (3.12) to that corresponding to the homogenized equation. One actually has

This property is essential to prove Theorem 3.3 (for the proof of this theorem, which is rather technical, we refer the reader to [14]).

3.2.3

The Two-Scale Convergence

This method is based on the notion of two-scale convergence introduced by Nguetseng [16] and developed by Allaire [1]. Definition 3.5. Let {vs} be a sequence of functions in L 2 (fl). One says that {v£} two-scale converges to VQ = vo(x,y) with VQ € L2(fi x Y) if for any function ip = V>(#, y) sufficiently smooth, one has

The main result concerning this convergence is the following. Theorem 3.6. (i) Let ve be a bounded sequence in L2(fl). Then, there exists a subsequence ve and VQ 6 L 2 (fi x Y) such that {ve } two-scale converges to VQ. (ii) If v€ 6 H1 (ft) is such thatv£ —^ VQ weakly in Hl(fl), then ve two-scale converges to VQ, and there exist a subsequence e' and v\ = v\(x,y) in L2(fi; V) (where V is defined by (3.19)) such that

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The proof of Theorem 3.1 by using the two-scale convergence is essentially based on this last theorem. Let us just point out that a priori estimate (3.3) implies that Theorem 3.6 applies to the sequence of functions ue, solutions of (3.2). Then, by using in (3.9) test functions of the form tpo• +e(tl) and tpi £ T>($l;C°°(Y)), F-periodic, one can pass to the limit and obtain the homogenized problem. It also should be mentioned that the two-scale convergence method provides a simpler proof of Theorem 3.3 than the preceding method. In the next section, we will see that the two-scale convergence is actually equivalent to a weak convergence in the space I/2(H x F).

3.3

The Periodic Unfolding Method

This new approach to periodic homogenization is elementary in nature and applies without difficulty to periodic multi-scale problems. It is based on a dilation technique transforming the domain fi in fi x F, a decomposition of any function in a main part without micro-oscillations and a remainder taking them into account. This decomposition is inspired by the method of finite element approximations. The results from this section are announced in Cioranescu, Damlamian and Griso [11].

3.3.1

The Periodic Unfolding Operator T£

Recalling (3.1), by analogy with the one-dimensional case, we introduce the integer part of x/e. With this notation, for each x & R", one has

Definition 3.7. Let w € L2(fi) be extended by zero outside of fi. The unfolding operator T£ is defined as follows:

Obviously, T£(w) (x,<-\

J = w(x) and

The unfolding operator doubles the dimension of the space and puts all the oscillations into the second variable, separating in this way the two scales of the problem. It is easy to check the following proposition showing that in fact, the two-scale convergence of a sequence of functions is nothing other than the weak convergence of their unfoldings in the dilated space L2 (fi x F).

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Proposition 3.8. Let we be a bounded sequence in L 2 (fi). The following are equivalent: (i) T£(we) weakly converges in L2(£l x Y) to w. (ii) ws two-scale converges to w. As it will be seen later, the use of the weak convergence of unfolded sequences is both simpler and more efficient than that of the two-scale convergence. In particular, it is well suited to cases where one has multi-scales. Let us list the properties of the unfolding operator TE. Proposition 3.9. (i) The following integration formula holds true:

(ii) I f w E L2(£l), one has Te(w) —> w strongly in L 2 (fi x Y). (iii) Let w£ € L2(fl). Then w£ -+ w weakly in L 2 (fi) => T£(w£) —>• w weakly in L 2 (fi x Y), T£(w£) —^ w weakly in L2(fl x Y) => w£ —*• MY(W) weakly in L 2 (Q). Now, we focus our attention on the gradients of unfolded functions. Notice that by the definition of Ts, for any w € H1 (SI),

Proposition 3.10 (T£ and gradients). Let {ws} C H1^) be a bounded sequence in L2(ft) such that Te(w£) – w in L 2 (f2 x Y). If furthermore e||Vtwe||i,p(n) < C, then The function w is Y-periodic, namely w € L 2 (fi; Hper(Y)) with Hper(Y) defined by (3.19).

3.3.2

Macro-Micro Decomposition of Functions: The Scale-Splitting Operators Qe and Ue

Assume that the boundary dfl is bounded and Lipschitz. Then, there exists a continuous extension or>erator P : H^tQ} H-> Hl(Kn] such that

where C is a constant depending only on d£l. We split every 0 € Hl(fl) as <j) = Q£() + 7Je(0), where Q£() is designed to not capture any oscillation of order e, contrary to fi£((j)). This decomposition will play an important role when studying the convergence of bounded sequences w£ in

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L 2 (f2). In fact, (see Theorem 3.11) w£, Qs(we) and Te(we) have the same limit w in H1^), respectively, L 2 (fi; V). So, the micro-oscillations provide no contribution in the limit at this level. The situation is completely different when dealing with Vwe. Indeed, while Vw£, V(Qe(«;e)) and Ts(VQs(we)) have again the same limit Vw in L 2 (Q), Te(Vws) converges in £ 2 (fi x Y) to a limit of the form Vw + r, where the extra term r comes from Te(V(n£w£)).

Definition of QE

We start by extending the domain fi to fi£ as depicted in Figure 3.5. For h € Z™, set £fc = X)™=i ^^j and

i.e., Q£ is the smallest finite union of eY cells containing fi. _^ For tp € L2(£l), one starts by defining Qs(
In the other points, Q£(4>) is the restriction to 0, of the Qi-interpolate of the discrete function Q e (0)(ff^fe), as in the finite element method (FEM). It is then obvious that for every cj> e H1^), one has the estimate ||Q £ (0)||/fi(n) < C\\(/)\\Hi(n)-

Figure 3.5. The domain fie.

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Definition of R.e We set TIE(4>) = (/> - Q£ ()• From the FEM one knows that

Theorem 3.11 (convergence of sequences in ff 1 (fi)). Let ws —*• w weakly in Hl(£l). Then, there exist a subsequence (still denoted s) and a w e L 2 (fi; V) such that

3.3.3

Periodic Unfolding and Homogenization

Let us now prove Theorem 3.1 in the framework introduced above that we call the periodic unfolding method. With this tool, the proof is elementary! To begin with, let us recall convergence (3.4). Then, according to Proposition 3.9 and Theorem 3.11, we have the following convergences:

with u €. L 2 (fi; V). The integration formula (3.35) used in (3.9) gives

where we also used (3.34). Here, we are allowed to pass to the limit because of (3.37) and Proposition 3.9. One gets

Next, taking in (3.9) the test function v£(x) = scf)(x)i(} (f) with <j> e T>(fl), ip € Hper(Y), one has successively v£ —^ 0 weakly in flo(fi), and T£(VxvE} -+ (x)Vip(y) uniformly over £1 x Y. Therefore, at the limit we have

and by density,

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Adding (3.38) and (3.39) shows that the pair (UQ,U) is the unique solution of the problem

This problem is a standard variational one on the space #o(O) x L2($l; V) (with V denned in (3.19)). Observe that from (3.38), one obtains u in terms of VUQ which, carried over to (3.36), yields the standard form of the homogenized equation, namely (3.12). Consequently, u(x,y) = ui(x,y), where u\ is defined in Theorem 3.2. The convergence of energies is also proved easily and implies in particular that the second weak convergence in (3.37) is actually strong:

3.3.4

Corrector Result and Unfolding

Let $ in L 2 (fi x Y) and let Ue be the following averaging operator: \I \ JY

^

L£J

Y

*• E J Y/

2

The operator Ke maps into the space L (£i). It allows us to replace the function x H-» $(x, {f }y)i which is meaningless in general, by a function which always makes sense. Observe that, by definition, Me(Te(
On the other hand, it is also obvious that, for every € L2(£l), Some other properties ofl4s, interesting for the periodic unfolding method, are listed in the next result. Proposition 3.12. Let $ in L 2 (fi x Y). Then,

Moreover, one has the equivalence of the following convergences: and a similar equivalence holds with strong convergence.

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This proposition, together with (3.42), allows us to prove a corrector result, analogous to Theorem 3.3, but without any additional regularity assumption on the corrector functions xi, a = 1,... , n. What is also striking is the fact that now the proof of this result is reduced to only a few lines. Theorem 3.13 (correctors). One has the following strong convergence:

Proof. We have already seen (see (3.41)) that

which by Proposition 3.12 is equivalent to

However, V x uo G L2(fl), so from (3.42) one has K£(Vxuo) —> V X UQ strongly in L 2 (fi), whence the desired result. D

3.4

Homogenization in Perforated Domains

The aim of this section is to apply the periodic unfolding method to perforated materials. Here, the holes play the role of inclusions so, for our model example, the equation in (3.2) is written in the matrix only. We have to add at (3.2) boundary conditions on the boundary of the holes in order to have a well-posed problem from a mathematical point of view. Notice that "hole" does not mean necessarily a physical hole. For example, in electrical problems, an isolating material plays the role of a hole since the equations are stated only in the conducting part. For composite materials, homogenization methods apply to perforated materials with a large number of holes, leading in general to good overall approximations. Let us state the equivalent of problem (3.2) in the case of a perforated domain. Let fl € R™ be a domain with a smooth boundary dfl and, as in Section 3.1, let Y be the reference cell. Let T be an open subset of Y with a smooth boundary dT, such that T CY. Set Y* = Y \ T; this is the part of Y occupied by the material. The set of all eT is denoted T£ and the perforated domain Oe is obtained by removing Ts from fl, The boundary of f2e is composed of two parts. The first part, the "exterior boundary" denoted dextfle can be nonsmooth (see Figure 3.6). It is formed by a part of dfi and a part of the holes that intersect it. The second part, its "interior boundary," is the union of the boundaries of the holes strictly contained in fi. It is denoted by dintQ£.

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Figure 3.6. A nonsmooth perforated domain. Let us consider in £2e the Neumann problem

The existence and uniqueness of the solution u£ are given by the Lax-Milgram Theorem in the space

equipped with the usual gradient norm (equivalent to the usual ff1-norm by the Poincare inequality). Moreover,

where C is a positive constant independent of e. This estimate does not imply any convergence for the sequence u£ since now, contrary to the situation in Section 3.1, the space V£ highly depends on e. This difficulty is specific to the case of perforated domains. It is overcome traditionally by introducing extension operators to the whole of f2 preserving the above a priori estimate. This is essential when applying Tartar's method (cf. [18]). Let us just mention that the multiple-scale method from Section 3.2 is carried out by making the same ansatz (3.18). The idea here is to apply the periodic unfolding method. Let us turn back to Section 3.3 and merely replace in the entire Subsection 3.3.1, Y by y*. Hence, we

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75

have the unfolding operator Te mapping in L 2 (fi x Y*) with the same properties as before. We now make use of operators Qe and RE. The only delicate point is when denning the values of Qep at the nodes of the e-paving for a function (p € Hl(Cie). In formula (3.36), we have taken the average on the whole of Y but observe that any average (i.e., on a part of Y) would lead to the same result. Here we can not apply formula (3.36), since (p is defined on Y* only. To simplify the presentation, let us suppose that the geometry is such that there is a ball B(0, r) such that the following formula makes sense:

(For the general case, without any additional assumption, we refer to [11].) Then, the whole procedure from Section 3.3 works, since by construction (using the Q\interpolate) Qsus is now defined on the fixed domain fl, is bounded, and we can speak about convergence. Therefore, by the periodic unfolding method we immediately get the following homogenization result for problem (3.43). Theorem 3.14. Let u£ € VE be the solution o/(3.43). One has

where u is Y-periodic in its second variable. The pair (u°,u) is the unique solution of the homogenized problem

With a special choice of test functions, one obtains easily the standard form of (3.46),

where

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where %> (j — 1,... , n) is the solution of the system

3.5

Homogenization of Truss-Like Structures

One can apply homogenization techniques to a class of complex perforated media: the truss-like structures. They are periodic materials with big holes and a very small amount of material concentrated along layers (honeycomb structures) or along bars (reinforcedstructures). Another small parameter, besides the period e, characterizes these kinds of structures: the small thickness 6 of the bars in the cell Y (hence, in the physical domain the thickness of the material is e6). Obviously, in the homogenized problem (3.47), the limit function u° depends on 6, via (3.48) and (3.49), since the thickness of the material in the reference cell is of size 6. To give the overall limit problem, we pass to the limit in the homogenized problem (3.47) as d —> 0. It was shown that for truss-like structures, the fact that the material is thin leads to a final limit problem whose coefficients are simple algebraic combinations of the original physical coefficients of the material. The case of grids and truss-like beams is even more complicated since a third small parameter is present, the global thickness e of the grid when considering it a plate, and the cross section e2 of the truss-like beam (when considering it globally a thin beam). We will give here some results concerning, in particular, the grid and truss-like beam from Figures 3.7 and 3.8 when studying the linearized system of elasticity.

Figure 3.7. A grid.

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Figure 3.8. A truss-like beam.

3.5.1

The Two-Dimensional Case

In this subsection, we return to our model example (3.2) written now in a "trusslike" domain. Its asymptotic behavior as 6 —> 0 is obtained by the reticulated structures method from Cioranescu and Saint Jean Paulin [12, 13]. This method was developed for the case of grids (time-dependent or not) in Banks, Cioranescu and Miller [2] (for other applications, see Banks, Cioranescu and Rebnord [3] and Banks, Smith and Wang [5]). Let fi C M2 be a bounded domain and Y =]0,1[ x ]0,1[ be the reference cell. Set

where Tg is the hole in Y and Yg denotes the part occupied by the material. With the notation from Section 3.4, the perforated domain £ls$ is defined by

where Tsg is the set of holes from fi. Notice that by this definition fie<5 is formed by bars of thickness eS. Assume that the holes do not intersect the boundary dfl. This assumption restricts the geometry of the fi. For example, fZ can be a finite union of cells homothetic to Y as in Figure 3.9. In this case, the assumption is true for every e of the form e = 1/2™.

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Figure 3.9. The perforated domain flegThen the model problem (3.43) is

Our aim is to pass to the limit as e —> 0 and 6 —> 0. To simplify the presentation, suppose that the coefficients a,ij axe constants. Let us fix S. Then, by Theorem 3.14 one has the homogenized system

The homogenized coefficients qfj are defined by

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where the function Xg, for i = 1,... ,n, is the solution of the system

We added the index S to the homogenized solution and to the homogenized coefficients because the geometry of the material in the period of reference depends on 6. Indeed, the auxiliary functions Xg are solutions of a system posed in Yg, the coefficients qfj are integrals computed on Yg and containing Xg an^ finally the solution us depends on 8 via qfj and the measure of Yg. We would now like to study the dependence of us on 5 and give its limit behavior as 6 —> 0. We have the following theorem (cf. [12]). Theorem 3.15. One has the strong convergence

where u* is the unique solution of the system

The coefficients q*j are simply

If (Aij)ij = ((aij)ij) also has

1

denotes the inverse matrix of the matrix A = (aij)itj, one

The fact that u* is a very good approximation of usS on Qsg is completely justified by the error estimate proved in [13]. Theorem 3.16. /// is sufficiently smooth, the following error estimate holds:

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Figure 3.10. The cross period Ys. with C independent of £ and 8, and where

with x§ and 0$e defined, respectively, by (3.49) and (3.17) written in Yg. Proof. Due to the periodicity, it is obvious that q^ do not depend on the choice of the representative cell, so one can choose the following particular one (see Figure 3.10): Y =]-l/2,+l/2[ x ]-l/2,+l/2[. Denote again by Ys the part occupied by the material and set 5^ = dYg \ dY, the boundary interior to Y. From (3.52), as |Yi| = 6(2 — 6), the following estimate is straightforward:

which, used in (3.51) yields

Using the geometry of YS, we are able to give explicit formulas for q*j. Observe that Yg is composed of a horizontal bar denoted by HS and of a vertical bar Vg, each having thickness 8. Their intersection is the central square denoted Kg. Then, (3.52) can be written in the form

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Prom estimate (3.58), one has immediately that Kg does not contribute to the limit because

To pass to the limit when 5 —> 0 in the two other integral terms in (3.60), the sets HG and Vg are dilated with a ratio l/g in the directions where they are thin: y{ = Vi, 3/2 = V2/8 for Hg, and y" = yi/S, y2" = y2 for Vs. By these dilations, Hg and Vs are transformed into Y. We denote by pH The transform by the first dilation of any function

and a similar one for xl v Consequently

where, due to the periodicity of xgs,

Consequently, from (3.60),

The values of Jy v\ dy and /y/i^ dy are derived from system (3.53) by choosing appropriate test functions. To do so, let p be a smooth function, periodic in Y and independent of y\. Then, using the former decomposition of Yg in (3.60) and the periodicity of (p on Vs, we get

where we use the technique of dilations. By construction one has pv =(p. This, together with (3.62), implies that at the limit

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Here we use Lemma 1.4 from Chapter 2 of [13] to derive

Analogously, with

Putting these values into (3.63) gives formula (3.56). Then, it is easily seen that, due to the ellipticity of the matrix A, the matrix (q*j)ij is elliptic too. As a consequence, we derive the following convergence:

where u* satisfies (3.55). The fact that the convergence is actually strong comes again from the ellipticity of (q*j)ij and from (3.55). D

3.5.2

Asymptotic Behavior of Grids

Let us consider an anisotropic grid, as in Figure 3.7, occupying the domain £l^s. It is contained in The set of holes in £le is denotec

The upper, respectively, lower, face of the grid is denoted Fe+, TeeS ; Fg is the exterior lateral boundary. Notation. Throughout this section, the Einstein convention of summation over repeated indices is adopted. Greek indices a,/3,£,£,T,6,p,i' take values in {1,2} and Latin indices i,j,k,£,m,p,r,s,t take values in {1,2,3}. The displacement Ugs = (ug\, Ugs2, u^3) of the grid is given by the linearized system of elasticity

where Fe = (Fe1, Fe2,Fe2) are the applied body forces and Ge± = (Ge1, Ge3, Ge3) the surface forces. There are no applied surface forces on the boundary of the holes dTeg and the grid is clamped on its lateral exterior boundary Fe0.

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We take successively e —> 0 and E —> 0 to recover a system of equations characterizing a thin homogenized grid. When doing so, we pass from a threedimensional problem to a two-dimensional one, written on the mid-surface w. Of course, this system depends on 8. The passage to the limit for e —> 0 is done by a typical domain reduction technique. It consists of transforming fte/gs, via dilations in the directions where one has a thickness of order e, into a fixed domain fi^. Then, a priori estimates independent of e are obtained, and the passage to the limit in the transformed system is standard. The second step is the homogenization and can be performed by any of the methods from Section 3.2 or by the periodic unfolding method (whose application for elasticity problems is in progress). For the sake of simplicity, let us consider here only the case where the elasticity constants aijkh are Lame coefficients, i.e., aijkh = yij^kh + fJu(5ik$jh + sihsjk)The results below (which show the complexity of the problem) are taken from [13]. For other applications, in particular to time-dependent grids, we refer the reader to [2]. The displacement of the thin grid, after having let e —> 0, E —> 0, is given by the limit function u*s of the form

where Ug satisfies the fourth-order homogenized system

and Yg is the part occupied by the material in the cell Y (see Figure 3.11). The limit function Wg = (W8/*1,Wg/*2) is the solution of the second-order homogenized system

In (3.64) and (3.65), F denotes some expressions in terms of the forces / and g, and 1aS/B0p an
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Figure 3.11. The period Ys. with

where E denotes the Young's modulus of the material and is given by

Furthermore, where U* is the solution of the limit equation

What is striking in this result is again the simplicity of the limit problem. As said above, this is the main feature of truss-like structures. Let us also mention another interesting property of the grid under consideration. We do not have a convergence result in [H1/0]2 for Wg*. This is due to the fact that the limit tensor obtained for q^B0p is no longer elliptic. Indeed, one has

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Figure 3.12. Period in a simple grid. with

The loss of ellipticity is related to the geometry of the grid from Figure 3.7, where the period is given in Figure 3.12. If we reinforce it by adding oblique bars (see Figure 3.13), the grid is more rigid. We show that in this case, the corresponding limit tensor q^*BQp is now elliptic.

3.5.3

Asymptotic Behavior of Truss-Like Beams

In this subsection, we give the limit result (obtained by passing to the limit as e —> 0, E —> 0 and S —> 0) for the beam of height L from Figure 3.8. The system to be studied is again (3.64), written in the corresponding domain fi^. One actually shows that we have

where V* is characterized by the following limit systems:

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Figure 3.13. Period in a reinforced grid. and

This type of result (always giving very simple limit equations) can be given for truss-like beams with very complicated design (cf. [13]).

3.6

Multi-Scale Periodic Unfolding Homogenization

As already said in Section 3.3, the periodic unfolding method works very easily in multi-scale homogenization problems. To show that, let us consider our model problem (3.2) in the domain from Figure 3.4. In this case, we have a partition of Y in two nonempty disjoint open subsets Y1 and Y2, such that Y = Y\ U Y%. Let uegEH1/0(o) be the solution of

where

Suppose thata1/ijare Y-periodic in the first variable and Z-periodic in the second, and thata2/ijare Y-periodic. Here, Y and Z are two reference cells, Y associated with the scale s and Z with the scale £<5.

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With standard .ellioticitv hvootheses. there is a subseauence such that Using the unfolding method for scale e, as in Section 3.3, we have the convergences

These convergences do not see the oscillations at the scale E6. In order to capture them, one considers the restrictions to the set fi x YI ,

Obviously, v£g — u^ in.L2(fi;.H'1(Yi)). Now we apply to ves a similar unfolding operation for the variable y, thus adding a new variable z € Z (x being here a mere parameter):

We use the macro-micro decomposition from Section 3.3, Then, again due to results from Section 3.3, one has the convergences

The following result holds. Theorem 3.17. The limit functions u0, u and u are the unique solutions of the variational problem

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Figure 3.14. Multi-scale domain. The proof uses test functions of the form

where U1, U2 are in 25(0), $lf $2 in H*er(Y) and O2 € H1 e r (Z). This theorem gives a rule to homogenize a multi-scale p.roblem. One has to start with the bigger scale and keep the unfolding procedure, adding at each step a new dimension. Therefore, the generalization of Theorem 3.17 for an N-scale problem is straightforward. In the right-hand side, we will have a sum of N integral terms. The first one is an N+1-tuple integral, the second an N-tuple integral, etc. Let us mention that there is no difficulty in mixing domains with or without holes and truss-structures at various micro-scales. Figure 3.14 depicts an example of this kind of general structure. Observe that in the right-hand part of the domain one actually has a 3-scale subdomain. It has a perforated zone at the third-order micro-scale with a period W. To give the limit problem, it is sufficient to replace W in the limit variational problem written with the rule described above, by W*, the part occupied by the material.

3.7

Homogenization of Rubber-Like Materials

In this section we present homogenization results for some variational problems coming from the modeling of nonlinear elastomers introduced by Treloar [19] and recently studied by Banks et al. [4]. These models lead to the introduction of energies characterized by the presence of pointwise constraints on the gradients of the admissible deformations, and the presence of singularities in the energy densities. The modeling of rubber-like elastomers has been developed following approaches based on molecular (polymer chain) statistical thermodynamic formulations and phenomenological (usually continuum) formulations involving stored energy or strain energy functions.

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In his classical book Treloar proposes the following strain-energy function for the statical analysis of rubber:

where W is the elastically stored free energy per unit volume, G is some characteristic of the material and i\ > 0, ^2 > 0, ^3 > 0 are the three principal extension ratios along three mutually perpendicular axes satisfying the condition i\i^i^ = 1 (i.e., the volume is constant). In the case of simple extension (4 = t, ti = 13 = ^~ 1/2 ), (3.69) reads

while in the case of simple shear (^i = i, (.% = 1, (.$ = j), one has

The following model is from Treloar [19] and is based on the qualitative Mooney model for large deviations:

where C1 and Cz are two elastic constants. For the case of simple extension, respectively, simple shear, (7.4) becomes

In more general cases (e.g., isotropic or viscoelastic materials), the constants Ci and C2 may depend on the space variable x. In models (3.70), (3.71) and (3.73), the energy density, as a function of l, is convex, has minima in l = 1 and diverges as i increases or as l vanishes. The study of these models leads to the minimization of functionals for convex integrands with singularities and defined on sets of deformations subject to constraints corresponding to A > 0. In the one-dimensional scalar case, such a functional is of the form

Equations linked to these integrands and their physical interpretation, in the framework of neo-Hookian materials, can be founded in Banks et al. [4].

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In practice, one deals with rubber composites filled with inactive (carbon black and/or silica) elements to gain rigidity. Other fillers, active elements (piezoelectric, magnetic or conductive particles) are used in order to control their properties. When the fillers are periodically distributed and sparse, homogenization phenomena appear. Obviously, in this case also the constraint set may quickly oscillate. Let us mention that in [9], homogenization results have been proved for the scalar case with inactive fillers. The framework is that of unbounded functionals (i.e., taking their values in Ru{+oo}) with different boundary conditions (Dirichlet or Neumann). We will just give a flavor of the abstract results from Carbone et al. [9] (see also [7] and [8]) and show an explicit computation of the homogenized problem for some of the examples mentioned above. Let Y =]0, l[x • • • x]0,1[, and suppose that f = f ( x , z) where

with / measurable in both variables, Y-periodic in the first variable and convex in the second. We consider, for every bounded open set g, /3 € L°°(£l) and h E N, the Dirichlet problem

the boundary condition being of the form uzo + c, where ZQ has some appropriate properties, and c £ R (uzo denotes the linear function with gradient ZQ). Problem (3.75) has solutions if f ( x , •) is lower semicontinuous for a.e. x € Rn, and the following coerciveness condition is satisfied:

for some for some c\ > 0, R > 0 and c2 > 0. As usual in homogenization problems (see [10]), the limit energy (as h —> 0) is related to the integrand fhom given by

We proved in [9] that fhom is convex, lower semicontinuous and satisfies a coerciveness condition similar to (3.76). Moreover, (m°(o,3)} converges to

If, for every h E N, Uh is a minimizer of rn°(Q,/3), then {uh} is compact in L°°(£l) and its converging subsequences converge to solutions of m^0po(Q,/?). Analogous

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results are also proved for the Neumann problem. As mentioned in Section 3.1, the proofs of these results are based on De Giorgi's F-convergence theory (see Carbone and De Arcangelis [10] and Dal Maso [15]). We now turn to the explicit computation of the homogenized integrand corresponding to the energy density (3.71) in the one-dimensional case. For the other models, we refer the reader to [9]. To simplify, we restrict ourselves to the case of simple shear (3.71), for which the energy density in (3.74) is given by

Observe that / satisfies (3.76). Moreover, f ( x , •) is lower semicontinuous for a.e. x € R, and

In this case, the function fhom is given by

where, for every z € R, the minimum exists due to the properties of /. Let us describe the behavior of fhom and compare it with that of /. To begin with, it is clear that fhom(z) < +°o if and only if z > 0, so problem (3.78) has a solution for every z > 0. Now, let 0 be the inverse function of

and c(z) € R be the unique solution of the equation

We point out that on the one hand, 0 is explicitly computable (with a complicated expression). On the other hand, c(z) exists since the function

is strictly increasing and, for every c E R,

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Let u E W££°(}Q, 1[) be the function

By (3.79), one has

for a.e. x €]0,1[. Moreover, it turns out that u is a weak solution of the Euler equation

whence u is actually a solution of (3.78). Consequently, for every z E

Let us now see what is the nature of fhom in the points of the biggest importance z = 0, z = 1 and z —> +00. Since <&(0) = 1, obviously c(l) = 0 and thus Then, taking into account the properties of c and G, it is easily seen that

Comparing this asymptotic behavior with that of the original / defined in (7.9), one can notice that the shape of / is not preserved in the homogenization process. As a matter of fact, one can check that the behavior of the homogenized function fhom in the neighborhood of z = 0+ and z = 1 is the same as that of the leading part of /, namely of 1/2G(x)z 2 . Similarly, for z —> +00, the behavior of fhom is the same as the leading part of /, namely 2G(x)(z — I).2.. In this weak sense, the model is stable with respect to the homogenization process. At this point, let us just notice that even in the above nonlinear case, the homogenization allowed us to give a limit overall behavior.

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Bibliography [1] G. Allaire, "Homogenization and two-scale convergence," SIAM Journal of Mathematical Analysis, 23, pp. 1482-1518, 1992. [2] H. T. Banks, D. Cioranescu and R. Miller, "Asymptotic study of lattice structures with damping," Portugaliae Mathematica, 53(2), pp. 1-19, 1996. [3] H. T. Banks, D. Cioranescu and D. Rebnord, "Homogenization models for 2-D grid structures," Asymptotic Analysis, 17, pp. 28-49, 1995. [4] H. T. Banks, N. J. Lybeck, B. Munoz and L. Yanyo, "Nonlinear Elastomers: Modeling and Estimation," in Proceedings of the Third IEEE Mediterranean Symposium on New Directions in Control and Automation, 1, Limassol, Cyprus, 1995. [5] H. T. Banks, R. C. Smith and Y. Wang, Smart Material Structures: Modeling, Estimation and Control, Masson/John Wiley, Paris/Chichester, 1996. [6] A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and Its Applications, 5, NorthHolland, Amsterdam, 1978. [7] L. Carbone, D. Cioranescu, R. De Arcangelis and A. Gaudiello, "An approach to the homogenization of nonlinear elastomers in the case of the fixed constraints set," Rendiconti dell Accademic della Scienze Fisiche e Matematiche, Serie LXVII Naples, 67, pp. 235-244, 2000. [8] L. Carbone, D. Cioranescu, R. De Arcangelis and A. Gaudiello, "An approach to the homogenization of nonlinear elastomers via the theory of unbounded functionals. The general case," Comptes Rendues Academie des Sciences, Paris, 332(3), pp. 283-289, 2001. [9] L. Carbone, D. Cioranescu, R. De Arcangelis and A. Gaudiello, "Homogenization of unbounded functionals and nonlinear elastomers. The case of fixed constraints set," Asymptotic Analysis, to appear. [10] L. Carbone and R. De Arcangelis, Unbounded Functionals in the Calculus of Variations, Monographs and Surveys in Pure and Applied Mathematics 125, Chapman and Hall/CRC, Boca Raton, FL, 2001. [11] D. Cioranescu, A. Damlamian and G. Griso, "Periodic unfolding and homogenization," Comptes Rendues Academie des Sciences, Paris, 335, pp. 99-104, 2002. [12] D. Cioranescu and J. Saint Jean Paulin, "Reinforced and alveolar structures," Journal de Mathematiques Pures et Appliquees, 65, pp. 403-422, 1986. [13] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Applied Mathematical Sciences 136, Springer-Verlag, New York, 1999.

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[14] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Its Applications 17, Oxford University Press, Oxford, UK, 1999. [15] G. Dal Maso, An Introduction to T-Convergence, Progress in Nonlinear Differential Equations and Their Applications, 8, Birkhauser, Boston, 1993. [16] G. Nguetseng, "A general convergence result for a functional related to the theory of homogenization," SIAM Journal of Mathematical Analysis, 20, pp. 608623, 1989. [17] L. Tartar, "Quelques remarques sur 1'homogeneisations" in Functional Analysis and Numerical Analysis, Proceedings Japan-France Seminar 1976, H. Fujita, ed., Japanese Society for the Promotion of Science, pp. 468-482, 1978. [18] L. Tartar, Cours Peccot au College de France, manuscript, 1977. [19] L. R. G. Treloar, The Physics of Rubber Elasticity, 3rd ed., Clarendon Press, Oxford, UK, 1975.

Chapter 4

Model Reduction for Control Design for Distributed Parameter Systems Ruth F. Curtain* 4.1

Introduction

During the past decades, considerable advances have been made in the numerical simulation of controlled distributed parameter systems (DPS). In the opinion of this author, this numerical sophistication has not been matched by the theoretical understanding of the approximation processes involved. The aim of this chapter is to shed a little light on some of the system theoretic properties which determine the suitability of an approximation scheme for control design of DPS. At the same time, a new robust control design is proposed which leads to robust, low-order controllers. It is shown that, at least for the class of exponentially stabilizable and detectable state linear systems with bounded and finite-rank input and output operators, this design always leads to a low-order controller which stabilizes not only the original system but also a large class of perturbations. This robustly stabilizing controller also guarantees bounds on the main performance indices. The class of systems considered in this chapter is that of the exponentially stabilizable and detectable state linear systems S(A, B, (7) on the Hilbert space Z, where A is the infinitesimal generator of the strongly continuous semigroup T(t) on Z, and the operators B and C are finite-rank and bounded; BE £(Cm,Z), C € £(Z, Ck). The basic properties required of a finite-dimensional controller for this system are: (P1) The controller stabilizes E(A, B, C). (P2) The controller is robust so that it will have a chance of stabilizing the actual physical plant. 'University of Groningen, Department of Mathematics, P.O. Box 800, 9700 AV Groningen, The Netherlands. E-mail: [email protected]

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(P3) The performance of the controller is reasonable, e.g., with respect to disturbance rejection, sensitivity of the output, etc. If one assumes that a transfer function model of the system is given, there exist several robust control designs for larger classes of systems than E(.A, B, C) which satisfy the above properties (P1)–(P3) (see Curtain and Zwart [7], Chapter 9 and other references in Section 9.7). However, the present mode of modeling physical linear DPS rarely leads to a nice compact transfer function model, but rather to a system of partial differential equations (PDE) or to a linear state model of extremely high dimensions (in finite-element form). This is the starting point taken in this chapter. It is appropriate to start with a review of two popular approaches used to design finite-dimensional controllers for state-space models of DPS. 4.1.1

Trotter-Kato Semigroup Approximations

The system £( A B, C) is approximated by a sequence of finite-dimensional approximating systems E(A n , Bn, Cn) which converges to the original system in some sense (see Theorem 4.10). One designs a controller Kn for Z(An,Bn,Cn) and uses it to stabilize the original infinite-dimensional system. Favorite choices for the controller design are the linear quadratic Gaussian (LQG) and min-max designs and various HOQ designs. Initially, the focus was on obtaining good numerical approximations of the operator solution to the standard Riccati equation. It is only relatively recently that researchers seriously addressed even the first requirement (PI) (see Ito [14, 15] and Morris [28, 29]). The approach of Morris is particularly nice in that, rather than listing various technical conditions, she emphasized the crucial property which is necessary (but not sufficient) for these approximations to satisfy the requirements (P1)-(P3); namely, the approximating transfer functions must converge in the gap topology (or graph topology, as in Vidyasagar [42]). If Gn is a sequence of stable transfer functions (i.e., G™ € HQQ), then convergence in the gap topology is equivalent to convergence in the Hoc-norm. For unstable state linear systems the definition is in terms of coprime factorizations (see Curtain and Zwart [7], Definition 7.2.7). The following is an equivalent definition from Vidyasagar [42] Lemma 20, p. 238, and for more background on the gap and the graph topology see Zhu [44]. Definition 4.1. Let E(A,B,C), E(An,Bn,Cn) have the transfer functions G and Gn, respectively, and suppose that G and Gn have left-coprime factorizations

Then Gn converges to G in the gap topology if and only if

The above definition is independent of the left-coprime factorizations chosen and, if (4.1) holds for one choice, it holds for all left-coprime factorizations. An

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equivalent definition can be given in terms of right-coprime factorizations. Sufficient conditions on the approximating systems 'E(An,Bn,Cn) to guarantee convergence in the gap topology are given in [29]. For properties (PI), (P3) to hold, even stronger conditions are needed, and these are discussed by Morris in [28]. As an application, she gives conditions under which the standard LQG design on the system E(An,Bn,Cn) will stabilize E(A,B,C). Similar results for the Hoo design are in Morris [27].

4.1.2

Proper Orthogonal Decomposition Reduced-Order Models

Proper orthogonal decomposition (POD) is a technique for obtaining reduced-order models from data collected from nonlinear partial differential equations. Initially it was applied with considerable success to obtain low-order models for uncontrolled dynamical systems, but more recently it has been applied to controller design. The controller is designed to control the reduced-order model with the hope that it will perform well on the original nonlinear partial differential equation. The literature on POD is extensive and there are several different approaches. One well-known approach is the "Method of Snapshots" (see Atwell and King [1, 2], Banks, del Rosario and Smith [3], Kepler, Tran and Banks [19], Ly and Iran [23, 22] and Kunisch and Volkwein [20]). Another approach stems from Principle Component Analysis, as in the recent paper by Lall, Marsden and Glavaski [21]. In their paper an attempt is made to justify a methodology of controller design based on empirically obtained POD reduced-order models. In particular, the important point is made that the requirements for POD models for control design are different than those for open-loop uncontrolled systems. The main theoretical result is that for the class of finite-dimensional linear systems their POD reduced-order modeling scheme is theoretically equivalent to obtaining balanced truncations (see Moore [26]). Although my knowledge of the POD literature is limited, my knowledge of the literature on balanced realizations and their truncations is considerable. So I focus my attention on the specific POD scheme treated in [21] applied not to nonlinear finite-dimensional systems as they do, but to linear infinite-dimensional systems (after all, partial differential equation models are infinite-dimensional). I analyze the use of POD reduced-order models for control design in light of the existing theory of balanced realizations and truncations for infinite-dimensional systems from Curtain and Glover [5] and Glover, Curtain and Partington [10]. The first thing which is clear is that the theory of balanced realizations exists only for linear stable systems and consequently this applies to the POD method as well - a significant limitation. Also note that balanced realizations are determined only by the transfer function and so are independent of the initial state. So to calculate POD models from data one should set the initial conditions to zero. Now suppose that E(A,B,C) is exponentially stable, and that E(An,Bn,Cn) corresponds to a sequence of balanced truncations. As already explained, for convergence in the gap topology of stable systems we require that

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Now in [10], various bounds on the HOC-errors of the balanced truncated approximations are obtained. These depend on the Hankel singular values o^i > 1, which are invariants of E(A, B,C) (see Section 4.2). Since E(A, B,C) is exponentially stable, it is known that

Even for finite-dimensional systems there are examples for which the tail does not drop off rapidly, which suggests that for a POD reduced-order model approach to controller design to have a good chance of success the original system should satisfy Y^iLi ai < °°; i- e -> the transfer function should be nuclear- another limitation. Systems with an A operator which is only strongly stable or with B or C unbounded may or may not be nuclear (see Sasane and Curtain [38]). It is known that many systems with infinitely many eigenvalues which asymptote to the imaginary axis at infinity will not be nuclear. This is often the case for many PDE models of undamped flexible systems (see Oostveen [34]). Still, the above discussion has illuminated one positive result: for an exponentially stable state linear system with finite-rank and bounded input and output operators, the balanced truncations converge in the gap topology and so do the POD approximations. Although the above discussion was limited to one POD scheme, in view of the underlying similarity of the approaches, it seems likely that many of the above comments may apply to other POD schemes. The implications for controller design are taken up in Section 4.2. Although the two above control design procedures have been successfully applied in many simulations of DPS, from a control theoretic viewpoint, they both have shortcomings. The connection of POD approximations to balanced truncations suggests that they are not suitable for unstable systems. In contrast, the TrotterKato approach is applicable to unstable systems, and there is a theory for testing whether properties (PI), (P3) hold. The weak point is, however, the robustness property (P2); it is known that, even in finite dimensions, the LQG design gives no guarantee of robustness (see Doyle [8]). Since no mathematical model can exactly match a physical model, if one is really interested in controlling the physical model, and not some sophisticated simulation of it, it seems to this author that the robustness issue is of paramount importance. Furthermore, the finite-dimensional theory demonstrates that the approximation procedure should be done in closed-loop; i.e., the type of approximations should match the robust control design. In my opinion, there is need for more research into designing robustly stabilizing finite-dimensional controllers for DPS incorporating the types of modeling errors and the choice of the approximating systems explicitly into the design procedure. In this chapter we elaborate further on some of the above issues for the special class of exponentially stabilizable and detectable state linear systems S(A, B, C) with bounded, finite-rank input and output operators. In Section 4.2 the theory of balanced truncations is reviewed and implications for control design are discussed. In Section 4.3, a sequence of approximations is proposed which is suitable for unstable systems. It is called the LQG-balanced truncations and it was introduced in the finite-dimensional literature by Jonckheere and Silverman [17] and other in-

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terpretations followed in Meyer [25] and Ober and McFarlane [33]. We show that LQG-balanced realizations and truncations exist for our special class of systems. In Section 4.4 the numerical computation of balanced and LQG-balanced truncations is treated. In Section 4.5, a new algorithm is proposed for designing a low-order finite-dimensional controller which has the properties (P1)-(P3). It is shown that it is always possible to find a robustly stabilizing, low-order controller for the original system and that this controller satisfies certain performance bounds. Finally, Section 4.6 contains my conclusions and suggestions for future research.

4.2

Approximating via Balanced Truncations

First we remark that only stable transfer functions (G € HOo) can possess balanced realizations. We review the theory of balanced realizations from Curtain and Glover [5] applied to the special class of exponentially stable state linear systems S(.A, B,C) on the Hilbert space Z, where A generates the exponentially stable semigroup T(t) on Z, and the operators B,C are finite-rank and bounded; B E £(Cm,Z), C E £(Z,Cck). E(A,B,C) has the transfer function G(s) = C(sl – A)~1B, but there are infinitely many other triples of operators A, B, C which define the same transfer function, i.e., infinitely many realizations. Many of these will have very unbounded B and C operators, but for our purposes, we only need to consider realizations which define a Pritchard-Salamon (PS) system with finite-dimensional inputs and outputs. Definition 4.2. PSE(.A, B, C) is a Pritchard-Salamon (PS) system with respect to the Hilbert spaces W, V if the following hold: (i) W«-» V.

(ii) A is the infinitesimal generator of a strongly continuous semigroup T(t) on V which restricts to a strongly continuous semigroup on W. (iii) Be £(Cm, V) and there exist ti,a>Q such that

(iv) C e £(W, Ck) and there exist t2,/3>0 such that

All of the nice system theoretic concepts and properties of state linear systems given in Curtain and Zwart [7] extend in a natural way to the PS-class (see van

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Keulen [41] and Curtain et al. [6]). In particular, the transfer function is given by G(s) = C(sl –A)- 1 B. We denote the PS-system by PSZ(A,B,C) to distinguish it from a state linear system with bounded input and output operators. By a PS realization of a transfer function G we mean a PS-system PSE(A, B, C) which has the transfer function G. For stable systems we shall be concerned with the balanced realization which was constructed by Curtain and Glover [5] who further studied it in [10] for a class of stable transfer functions with finite-dimensional inputs and outputs. Later, Ober and Montgomery-Smith [32] showed that most stable systems possess (par)balanced realizations. Definition 4.3. Let G E H 00 (C /sxm ). A PS-system PSZ(A,B,C) is called a balanced realization of G if its transfer function is G and the controllability and observability gramians are both equal to the same diagonal operator. The controllability gramian LB and the observability gramian LC of an exponentially stable state linear system E(A, B, C) are denned in Definition 4.1.20 of [7], and in Lemma 4.1.24 of [7] it is proven that they are the unique self-adjoint, non-negative definite solutions of their respective Lyapunov equations

The above-mentioned theory extends to a much larger class of systems, but here we restrict our remarks to PS-systems. In particular, the theory extends to allow for PS-systems which are not necessarily exponentially stable, but B and C are infinite-time admissible, i.e., conditions (iii) and (iv) of Definition 4.2 hold with t\=ti = oo. This is sufficient to guarantee the existence of bounded controllability and observability gramians which satisfy their respective Lyapunov equations. If, however, no explicit assumptions on the stability of the semigroup are made, the gramians are not necessarily the only solutions of the Lyapunov equations (see Grabowski [12] and Hansen and Weiss [13]). In the special PS-case, the gramians are well-defined bounded operators in l ( V ) fl C(W) which satisfy their respective Lyapunov equations considering either W or V as the state space (see van Keulen [41], Chapter 2). The construction of the balanced realizations in [5, 10] is based on the singular values and Schmidt vectors of the Hankel operator of the system. Suppose that the transfer function G is the Laplace transform of h £ Li(0,oo;C mxfe ). The Hankel operator with symbol G is the bounded operator F : L2(0, oo; Cm) —> L 2 (0, oo; C fc ) defined by

for all u E L 2 (0, oo;Cm). F is compact and has countable many singular values {
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The balanced realization has the controllability and observability gramians both equal to diag (o1,o 2 ,... , o y , . . . ) . In Curtain and Sasane [39] it is shown that an exponentially stable state linear system with finite-rank input and output operators has a nuclear Hankel operator; i.e., its singular values satisfy

In the following lemma, we collect the results on the existence (via construction) of a balanced realization from [5, 10] applied to our special class. Lemma 4.4. Let E(A, B, C) be an exponentially stabilizable and detectable state linear system with bounded, finite-rank inputs and outputs. The transfer function G(s) = C(sl- A)-1B has a balanced realization PSZ(Abal,Bbal,Cbal) on the state space /2 which is a PS-system with respect to the spaces V, W, where

Its controllability and observability gramians both equal S = diag(aj) and they satisfii their rp.snp.r,tivp. Liiami.nnii emi.atinns. isfy respective Lyapunov equations Proof. Most of the results are proven in [5], but others are more easily deduced from results on the output normal realization PSH(A°ut, Bout, Cout) which is constructed in [10]. The output normal realization has diagonal gramians and the observability gramian is the identity. It is closely related to the balanced one via

Notice that the extension of the balanced semigroup on V corresponds to the output normal semigroup on 12, while the restriction of the semigroup to W corresponds to the input normal semigroup on l^. (The input normal realization is the dual of the output normal one; it has the controllability gramian / and the realization PSE(£-?AbalE?,'£-5Bbal,CbalE?).) D We note that balanced realizations are unique up to a state space transformation by a unitary operator [32]. When we write the balanced realization we refer to the particular one constructed in [5]. A rather strange feature is that even though the original realization has bounded input and output operators, in the balanced realization they will in general be unbounded. In the output normal realization the input is bounded, but the output is usually unbounded. Moreover, the semigroups are usually not exponentially stable; Abal and Aout both generate contraction semigroups, but only A°ut is guaranteed to generate a strongly stable semigroup. Nonetheless, both realizations have well-defined controllability and observability gramians which satisfy their Lyapunov equations. Balanced truncations are constructed from the balanced realizations in the following manner. Choose an integer r such that ar > oy+i and partition PSE(.A, B, C)

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compatibly with the partition LB = LC = diag(Er, *), where Er = diag(oi,... , or):

The rth order balanced truncation of ~S(A, B, C) (or of G) is the finite-dimensional system E(A11(r),Bi(r),.C'i(r)). Note that this system is stable and balanced and its Hankel singular values are < o , 2 , . . . ,oy. Denote its transfer function by Gr. There holds

and so the transfer functions of the balanced truncations converge in the Hocnorm and so in the gap topology as n —» oo. In fact, the convergence holds in the stronger nuclear norm (see [10]). Convergence in the Hoc-norm is a very strong type of convergence which is only guaranteed to hold for nuclear systems, a restricted class of infinite-dimensional systems. In particular, systems with C = I (the whole state is measured or estimated) are unlikely to be nuclear. This does not mean that the balanced truncations will not converge in some weaker norm; simulations of the open-loop balanced truncations can still produce good approximations to the original system. Convergence in the Hoc-norm becomes important when one uses the balanced truncations to design controllers to achieve some performance objective, such as tracking, reduction of sensitivity to disturbances or increasing the stability margin. There are many finite-dimensional designs which produce a controller Kr which achieves the chosen performance objective for Gr and produces a stable closed-loop system. However, to ensure that Kr applied to G will have the same effect requires a careful analysis. For example, in Morris [28] it is shown that this approach succeeds for one Hoc-design (weighted mixed sensitivity), but fails for another. Two other control designs which will be successful are the robust stabilization under additive uncertainty in Curtain and Zwart [7], Section 9.3 (for increasing the stability margin) and robust stabilization with respect to normalized left-coprime factor perturbations in Section 9.4. Note that in the above discussion we have tacitly assumed that the truncated balanced realizations are available. Of course, one needs to compute these numerically. If one has available an explicit expression for the transfer function, then a first step would be to approximate it closely in the Hoc-norm by a high order rational transfer function and then use MATLAB® to compute its balanced truncations. These will be good low-order approximations to the original transfer function in the Hoo-norm. Most often the transfer function is not readily available and so POD methods are used to estimate the system. In Corollary 4.11 of Section 4.4 we give an alternative approximation approach based on a state-space description of the system.

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4.3

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LQG-Balancing

As we have already emphasized, balanced realizations and truncations only make sense for stable systems. In this section, we investigate the existence of an LQGbalanced realization of the exponentially stabilizable and detectable state linear system £(.A, B, C) on the Hilbert space Z, where A is the infinitesimal generator of the strongly continuous semigroup T(t) on Z, and the operators B and C are finite-rank and bounded; B E £(Cm, Z), C € £(Z, Ck). To do this we make use of the unique self-adjoint, nonnegative definite, stabilizing solutions Q and P of the control and filter Riccati equations, respectively,

"Stabilizing" refers to the fact that AQ = A–BB*Q and AP = A-PC*C generate the exponentially stable semigroups TQ(£) and Tp(t), respectively, on Z (see Curtain and Zwart [7], Chapter 6). The linear quadratic Gaussian (LQG) control design is based on P and Q, although one usually includes weighting matrices in the quadratic terms, e.g., QBR~1B*Q. It is easy to allow for this modification by redefining B asBR-2. Definition 4.5. The PS-system PSE(A, B, C) is called an LQG-balanced realization (of its own transfer function) if there exist bounded, self-adjoint, nonnegative solutions Q, P to its control and filter Riccati equations such that Q = P = A is a diagonal operator. In finite dimensions it is known that every transfer function possesses an LQGbalanced realization and the eigenvalues of QP, {U2} are invariants of the transfer function and A = diag(/i,) (see Jonckheere and Silverman [17]). It is our aim to investigate the viability of this concept in infinite dimensions (we are unaware of any theory for the infinite-dimensional case). The key to our analysis is to exploit the one-one relationship between the exponentially stabilizable and detectable state linear system E,(A,B,C) and the exponentially stable state linear system E(Ap, [B, — PC*],C, [0, /]), as is done in Meyer [25]. E(Ap, [B, —PC*],C, [0,/]) is a state-space realization of the normalized leftcoprime factor system of G, i.e., of the transfer function [N(s),M(s)] given by

where G(s) = M(s) –1N(s) and the coprime property is with X(s) = / - C(sl - AQ^PC*, Y(s) = B*Q(sI - AQ)~1PC*. The normalization property is

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For background theory of coprime factorizations for state linear systems (bounded B, C) consult Curtain and Zwart [7] - in particular, Section 7.3, Example 7.29 and Lemma 9.4.10; for the PS-case see Curtain [4]. Since we shall repeatedly appeal to known properties of the normalized left-coprime factor system and its Hankel singular values, we collect them here. Lemma 4.6. IfS(A, B, C) is an exponentially stabilizable and detectable state linear system with finite-rank input and output operators, its normalized left-coprime factor system £(Ap, [B, — PC*],C, [0, /]) with transfer function [N,M] is an exponentially stable state linear system with a nuclear Hankel operator. Its singular values {o~i;i = 1,... ,00}, ordered according to decreasing magnitude, satisfy

The singular values are system invariants; i.e., they comprise a property of the transfer function only and are independent of the realization. Furthermore, the controllability and observability gramians Lsnic and Lcnlc, respectively, of £(Ap, [B, —PC*],C, [0,/]) are nuclear and satisfy

where Q,P are the stabilizing solutions of the Riccati equations (4-8), (4-9), respectively. The nonzero eigenvalues of LiBnlcLcnla (LcnicL>Bnlc) are {o~?\i = I,... , oo}. Proof. Hankel operators have been defined in Section 4.2. The nuclearity of Lb n l c , Lcnlc and the Hankel operator F is shown in Curtain and Sasane [39]. This implies the second part of (4.13). The first part follows from Lemma 9.4.7 in Curtain and Zwart [7], since the coprime factors are normalized. The connection between the controllability and observability gramians of S(Ap, [B, — PC*],C, [0,/]) in (4.14) and P, Q, was proven in Lemma 9.4.10 of [7]. The relationship between the eigenvalues of LcnlcL
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Figure 4.1. Normalized left-coprime factor system. of LBnlc, Z/cnic was proven in Lemma 4.6 and the nuclearity of P, Q follows from (4.14). The relationship (4.14) and the one-one relationship between H(A,B,C) and £(Ap, [B, —PC*], C, [0, /]) show that the eigenvalues of QP are positive and depend only on the transfer function. D Following the finite-dimensional terminology we call the square roots of the eigenvalues of PQ the LQG-characteristic values of G or of E(A, B, C). From (4.14) we deduce the following simple relationship between the Hankel singular values of [N,M], {oi,i = 1,... ,00}, and the LQG-characteristic values of G, {/J,i,i = 1,... , oo}, where they are ordered according to decreasing magnitude:

The diagram depicted in Figure 4.1 may help us to understand the one-one relationship between the exponentially stabilizable and detectable state linear system E(A, B, C) and its normalized left-coprime factor system £(Ap, [B, —PC*], C, [0, /]). Notice that there are two ways of connecting the two systems: one with state-space descriptions using the one-one relationship between the solutions of the LQG-Riccati equations of £(.4, B, C) and the solutions of the Lyapunov equations of £(.Ap, [B, —PC*],C, [0,/]). The other is via the transfer function relationship G = M-1N, although we note that normalized left-coprime factorizations are only unique up to left multiplication by a constant unitary matrix. There is a complete generalization of these one-one relationships to the exponentially stabilizable and detectable PS-class; i.e., the unique solutions to the LQG-Riccati equations (4.8), (4.9) are related to the observability and controllability gramians of the normalized left-coprime factor system via (4.14); see Curtain [4]. In fact, not only the stabilizing solutions but any pair of solutions Q, P to (4.8), (4.9) correspond to a pair of solutions to the Lyapunov equations of the normalized left-coprime factor system in a one-one way. In Meyer [25] the next step was to find a balanced realization of the stable system [N, M] and, using (4.14), prove that PQ can be diagonalized. This approach

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carries over to the infinite-dimensional case. Theorem 4.8. Let E(A, B, C) be exponentially stabilizable and detectable with bounded, finite-rank inputs and outputs. The transfer function possesses an LQGbalanced realization on the state space 1-2 which is a PS-system with respect to the spaces W, V, where W = E~2/ 2 ^-* 1% <—* £2/2 = V. The Riccati equations (4-8), (4-9) both have the solution

Proof. We apply Lemma 4.4 to obtain a balanced realization of £(Ap, [B, —PC*], C, [0,1]) on 12. This is a PS-system with controllability and observability gramians both equal to £ and its transfer function is [N, M]. We now apply a new coordinate t.ra.nsfnrma.t.inn to obtain yet another realization of [N, M] on 12, the system PS£(^4', Bl,Cl, [0,/]), where

S is boundedly invertible. Thus it is easy to see that PSH(Al,Bl,Cl, [0,1]) is also a PS-system on 12, its controllability gramian is E52 and its observability gramian is ES2. Considering Figure 4.1, we wish to identify PSZ(Al,Bl,Cl,[Q,I]) as a normalized left-coprime factor system of a system on the left. Our candidate is the PS-system PSS(^iw,Bi",C'«ff) on 12 given by

To see that this is a well-defined PS-system, we recall from Curtain et al. [6], Theorem 4.1, that it is sufficient that A(C')* 6 L(Ck, W). Now A < 1/^/1 -
The obvious next step is to verify that Q and P are solutions to the LQG-Riccati equations (4.8), (4.9), respectively, of PS£(4'9ff, B1™, C1™). For bounded input and output operators, this was done in Lemma 9.4.10 of [7] by direct verification. It is tempting to appeal to the generalization of the smooth PS-case in [4]. However, the smooth property is used in an essential way in the proof and it is unlikely that our PS-realization will be smooth. Fortunately, using a different approach, the result can be shown to hold for nonsmooth PS-systems (see [35]). Note that we do not require that the solutions be unique, since there is a one-one relationship between the pairs {LB,LC} and {P,Q}. Next we verify that PSE(Alq<>, Bl™, C1*9) has the transfer function M^N — G as in Exercise 7.29 of [7], noting that this also extends

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to the PS-class (see [4]). So PSE(Alw,Blg9,Cl119) is an LQG-balanced realization of the transfer function G. d We remark that an alternative route would be to use normalized right-coprime factorizations. We continue with the approach of Meyer [25] and introduce a sequence of truncations of the LQG-balanced realization which are themselves LQG-balanced. Henceforth we use the notation PSE(A, B,C) for the LQG-balanced realization constructed in Theorem 4.8. We use the infinite matrix representation for operators on /2 with respect to the usual orthonormal basis. Thus PSE(Al, Bl, Cl, [0,/]) has the observability gramian E(I — E 2 )~3 and the controllability gramian S(7 — E2) 2 and the corresponding Riccati equation solutions for PSS(.A, B, C) are given by P = Q = A = £(/ — E 2 )~3. Choose a positive integer r such that fj,r > /zr+i and partition PS£(A, B, C) and PSS(A*, Bl,Cl, [0,1]) compatibly with the following partition of P = Q = A = diag(Ar, *), where Pr = Qr = Ar = diag(/ii,... ,/v). PSE(i, B, C) and PSE(A!, B1, Cl, [0, /]) become

For simplicity of notation we henceforth assume that the /^ are all distinct. The rth order LQG-balanced truncation of PSS(A, B, C) is defined to be the finite-dimensional system S(An(r),Bi(r),C'i(r)) and the rth order truncation of PSE(A', Bl, C1, [0, /]) is E(A'u(r), [Bi1(r),_5i2(r)I, C{(r)_, [0, Ir]). The following result shows that E(An(r),Bi(r),C'i(r)) is a good candidate as a reduced-order model for PSE(.A, B, C) and hence for G. The approximations converge in the gap topology. Theorem 4.9. The transfer functions Gr of the LQG-balanced truncations S(An(r), Bi(r),Ci(r|) converge in the gap topology as r —> oo to G, the transfer function of PSE(A, J5, C) and of E(A, B, C) . Proof, (a) First we show that S(A'n(r), [B[1(r),B[t(r'j[,C[(r'), [0,/r]) is the normalized left-coprime factor system of Y,(An(r),Bi(r),Ci(r)). The LQG-Riccati equations have solutions diag (/ij,... ,/i r ) which correspond to the observability and controllability gramians of S(A'n(r), [B{1(r),B(2(r)],C'i(r), [0,/r]) via (4.14). Moreover, from the block structure it is clear that

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So recalling Figure 4.1 and arguing as in the proof of Theorem 4.8, it is easy to see that E(A l n (r), [Bl^r), B { 2 ( r ) ] , C { ( r ) , [0,/r]) is the normalized left-coprime factor system of Z(A11(r), B\(r), Ci(r)) on 12 and Gr = M^N,. for each r < n. (b) Note that [Nr, Mr] is also the transfer function of the rth order truncations of the balanced and output normal realizations of [N, M]. So appealing to the theory in [10], Theorem 5.1, we conclude that

By (4.13) Y^i o~i < oo, and so the normalized left-coprime factors [Mr,Mr] of Gr converge to those of G in the Hoc-norm, i.e., we have convergence in the gap topology. n

4.4

Numerical Computation of LQG-Balanced Truncations

Here we address the question of computing the balanced and LQG-balanced truncations from a given state-space description S(A, B,C). Since the balanced case can be seen as a special case of the LQG-case, we consider the latter. In the finitedimensional case studied in Mustafa and Glover [30], it is very easy to calculate P and Q and then diagonalize PQ. In infinite dimensions, the best you can hope for is to obtain a good numerical approximation to the operator solutions to the LQGRiccati equations. As pointed out in Morris [29], if we intend to use the truncations for control design, we shall require not only sufficient conditions on a sequence of systems E(An,Bn,Cn) approximating Y;(A,B,C) to ensure convergence of the solutions to the LQG-Riccati equations, but additional ones to ensure convergence of the transfer functions in the gap topology. Theorem 4.10. Suppose that the exponentially stabilizable and detectable state linear system £(.4, B, C) with finite-rank and bounded input and output operators has transfer function G and denote the solutions to the control and filter Riccati equations (4-8) and (4-9) by Q and P, respectively. Let S(A™, Bn, Cn) be a sequence of finite-dimensional linear systems which satisfies the following assumptions: (Al) Zn is a sequence of finite-dimensional subspaces of Z andII 1 is the orthogonal projection of Z into Zn such that

(A2) An e C(Zn] and for each z^Z there holds (i) eAntUnz -> T(t)z, (ii) (eJ*B*)*IInz -» T(t)*z uniformly in t on bounded intervals as n —» co.

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(A3) (An,Bn) is uniformly exponentially stabilizable; i.e., there exists a uniformly bounded sequence of operators Fn £ £(Zn,
for some positive constants MI > 1 and a; (A4) (An,Cn) is uniformly exponentially detectable; i.e., there exists a uniformly bounded sequence of operators Ln € >C(Cfe, Zn) such that

for some positive constants M% > 1 and (3. Let Qn,Pn be the unique stabilizing solutions to the LQG-Riccati equations (4-8) and (4.9) corresponding to E(An,Bn, Cn) with transfer function G n . Then Qn and Pn converge in the nuclear norm to Q, respectively, P as n —> oo. TP(t) = n n e(A -p"(c ) c )t converges strongly to Tp(t) uniformly on compact intervals and there exist constants MS > 1,7 > 0 such that

Similarly, Tp(t)* converges strongly to Tp(t)* as n —> oo. Let [N™,M n ] denote the transfer function of the normalized left-coprime factor system S(^4p, [Bn, —Pn(Cn)*], Cn) ofE(An,Bn,Cn). Then

i.e., G™ converges to G in the gap topology. Let F, F™ denote the Hankel operators o/[N, M], [Nn,Mn], respectively, and their singular values, ordered according to decreasing magnitude, by {o~i; i = 1,... , oo}, {<7™; i = 1,... , s(n)}, respectively. We have a\ < 1, erf < 1 for all n and the singular values and Schmidt vectors of Fn converge to those of T according to

Proof, (a) The proof of the strong convergence of Pn, Qn, T£(i), T£(i) and (4.17) can be found in Ito [14] and Kappel and Salamon [18]. Equation (4.17) and the fact that B, C have finite rank can be used to show uniform convergence of Pn and Qn (see the remarks in [18] on p.1143). In part (d) we prove convergence in the nuclear norm.

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(b) The convergence in the gap topology is given in Morris [29], but for completeness we give a simpler proof. We consider the impulse responses

corresponding to [Nn,Mn], [N, M] minus the constant terms. Now since Pn,Bn, Cn,Tp(t) all converge strongly as n —> oo, they are uniformly bounded in norm in n and, moreover, hn(t) —> h(t) pointwise as n —>• oo. Using (4.17) we see that

and from the Lebesgue dominated convergence theorem we have that hn —>• h in the Li(0,oo;C mxfe ) norm as n -> oo. This implies (4.18). (c) SinceCTJand a? are singular values of a normalized left-coprime factor system, the largest is strictly less than 1 (Lemma 9.4.7 of [7]). (d) To establish the convergence of the Schmidt vectors and of the singular values of Tn according to (4.19), (4.20), we show that Fn converges in the nuclear norm to F. Recall from Lemma 8.2.2 in [7] that T = CnicBnlc, where Cnic e £(Z, L2(0, oo; C fe ; and Bnic 6 £(1/2(0, oo;C m ),/?) are the observation and control maps, respectively, of E(AP, [B, -PC*}, C, [0, /]) defined by

Similarly, Tn = C™lcB™lc, where C™lc, B™lc are the observability, respectively, controllability maps of S(A£, [Bn,-Pn(Cn)*],Cn). Recall from Partington [36], Corollary 1.4, that where N, HS denote the nuclear and Hilbert-Schmidt norms, respectively. So, if we show that

then

Prom the duality between the observability and controllability maps, it suffices to prove that

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To do this we recall from Weidmann [43], Theorem 6.12 that

where the second norm is in £(Z,Ck). So we need to show that

Now Cn, Tp(<) and Tp(t)* all converge strongly to C, TP(i) and TP(t)*, respectively, as n —> oo, and since C has finite-rank, we have

Prom (4.17) and (Al) we have

So applying the Lebesgue dominated convergence theorem we obtain

and F™ converges in the nuclear norm to F. (e) Since LCnlc — C*/cCn;c, LBnlc — BnicB^lc and similar expressions hold for the approximating sequences L/cnl(,, £j3nlc, it is clear from the arguments in (c) that £j3n!(. —» Lsnlc and £<7nlc —» isnlc in the nuclear norm as n —> oo. So (4.14) shows immediately that Pn —> P and Qn —> Q in the nuclear norm as n —> oo. D We remark that the assumption that 11" are orthogonal projections is not essential and it can be relaxed to allow for more general Galerkin approximations (see Ito and Kappel [16]). An earlier result in Gibson [9] on retarded systems showed that Qn converges to Q in the nuclear norm for the special case that Cn = C for all n. The convergence of the solutions to the Riccati equations in the nuclear norm in Theorem 4.10 appears to be a new result. We shall see in Section 4.30 that it is necessary for a successful controller design. One can use the same approach to prove convergence for Lyapunov equations as promised at the end of Section 4.2. Corollary 4.11. Suppose that the exponentially stable state linear system £(.A, B, C) with finite-rank and bounded input and output operators has the transfer function G and Hankel operator F. Denote the Hankel singular values (ordered according to decreasing magnitude) by {cr^i = 1,... ,00} and the solutions to the Lyapunov equations (4-3) and (4-4) byLc, Lg, respectively. Let Yl(An, Bn, Cn) be a sequence of finite-dimensional linear systems which satisfies the assumptions (Al), (A2), and An is uniformly exponentially stable; i.e., there exist positive constants a,M such that

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Let LC, U^ denote the unique solutions to the Lyapunov equations (4-4)> (4-3), respectively, corresponding to !£(An,Bn,Cn) with transfer functions G™. Then L%, U^ converge in the nuclear norm to LC, LB> respectively, and G™ converges to G in the Hoc-norm as n —> oo. Let F™ denote the Hankel operators and erf the Hankel singular values of S(j4™, Bn,Cn) (ordered according to decreasing magnitude: a? < crf+1). Then Tn converges to T in the nuclear norm

4.5

Robust Controller Design via LQG-Balanced Truncation

Let us first clarify that what we mean in this section by stability of a system is the concept of input-output stability defined in Definition 9.1.2 in [7]. At the same time, we recall that if S(A, B, C) is exponentially stabilizable and detectable and the input and output spaces are finite-dimensional and it is stabilized in the input-output sense by a controller with an exponentially stabilizable and detectable realization E(AK,BK,CK,DK), then the semigroup of the resulting closed-loop system is exponentially stable (Exercise 9.6.2 in [7]). Of course, it is always possible to find a stabilizable and detectable (actually controllable and observable) realization of a rational transfer function, and so we can achieve exponential stability of the closedloop system by a suitable implementation of the controller. Our aim is to design a finite-dimensional controller that stabilizes an exponentially stabilizable and detectable state linear system S(A, B, C) with bounded finite-rank input and output operators. As many people do, we shall start with a state-space description ~E(A, B, C) and approximate it by a sequence E(.An, Bn, Cn) satisfying the conditions in Theorem 4.10. In Morris [28] it is shown that for sufficiently large n the popular LQG-controller with transfer function K™ designed to stabilize the reduced-order model H(An,Bn,Cn) will also exponentially stabilize S(.A, B, C). Although this is a pleasing result, it is known that LQG-controllers do not have very good robustness properties, even in finite dimensions. So even if it stabilizes E(A, B,C), it may not stabilize the physical plant. We propose an alternative robust controller design based on reduced-order models of £(^4™, Bn, Cn) obtained using LQG-balanced truncations. Since the controller is designed to be robust, it will stabilize not only £(.4, B, C) but other neighboring ones as well. The first step will be to compute the numerical approximations Pn,Qn to the LQG-Riccati equations corresponding to ~S(An, Bn, Cn) as outlined in Section 4.4 and then to compute the LQG-balanced truncations of this finite-dimensional system (a finite-dimensional computation in MATLAB). These two approximation steps are entirely different approximation procedures with different types of errors involved. Nonetheless, we shall show how, starting from our DPS E(A,B,C) we obtain a sequence of reduced-order models with transfer functions Gq which converge to the transfer function of G in the gap topology as q —> oo. We propose a

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controller design based on a reduced-order model Gq and prove that it is guaranteed to stabilize the original system. Moreover, we give a priori computable estimates for q which determines the order of the controller.

4.5.1

Robust Controller Design via LQG-Balanced Truncation

Step 1: Using your favorite approximating sequence S(A™, Bn, Cn) to E(.A, B, C), which satisfies the assumptions in Theorem 4.10, find numerical approximations Pn and Qn for sufficiently large n. Calculate and order the eigenvalues values of PnQn, {(M?) 2 )* = 1, • • • ,s(n)} in decreasing magnitude. Calculate the related Hankel singular values (erf)2 = ^rf n\t and the sum J^l™ cr". An indication of "large enough" is that this sum appears close to a limit. Step 2: Obtain the LQG-balanced realization of 2(An,Bn,Cn) on /2 as explained in Section 4.3. Step 3: Choose r so that Eil™+iCT™is smail compared with ^i^ a f . Form the rth order LQG-balanced truncation of £(^4™, Bn,Cn) which we denote by E(An(r),Bn(r),Cn(r)) and its transfer function by G?. Step 4: Design a controller for E(j4 n (r), Bn(r), Cn(r)) which is robustly stabilizing with respect to normalized coprime factor perturbations as outlined in Chapter 9.4 of [7]. The central controller (Theorem 9.4.16) has the transfer function

where

and e is chosen strictly less than the maximal robustness margin attainable for GTr which is

Note that the maximal robustness margin is independent of r. This controller is robust in the sense that it stabilizes G™ and all perturbed systems GA with a left-coprime factorization of the form GA = (Mn(r)+&i)~l(Nn(r)+ A2) and such that

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So it will stabilize G provided that

Step 5: Tune r in Step 4 to obtain a controller which achieves a satisfactory level of robustness and performance with respect to the original system 1^(A, B, C). The idea is to choose n large and r as small as possible so as to obtain a low-order controller; the order of the controller is equal to the order of Gn(r). We now prove that we can always choose n » r so that this controller robustly stabilizes our original system. Theorem 4.12. Under the assumptions of Theorem 4-10, given a positive e < ^/l — a\, we can always find two integers n » r such that the controller K" given by (4-23) stabilizes G with a robustness margin with respect to left-coprime factor perturbations of s — -\/l — a\. Proof, (a) Given 5 > 0, (4.18), (4.19) and (4.20) show that we can always find a sufficiently large N = N(6) such that for all n > N(S) there holds

Now using the above inequalities, we estimate the "gap" between G and G ra (r):

For a fixed n > N(6) the maximum robustness margin is ^/l — (cr™)2 and so we can design K£ with a robustness margin of £ = -^/l — a\ — 6 provided that

Equation (4.28) shows that this is satisfied and from (4.26) and (4.30), K™ will stabilize G if 26 + 2 £i^+i
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We can always choose r to satisfy this (see (4.13)) and so ensure that K™ stabilizes G. (b) We now estimate the robustness margin of this controller with respect to G. Let GA be a perturbation of G with a left-coprime factorization G = (M + Ai)~ 1 (N + A2). We need to show that the gap with respect to G"(r) satisfies (4.26). We estimate the gap

provided that ||[Ai, AaJUoo < £ ~ \A — a\ + 5, and this establishes the robustness margin. D To discuss the performance we need to introduce the following index for the transfer function G in closed-loop with a controller K:

The various components have the following interpretations: • (/ — GK)-1G corresponds to additive uncertainty in the controller, • K(7 — GK)"1 corresponds to additive uncertainty on the plant, • (/ — GK)^ 1 is the sensitivity, • K(/ — GK)-1G corresponds to input multiplicative uncertainty on the plant. While we have no bounds on these components individually, they are each bounded by the bound on M(G, K). Applying the results from [7], Exercise 9.14 b, we deduce the following bounds on the performance of the controller K™ on G" and G:

It is important to realize that the choice of the controller that is robustly stabilizing with respect to coprime factor perturbations was crucial. The above conclusions would not hold for other types of controllers which stabilize G™. In particular, we can make no conclusions about the success of an LQG design based on G™, as LQG designs have no guaranteed robustness margin.

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Conclusions

In Section 4.2, I summarized the known theory on balanced realizations and truncations for the class of exponentially stable systems with bounded finite-rank input and output operators. This gives a theoretical basis for balanced truncations and, to a certain extent, POD approximations. However, the important point was made that nuclearity is an essential property if one intends to use balanced truncations for controller design. Moreover, for controller design, one should obtain balanced truncations (or POD bases) using the measured output and not the state. This has been observed in some numerical experiments (via private communication with H.V. Ly). Nuclearity is not sufficient to ensure that the finite-dimensional controller will achieve the desired performance on the original system; extra analysis is required. Since balanced realizations only exist for stable systems, I introduced in Section 4.3 the concept of LQG-balancing and LQG-balanced truncations for the class of exponentially stabilizable and detectable state linear systems with bounded and finite-rank input and output operators. I synthesized known results on balanced realizations and on normalized left-coprime factorizations to obtain the main new result on the existence of LQG-balanced realizations (Theorem 4.8). An interesting feature was that the LQG-balanced realizations will not be in the original class but in the much larger Pritchard-Salamon class. The nice properties of this class played a key role in the proofs. Finally, in Section 4.4 I addressed the question of the numerical computation of balanced and LQG-balanced truncations from a given DPS E(A, B, C). In Theorem 4.1, I extended known results on the convergence of numerical approximations of solutions to the LQG Riccati equations and on the convergence of the approximating transfer functions. Some proofs are new and the nuclear convergence of the approximating LQG solutions is a new result. In Section 4.5 I presented a low-order, robustly stabilizing controller design based on the LQG-balanced truncations. More important, in Theorem 4.12, I proved that this design always leads to a low-order controller which is guaranteed to stabilize the original DPS with a certain robustness margin. The surprising feature is that the previous theory followed fairly easily by weaving together existing results from the systems theory of Pritchard-Salamon systems. Although the theory sounds promising, the real test is whether the LQGbalanced truncation design works well on real physical plants and on sophisticated simulations. In particular, comparisons with the standard LQG design should be made. Does this design lead to significantly lower-order controllers with a better robustness margin? There are many interesting possibilities for research in these directions. Since the class considered here is restricted, it is interesting to ask whether the design can be shown to yield similar results for systems with unbounded inputs and outputs. It is striking how, in all the theorems, the nuclearity of the normalized coprime factor system played a key role. This suggests that the proposed controller design will only work with systems having this property. In this context, it is interesting to know that exponentially stable analytic systems are nuclear (see

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Curtain and Sasane [39]). Since such models occur frequently in the applications, an interesting open problem for future research is to investigate whether the theory extends to exponentially stabilizable and detectable regular analytic systems. It is also known that many delay systems are nuclear (see Glover, Lam and Partington [11] and Partington, Glover and Zwart [37]). So extensions in this direction may also be possible. Since nuclearity appears to be so crucial, what can one do if the normalized coprime factor system is not nuclear? One could try precompensation, e.g., use a smoother control action u = —u + v, and then v(i) is the new control variable. This will have the effect of producing a normalized coprime factor system with a smoother Hankel operator. Other closely related controller designs which have been developed for finitedimensional systems are those based on Hoc-balancing (see Mustafa and Glover [31, 30]). These theories can also be generalized to the class of exponentially stabilizable and detectable state linear systems with bounded and finite-rank input and output operators. In some applications the performance requirements are frequency dependent. It should be possible to design controllers which take these into account using the loop-shaping design procedure in Chapter 6 of McFarlane and Glover [24]. Of course there are many other possibilities for future research motivated by the topics touched on in this chapter: • LQG-balancing for well-posed linear systems, • (LQG-) balancing for nonlinear infinite-dimensional systems, • control design for specific performance objectives, • robustness with respect to nonlinear perturbations, • control design for nonlinear infinite-dimensional systems. Some preliminary results on balancing of finite-dimensional nonlinear systems can be found in Scherpen [40] and Lall, Marsden and Glavaski [21]. Research into some of the above problems will lead to new theoretical results, but many will not be amenable to theory alone. The way to gain real insight into the control of physical DPS is to couple theory with well-designed numerical experiments on simulations and on real physical systems.

Acknowledgements The idea for this chapter arose from many stimulating discussions with Belinda King and John Burns during a visit to I.C.A.M. at the Virginia Polytechnic University in Blacksburg in April 2000. I am grateful to Belinda, John, Mark Opmeer and Hans Zwart for their suggestions for improving this manuscript.

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Bibliography [1] J.A. Atwell and B.B. King, "Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations," Mathematics and Computer Modelling, 33, pp. 1-19, 2001. [2] J.A. Atwell and B.B. King, "Reduced order controllers for spatially distributed systems via proper orthogonal decomposition," SI AM Journal on Scientific Computing, to appear. [3] H.T. Banks, R.C.H. del Rosario and R.C. Smith, "Reduced order model feedback design: Numerical implementation in a thin shell model," IEEE Transactions on Automatic Control, 45(7), pp. 1312-1324, 2000. [4] R.F. Curtain, "Robust stabilizability of normalized coprime factors: The infinite-dimensional case," International Journal of Control, 51, pp. 1173-1190, 1990. [5] R.F. Curtain and K. Glover, "Balanced realisations for infinite-dimensional systems," in Operator Theory and Systems (Amsterdam, 1985), Birkhauser, Basel, pp. 87-104, 1986. [6] R.F. Curtain, H. Logemann, S. Townley and H.J. Zwart, "Well-posedness, stabilizability and admissibility for Pritchard-Salamon systems," Journal of Mathematical Systems, Estimation and Control, 7, pp. 439-476, 1997. [7] R.F. Curtain and H.J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, New York, 1995. [8] J.C. Doyle, "Guaranteed margins for LQG regulators," IEEE Transactions on Automatic Control, 23, pp. 756-767, 1978. [9] J.S. Gibson, "Linear-quadratic optimal control of hereditary differential systems: Infinite-dimensional Riccati equations and numerical approximations," SIAM Journal on Control and Optimization, 21, pp. 95-139, 1983. [10] K. Glover, R.F. Curtain and J.R. Partington, "Realization and approximation of linear infinite-dimensional systems with error bounds," SIAM Journal on Control and Optimization, 26, pp. 863-898, 1988. [11] K. Glover, J. Lam and J.R. Partington, "Rational approximation of a class of infinite-dimensional systems, I: Singular values of Hankel operators," Mathematics of Control, Signals and Systems, 3, pp. 325-344, 1990. [12] P. Grabowski, "On the spectral-Lyapunov approach to parametric optimization of distributed parameter systems," IMA Journal of Mathematical Control and Information, 7, pp. 317-338, 1990. [13] S. Hansen and G. Weiss, "New results on the operator Carleson measure criterion," IMA Journal of Mathematical Control and Information, 14, pp. 3-32, 1997.

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[14] K. Ito, "Strong convergence and convergence rates of approximating solutions for algebraic Riccati equations in Hilbert spaces," in Distributed Parameter Systems, F. Kappel, K. Kunisch, and W. Schappacher, eds., pp. 151-166, SpringerVerlag, Berlin, 1987. [15] K. Ito, "Finite-dimensional compensators for infinite-dimensional systems via Galerkin-type approximation," SIAM Journal on Control and Optimization, 28, pp. 1251-1269, 1990. [16] K. Ito and F. Kappel, "To families of approximation schemes for delay systems," Results in Mathematics, 21, pp. 93-137, 1992. [17] E.A. Jonckheere and L.M. Silverman, "A new set of invariants for linear systems—application to reduced order compensator design," IEEE Transactions on Automatic Control, 28, pp. 953-964, 1983. [18] F. Kappel and D. Salamon, "An approximation theorem for the algebraic Riccati equation," SIAM Journal on Control and Optimization, 28, pp. 1136-1147, 1990. [19] G.M. Kepler, H.T. Tran and H.T. Banks, Reduced Order Model Compensator Control of Species Transport in a CVD Reactor, Technical Report CRSC-TR9813, Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC, 1999. [20] K. Kunisch and S. Volkwein, "Control of Burger's equation by a reduced order approach using proper orthogonal decomposition," Journal of Optimization Theory and Applications, 102, pp. 345-371, 1999. [21] S. Lall, J.E. Marsden and S. Glavaski, Empirical Model Reduction of Controlled Nonlinear Systems, Technical Report CIT-CDS-98-008, Caltech, Pasadena, CA, 1998. [22] H.V. Ly and H.T. Tran, "Modeling and control of physical processes using proper orthogonal decomposition," Mathematical and Computer Modelling, 33, pp. 223-236, 2001. [23] H.V. Ly and H.T. Tran, "Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor," Quarterly Journal of Applied Math, 60(4), pp. 631-656, 2002. [24] B.C. McFarlane and K. Glover, Robust Controller Design Using Normalized Coprime Factor Plant Descriptions, Lecture Notes in Control and Information Sciences, 138 Springer-Verlag, Berlin, 1989. [25] D.G. Meyer, "Fractional balanced reduction: Model reduction via fractional representation," IEEE Transactions on Automatic Control, 35, pp. 1341-1345, 1990.

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[26] B.C. Moore, "Principal component analysis in linear systems: Controllability, observability and model reduction," IEEE Transactions on Automatic Control, 26, pp. 17-32, 1981. [27] K.A. Morris, "Hoc-output feedback of infinite-dimensional systems via approximation," Systems and Control Letters, 44, pp. 211-217, 2001. [28] K.A. Morris, "Convergence of controllers designed using state-space methods," IEEE Transactions on Automatic Control, 39, pp. 2100-2104, 1994. [29] K.A. Morris, "Design of finite-dimensional controllers for infinite-dimensional systems by approximation," Journal of Mathematical Systems, Estimation and Control, 4, pp. 1-30, 1994. [30] D. Mustafa and K. Glover, Minimum Entropy Control, Springer-Verlag, Berlin, 1990. [31] D. Mustafa and K. Glover, "Controller reduction by /f00-balanced truncation," IEEE Transactions on Automatic Control, 36, pp. 668-682, 1991. [32] R. Ober and S. Montgomery-Smith, "Bilinear transformation of infinitedimensional state-space systems and balanced realization of nonrational transfer functions," SIAM Journal on Control and Optimization, 28, pp. 438-465, 1990. [33] R.J. Ober and D.C. McFarlane, "Balanced canonical forms for minimal systems: A normalized coprime factor approach," Linear Algebra and Its Applications, 122, pp. 23-64, 1989. [34] J.C. Oostveen, Strongly Stabilizable Infinite-Dimensional Systems, Ph.D. thesis, Rijksuniversiteit Groningen, The Netherlands, January 1999. [35] M.R. Opmeer, LQG Balancing for Continuous-Time Infinite-Dimensional Linear Systems, manuscript. [36] J.R. Partington, An Introduction to Hankel Operators, London Mathematical Society Student Texts, Cambridge University Press, Cambridge, UK, 1988. [37] J.R. Partington, K. Glover, H.J. Zwart and R.F. Curtain, "Loo approximation and nuclearity of delay systems," Systems and Control Letters, 10, pp. 59-65, 1988. [38] A.J. Sasane and R.F. Curtain, "Inertia theorems for operator Lyapunov inequalities," Systems and Control Letters, 43, pp. 127-132, 2001. [39] R.F. Curtain and A.J. Sasane, "Compactness and nuclearity of the Hankel operator and internal stability of infinite-dimensional linear systems," International Journal of Control, 74, pp. 1260-1270, 2001. [40] J.M.A. Scherpen, "H-infinity balancing for nonlinear systems," International Journal of Robust and Nonlinear Control, 6, pp. 645-668, 1996.

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[41] B. van Keulen, Hoc-Control for Distributed Parameter Systems: A State-Space Approach, Birkhauser, Boston, 1993. [42] M. Vidyasagar, Control System Synthesis, MIT Press, Cambridge, MA, 1985. [43] J. Weidmann, Linear Operators in Hilbert Spaces, Springer-Verlag, Berlin, 1980. [44] S.Q. Zhu, "Graph topology and gap topology for unstable systems," IEEE Transactions on Automatic Control, 34, pp. 848-855, 1989.

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Chapter 5

Max-Plus Linear Partial Differential Equations

Wendell H. Fleming* 5.1

Introduction

This survey chapter considers a class of first-order partial differential equations (PDEs) which are nonlinear in the usual sense but linear when considered with respect to "max-plus" operations of addition and scalar multiplication. The PDEs considered have either the form (5.2) or (5.5) in the time-dependent case. The max-plus sum and product of a and 6 are, respectively,

Included among max-plus linear PDEs are the Hamilton-Jacobi-Bellman (HJB) equations of control theory and calculus of variations (Section 5.3). Solutions of HJB equations are understood to be in the viscosity sense, since typically there do not exist classical (smooth) solutions of the PDE with given initial or boundary conditions. For general background on viscosity solutions and control theory see [4, 11]. Equations of HJB type also arise in asymptotic analyses of problems from physics and probability (for example, semiclassical limits in quantum mechanics and large deviations for small random perturbations of dynamical systems). There are also interesting applications in technology, such as shape-from-shading problems in computer vision and robot path planning [25]. In addition, max-plus algebra is of interest in discrete mathematics and computer science [3, 17, 19]. Semiconvex functions play a natural role in the study of max-plus linear PDEs, analogous to the role of smooth functions in the study of second-order linear PDEs of elliptic or parabolic type. In Section 5.3 we note that the solution operator associated with 'Division of Applied Mathematics, Brown University, Providence, RI 02912. E-mail: whf@cfm. brown.edu

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time-dependent HJB equations maps the class S of semiconvex, bounded above functions into itself. In Example 5.2, we give an illustration of the appearance of a max-plus linear PDE in an asymptotic problem under small random perturbations. This is a result in the Freidlin-Wentzell theory of large deviations [12]. Under a nondegeneracy assumption, there is a representation (5.26) for the solution V(t,x) of an initial value problem for HJB equations in terms of the max-plus fundamental solution. Small time asymptotics of the fundamental solution give additional information about V for small t and smooth initial data. The Maslov idempotent calculus provides a "max-plus" probability framework in which deterministic optimal control and HJB equations can be considered. In Section 5.6 we give a brief introduction to some elements of max-plus stochastic calculus, including max-plus martingales and change of drift. Finally, Sections 5.7 and 5.8 concern max-plus expansions of semiconvex functions in terms of quadratic basis functions, and an application of that method in deterministic nonlinear filtering.

5.2

Max-Plus Linearity

To introduce the ideas with fewer technical complications, we first consider timeindependent PDEs of the following form:

where G C R" is an open set and Vx is the gradient vector. Boundary conditions for the solution V(x) are imposed upon. Typically, there is no classical solution of (5.2) and the boundary conditions. In this paper we assume that V is locally Lipschitz; i.e., the restriction of V to any compact subset of G satisfies a Lipschitz condition. Then V is differentiable at almost every x £ G. If (5.2) holds for almost all x, then V is called a generalized solution. Unfortunately, there is a severe lack of uniqueness among generalized solutions, as seen from the following simple example. Example 5.1. Let n = l,H(p) = \p\2 — 1,G — (—1,1) with the boundary condition Vx(±l) = 0. If V is a generalized solution, then Vx(x) = ±1 for almost all x & G. There are infinitely many piecewise-linear generalized solutions and no classical solution. Viscosity solutions. A good introduction to this topic is found in [4], Chapters 1 and 2. A function V is a viscosity solution of (5.2) if the following pair of inequalities holds for all x € G:

where D+V(x),D V(x) are, respectively, the superdifferential and subdifferential at x. Every locally Lipschitz viscosity solution V is a generalized solution. Intuitively, the unique viscosity solution of (5.2) and the boundary conditions should be regarded as the "relevant" generalized solution. In Example 5.1 this solution is V(x) = \x\ — 1 which is convex on the interval (—1,1).

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Remark 5.1. In the viscosity solution literature, the inequalities in (5.3) are typically reversed. See [4] p.31. Thus a viscosity solution of H = 0 as we have defined it is a viscosity solution of — H = 0 according to [4]. The convention which we chose seems slightly more natural from a control theory viewpoint. Semiconvex solutions. In order for equation (5.2) (or (5.5) in the timedependent case) to be max-plus linear, the following assumption is needed: A function is called semiconvex on G if, for every compact set K C G, there exists a constant C (which may depend on K) such that (x) + \C\x 2 is convex on K. Every semiconvex function is locally Lipschitz. When (5.4) holds, every semiconvex generalized solution of (5.2) is a viscosity solution. Conversely, under some additional technical assumptions on H, every viscosity solution is a semiconvex generalized solution [4], p.78. Max-plus linearity. If Vi and V% are semiconvex on G, then is semiconvex on G. Similarly, for any constant c,

is semiconvex if V is semiconvex. Thus the class of semiconvex functions on G is closed under max-plus addition and scalar multiplication. If V\ and Vg are semiconvex generalized solutions of (5.2), then V\ ® V-2 can be shown to be a semiconvex generalized solution. Since (5.2) involves Vx and not V, c <8> V is a semiconvex generalized solution if V is. Thus, the PDE (5.2) is max-plus linear on the class of semiconvex functions. Time-dependent PDEs. In the sections to follow, we will encounter PDEs of the form

which are of Hamilton-Jacobi-Bellman (HJB) type (Section 5.3.) We take G = . Thus the only boundary condition imposed is an initial condition We will be concerned with viscosity solutions V(t, x) of (5.5)-(5.6) which are locally Lipschitz in (t, x) and semiconvex in x uniformly for t in any finite interval.

5.3

Time-Dependent HJB Equations

Consider a deterministic optimal control problem on the time interval 0 < s < t, with states xs 6 1R", controls vs € U and state dynamics

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Given an initial state XQ = x, the goal is to choose a control function v. to maximize

In dynamic programming, the value function V has a key role:

Under suitable assumptions on F, L, and the control set U, V(t, x) is finite. Moreover, V is a viscosity solution to the corresponding HJB equation (5.5) where

see [4], Chap. 3. Since F(x, v) • p + L(x,v) is a linear function of p, H(x,p) is a convex function of p, as in (5.4). In this paper, we shall consider the case when U = Wn and

Then the HJB equation (5.5) has the form

where a = era'. The function <j) in the initial data (5.6) is the same as the "terminal cost" function in (5.8). Let us assume that /,&,£ are smooth (class C2) and that their first-order partial derivatives are bounded. We also assume that a(x) is bounded, t(x) is bounded above and 4>(x) is semiconvex and bounded above. The value function V is bounded above, for t in any finite interval, and V is a locally Lipschitz viscosity solution of the HJB equation (5.12) with initial data <j>. Under some additional assumptions, V is the unique viscosity solution to (5.12)-(5.6) such that Vx\ grows at most linearly with x\ as \x\ —» oo. See [21], Sec. 4 for the case when a is constant, and for related results see [5, 14]. A "range of dependence" property [9], Lemma 2.1 implies that, for (t,x) in any given compact set K, there exists RI (depending on K) such that V(t, x) is unaffected by values of .£(£) and (£) for |£| > RI. By modifying ^(£)>(£) f°r |£| large so that £, are bounded functions, a "standard" uniqueness theorem [4], p. 56 for bounded continuous viscosity solutions applies. The value function V is a semiconvex function of x. See [9], Thm. 4.1 and the related result in [4], p. 163. Let us merely indicate a method of proof. It suffices to assume that is smooth and that for every RI > 0 there exists F such that

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127

The matrix of partial derivatives Jxx(t, x; v.) is found by differentiating (5.8) twice with respect to the components of x = XQ. For |z < R, it suffices to consider controls v with L2-norm bounded by some M, depending on R and t. Then there exists C such that

By taking the sup over v., V(t,x) + |C"|x|2 is convex on {\x\ < R}. The solution operator. Let S denote the class of functions (j> on R™ which are semiconvex and bounded above. We recall that S is closed under the operations of max-plus addition and scalar multiplication. For t > 0, the solution operator St is defined by

where V is the value function in (5.9). As noted above, St € S for every (j> e <S. The dynamic programming principle implies the semigroup property

Moreover, the solution operator St is max-plus linear:

(b)

St(c®(l>) = c®St.

The max-plus additivity of St in (5.15)(a) is an easy consequence of (5.13), and (5.15)(b) is immediate.

5.4

Vanishing Viscosity, Small Random Perturbations

Instead of the time-dependent first-order PDE (5.5), consider the "perturbed" equation

with initial data (as in (5.6))

Equation (5.16) is of second-order parabolic type. Under suitable assumptions, (5.16)-(5.17) has a unique classical (smooth) solution with appropriate boundedness or growth conditions as \x\ —» oo. The parameter e has a role similar to a viscosity parameter in PDEs of fluid flow. By viscosity solution methods it can be shown that V£ tends to V as e —> 0, where V(t, x) is the unique viscosity solution to (5.5)-(5.6) satisfying the corresponding boundedness or growth conditions. See [4], Sec. 6.3 and [11], Sec. 7.3.

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Example 5.2. Let us now consider the special case

Then (5.16) is quadratic in V£ and can be linearized by the following exponential transformation:

The linear narabolic PDE satisfied bv u£ is

The partial differential operator on the right side of (5.20) is the generator of a Markov diffusion process if, governed by the stochastic differential equation

where ws is a Brownian motion. If (f> is continuous and bounded above, then u£(t, x) has the stochastic representation

where E denotes expectation and the subscript refers to the initial data #§ = x. As e —> 0, Vs (t, x) tends to the value function V(t, x) in (5.9) for the case when a(x) is the identity matrix and (.(x) = 0 in (5.10)-(5.11). Equation (5.22) is a small random perturbation of the ODE

In this special case, the dynamics (5.7) of the control problem become

If the control vs is regarded as a disturbance, then (5.24) is the "undisturbed" system (with vs = 0). By taking small stochastic disturbance limits as e —> 0 of this kind, a connection is made between stochastic (risk sensitive) and deterministic (robust/floo-control) approaches to disturbance attenuation problems in control theory. See [8, 15, 16].

Chapter 5. Max-Plus Linear Partial Differential Equations

5.5

129

Max-Plus Fundamental Solutions

Let us assume that the matrix a(x) in (5.10) has a bounded inverse (x) is semiconvex for t > 0, even though <j) is merely continuous and bounded above ( is not necessarily semiconvex). This is seen by representing V(t, •) — St4> as a max-plus convolution of (j> and the max-plus fundamental solution I(t, x, £), denned as follows:

with L(x,v) as in (5.11). Prom (5.8), for t > 0

By an argument similar to [9], Sec 4.1), I(t, •,£) is semiconvex. By using a "range of dependence" argument, semiconvexity of St4> follows. Small time asymptotics. The behavior of / for small t is of interest. A good asymptotic approximation for I as t —> 0 should provide good approximations to St<j> for small t if cj) is smooth. In particular, (f> may be one of the quadratic basis functions V'i to be considered in Section 5.7. For simplicity, let us assume as in Example 5.2 that cr(x) is the identity matrix and that f.(x) = 0. Then the max-plus fundamental solution is a deterministic analogue of the fundamental solution pe(t, x,£) of the linear parabolic PDE (5.20). Let us write xs = xs + £s, where

Also, let

A calculation gives

where £s = d£s/ds. A calculus of variations argument shows that I(t, x, £) is of order t as t —> 0. Thus (5.27) gives the first two terms in an asymptotic series expansion for I.

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Wendell H. Fleming

Max-Plus Probability and Stochastic Calculus

The Maslov idempotent calculus provides a framework in which a variety of asymptotic problems, including large deviations of stochastic processes, can be considered. The asymptotic limit is often described by a deterministic problem of calculus of variations or control. However, the limit still retains a "stochastic" interpretation if the traditional probabilistic and stochastic calculus frameworks are replaced by "max-plus" probability and stochastic calculus. In the max-plus framework, firstorder HJB equations play a role similar to that of second-order linear parabolic PDEs in the theory of Markov diffusion processes. For an introduction to max-plus probability see [1, 2, 7, 24] and references therein. A good introduction to the role of max-plus probability in nonlinear Hx control theory appears in Appendix C of [13]. Let us begin with the idea of max-plus expectation. Let v denote an "uncertainty," with v E fi and Q a function on fi with

We call — Q(v) the likelihood of v. The max-plus expectation of a function J on fi which is bounded above is

Max-plus expectation is easily seen to be a max-plus linear operation. Max-plus expectations often turn out to be limits of ordinary expectations of exponential functions of J, under suitable scalings. We illustrate this with two examples. that

Example 5.3. Let 0 be a finite set, and let v have probability pe(v), such

Then

where Ee is expectation under probability p e (-)Example 5.4. Let ft = L 2 ([0,i];E m ) and for v. e ft let

Let J(v.) = J(x.), where

Chapter 5. Max-Plus Linear Partial Differential Equations

131

for given t and x — x0, where

as in (5.8)-(5.10). Prom (5.11), E+J = V(t,x) with V(t,x) the value function in (5.9). The analogue of (5.30) is a Freidlin-Wentzell type large deviations result in which the deterministic perturbation a(xs)va to (5.24) is formally replaced by the small random perturbation s^a(xa)dws/ds. See [12]. In the rest of this section we indicate some elements of a "max-plus stochastic calculus." Max-plus analogues of stochastic integrals were considered in [6] and max-plus martingale techniques were used in [22, 23]. In the terminology of [2] Example 5.4 provides an example of a Bellman process. We consider only the case when the likelihood of a disturbance v. is — Q(v.}, where

We assume that q is of class C2. Moreover, \v\ 1q(v) —> +00 as |i;| -> oo and the eigenvalues of the Hessian matrix qvv(v) are bounded below by a positive constant. Let q*(0) be the convex dual

The max occurs at v*(0) = (qv}~1(e). In Example 5.4, q(v) = \\v\* and q*(ff)

-

\\W-

We begin with max-plus conditional expectations. See [7], Sees. 2.2, 2.4 for a related discussion. In this brief sketch, we omit some technical details needed to make statements completely precise. For 0 < T < t we identify v. with the pair (VT, v') which are the restrictions of vs to the intervals [0, T) and [T, t), respectively. The max-plus conditional expectation of J given VT is

From (5.34) we then have In the special case considered in Example 5.4, (5.37) is equivalent to the dynamic programming principle. Max-plus martingales. In analogy with the usual definition of martingale, let us call Mt = Mt(v.} a max-plus martingale if

Example 5.5. Let 0(v.) be progressive in the sense that vr = vr for almost all r e [0, s] implies 6(v.)r = 0(v.)r for almost all r 6 [0, s], 0 < s < t. Let

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Wendell H. Fleming

where 6S = 6(v.)s. Then

The integrand is nonpositive and is 0 when vs = qg(&s)- Thus, Mt is a max-plus martingale provided that (5.33) with initial data x0 = x has a unique solution with this choice of vs. In particular, this is true if 6(v.)s = 0(xs) where #(•) is bounded and Lipschitz. The following analogue of the Ito stochastic differential rule holds. Let

Then for every g of class C2 with gx,gxx bounded,

where Mt is the max-plus martingale obtained by taking Os = cr(xs)gx(xs) in (5.39). Change of drift. The function / in (5.33) plays a role similar to the drift in an Ito-sense stochastic differential equation. Let us suppose that xt in (5.33) also satisfies a similar equation with drift /:

If we assume that cr(x) has a bounded inverse a Vt are related by

1

(a;), then the disturbances vt and

Let Q(v) = ±H 2 . Then q*(6) = ±|0|2 and

with Mt as in (5.39). Let J = J(x.) as in (5.32) and E+J = sup5_ [J(x.) - Q(v.)]. Then

This is a max-plus analogue of a Ginsanov transformation in Ito stochastic calculus.

Chapter 5. Max-Plus Linear Partial Differential Equations

5.7

133

Max-Plus Basis Expansions

We recall that 0 is semiconvex on Rn if for every R > 0 there exists CR such that (j>(x) + ^CR\X\Z is convex on the ball {|z| < R}. To simplify the discussion, let us assume that C = CR does not depend on R. Thus tp(x) = (p(x) + ^C^a;!2 is convex on Rn. Moreover, by replacing C with C + 6 for any S > 0, we may assume that \x\~1ip(x) —> +00 as \x\ —> oo. Prom the well-known convex duality representation of if), the following semiconvex duality representation of is obtained [9], Sec. 4. For each £ £ R", let

We call —o(-) the semiconvex dual of . The dual formula to (5.43) is

Let {&},i — 1,2,..., be a countable dense subset of R", and write ipi = ip£nai = a (&)- By (5.44), cf> has the max-plus series representation in terms of the quadratic basis functions j: By truncating this series after N terms, <j> is approximated for large AT by a finite max-plus linear combination of basis functions. Since the solution operator is maxplus linear, for large N,

Similarly, Sti}>i is approximated by a max-plus linear combination

We omit details which make these statements more precise. After substituting in (5.46),

This procedure reduces the solution of the HJB equation (5.12) with initial data to two steps: (1) Find the max-plus basis coefficients for ; (2) find the coefficients bij(t) in the max-plus expansion (5.47). This can be done "off-line" before the initial data (j>(x) are known. In the next section, this idea is applied to obtain an algorithm for an approximate solution of some HJB equations which arise in deterministic nonlinear filtering.

134

5.8

Wendell H. Fleming

Deterministic Nonlinear Filtering

In filtering theory the goal is to obtain good estimates for the state xs of a system based on partial, disturbance error corrupted observations of states at times before s. Let us assume that observations are made at discrete times 0 < t\ < i2 < • • • • Let Zj denote the observation at time tj and es denote an estimate for xa. In nonlinear filtering theory, xs is often modeled stochastically as a Markov process governed by a stochastic differential equation

with ws a brownian motion independent of the (random) initial state x0. The observations are of the form

where z/,, Cfc € Km and {£fc} are independent Gaussian random vectors with mean 0 and where the covariance matrix is the identity matrix. Typical criteria considered for choosing an estimate eg are minimum mean square error or maximum likelihood. To obtain such estimates, the conditional distribution of xs given observations Zk for tk < s is needed. The conditional distribution satisfies a forward Kolmogorov PDE between observations and is updated by Bayes' formula when a new observation z^ is obtained. The filter estimates must be made in real time. In [18] a technique was introduced according to which the conditional density of xs is expanded in terms of basis functions, and the Kolmogorov PDE is solved "off-line" for basis function initial data. This procedure moves a substantial part of the calculations "off-line." There is an alternative deterministic approach to filtering, in which the term cr(xs)dws in (5.49) is replaced by
where xs is the solution to (5.33) with Xt = x. Note that the initial state XQ is unknown and xs is given at the final time s = t, in contrast to Section 5.3. Given nhsprvat.inns y.t. fnr ft. •f f

Ipt

Then V(t, Xt) can be regarded as the likelihood of Xt. Various estimates es for the state xs based on this likelihood function can be defined. The Mortensen estimate chooses es to maximize V(s, a;) as a function of x. In [10] a minimax estimate es is chosen to minimize the maximum over x of fj,\x — e\2 + V(s,x), where /x > 0 is a parameter. For small enough /i, the minimax estimate is a robust filter in a sense defined in [20].

Chapter 5. Max-Plus Linear Partial Differential Equations

135

The likelihood function V has the following properties:

If we set Vk(x) = V(f£,x) for k - 1,2,... and VQ(x) = $(x), then

where St is the solution operator. By using max-plus basis expansions as in Section 5.7, one obtains the following analogue of the computational algorithm in [18] for the stochastic model. For simplicity, let tk = 6k for fixed 6. For the basis functions tpi, Stipi is precomputed off-line for 0 < t < 6. At each step k, Vk is approximated by a max-plus linear combination of basis functions fa, and then V(£j~+1,-) — SgVk is found approximately from (5.46) and (5.55). Finally, Vfc+i is obtained by (5.54)(iii).

Dedication To H.T. Banks on the occasion of his 60th birthday.

Bibliography [1] M. Akian, "Densities of idempotent measures and large deviations," Transactions of the American Mathemetical Society, 351, pp. 4515-4543, 1999. [2] M. Akian, J.-P. Quadrat and M. Viot, "Bellman processes," in Lecture Notes in Control and Information Science, No. 199, G. Cohen and J.-P. Quadrat, ed., Springer-Verlag, Berlin, pp. 302-311, 1994. [3] F. Baccelli, G. Cohen, G.J. Olsder and J.-P. Quadrat, Synchronization and Linearity: An Algebra for Discrete Event Systems, John Wiley and Sons, New York, 1992. [4] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhauser, Boston, 1997.

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[5] M. Bardi and F. Da Lio, "On the Bellman equation for some unbounded control problems," Nonlinear Differential Equations and Applications, 4, pp. 491-510, 1997. [6] F. Bellalouna, Processus de decision min-markoviens, Ph.D. thesis, University of Paris-Dauphine, 1992. [7] P. Del Moral and M. Doisy, "Maslov idempotent probability calculus, I," Theory of Probability and Its Applications, 43, pp. 562-576, 1998. [8] W.H. Fleming and W.M. McEneaney, "Risk sensitive control on an infinite time horizon," SIAM Journal on Control and Optimization, 33, pp. 1881-1915, 1995. [9] W.H. Fleming and W.M. McEneaney, "A max-plus based algorithm for a Hamilton-Jacobi-Bellman equation of nonlinear filtering," SIAM Journal on Control and Optimization, 38, pp. 683-710, 2000. [10] W.H. Fleming and W.M. McEneaney, "Robust limits of risk sensitive nonlinear filters," Mathematics of Control Signals and Systems, 14, pp. 109-142, 2001. [11] W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions, Spring-Verlag, Berlin, 1993. [12] M.I. Freidlin and A.D. Wentzell, Random Perturbations of Dynamical Systems, Springer-Verlag, Berlin, 1984. [13] J.W. Helton and M.R. James, Extending .H"00 Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives, SIAM, Philadelphia, PA, 1999. [14] H. Ishii, "Comparison results for Hamilton-Jacobi equations without growth condition on solutions from above," Applicable Analysis, 67, pp. 357-372, 1997. [15] M.R. James, "Asymptotic analysis of nonlinear risk-sensitive control and differential games," Mathematics of Control Signals and Systems, 5, pp. 401-417, 1992. [16] H. Kaise and H. Nagai, "Bellman-Isaacs equations of ergodic type related to risk-sensitive control and their singular limits," Asymptotic Analysis, 16, pp. 347-362, 1998. [17] G.L. Litvinov and V.P. Maslov, Correspondence Principle for Idempotent Calculus and Some Computer Applications, in Idempotency, J. Gunawardena, ed., Publications of the Newton Institute, 11, Cambridge University Press, Cambridge, UK, pp. 420-443, 1998. [18] S. Lototsky, R. Mikulevicius and B.L. Rozovskii, "Nonlinear filtering revisited: A spectral approach," SIAM Journal on Control and Optimization, 35, pp. 435461, 1997.

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[19] V.P. Maslov and S.M. Samborskii, eds., Idempotent Analysis, Advances in Soviet Mathematics, 13, AMS, Providence, HI, 1992. [20] W.M. McEneaney, "Robust//^ filtering for nonlinear systems," Systems Control Letters, 33, pp. 315-325, 1998. [21] W.M. McEneaney, "Uniqueness for viscosity solutions of nonstationary Hamilton-Jacobi-Bellman equations under some a priori conditions (with applications)," SIAM Journal on Control and Optimization, 33, pp. 1560-1576, 1995. [22] A.A. Puhalskii, "Large deviations of semi-martingales: A maxingale approach," Stochastics and Stochastics Reports, Part I, 61, pp. 141-243, 1997; Part II, 68, pp. 65-143, 1999. [23] A.A. Puhalskii, Large Deviations and Idempotent Probability, Chapman and Hall/CRC Press, Boca Raton, FL, 2001. [24] J.-P. Quadrat, Min-plus probability calculus, Actes 26 erne Ecole de Printemps d'Informatique Theorique, Noirmoutier, 1998. [25] J.A. Sethian, "Fast marching methods," SIAM Review, 41, pp. 199-235, 1999.

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Chapter 6

Geometric Theory of Output Regulation for Linear Distributed Parameter Systems C. I. Byrnes,* D. S. Gilliarrd and V. I. Shubov^ Abstract Our first main objective is to present a brief introduction to the geometric theory of output regulation for distributed parameter systems. In this introduction we include references to some of the literature, as well as a survey of our recent work in this area. In particular, we describe our extension of the characterization, well known in finite-dimensional theory, of solvability of the state and error feedback regulator problems in terms of solvability of a pair of operator equations, referred to as the regulator equations. We present our main results for bounded input and output operators and finite-dimensional exosystems. Next we present an extension of these results to the class of regular linear systems with unbounded input and output operators obtained in our most recent work. We also present a result establishing that a class of boundary control systems governed by the heat equation on a bounded domain belongs to the well-known class of regular linear systems. Thus we provide a large class of systems for which our regulator theory applies. Next, the results for bounded input and output operators are applied to derive a simple formula for the solution of the regulator equations for retarded systems. Finally, we discuss several directions of future research in this area.

'Department of Systems Science and Mathematics, Washington University, St. Louis, MO 63130. E-mail: [email protected] t Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409. E-mail: [email protected], [email protected]

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C. I. Byrnes, D. S. Gilliam and V. I. Shubov

Introduction

This chapter is concerned with the development of a systematic methodology for the design of feedback control schemes capable of shaping the response of infinitedimensional dynamical systems. Among the most important design objectives that this entails is the problem of output regulation: asymptotic tracking, stabilization or disturbance attenuation or rejection for distributed parameter systems. In particular, one of the central problems in control theory we address is the control of a fixed plant in order to have its output track a reference signal (and/or reject a disturbance) produced by an external generator or exogenous system, thereby developing an extension of the geometric theory constructed in [24] for lumped nonlinear systems. Generally two versions of this problem are considered. In the first, the state feedback regulator problem, the controller is provided with full information of the state of the plant and exosystem, while in the second, only the components of the error are available for measurement. Our approach in analyzing these problems for linear distributed parameter systems follows the pioneering work for linear finite-dimensional systems carried out by numerous authors during the 1970s and 1980s (cf. Davison [15], Francis and Wonham [17], Francis [16], Wonham [42]). In particular, Francis [16] showed that the solvability of a multivariable linear regulator problem corresponds to the solvability of a system of two linear matrix equations, called the regulator or Francis equations. Hautus [19] gave necessary and sufficient conditions for solvability of the Hautus equations which contain the regulator equations as a special case. Indeed, for finite-dimensional linear systems, the Hautus conditions given in [27] state that no eigenvalue of the exosystem is an invariant zero of the plant. In 1990, Byrnes and Isidori [24] extended the results of Francis to finite-dimensional nonlinear systems for the case when the plant is exponentially stabilizable and the exosystem has bounded trajectories that do not trivially converge to zero. In particular, they give necessary and sufficient conditions for solvability of the regulator problem in terms of solvability of a pair of nonlinear regulator equations. The results in [24] are primarily based on geometric methods appealing to the center manifold theorem (see also [3, 32, 36]). There are also some early works on the regulator problem for distributed parameter systems published in the early 1980s [29, 30, 33, 34]. Of these works the most closely related to our work is that of Schumacher [33, 34]. Schumacher considers plants whose dynamics are governed by discrete spectral operators whose generalized eigenvectors form a complete set. In particular, the proof employed by the author requires that the state operator satisfies the spectrum decomposition property (cf. [25, 14]), the spectrum determined growth condition and a controllability condition that implies the stabilizability of the plant with a finite-dimensional controller. Just as in [7], the reference signals and disturbances considered in [33, 34] are assumed to be generated by a finite-dimensional exosystem. A further observability condition is imposed on the composite system consisting of the plant and exosystem (equivalent to our condition of detectability in [7]). Under these assumptions, and in the case of bounded input and output operators, it is shown that a sufficient condition for the design of a finite-dimensional controller that solves the error feedback regulator problem is that there exists a solution to a certain set of

Chapter 6. Geometric Theory for Linear Distributed Parameter Systems

141

operator equations which we call the regulator equations. Construction of a finitedimensional controller, as opposed to an infinite-dimensional controller, is simply a matter of applying all the assumptions imposed on the system to obtain finitedimensional approximations to infinite-dimensional operators using eigenfunction approximation (see for example [14], problems 5.22, 5.23, page 261 which are based on the work in [34, 35]). We also mention the recent work of Pohjolainen [31] who provides sufficient conditions for the existence of finite-dimensional controllers solving problems of tracking and disturbance rejection for stable parabolic systems (the system operator generates a holomorphic semigroup) in the case where the exosystem is a finite-dimensional linear system with a complete set of eigenfunctions (a matrix representation is diagonalizable). As pointed out in [31] this does not account for disturbances or signals to be tracked that are of the form tp sm(at) for p > 1. Not surprising, hidden in this work is also a solution based on the solvability of the regulator equations. Consider, for example, page 486 of [31], the discussion from formula (15) up to (17). In our work [7] we have extended the geometric methods introduced in [16] and [24] for solving the state and output feedback regulator problems for infinitedimensional linear control systems, assuming that the control and observation operators are bounded on the Hilbert state-space. As we have already mentioned, our objective in this work is to develop a systematic approach to the design of feedback control schemes. In [7] we derive the regulator equations for a class of distributed parameters systems, obtaining an operator Sylvester equation, and characterize the solvability of both state and error feedback regulator problems in terms of solvability of these regulator equations. For systems described by partial differential equations the regulator equations typically reduce to elliptic boundary value problems that can be solved off-line and, as discussed in Section 6.6, for retarded functional differential equations the regulator equations reduce to a finite-dimensional linear system of equations. There are, of course, several technical difficulties that arise in extending the work in [16, 24] to the distributed parameter case. These difficulties include the fact that the phase space is infinite-dimensional; the state operator is unbounded and consequently only densely defined; there is no direct analogue of the Jordan decomposition and consequently care had to be exercised in dealing with the spectra of certain composite systems; and the usual invariance concepts which are all equivalent in the finite-dimensional linear case are no longer equivalent (cf. [11, 12, 43]). The chapter is organized as follows. In Section 6.2 we present a brief overview of the results found in [7] for the state and error feedback regulator problems for bounded inputs and outputs. In Section 6.3 we take a closer look at the regulator equations, showing how the first regulator equation can always be solved, so that the main ingredient is the error zeroing described by the second regulator equation. We give explicit formulas for the solution of the first regulator equation. This section also contains a simple proof of the main theorem of [7]. Section 6.3 also contains a statement of the error feedback problem and the corresponding characterization and also contains a characterization of solvability for the regulator equations. Namely, under an additional condition of detectability, it is shown in [7] that the regulator equations are solvable, for a fixed plant and exosystem and all disturbance and

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C. I. Byrnes, D. S. Gilliam and V. I. Shubov

reference signals, if and only if no eigenvalue of an exosystem is a transmission zero of the plant (see also [33, 34]). For general distributed parameter systems the concepts of transmission and invariant zeros do not coincide (see [43]). Section 6.4 contains a discussion of our recent results, contained in [5], extending the results of Section 6.2 to the class of regular linear systems corresponding to the case of unbounded input and output operators. This work relies heavily on the development of system theoretic constructs, such as feedback theory for systems with unbounded perturbation terms as can be found in [12] and [38]-[41]. A difficulty with applying these results is that one must verify that the plant corresponds to a regular linear system. A substantial research effort needs to be carried out to verify that many important boundary control problems are given by regular systems. A step in this direction is outlined in Section 6.4 which contains a statement of some of the main results from the recent work [18] in which we show that a large class of boundary control problems for the heat equation on bounded domains in higher-dimensional Euclidean spaces is given by regular linear systems. In Section 6.6 we apply the results of Section 6.2 to a class of systems governed by retarded delay differential equations. We show, for example, that the regulator equations, in this case, reduce to a finite-dimensional linear system of equations which can be solved off-line. Finally in Section 6.8 we provide a discussion leading to several important future directions for research in regulator theory for linear distributed parameter systems.

6.2

Bounded Input-Output

In this section we outline our results from [7] which provide necessary and sufficient conditions for solvability of the output regulation problem for linear distributed parameter systems with bounded input and output operators in terms of solvability of a pair of regulator equations. We also present frequency domain solvability criteria for the regulator equations. In [7], we considered systems of the form

with finite- or infinite-dimensional input space U, output space Y and infinitedimensional separable Hilbert state-space Z. A is assumed to be the infinitesimal generator of a strongly continuous semigroup T(t) on Z. It is assumed that the reference signal (signal to be tracked), as well as the disturbance D(t), is generated by a finite-dimensional exogenous system acting in a finite-dimensional vector space W:

where Q e £(W, Y) and 7 € £(W, Z).

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143

In this section we assume that B and C are bounded operators, i.e., B G £(U,Z) and C € £(Z, Y). We also consider the following standard assumptions.

Problem 6.2.1. State Feedback Regulator Prvblem. Find a feedback control law in the form

such that K € £(Z, U),L& C(W, U) and (l.a) the system z(t) = (A + BK)z(t) is stable, i.e., (A + BK) is the infinitesimal generator of an exponentially stable Co semigroup, and (l.b) for the closed-loop system

the error for any initial conditions ZQ e Z in (6.3) and WQ € W in (6.5). Assumption 6.2.1. HI. For the finite-dimensional exosystem o~(S) C C£ (the closed right half-plant Here and below we use the notation o~(M) for the spectrum of an operator A Also, by p(M) we will denote the resolvent set of M. H2. The pair (A,B) is exponentially stabilizable; i.e., there exists K € £(Z,l such that A + BK is the infinitesimal generator of an exponentially stable (. semigroup TAK(t). H3. The pair

is exponentially detectable; i.e., there exists G 6 JC(Y, Z x W) with

such that

is the infinitesimal generator of an exponentially stable Co semigroup.

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The first main result from [7] concerning the solvability of the regulator problem is contained in Theorem 6.1 which gives necessary and sufficient conditions for the solvability of the state feedback regulator problem. Theorem 6.1. Let HI and H2 hold. The linear state feedback regulator problem is solvable if and only if there exist mappings II € £(W, Z) with Ran(H) C T>(A) and F € £.(W, U) satisfying the "regulator equations,"

In this case a feedback law solving the state feedback regulator problem is given by where 3C is any exponentially stabilizing feedback for (A, B). The regulator equations are a system of Sylvester-type operator equations. For the examples considered in [7], these operator equations boil down to a coupled system of two-point boundary value problems subject to extra constraints. For boundary control systems as considered in Section 6.4 the regulator equations provide an elliptic boundary value problem typically having a distributional forcing term. For delay systems these equations reduce to solving a linear finite-dimensional system of equations.

6.3

The Regulator Equations and Proof of the Main Theorem

Rather than seek u = Kz + Tw, it is convenient to replace A by AK = (A + BK) (for any stabilizing feedback K) and seek u = Tw. In this case the regulator equations become

It is easy to see that there is no loss of generality in making this change. The main advantage of this change is that now the state feedback is given only in the exosystem variable w. The main point here is that in this development we are not concerned with the problem of stabilizing the plant. We do not suggest that this is an unimportant part of the problem; rather, we note that there is a vast literature available for obtaining stabilizing state feedback laws. For us any such stabilizing feedback will suffice. With this understanding consider the closed-loop composite system

Chapter 6. Geometric Theory for Linear Distributed Parameter Systems

145

Here A generates a Co semigroup

in the Hilbert space X = Z x W and the spectrum of A (in X) decomposes as

Furthermore, A satisfies the spectrum decomposition condition at ß — 0. Namely, we have Here At this point there are (at least) three different but related directions that we can take to prove the main Theorem 6.1: (1) Apply spectrum decomposition to get invariant subspaces; (2) directly solve the Sylvestor equation; (3) diagonalize the composite operators in (6.15). Let us briefly describe what we mean by this and then give a simple proof of Theorem 6.1. Remark 6.3.1. The First Regulator Equation. 1. Decomposition of Spectrum. The composite state operator A satisfies the spectrum decomposition condition at ß = 0, as defined in [14], pages 71 and 232. Thus we can conclude that X decomposes into the direct sum

where V+ are invariant subspaces under the corresponding Co-semigroup TA (t) and also under (SI - A ) – 1 for s e p(A). Also V+ C D(A), AV+ C V+, A(D(A) n V–) C V– and dimV+ = dim(W). A restricted to V+ has all its eigenvalues in C+ (i.e., they coincide with the eigenvalues of S), while JA restricted to V– is exponentially stable. Therefore we can define a linear operator II € C(W, Z) by the condition

and we have Ran(H) C D(A). From the structure of A it is easy to see that

For every WQ 6 W, from the A invariance of V+ we can write

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C. I. Byrnes, D. S. Gilliam and V. I. Shubov

This implies and therefore the first regulator equation, (6.10), holds with F = (L + KIT). 2. Direct Solution of Sylvester Equation. Recall that our assumptions are B, P, C are bounded, the exosystem is finite-dimensional, and since AK is stable, the spectrum of A satisfies a(A) =
where 7 C C is a simple closed positively oriented curve in P(AK) and cr(S) C Int(7). This formula is readily verified directly using the Cauchy integral formula and the fact that (XI — AK}~I is analytic on and inside the curve 7. Under our assumptions we can also express the solution as in [7]

This formula is useful for the case of an infinite-dimensional exosystem. 3. Diagonalization of A. Next we show that the first regulator equation is related to the diagonalization of the upper triangular block matrix operator A. To this end let us define the operator

We have

where the last equality holds if and only if the 2 x 2 position of the middle term is zero. Thus 3 diagonalizes A if and only if II satisfies the first regulator equation.

Proof of Theorem 6.1 We now present a proof of Theorem 6.1. Based on the fact that solutions II solving the first regulator equation (6.13) exist for all F we need only show that the regulator problem is solvable if and only if we can find F so that the second regulator equation is solvable; i.e., there is a F so that Cli = Q.

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147

Proof. From part 3 of Remark 6.3.1 (or even from the 7j\.(t) invariance of V + ) we see that

Thus for any initial condition

° € X = Z x W, we have the solution

Applying [C, —Q] we have

Prom this equation and the fact that Tjf(£) = exp(-Axi) is an exponentially stable semigroup, it is easy to see that

The theorem is proven. Error Feedback Regulator Problem In practice one usually does not have the entire state of the system to build a static feedback as in the last section. In this case a more general version of the regulator problem is considered - the error feedback regulator problem. This problem can be stated as follows. Problem 6.3.1. Error Feedback Regulator Problem. Find an error feedback rnrt.trnllpr nf th.p fnrm.

where X(t) e 3t for t > 0, X is a Hilbert space, G 6 £(Y,X), H e £(X,U) and F is the infinitesimal generator of a Co-semigroup on £ with the properties that (2.a) The system

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C. I. Byrnes, D. S. Gilliam and V. I. Shubov

is exponentially stable when w = 0, i.e.,

is the infinitesimal gen-

erator of an exponentially stable Co semigroup. (2.b) For the closed-loop system

the error for any initial conditions Zo e Z, X(0] € X and Wo e W. Theorem 6.2 (Error Feedback Problem). Assume C, B, P, Q are bounded operators and conditions H1, H2 and H3 are satisfied. Then the Error Feedback Regulator Problem is solvable if and only if there exist mappings

solving the "Regulator Equations"

A controller in X e x = Z x W

in terms of H and F is given by

Here G is an exponentially stabilizing output injection.

Remark 6.3.2. 1. In other words, under the additional detectability condition, we obtain exactly the same necessary and sufficient criteria for solvability of the error feedback regulator problem as we have for the state feedback problem. The proof can be found in [7].

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149

2. We also point out that, just as in [33, 34], with additional assumptions on the state operator A (e.g., a discrete Riesz spectral operator) we can provide a finite-dimensional stabilizing feedback K, a finite-dimensional stabilizing output injection G and obtain a finite-dimensional controller (6.26).

Nonresonance Solvability Criteria for Regulator Equations We have seen that, under the hypotheses HI and H2, the first regulator equation is solvable for every T 6 £(W, U). In this section we show that for systems satisfying the extra hypothesis H3 (imposed also in Theorem 6.2) the second regulator equation is solvable if and only if a frequency domain nonresonance condition is satisfied. For simplicity in dealing with various properties of the transfer function, let us consider the case In this case the transfer function is an M x M matrix given by

We shall also assume that det G(s) ^ 0. In this case we can easily define the concept of transmission zero. Definition 6.3. SQ € C is a transmission zero «/detG(so) = 0. Theorem 6.4. Assume C, B, P, Q are bounded operators and HI, H2 and H3 hold. Then the Regulator Equations are solvable if no natural frequency of the exosystem is a transmission zero of the plant, i.e.,

We present a short proof of Theorem 6.4 for the case when S is diagonalizable. The result is valid for the more general case but the proof is a bit more lengthy. Remark 6.3.3. Under the additional assumption that S is diagonalizable in W with eigenvalues \j, eigenvectors $j and biorthogonal sequence Wj, we have

In this case the operator II has the representation

This formula follows immediately from the residue theorem. We note that this formula also can be extended to infinite-dimensional exosystems governed by Riesz spectral operators.

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C. I. Byrnes, D. S. Gilliam and V. I. Shubov

Proof. Recall the regulator equations (6.13), (6.14) and our representation (6.29) for II. Let us suppose that II is given by (6.29) and, in order to find F satisfying (6.14), we apply these expressions to w = $^ and then apply C to II and apply this to an eigenvector <&^ (in (6.28)). Then we set the result equal to Q$t. Thus we obtain the equation

forr$ f , forf = 1 , . . . ,fc. This equation can be written as

where G(\e) is the transfer function G(s) = C(sl — A) 1B evaluated at A£. The result follows immediately since we can solve for F$^ for all I provided det G(Xt] =£ 0. i = 1 k. In fact we have

6.4

Unbounded Inputs and Outputs

For unbounded B, even if A generates an analytic semigroup it may happen that (A + B) is not a generator. Further, for unbounded B and C (and even possibly K) expressions such as CB or BKC may make no sense. On the other hand there is considerable interest in the case of unbounded inputs and outputs that arise, for example, in the study of boundary control systems governed by partial differential equations. Typical applications include actuators and sensors supported at isolated points or on lower-dimensional hypersurfaces in, or on the boundary of, a spatial domain. Recently we have extended the results of Section 6.2 to the class of Regular Linear Systems developed in [13] and in [38]-[41]. In this case we consider regular linear systems

where CA is the A-extension of the observation operator C (see [39]) defined for z € P(CA) by A system is called regular provided the system is well-posed and satisfies the Regularity Condition. For complete details of what these concepts involve we refer the reader to [38]-[41] and present a short overview.

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151

1. Well-Posedness: System (6.31) is well-posed provided that B and C are admissible and there exists a transfer function G(s) = C&(sl — A)~*B for some (hence, for every) s € p(A) — this means that

2. Regularity: A well-posed system is called regular provided there exists a feedthrough term D g £(U, V), such that

While we will not go into detail concerning the questions of admissibility, etc., we still need to introduce certain terminologies. Let us define the space

and the space Z-i the completion of Z with respect to

\\z\\-i = \\(pl — A)

Then there are the dense embeddings

Assumption 6.4.1. 1. We assume that B € £(U, Z-\), C € £(Zi,Y) are admissible. 2. G(s) = Cj^(sI-A)~lB

exists for s e p(A).

3. (A,B) stabilizable; i.e., there exists K e £(Zi,U) so that (A + K^B) is a stable generator. In this setting we have extended the solvability results for both the state and error feedback regulator problems and, for example, after several important adjustments due to unbounded B and C, we have obtained the following analogue of Theorem 6.1. Problem 6.4.1. State Feedback Regulator Problem for Regular Systems. Find a feedback control law in the form

such that K 6 £(Z, U), L € £(W, U) and (l.a) the system z(t) = (A + BK/C)z(t) is stable; i.e., (A + BK\) is the infinitesimal generator of an exponentially stable Co semigroup, and

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C. I. Byrnes, D. S. Gilliam and V. I. Shubov

(1 .H^ fnr f.h.p rlnspA-lnnn siist.p.m.

the error where for some a < 0

We give the following result from [5]. Theorem 6.5. Under the above assumptions, the state feedback regulator problem is solvable if and only if there exist mappings II 6 C(W, Z C Z) and F s £(W, U) satisfying the "Regulator Equations"

Here the space Z is given by

If II and T satisfy the regulator equations, then a feedback law solving the problem of output regulation is given by u = K\z + (T — K^Ti^w.

6.5

Examples of Regular Linear Systems

One of the main difficulties in applying the results of the previous section is that one must show that all the hypotheses are satisfied. A particular problem is the lack of interesting examples of systems which have been shown to be regular. Recently, in [18] we were able to describe several general classes of boundary control problems for the heat equation in bounded domains in W1. In this section we briefly describe these systems. Let x = (x\, . . . , xn) denote a point in M" and 0 C R™ denote a bounded domain with piecewise C2-smooth boundary §. We consider a control system governed by the heat equation

For the equations (6.33), we also include initial conditions and associate boundary conditions and actuators and sensors described by unbounded operators in the Hilbert state-space. We consider two types of inputs and two types of outputs.

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153

a) Type 1 Input: Assume d£l = So U S, where So and S are piecewise (in a sense similar to our earlier definition of a C2-domain) (72-smooth (n — 1)hypersurfaces with piecewise C1-smooth (n — 2)-dimensional boundaries (S is assumed to contain the common boundary). We impose the boundary conditions

Here u(x, t) for x € S is the input function at time i, evaluated at the point x, and ^(x, t) = n(x) • Vz(x, t) is the derivative in the direction of the outward normal n(x) to dfl. It is possible that So = 0 and S = 9fi so that the control acts on the entire boundary. h} Tvnfi 2 Innnt: Let <9Q consist of a union of nnnovfirlanniner surfaces.

Each Sj is a (piecewise) smooth (n — 1)-dimensional hypersurface with a (piecewise) smooth (C2) boundary. We assume, in addition, that |Sj > 0 for j = 1,... , M, where |Sj| is the (n — l)-dimensional hypersurface measure of S j . The inputs are implemented through the boundary conditions as

In this case we have a finite-dimensional control space and

Type 1 Output: Let <9fi be represented as a union: d£l = So U S, where SQ and S have the same properties as SQ and S above. They may be the same subsets in the case of co-located actuators and sensors, or they may be different, representing non-co-located sensors and actuators. The output in this case is the trace of the state function on S:

d) Type 2 Output: Let where the partition has the same properties as in the Type 2 input above. We define the output vector

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C. I. Byrnes, D. S. Gilliam and V. I. Shubov by the formula

where by dax we denote the natural hypersurface measure on d£l. Remark 6.5.1. We point out that the output (6.40), (6.41) is physically natural. Indeed, measuring the "temperature" z(x, t) at the points x € <9fi would produce the trace considered in (6.39). However, in reality a measuring device could only measure the average value of the "temperature" over a small subregion §m C d£l. Therefore, instead of the function y(x, t) defined on part of the boundary, it produces a discrete set of values as given in (6.41). Combining the heat equation (6.33) with any of the above pairs of inputs and outputs we obtain four possible types of distributed parameter boundary control systems. In order to easily identify a pairing of Type 1 or Type 2 input with a Type 1 or Type 2 output we will use the notation (i, j), where i refers to the type of input and j refers to the type of output. Thus we have four possible systems denoted by (1,1), (1,2), (2,1) and (2,2) corresponding to the types of inputs and outputs. The most important system obtained in this way is the system (1,1). Theorem 6.6. With the choices U = ff~1/2(S) and Y = ff1/2(S) the system (1,1) is well-posed. Theorem 6.7. The system (1,1) is a regular linear system with feedthrough term T> = 0 provided

Moreover, the following estimates hold for the norm of the transfer function G(s). In Case (i), we have where CQ depends on SQ. In Case (ii), we have

where Co depends on SQ • It is clear from these estimates that

as long as a > 0, /? > 0 and, therefore, D = 0.

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155

The situation with the other three types of systems is less complicated. Namely, define 1. U = H~1/2(§),

Y = RK for (l,2)-type systems;

2. Y = M M , Y = H1/2^)

for (2, l)-type systems;

3. U = R M , Y = RM for (2,2)-type systems. Then we have the following result. Theorem 6.8. With the above choice of input and output spaces each of the systems (1,2), (2,1) and (2,2) is a regular linear system with a feedthrough operator T> = 0. Remark 6.5.2. In the proof of Theorem 6.6 presented in [18], it was only necessary to consider the case of system (1,1). The statement of Theorem 6.6 for all other combinations of inputs and outputs is an easy corollary of the main result for this case. This is because the second output is obtained from the first one via a bounded operator (mapping of ff~1/2(§) into MM denned in (6.41)) and the dual statement holds for the control (the second input space R^ is embedded into the first input space Hl'2(§) via the bounded operator, which maps (ui, ... ,UK) £ K^ into the function £}jLi ujXSj where xs, is the characteristic function of Sj). We introduced all four types of systems since they are all meaningful from the point of view of applications of distributed parameter systems. Remark 6.5.3. If for the Type 1 Input-Type 1 Output we define the corresponding input and output spaces as U = L2(B) and Y = L2(S), then the statement of Theorem 6.6 is correct and is an obvious corollary of the above stronger statement. This is due to the trivial bounded embeddings

Our assumptions about the input and output spaces lie on the borderline: we cannot increase the input space and reduce the output space so that the regularity property still holds (i.e., we cannot take U = H~1/2-0(S) and Y = H1/2+a(§) with a > 0 and (3 > Q).

6.6

Systems with Delays

As an important example of the methodology given in [7], in this section we describe some of the main points from our recent work [4]. Namely, in this section we specialize the results of Section 6.2 to the class of retarded delay differential systems

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of the form

In this presentation we adopt the notation and terminology presented in [14]. Namely, we denote by x(t) e R™ the state of the system, Aj e £(M"), for j = 1,... ,p. The input operator is given by B0 € £(Cm, C n ) for inputs u £ £2([0, r], C m ) for all r > 0, and the output operator is given by Co 6 £(C", C m ). It can be shown (see, for example, [14]) that the solution to (6.42)-(6.44) with u = 0 and D0 = 0 can be expressed as

We can also formulate this problem in a standard state-space format in the infinite-dimensional state-space (cf. [1, 2, 14])

with the inner product in Z given by

For u = 0, the solution Z(t) can be expressed in terms of a Co-semigroup of bounded operators T(t) in Z,

where x(-) is the solution in (6.45) and x(—s) = f(—s) for 0 < s < hp. The infinitesimal generator of 7(t) is the unbounded state operator A, given as

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157

If we also define the input operator T> € £(Cm, L2([-hp, 0], C")) by

the output operator 6 € C(Z, C fc ) by

and the disturbance operator T> 6 L2OC([Q, oo), Z) by

then with this notation we can write the retarded system as

Along with (6.55) and (6.56) we are given an exosystem

where Q e £(R fe ,R m ), 7 = \P*\ € £(R fe ,Z) and P0 e £(R fc ,R n ). In this case we want to examine more closely the regulator equations given in (6.10) and (6.11). For this development we will assume that dim(f7) = Nu < oo and dim(Y) = Ny < oo. We consider the natural decomposition of II from the form of the statespace 2. and under the assumption that we have fixed a basis in W with dim(W^) = k and R n :

where R € £(R fe ,R") so that Rj € R™ for j = 1,... , k and $ e L2([-hp, 0),R™ R fe ) so that $_,- 6 L2([-hp, 0), R") for j = 1,... , k and

Forre£(R f c ,M m ) we set

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C. I. Byrnes, D. S. Gilliam and V. I. Shubov

Using this notation the first regulator equation can be written as a system of equations for $ and R as

The requirement that II : Mfc —> T>(A) (the domain of A) imposes the further requirement that

The second regulator equation says that

The equation (6.62) subject to the initial condition from (6.63) can be solved to obtain x*rn\

r>JS8}

Thus we arrive at the important fact that the regulator equations for the retarded delay differential systems (6.42)-(6.45) reduce to a finite-dimensional linear system of equations for the unknowns R and F. Namely, we have

Under the additional assumption that the nonresonance condition in Theorem 6.4 is satisfied, it is possible, using hadamard and tensor products, to give an explicit formula for the solution of (6.65), (6.66) and hence give explicit formulas for the feedback thus solving the problem of output regular. Since these formulas are somewhat complicated we have, instead, decided to present the much simpler formulas in the case in which S is diagonalizable and use the notation introduced in Remark 6.3.3. In particular, we use the spectral representation for 5,

to write

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159

Then we use the biorthogonal family {$j,^j} and the formulas (6.67), (6.68) to obtain a representation for F defined on the basis {3>j}. To this end, we apply (6.65) and (6.66) to a basis function $g to obtain

The advantage to this formula is that both occurrences of S are replaced by a scalar involving Xq, and the equation (6.69) gives rise to the k equations

As in [14] we define the matrix valued function A(A) by

with the property that the open-loop poles are the roots of the characteristic equation Thus from (6.71) under the assumptions that a(A) n o~(S) = 0,

We now apply CQ to both sides, use (6.70) and, in a standard notation (see [14]), introduce the transfer function

to obtain

As shown in [7] the detectability condition H3 in Assumption 6.2.1 implies that none of the right-hand sides in (6.74) are zero. Lemma 6.9. Under assumptions til, H2, H3 we see that the regulator equations are solvable if and only if no eigenvalue of S is a transmission zero of the plant. If no eigenvalue of S is a transmission zero of the plant, then for j = 1,2,... , k, G(Aj)" 1 exists and we can solve each of the equations in (6.74) to obtain Tj. Then under Assumptions 6.2.1 we can solve (6.74) and get

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C. I. Byrnes, D. S. Gilliam and V. I. Shubov

Let us define

Using (6.75) we can write

so that by (6.60) we have

and

6.7

Tracking and Disturbance Rejection for an Oscillator with Delayed Damping

This example has been considered in several works [20, 21, 22] and provides an interesting example in which the uncontrolled open-loop system has two unstable poles. As we have already noted, in order to solve this problem we introduce a proportional error velocity feedback which, in the high gain limit, again produces an unstable system due to a single negative real eigenvalue which tends to zero as the gain tends to infinity. The control system, as considered in [22], is a harmonic oscillator with a delay in the damping term. Specifically, we have

We can write this equation in the form (6.42)-(6.45) by introducing the new variables

as

where

We also introduce an output consisting of

Chapter 6. Geometric Theory for Linear Distributed Parameter Systems

161

Next we reformulate the problem in an infinite-dimensional state-space Z = C eL 2 ([-/i,0],C 2 ) with the delay h = 1 in the form (6.55)-(6.56). As in (6.73) the transfer function can be readily computed, and the open-loop poles are the zeros of 2

There are two unstable poles A±i with approximate values

Following the discussion in (6.6) we introduce a stabilizing state feedback

The result of adding this feedback is to replace the matrix AQ by

and the resulting closed-loop transfer function is

The closed-loop poles are the zeros of the so-called return difference equation,

Since the system is real, complex poles must occur in conjugate pairs. It can be shown that all the closed-loop poles lie in the left half-plane for k > 0.05, but for larger values of k the single negative real closed-loop pole Ao(A;) approaches zero. For this reason only moderate values of k should be used; i.e., the system is not minimum phase. Our objective in this regulator problem is to attenuate the constant disturbance d and force the output of the system to track a sinusoid of given amplitude M and frequency a. To this end we introduce an exosystem

The system generates both the disturbance d and the required signal to be tracked. Namely, we have

We note that the spectrum of S is

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C. I. Byrnes, D. S. Gilliam and V. I. Shubov

Straightforward calculations from (6.75) yield

where <7fc(za) 1 = (l — o? + asin(o!)) + (ka + acos(a)) i. For the numerical example we have set

In Figure 6.1, we have plotted the reference signal yr(t) and output of the plant y(t) on the left and the error e(t) = y(t) — yr(t) on the right.

6.8

Future Directions

The problem of having the output of a system asymptotically tracking prescribed trajectories and/or asymptotically rejecting undesired disturbances, in the presence of (possibly large) uncertainties, is ubiquitous in control theory. Depending on the dominant kind of uncertainty that is to be addressed, the problem can be solved in different ways. For instance, if the trajectory to be followed (or the disturbance to be rejected) is known, but the parameters of the mathematical model of the plant to be controlled are known only within certain tolerance bounds (structured uncertainty), feedback schemes based on the so-called "certainty equivalence principle," which incorporate the on-line adaptation of the parameters of a control law having a fixed structure, are known to address the issue in a very efficient manner. If the uncertainty on the plant not only concerns the parameters of a given model but also includes neglected (usually high-frequency) dynamic phenomena (unstructured uncertainty), then robustness with respect to these uncertainties can be successfully achieved by means of improved adaptive schemes that exploit the "passivity" and/or the "finite-gain" properties of these neglected parts.

Figure 6.1. Outputs and error 0 < t < 4-7T.

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163

However, if the trajectory to be followed, or the disturbance to be rejected, is not known, then the control strategy must reconstruct in some way this information also, which is usually done by means of feedback schemes based on the so-called "internal-model principle." It has been well known for quite a long time in the case of linear systems [17], and for about ten years or so in the case of nonlinear systems [24], that if the trajectories/disturbances in question are not known but are known to belong to a specified class of functions of time (such as in the classical problem of set point control), then asymptotic tracking can be successfully achieved if the feedback mechanism incorporates a device that is able to "generate" all such trajectories/disturbances. A remarkable feature of the internal-model based control schemes for lumped linear systems [15, 16] is that they guarantee asymptotic decay of the tracking error also in the presence of plant parameter uncertainties (so long as the stability of the overall feedback loop is preserved), a property of paramount importance that has been recently extended also to internal-model based schemes for lumped nonlinear systems [23, 26, 10]. For linear systems, the role of the internal model is to ensure that the corresponding closed-loop system has a transfer function between the exogenous input (the uncertain trajectory/disturbance to be followed/rejected) and the tracking error and has a set of transmission zeros that includes all the eigenvalues of the autonomous dynamical system which is capable of generating all trajectories/disturbances (the exosystem). The simple consequence of this is that all such exogenous inputs are asymptotically blocked in their transit through the system, and this remarkable feature remains in effect despite plant parameter uncertainties. Internal-model based control schemes efficiently address the problem of tracking/rejecting families of exogenous inputs that can be generated by a fixed autonomous finite-dimensional dynamical system (the initial condition of which identifies the actual, unknown, exogenous input affecting the plant). In this sense, they generalize the classical way in which integral-control based schemes cope with constant but unknown disturbances. However, the main limitation of these schemes is that a precise model of the system that generates all exogenous inputs must be available so that it can be replicated in the control law. This limitation is not evident in the problem of set point control, where the uncertain exogenous input is constant and thus obeys a trivial, parameter-independent, differential equation. However, it is immediately evident in the problem of rejecting, e.g., a sinusoidal disturbance of unknown amplitude and phase. An internal-model based controller is able to cope with uncertainties in amplitude and phase of the exogenous sinusoid, but the frequency at which the internal-model oscillates must exactly match the frequency of the exogenous sinusoid: any mismatch in such frequencies results in a nonzero steady-state error. Some recent research in the lumped case addresses the problem of overcoming this limitation of the (otherwise very appealing) internal-model based control. This work includes the development of new methodologies for the design of internalmodel based control schemes which do not necessarily contain a complete model of the exogenous signal generator, but rather a copy of the asymptotic behavior of the signal generator. One approach that has proven useful is a method based on asymptotic internal models. Another is based on the design of adaptive internal

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models in which the internal model is not fixed once and for all, but is allowed to change as soon as the structure of exogenous inputs changes, so as to match an exosystem which is possibly totally unknown (except for an upper bound on its dimension). In these contexts, some very promising preliminary results have been obtained that show how it is possible, in certain circumstances, to successfully tune the parameters of an internal model by appealing to some basic concepts and techniques in adaptive control (see, e.g., [36]) or from nonlinear dynamics [32]. For robustness against unstructured uncertainty, robustness results have been obtained in [34] for lumped linear systems. These results represent the beginning of a new area of research in robust control, a direction which will be important to develop for distributed parameter systems. The explosive merger of computers and communications has resulted in the opportunity for an increased use of communications in control. Examples of remote control include the control of robots via the internet and the pointing of a laser beam toward a (master) telecommunications satellite via a beam toward the ground station from a (slave) satellite whose trajectory the master satellite tracks by a fixed lag. When communications are integrated into a control design, we believe that it will be important to understand the incorporation of delays, and small variations in a fixed delay, in plant models and their effect on the performance of controllers. Using a small gain theorem one can show that, for a certain class of nonlinear systems, stability is not necessarily destroyed by the incorporation of a sufficiently small delay, an avenue which has been studied extensively in the linear case. Nonetheless, using an explicit solution to the output regulation problem for a simple linear system having a pure point delay in its state equation illustrates the fact that while for certain parameter ranges the appropriate regulator equations exist whose solvability implies the solvability of the output regulator problem, the feedback law is in fact necessarily dependent on the delay. This motivates the development of an alternative output regulation design which is robust with respect to variation in the delay. In general, one would also like to develop a similar theory for linear distributed parameter systems having real parametric uncertainty.

Acknowledgement We would like to express our gratitude to Professor Ruth Curtain who, at the Conference on Future Directions in Distributed Parameter Control, brought to our attention the work of J. M. Schumaucher and, subsequently, the work of S. A. Pohjolainen in which sufficient conditions for output regulation are derived for certain types of distributed parameter systems.

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Bibliography [1] H.T. Banks, "Representation for solutions of linear functional differential equations," Journal of Differential Equations, 5, pp. 399-410, 1969. [2] H.T. Banks and J. Burns, "An abstract framework for approximate solutions to optimal control problems governed by heriditary systems," in Proceedings of the International Conference on Differential Equations, University of Southern California, 1974, Academic Press, New York, pp. 10-25, 1975. [3] C.I. Byrnes, A. Isidori and F. Belli Priscoli, Output Regulation for Uncertain Nonlinear Systems, Birkhauser, Boston, 1997. [4] C.I. Byrnes, D.S. Gilliam and V.I. Shubov, Output Regulation for Delay Systems, preprint, Texas Tech University, Lubbock, TX, 2000. [5] C.I. Byrnes, D.S. Gilliam, V.I. Shubov and G. Weiss, The Output Regulator Problem for Regular Linear Systems, preprint, Texas Tech University, Lubbock, TX, 2000. [6] C.I. Byrnes, D.S. Gilliam and V.I. Shubov, "Example of output regulation for a system with unbounded inputs and outputs," in Proceedings of the 38th IEEE Conference on Decision and Control, pp. 4280-4284, 1999. [7] C.I. Byrnes, D.S. Gilliam, I.G. Lauko and V.I. Shubov, "Output regulation for linear distributed parameter systems," IEEE Transactions on Automatic Control, 45, pp. 2236-2252, 2000. [8] C.I. Byrnes, D.S. Gilliam, I.G. Lauko and V.I. Shubov, "Output regulation for parabolic distributed parameter systems: Set point control," in Proceedings of the 36th IEEE Conference on Decision and Control, December, 1997. [9] C.I. Byrnes, D.S. Gilliam and V. Shubov, "On the global dynamics of a controlled viscous Burgers' equation," Journal of Dynamical and Control Systems, 4, pp. 457-519, 1998. [10] C.I. Byrnes, F. Delli Priscoli, A. Isidori and W Kang, "Structurally stable output regulation of nonlinear systems," Automatica, 33, pp. 369-385, 1997. [11] R.F. Curtain, "Invariance concepts in infinite dimensions," System and Control Letters, 5, pp. 59-65, 1984. [12] R.F. Curtain, "Invariance concepts in infinite dimensions," SIAM Journal of Control and Optimization, 24(5), pp. 1009-1030, 1986. [13] R.F. Curtain and G. Weiss, "Well posedness of triples of operators (in the sense of linear systems theory)," in Control and Estimation of Distributed Parameter Systems, F. Kappel, K. Kunisch and W. Schappacher, eds., pp. 4159, Birkhauser-Verlag, Basel, 1989.

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[14] R.F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems, Springer-Verlag, New York, 1995. [15] E.J. Davison, "The robust control of a servomechanism problem for linear time-invariant multivariable systems," IEEE Transactions on Automatic Control, 21, pp. 25-34, 1976. [16] B.A. Francis, "The linear multivariable regulator problem," SIAM Journal on Control and Optimization, 15, pp. 486-505, 1977. [17] B.A. Francis and W. M. Wonham, "The internal model principle of control theory," Automatica, 12, pp. 457-465, 1976. [18] D.S. Gilliam, V.I. Shubov and G. Weiss, Regular Linear Systems Governed by a Boundary Controlled Heat Equation, preprint, Texas Tech University, Lubbock, TX, 2000. [19] M. Hautus, Linear matrix equations with applications to the regulator problem, in Outils and Modeles Mathematique pour 1'Automatique, I.D. Landau, ed., C.N.R.S., Paris, pp. 399-412, 1983. [20] H.T. Banks and F. Kappel, "Spline approximations for functional differential equations," Journal of Differential Equations, 34(3), pp. 496-522, 1979. [21] H.T. Banks, J.A. Burns and E.M. Cliff, Spline-Based Approximation Methods for Control and Identification of Hereditary Systems, Preprint: Invited Lecture, International Symposium on Systems Optimization and Analysis, INRIA, France, 1978. [22] H.T. Banks, Approximation of Delay Systems with Applications to Control and Identification, Invited Lecture, Conference on FDE and Approximation of Fixed Points, Universitat Bonn, 1978. [23] J. Huang, "Asymptotic tracking and disturbance rejection in uncertain nonlinear systems," IEEE Transactions on Automatic Control, 40, pp. 11181122, 1995. [24] A. Isidori and C.I. Byrnes, "Output regulation of nonlinear systems," IEEE Transactions on Automatic Control, 35, pp. 131-140, 1990. [25] T. Kato, Perturbation Theory of Linear Operators, Springer-Verlag, Berlin, 1966. [26] H.K. Khalil, "Robust servomechanism output feedback controllers for feedback linearizable systems," Automatica, 30, pp. 1587-1599, 1994. [27] H.W. Knobloch, A. Isidori and D. Flockerzi, Topics in Control Theory, Birkhauser-Verlag, Basel, 1993. [28] S.A. Pohjolainen, "Computation of transmission zeros for distributed parameter systems," International Journal of Control, 33(2), pp. 199-212, 1981.

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[29] S.A. Pohjolainen, "Robust multivariable Pi-controller for infinite dimensional systems," IEEE Transactions on Automatic Control, 27(1), pp. 17-30, 1982. [30] S.A. Pohjolainen, "On the asymptotic regulation problem for distributed parameter systems," in Proceedings Third Symposium on Control of Distributed Parameter Systems, Toulouse, France, pp. 197-201, July, 1982. [31] S.A. Pohjolainen, "Robust control and tuning problem for distributed parameter systems," International Journal of Robust and Nonlinear Control, 6, pp. 479-500, 1996. [32] J. Ramsey, A Nonequilibrium Theory for Output Regulation of Nonlinear Systems, Ph.D. Dissertation, Washington University, St. Louis, MO, November 2000. [33] J.M. Schumacher, "Finite-dimensional regulators for a class of infinite dimensional systems," Systems and Control Letters, 3, pp. 7-12, 1983. [34] J.M. Schumacher, Dynamic Feedback in Finite- and Infinite-Dimensional Linear Systems, Mathematical Centre Tracts No. 143, Mathematical Centre, Amsterdam, 1981. [35] J.M. Schumacher, "A direct approach to compensator design for distributed parameter systems," SIAM Journal on Control and Optimization, 21, pp. 823-836, 1983. [36] A. Serrani, A. Isidori and A. Marconi, "Semiglobal nonlinear output regulation with adaptive internal model," submitted. [37] R. Sedwick, D. Miller and E. Kong, "Mitigation of differential perturbations in clusters of formation flying satellites," AAS/AIAA Space Flight Mechanics Meeting, American Astronomical Society, Washington, DC, pp. 99-124. [38] G. Weiss, "Admissible observation operators for linear semigroups," Israel Journal of Mathematics, 65, pp. 17-43, 1989. [39] G. Weiss, "Transfer functions of regular linear systems, part I: Characterizations of regularity," Transactions of the AMS, 342, pp. 827-854, 1994. [40] G. Weiss, "Regular linear systems with feedback," Mathematics of Control, Signals and Systems, 7, pp. 23-57, 1994. [41] G. Weiss and R.F. Curtain, "Dynamic stabilization of regular linear systems," IEEE Transactions on Automatic Control, 42, pp. 4-21, 1997. [42] W.M. Wonham, Linear Multivariable Control: A Geometric Approach, 2nd Edition, Springer-Verlag, Berlin, 1979. [43] H.J. Zwart, Geometric Theory for Infinite Dimensional Systems, Ph.D. thesis, Rijksuniversiteit Groningen, 1988; also Lecture Notes in Mathematics, Vol. 115, Springer-Verlag, 1989.

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Chapter 7

Smart Structures, Structural Health Monitoring and Crack Detection Daniel J. Inman* and Sergio H. S. Carneircfi Abstract Smart materials, or active materials, offer a host of new solutions to engineering problems in control and identification. In addition they offer new modeling and analysis challenges. Here we examine the use of smart materials in structural applications, called smart structures, focusing on their use in damage detection and structural health monitoring. Identification and parameter estimation are at the root of the structural health monitoring problem. Both identification and monitoring are generally improved with increased numbers of sensors and actuators. Hence smart materials, which allow almost distributed arrays of sensing and actuation form a natural tool for structural health monitoring. A summary of the issues and problems of structural health monitoring using smart materials is presented. The chapter concludes with some thoughts on open problems in smart structures and structural health monitoring.

7.1

Smart Materials and Structures

Smart materials is the name given to a class of materials that exhibit the ability to change mechanical force and motion into some other form of energy and vice versa. On a simple transducer, however, many of these materials are able to transform * Center for Intelligent Material Systems and Structures, Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061. E-mail: dinman@vt. edu tlnstituto de Aeronautica e Espaco, Centre Tecnico Aeroespacial, Sao Jose dos Campos, SP 12228, Brazil. E-mail: [email protected]

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energy into motion and mechanical motion into energy. Examples of such materials are those that exhibit the piezoelectric effect, shape memory alloys, magnetorheological fluids and ionic polymers. The most common examples of such materials are those based on the familiar piezoelectric effect. Piezoelectric materials produce a voltage when strained and a strain when an electric field is applied across them. This means in particular that a piezoelectric material can be used both as a sensor and as an actuator. Most of the examples used in this chapter employ piezoceramic materials know as PZT. However, piezoelectric polymers are also very useful. More detailed descriptions of smart materials can be found in Banks et al. [3] and Culshaw [9]. A smart structure is a structure that hosts smart materials in some embedded or layered way and performs some sort of control function of sensing and actuating. A precise definition is not appropriate and many examples are given by Banks et al. [3] and Culshaw [9]. The word smart is an overstatement but the field remains meaningful because of the ability to create structures with highly integrated sensing and actuation functions. Numerous other names have been attached to such systems including active structures, adaptive structures, intelligent structures, multi-functional structures and structronics. The algorithms for control and sensing, as well as the associated hardware and power considerations, are all included within the discipline. An important characteristic of smart materials is that they provide an unobtrusive, integrated and distributed way to add actuation and sensing to a structure. This characteristic is a perfect match for use in health monitoring and in structural control applications where space and mass provide heavy penalties. In the following, several examples are given of using smart structures to solve both control and health monitoring problems.

7.2

Structural Health Monitoring

Structural health monitoring (SHM), also called damage detection or diagnostics, is the concept of using structural measurements to determine the condition, integrity or state of a structure. A basic goal of SHM is to determine if a structure is in danger of failing or not. Examples of common objectives of SHM are to determine if any cracks exist in the structure, if bolted or welded joints are intact and up to specification or if there are any holes or other structural damage present in the structure. In some sense one might fit this topic in with nondestructive evaluation (NDE) methods. The difference between classic NDE methodology and SHM is that of immediacy. In NDE, parts or systems are taken out of service to be inspected while a goal of SHM is to leave the structure in service while the analysis is performed. An example of current SHM is the NASA space shuttle that undergoes a vibration test (modal test) before and after each flight. Differences in modal information are then used to determine damage incurred during the flight. While not quite an "on-line" system, it does capture the function of SHM. The problem of damage analysis can be sub-divided into four subproblems or levels of health monitoring (Doebling et al. [10]):

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1. Detect the existence of damage. 2. Detect and locate the damage. 3. Detect, locate and quantify the damage. 4. Detect, locate, and quantify the damage, then estimate its remaining service life. Each level requires more modeling and hence more mathematics in order to solve it. The Level 1 problem has numerous solutions listed in the literature and in fact has been implemented in practice. Fewer solutions are available for the remaining topics. Level 4 leads to the emerging area called "Prognosis" which has few solutions and is motivating several large programs in government laboratories. Here we propose to expand this list to include several more possibilities: 5. Combine Level 4 with smart structures to form Self-Diagnostic Structures. 6. Combine Level 4 with smart structures and control to form Self-Healing Structures. 7. Combine Level 1 with active control and smart structures to form simultaneous control and health monitoring. Taking the simplified point of view that health monitoring involves looking for changes in a system's physical parameters (such as mass, damping or stiffness) leads to the observation that much of the mathematics associated with SHM will come from the field of parameter identification. This is clear in the work of Banks et al. [2] who solve a Level 3 health-monitoring problem by adapting techniques from identification theory to solve a damage detection problem for simple beams using piezoceramic actuators and sensors as well as traditional instrumentation. Obviously, each level problem becomes more difficult to solve. The problem of Level 1 can be solved without any reference to a structural model by using simple signal analysis. One example of such a solution is given by Cattarius and Inman [8]. In this technique the idea that a defect will produce a small change in stiffness and/or mass, and hence frequency, is often used in damage detection. Early vibration-based damage detection methods often looked at frequency response function (FRF) data for a small change in frequency. However, small changes in frequency are generally difficult to measure using FRF data. As an alternative, Cattarius and Inman [8] continuously compare the healthy time signal of the structure under study to the current time signal of the structure. If a small difference in frequency exists between these two signals, then when they are combined they will produce the beat phenomena that serves to magnify small differences in frequencies. This effect is used detect the presence of damage in plates, such as a helicopter blade, by using internal piezoceramic materials to both excite and sense the various time histories. This is an example of a time domain procedure that is totally self-contained in a moving structure capable of self-diagnostics using embedded piezoceramic sensing and actuation at relatively low power costs. The procedure depends only upon

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Figure 7.1. Impedance-based damage detection of a piping system using piezoceramics. subtracting two signals, does not require any modeling of the structure and hence is simple enough for on board use. Another diagnostic procedure not relying on any mathematical model is impedance based and may also be used in a self-diagnostic configuration by incorporating local, embedded and/or surface mounted piezoceramic sensors and actuators. This approach is a high frequency impedance-based method that looks for a shift in electrical impedance measurements as an indicator of damage. In the impedance-based qualitative health monitoring technique, real-time implementation relies on a simple scalar damage index that can be easily interpreted. Using this damage index in conjunction with a damage threshold value, the approach can warn an operator in a green/red light form, whether or not the threshold value has been reached. A damage index that is not affected by environmental effects was established based on statistical analysis and signal processing, making the damage index more stable under all types of environmental conditions (Park et al. [15, 16]). Figure 7.1 illustrates the use of a self-contained impedance-based system of a frame with bolted joints. Such a system represents the complexity of a real service structure, yet allows for systematic "damage" in the structure by adjusting the torque on the joints. In Banks et al. [2], a method which takes advantage of knowledge of the structure is presented to provide experimental proof that a PZT-based diagnostic system can not only determine the existence of damage, as can the previous methods, but is also able to determine the size and location of the damage (Levels 2 and 3). The method is based on using a partial differential equation model of a structure that is partially layered with PZT patches which again forms a "self-diagnostic" structure. The algorithm uses a spline-based approximation of the equations of vibration and successfully identifies the existence, size and location of holes in a beam. The method works by estimating functions of the longitudinal direction of the beam corresponding to the damping parameters (both Kelvin-Voigt and air damping), the modulus and the density. Each of these is allowed to be discontinuous in order to allow holes to be included in the solution set of functions.

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It is important to note that while this method works very well, it has the disadvantage of requiring a very detailed model of the structure, including a model of the internal damping mechanisms. The parameters are allowed to have some measure of uncertainty as they are estimated in the inverse procedure used to identify the damage. However, the form of the governing differential equation must be known in substantial detail. With this noted, the method very effectively identifies the damage. Furthermore, the results are consistent across several different types and sets of measurements. Another important feature of the estimation-based method is that it does not use modal data, but rather is based on time domain measurements. The experiments were repeated with traditional excitation means (an instrumented hammer) and response measurement (accelerometer and position probe) as well as with the internal self-sensing actuation scheme offered by the PZT patch. Besides verifying a damage detection algorithm, the tests show conclusively that it is possible to use piezoceramic materials to form self-monitoring devices. In the following the impedance method is used to illustrate Level 6 and Level 7 solutions. The main point of using the impedance method here is that the method works off of a very low voltage (1 volt) excitation in the kilohertz (kHz) range so that it does not interfere with the control law for active vibrations suppression that typically takes place at low frequency. Furthermore, it is shown that this method detects damage while the control law is turned on so that vibration suppression is occurring simultaneously. This works because the high frequency impedance signal is not part of the control bandwidth. The basic concept of this impedance-based structural health monitoring technique is to monitor the variations in the structural mechanical impedance caused by the presence of damage. Since structural mechanical impedance measurements are difficult to obtain, this nondestructive evaluation technique utilizes the electromechanical coupling property of piezoelectric materials. This method uses one piezoceramic (PZT) patch for both sensing and actuating. The PZT patch is considered as a thin bar undergoing axial vibrations in response to the applied sinusoidal voltage. Assuming that the PZT parameters remain constant, any changes in the mechanical impedance (Zs) alter the overall admittance of the combined structure and PZT system. Previous experiments have shown that the real part of the overall impedance contains sufficient information about the structure and is more reactive to damage than the magnitude of the imaginary part. Therefore, the impedance analysis is confined to the real part of the complex impedance. The actual health monitoring is performed by saving a healthy impedance signature of the structure and comparing the signatures taken over the structure's service life. The impedance measurements are taken with an HP 4194A Impedance Analyzer; however, a simple measurement across a resistor would do. A frequency range from 45 kHz to 55 kHz proved to be an optimum for this structure. To simulate damage on a plate bolted on all four sides to a frame, one or two bolts of the clamping frame were loosened from 25 ft-lb to 10 ft-lb. Note that at this level of torque the bolt is still very tight with no slip occurring at all. For comparing impedance signatures, a qualitative damage assessment has been developed. The assessment is made by computing a scalar damage metric, defined as the sum of the

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Figure 7.2. Damage metric chart of different impacts to a plate.

squared differences of the real impedance at every frequency step. Equation (7.1) gives the damage metric M in a mathematical form. The variables include Y^i, the healthy impedance at the frequency step i; Yi$, the impedance of the structure after the structure has been altered; and n, the number of frequency steps:

The damage metric simplifies the interpretation of the impedance variations and summarizes the information obtained by the impedance curves. Different damage metric values of the plate are depicted in Figure 7.2. Note the difference in the metric between one bolt loosened and two bolts loosened.

7.3

Vibration Suppression

Vibration of aircraft panels is a major source of fatigue and requires the addition of damping material to increase fatigue life and reduce noise transmitted to the interior. The added damping treatments bring a substantial weight penalty. In addition, passive damping treatments are limited in frequency range and subject to variation with temperature. In this section, we examine via an experimental approach, the possibility of using active vibration suppression implemented through smart materials to perform damping in structural plates subject to both structural and acoustic loading. Results are presented indicating a high level of damping

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available through a single piezoceramic patch serving as an actuator and a fiber optic sensor. Because modeling and boundary conditions in real aircraft panels have significant variance, a control method is chosen that is based only on knowing experimentally determined frequency response data for the structure. Positive position feedback (PPF) control based on a measured frequency response to the structure is chosen as the control law (Fanson and Caughey [12]). This technique combined the technology of optical fiber sensors with piezoelectric actuators to minimize the vibration levels in the test article. This PPF control law uses a generalized displacement measurement from the test article to accomplish vibration suppression. The experiments were performed in a standard test fixture commonly used in industry for evaluation damping materials. The experiments show that smart damping materials have substantial performance benefits in terms of providing effective noise and vibration reduction at a frequency range that is often outside of the effective range of passive damping materials. Further, judging by vibration reduction per added weight, the test results indicate that the smart damping materials can provide substantial vibration reduction at selected frequencies, without adding any appreciable amount of weight to the substrate structure. For example, smart damping can decrease the vibration peak of a steel panel at 47 Hz by up to 20 dB with an additional mass of only 50 grams. This feature of smart damping materials is particularly useful for applications that involve vehicles, where the constraint requires a particular noise or vibration cancellation at a specified frequency, without adding any weight to the vehicle or requiring any change to the vehicle structure. Overall, the test results show that the application of smart damping materials and fiber optic sensors can be used for active control of aircraft style panels. The smart damping materials combined with fiber optics can be viewed as a new technology that, once developed and optimized, can extend panel life by providing more effective noise and vibration solutions for new or existing aircraft structures. Fiber optic sensors are widely used as physical parameter gauges in various structural applications, such as strain and vibration sensing and damage detection. The Fabry-Perot (FP) interferometric strain sensors can be classified into two main types: intrinsic FP interferometers (IFPI) and extrinsic FP interferometers (EFPI). The EFPI sensor is constructed by fusion splicing a glass capillary tube with two optical fibers. Compared to the IFPI sensor, the EFPI-based sensor is relatively simple to construct and the FP cavity length can be accurately controlled. The EFPI sensor can also be easily configured to suit different applications with desired strain range and sensitivity by altering the type of fibers, the capillary tube, air-gap distance and the length of the sensor. In addition, a major advantage of the EFPI sensor is its low temperature sensitivity, which makes it possible to interrogate the EFPI sensor with simple signal processing techniques. An EFPI sensor can be constructed using either single-mode or multi-mode fibers. The single-mode design offers higher accuracy and low insensitivity to unwanted disturbance, while the multi-mode design offers higher power coupling efficiency. In our experiment, single-mode fibers are used to deliver and collect the light and are used as an internal reflector as well.

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The test panel used for the active control tests is a 500mm x 600mm, 20guage, galvanized steel plate with two 72.4 mm x 72.4 mm piezoelectric actuators bonded to its surface. These two locations were chosen for the actuators because the plate has a large amount of strain energy in these regions for the modes that needed to be controlled. In general, the control authority of an actuator is increased when it is placed in a region of high strain energy. Although the test plate has two piezoelectric actuators attached to its surface, it was determined that the center PZT was more effective at minimizing the levels of vibration. The sensor used for control was a modal domain optical fiber sensor for vibration monitoring. This sensor is based upon a laser that focuses coherent light through a lens into one end of a multi-mode optical fiber. One end of the fiber was attached to the plate and the other end passed through a spatial filter and into a photo detector. The output of the photo detector is a variable voltage that is fed into a monitoring unit such as an oscilloscope. The control system was set up such that the optical fiber sensor sends a signal through a signal conditioner and amplifier into the dSPACE board. Once the signal was processed through the PPF control law it was sent out to an amplifier to drive the PZT actuator. The system's parameters and the overall control system's performance were measured using a two channel HP Dynamic Signal Analyzer. This signal analyzer was used to get the frequency response function between the plate and the clamping frame using two accelerometers, one on the bottom center of the plate and one on the clamping frame. A series of tests were run to determine the optimum parameters for the active vibration suppression of a representative aircraft panel. These parameters, ranging from the gains assigned to the various amplifiers to the damping ratios of the PPF filters, were based upon the tuned frequency and the number of modes assigned to each PPF filter. The final Simulink Model used for the active control tests uses three PPF filters to provide active damping to most of the modes from 0 to 400 Hz. However, significant active damping of a mode or modes can be achieved with one PPF filter tuned properly. There were two different means of supplying disturbance energy to the plate for the active control tests. The first method used a 100 Ib shaker to excite the plate mechanically, and the second method used a 10" sub-woofer to excite the plate acoustically. Both the shaker and the speaker were driven with a 0 to 400 Hz periodic chirp input signal. The initial design procedure for the active control system was to create a Simulink model with one PPF filter to control as many modes as possible. This design method involved determining the frequency range at which the system had control authority over multiple modes. The test results show that the control system with one PPF filter was most effective when tuned to a frequency between 60 and 70 Hz. Figure 7.3 shows some of the best results obtained with one PPF filter controller. In each of these tests the 100 Ib shaker excited the plate mechanically. The active control test was run with a PPF filter damping ratio of 0.02, a Simulink gain of —80,000, a sampling time of 0.0001 seconds, and a frequency of 60 Hz. Additional tests were run using pure tone inputs at the resonant frequencies of the plate. These tests were performed to determine the effectiveness of the controller in the suppression of the structure-born noise resulting from the vibration of the

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Figure 7.3. Broadband vibration suppression using five PPF controllers. test plate. Most of these tests were run at the first resonant frequency of the plate (47 Hz) because this mode produced the most audible structure-born noise. The test results proved that the controller reduced the noise from the plate significantly. Figure 7.3 shows the best reduction achieved by the controller with a 47 Hz pure tone input signal. This experiment represents the first use of a fiber optic sensor in an active vibration suppression system with piezoceramic actuation. The controller is not based on an analytical model but rather on an experimentally determined modal model of the structure. The performance obtained via the active control system is comparable to, and in some cases better than, that obtained by passive damping treatments. The piezoceramic actuator and fiber optic system weigh much less than a constrained layer damping treatment for similar performance. However, the mass of the control hardware must also be considered. While the control system here was obtained using off the shelf components (dSpace, MATLAB) which are of substantial size and weight, it is possible to put this entire control system, signal conditioning, etc. onto a single chip weighing just a few grams (Inman [13]).

7.4

Simultaneous Diagnostics and Control (Level 7)

The problem of simultaneously monitoring the health of a structure and active vibration suppression on the same structure represents opposing goals. The goal of active vibration suppression is to reduce the structure's response as quickly as possible. On the other hand, diagnostic, or health monitoring, generally requires that the response be examined as long as possible. Furthermore, if the same hardware is to be used for both vibration suppression and health monitoring, the possibility of signals interfering with each other presents itself. To solve these problems the impedance method of health monitoring it used. The impedance method uses a high frequency, low voltage, self-sensing signal to determine small changes in stiffness, damping and mass of a structure.

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The control law chosen must not interfere with the diagnostic measurements but at the same time provide robustness to analysis. In particular the control must not depend on an analytical model and must continue to provide vibration suppression as the panel heats up. In the case of the aircraft panel demonstrated here, the control hardware and diagnostic hardware were required to be the same in order to minimize the amount of hardware in an aircraft. A piezoceramic patch was chosen for the actuator and sensor, building on the concept of a self-sensing actuator (Dosch, Inman and Garcia [11]). The control law was chosen to be Positive Position Feedback (PPF) because it can be applied to an experimental model and because its closed-loop stability depends only on knowing the system's natural frequencies. Thus, the PPF control law may be applied with the hope of success as long as experimentally determined frequencies are available. Since the task consisted of simultaneous health monitoring and active control with the same actuators for both the control and the health monitoring, the system needed to be de-coupled. The impedance method is very sensitive to disturbing voltages in the measuring circuit. The controller, however, creates exactly those disturbances by generating the control signal. A simple capacitor of 390 nF in series with the impedance analyzer efficiently blocked the control signal from the impedance analyzer. All health monitoring data was taken while the shaker was exiting the plate with a periodic chirp signal from 0 to 200 Hz. The active controller was also switched on and increased the damping of the first few modes of the plate significantly. The control results are straightforward. First an experimental result is given for the case of constant temperature. Figure 7.3 illustrates the open-loop transfer function along with the closed-loop control provided by five PPF controllers. The controller is able to suppress all the modes in the bandwidth from 0 to 400 Hz. The ability to perform simultaneous health monitoring and active vibration suppression has been illustrated and experimentally verified. Furthermore, the active control of panel vibrations in the presence of temperature changes has been illustrated in Figures 7.2 and 7.3, which were made simultaneously. These experimental results do not depend on any model of the system and hence are applicable to complex structures. The control method and the health monitoring method were performed with the same hardware at the same time. These results indicate that it may be possible to perform simultaneous health monitoring and control on real structures outside of the laboratory setting. Applications of smart materials to problems of interest to the private sector are now fairly routine to solve. In addition, the use of smart materials has been shown to be extremely beneficial in numerous cases. The only thing that holds back the wide spread use of smart materials in industry is the lack of good, design-based models that are easily accessible to industrial designers.

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179

Self-Healing (Level 6)

This section illustrates the feasibility of creating smart structural bolted connections, which consist of structural members joined together by bolt and nut combinations equipped with piezoceramic and shape memory alloy (SMA) elements. These combinations can be used to monitor bolt tension and connection damage. When damage occurs, temporary adjustments of the bolt tension can be achieved actively and remotely in order to restore lost torque for continued operation, thus illustrating a self-healing systems. The system is illustrated in Figure 7.4. The piezoceramic material is used to detect the existence of damage (an out of torque bolt in this case) and the shape memory alloy is then used to regain the lost torque by applying an increased normal force when activated. The impedance for the various levels of torque is illustrated in Figure 7.5 which shows that the torque is recovered, at least partially. A test specimen consisting of two aluminum beams was constructed with a bolted joint. The bolted-joint structure was hung vertically by a string. One PZT patch bonded to one of the members was used to measure the electrical impedance. An SMA washer (Intrinsic Devices AHE 0957, Figure 7.1) was inserted between the bolt and the nut, as illustrated in Figure 7.2. Initially, the bolt was tightened to 30 ft-lb, and the torque was reduced to 10 ft-lb to introduce a loosening mode of a bolt failure. This damage, however, can be considered in its incipient stage, which still maintains the integrity of the joint. This level of damage corresponds roughly to a quarter turn of the bolt so that the joint is still very tight but not up to the design specified torque. Figure 7.5 summarizes the basic proof of concept experiment. The impedance monitoring system is activated while the bolt is at its design specified value of torque. The PZT patch applies a 1-volt excitation signal in the kHz frequency range that continuously monitors the system by comparing the impedance signal of a healthy system to the impedance of the current system. When the impedance signal changes, the bolt is then indicated to be out of torque. This initiates a current to the SMA washer, resulting in a recovery of the lost torque (Inman, Muntges and Park [14]).

7.6

Diagnostics of Cracked Beams via Genetic Algorithms (Level 3)

Most of the health monitoring techniques proposed in the literature are based on frequency domain data, usually modal parameters. However, there are two main reasons why the use of modal parameters may pose a serious limitation for the case of crack-type damage: first, the effect of small flaws on modal quantities, such as natural frequency shifts and local changes in the mode shapes, are usually very small and likely to be masked by experimental uncertainties and data reduction; and second, the assumption of linearity inherent in modal methods may introduce errors in the identification procedure.

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Figure 7.4. The proof of concept lap joint assembled with SMA washer and PZT impedance sensor and actuator.

Figure 7.5. The impedance showing the deterioration of impedance as the torque is reduced, followed by the recovery of impedance as the SMA is activated.

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Figure 7.6. Typical geometry of a cracked beam.

A promising alternative is the use of time responses in a model-based, iterative parametric identification procedure proposed by Banks et al. [1, 2], proven to be capable of detecting fairly small defects modeled as geometrical imperfections in the cross section of beams. The methodology involves a theoretical model based on a b-spline Galerkin approximation of the differential equations of equilibrium that provides both the accuracy and the simplicity required for a successful damage diagnosis. The use of time responses is less likely to be limited by data reduction problems, since the only differences from raw, analogue data are due to the digital data acquisition process. Besides, the nonlinear behavior is also preserved, increasing the chances of a more realistic characterization of crack-type damage. In order to preserve the mathematical framework developed by Banks et al., which guarantees that the inverse problem to be solved is well-posed, Carneiro and Inman [5, 6] developed a continuous model for the vibration of cracked beams and used this model as the basis for crack diagnostics. The proposed diagnostic technique is based on a parameter identification procedure. More specifically, it is an enhanced nonlinear least-squares estimation of the variables representing damage site and severity. The method consists of minimizing a defined cost function that represents some norm of differences between the measured and numerical responses. The identification problem for a beam with a crack of depth a located at xc, as depicted in Figure 7.6, can be mathematically stated as the minimization of the cost function

where w^' and w^ are the Nm experimental and numerical responses, respectively, measured at a total of Nt discrete times tk • The weights c, are introduced to account for differences in amplitudes of responses to different excitations.

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The minimization process was successfully performed for the case of a beam with hole-shaped flaws in [2] by a traditional gradient-based method. However, it has been shown that the denned cost function is multimodal, i.e., it presents numerous local minima [4], which renders the identification procedure extremely dependent on a good initial guess. Additionally, for the case of crack diagnostics, the choice of record length must satisfy two conflicting demands: it must be long enough to reveal subtle differences in the responses caused by small flaws, but not so long that it introduces further artificial local minima. In summary, traditional gradient-based minimization methods have very little chance of success in finding the global minimum and hence in identifying the desired damage parameters. In order to overcome those intrinsic limitations of the problem, Carneiro and Inman [7] proposed the use of genetic algorithms (GAs) in a scheme as depicted in Figure 7.7. The first step is the collection of experimental data via measurement of time responses of the damaged structure due to the chosen excitations. Next, an initial population is randomly created, consisting of different combinations of possible crack parameters within defined bounds. For each individual (a, xc)i, numerical time responses are computed using the numerical model, and the corresponding fitness is calculated as the negative value of the objective function in (7.2). The GA is then initialized and successive generations are created through selection and reproduction until a termination criteria is achieved, most commonly based on a maximum number of generations. The process is computationally intensive due to the numerous evaluations of time responses, but it has the advantage of allowing straightforward parallelization since the computation of each individual's fitness is completely independent from other computations. The identification problem defined in (7.2) was successfully solved with the help of GAs. Several numerical experiments were performed in order to investigate the robustness of the discussed methodology to measurement noise, modeling uncertainties and choice of excitation signal (Carneiro and Inman [7]). One of the most important results is related to the importance of including the nonlinear behavior due to the opening-closure cycles of a cracked vibrating structure, the so-called breathing crack model. Figure 7.8 shows that the crack parameters estimated by means of a linear approximation predict reasonable crack location but consistently underestimate crack depth. This result is nonconservative and could lead to erroneous maintenance decisions based on a poor estimation of remaining service life, i.e., when attempting Level 4 diagnostics.

7.7

Future Directions

It is clear that the ultimate goal of structural health monitoring and diagnostics will be best achieved by (a) integrating both the sensing and actuation aspects of smart materials into the process, (b) incorporating more sophisticated modeling methods and (c) recognizing the connection between health monitoring and the mathematics of parameter estimation and identification algorithms. The future goals of many industries involve selling service rather than products, and this requires a much better understanding of structural health monitoring than is currently provided by

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Figure 7.7. Flow chart of the time domain, model-based crack diagnostic procedure using Genetic Algorithms.

the academic community. In particular, many agencies and companies are taken with the notion of "prognosis," the ability to assess damage and predict remaining life subject to a variety of different inputs. Prognosis is essentially Level 4 and requires a great deal of advanced modeling of the structures and systems in question. Modeling of various effects (nonlinearities, hysteresis, temperature and humidity) become increasingly important. Many of these effects have been modeled in the literature but have not yet made their way into structural health monitoring. The invention of new materials of course requires new modeling, experiments and verification. Many more advanced models of smart materials than used here exist in the literature and are yet to be integrated into health monitoring schemes. Clearly the integration of smart materials into identification and structural health monitoring offers new benefits and new research topics. Hence, one sure way to move forward is by the continued integration of these disciplines. The examples given here are already somewhat dated, as they depend largely on the piezoelectric effect, and the field oF'smart materials is much broader than this. A great increase in the ability and success of identification and structural health monitoring algorithms

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Figure 7.8. Comparison of diagnostic results using the open crack model and the breathing crack model. Damage case: xc/L = 0.15, a/Id = 0.1.

would result if sensors could be developed that more directly measure the physical quantities of interest. For instance, if one could directly measure corrosion, then new, more precise identification and monitoring would result. Perhaps one of the many existing smart material effects could be used to more directly sense required information. The march towards nano-technology (i.e., engineering at the atomic level) may provide smart material sensors and exciters that are able to directly address quantities of interest. The availability of a variety of new sensing and exciting abilities, and the advent of such ideas as nano-technology, multi-functional materials, etc., may have a large impact on the types of algorithms and schemes developed for identification and monitoring in the future.

7.8

Summary

A variety of applications of smart materials to vibration suppression and structural health monitoring have been presented. As the level of diagnostics increases, more sophisticated modeling is needed and the more mathematical modeling plays a key role, as illustrated by crack diagnostics. In each case the u§e of smart materials integrated into a smart structure forms a unique solution to an important problem.

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It is hoped that the information in this chapter will stimulate others to attempt using smart materials to solve other parameter estimation, identification and health monitoring problems.

Acknowledgements The first author would like to thank the Flight Sciences Department, Raytheon Systems Company for both suggesting and funding the work on simultaneous control and monitoring (and for the excellent technical monitoring of Richard A. Ely). The second author would like to thank the Brazilian Air Force for funding his research on structural health monitoring. In addition, thanks are due to the National Science Foundation (CMS-9713453-001, CMS-0120827) and the Air Force Office of Scientific Research (F49620-99-1-0231) for funding our work in control, smart structures and health monitoring used in this chapter. Dr. Guyhae Park provided expertise in health monitoring using the impedance-based method and with the self-healing bolt.

Bibliography [1] H. T. Banks and P. Emeric, "Detection of non-symmetrical damage in smart plate-like structures," Journal of Intelligent Material Systems and Structures, 9, pp. 818-828, 1998. [2] H. T. Banks, D. J. Inman, D. J. Leo and Y. Wang, "An experimentally validated damage detection theory in smart structures," Journal of Sound and Vibration, 191, pp. 859-880, 1996. [3] H. T. Banks, R. Smith and Y. Wang, Smart Materials and Structures: Modeling, Estimation and Control, Wiley/Masson, Chichester/Paris, 1996. [4] S. H. S. Carneiro, Model-Based Vibration Diagnosis of Cracked Members in the Time Domain, Ph.D. thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, August, 2000. [5] S. H. S. Carneiro and D. J. Inman, "A continuous model of a cracked Timoshenko beam for damage detection," in Proceedings of the 18th International Modal Analysis Conference, pp. 194-200, February 2000. [6]

, "Comments on the free vibrations of beams with a single edge crack," Journal of Sound and Vibration, 244, pp. 729-737, 2001.

[7]

, "Vibration diagnostic of cracked beams using genetic algorithms," in Proceedings of the 16th COBEM — Brazilian Congress of Mechanical Engineering, November 2001.

[8] J. Cattarius and D. J. Inman, "Experimental verification of intelligent fault detection in rotor blades," International Journal of Systems Science, 31, pp. 13751379, 2000.

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[9] B. Culshaw, Smart Structures and Materials, Artech House, Norwood, MA, 1996. [10] S. W. Doebling, C. R. Farrar, M. B. Prime and D. W. Shevitz, "A review of damage identification methods that examine changes in dynamic properties," The Shock and Vibration Digest, 30, pp. 91-105, 1998. [11] J. J. Dosch, D. J. Inman and E. Garcia, "A self-sensing piezoelectric actuator for collocated control," Journal of Intelligent Material Systems and Structures, 3, pp. 166-185, 1992. [12] J. L. Fanson and T. K. Cauhey, "Positive position feedback control for large space structures," AIAA Journal, 28, pp. 717-724, 1990. [13] D. J. Inman, "Vibration suppression through smart damping," in Proceedings of the 5th International Conference on Sound and Vibration, Vol. 1, pp. 115132, 1997. [14] D. J. Inman, D. E. Muntges and G. Park, "Investigation of a self-healing bolted joint employing a shape memory actuator," in Proceedings of the SPIE 8th Annual International Symposium on Smart Structures and Materials, Vol. 4327, Newport Beach, California, March 2001. [15] G. Park, H. Cudney and D. J. Inman, "An integrated health monitoring technique using structural impedance sensors," Journal of Intelligent Material Systems and Structures, 2000. [16] G. Park, K. Kabeya, H. Cudney and D. J. Inman, "Impedance-based structural health monitoring for temperature varying applications," JSME International Journal, Series A, 42, pp. 249-258, 1999.

Chapter 8

Survey of Research in Modeling the Human Respiratory and Cardiovascular Systems F. Kappel* and J. J. Batzetf Abstract In this chapter we examine some major issues concerning the modeling of the respiratory and cardiovascular control systems as well as review some of the important models which have been developed to address these issues. The physiological processes that control the ventilation rate VA are fairly well understood and we will look at a number of mathematical models that have been devised to describe this control. The cardiovascular control system, on the other hand, involves a more complex set of interrelationships involving a number of factors including heart rate, blood pressure, cardiac output and blood vessel resistance. At the current time the description of these control relationships is far from complete. We will consider several important areas in cardiovascular control and some of the modeling approaches that have been used to understand the control process. One approach to studying the cardiovascular and respiratory systems considers the introduction of optimality conditions into the design of these systems and application of optimal control theory to the modeling process. Many physiologists assume that optimization is a basic feature in the evolution of biological systems (e.g., see Kenner [42], Swan [79] or Taylor [81]). It is reasonable to consider, for example, whether such systems have evolved in such a way as to minimize energy expenditure. Finally, after we have discussed the respiratory and cardiovascular systems separately, we will consider the interaction and interrelations of these two systems. 'Department of Mathematics, Karl-Franzens-Universitat, Heinrichstrafle 36, 8010 Graz, Austria. E-mail: [email protected] tSFB "Optimierung und Kontrolle," Karl-Franzens-Universitat, Heinrichstrafie 22, 8010 Graz, Austria. E-mail: [email protected]

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Respiratory System Modeling Introduction

We consider first the respiratory system. The purpose of the respiratory system is to exchange carbon dioxide (CO2), which is produced by the various tissues in the body, for oxygen (©2), which is a fundamental requirement of metabolism. When not influenced by voluntary or neurological changes in breathing, the respiratory control system varies the ventilation rate in response to the levels of CC>2 and C>2 in the blood. This is often referred to as the chemical control system. The control mechanism that responds to changing requirements to acquire C>2 and expel CC>2 acts to maintain the levels of these gases within very narrow limits. The mechanism by which this is accomplished has been extensively studied and consists of three components: • sensors that gather information about blood gas levels of CO2 and 02; • effectors that are nerve-muscle groups which control ventilation; • the central control processor, located in the brain, that organizes information and sends commands to the effectors. The sensory system consists of two main components: central sensors located in the brain that respond exclusively to CO2 and peripheral sensors located in the carotid artery (and perhaps aortic arch) that respond to both arterial CO2 and arterial C>2. The tissue site for the peripheral sensors is the carotid bodies while the aortic bodies may also play a role. In general, it is assumed the sensors respond to the partial pressures Pco2 and Po2 °f these blood gases. The information gathered by the sensors on blood gas levels is sent to the central control processor in the brain. The central control processor synthesizes this information and modulates the ventilation rate (VE). For convenience, the peripheral sensory site and its contribution to the overall ventilation rate are often referred to as the peripheral controller, while the central sensor site and its contribution are referred to as the central controller. This system represents a negative feedback control system. The sites at which the blood gas levels are measured are physically distant from the lungs, where the blood gas levels are adjusted. Any alterations in blood gas levels by the lungs cannot be sensed until the general blood flow transports these gas level changes to the sensors. Hence there is an inherent time delay in the control process which can produce instabilities in the system. A model of the respiratory control system must include metabolic rates for the production of CC>2 and consumption of C>2 in the tissues as well as describe uptake of Q% and expulsion of CC>2 by the lungs. The balancing of these exchange processes must be accomplished by the control of minute ventilation (VE) via the modulation of the rate and depth of breathing. The variation in the flow of fresh air through the lungs acts to maintain sufficient pressure gradients between the blood capillary and alveolar wall. This is essential since the exchange of gases is by passive diffusion alone. Clearly, knowledge about blood flow to tissue compartments and the lungs is also necessary for one to construct accurate models. Typically, such

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models consist of a set of mass balance equations describing inflow and outflow of a compartment for relevant state variables, plus auxiliary equations for the control mechanism, cardiac output, cerebral blood flow and other relationships.

8.1.2

Survey of Modeling Developments

Respiratory control system models date back to 1905 when Haldane and Priestley discussed the negative feedback nature of ventilation control and CC>2 regulation [30]. Other important milestones include Gray's multiple factor theory of the control of ventilation in 1946 [22] and the seminal paper of Grodins et al. utilizing control theory published in 1954 [27] and expanded in 1967 [26]. While neural control issues are important, most attention and effort has been directed toward modeling the chemical control of respiration. Since the respiratory system is quite complex, simplifying models have been devised to study specific functions of this system. Models can be broadly classified functionally as follows (see Khoo and Yamashiro [47]): • models of respiratory system response to hypoxia and hypercapnia; • models of stability, periodic breathing (PB) and apnea; • models of hyperventilation and exercise. Models can also be classified as minimal (simple) or comprehensive (complex). Examples of each type will be given below. Minimal models are important in clinical settings where it is feasible or possible to measure only a few physiological parameters. They can be used as tools to estimate parameters and to explore the stability properties of more complex models whose stability properties must be studied numerically (see, e.g., Cooke and Turi [11]). Comprehensive models seek to incorporate a large number of features of the respiratory system, including breathing patterns, cerebral blood flow, cardiac output, mechanical features of the respiratory system, metabolic features and differentiated compartment volumes. In general, control models view blood gas levels as the controlled quantities and the ventilation rate as the controller. To illustrate some of these ideas, the model given by Khoo et al. [46] is described in the Appendix. Models of the Control Mechanism

A number of models focusing on the respiratory control mechanism have been proposed. Among these are the important Gray controller [22] which has been incorporated into many models and the control proposed by Lloyd and Cunningham [53]. These control models differ in their identification of the source of the control sensory signals and of how the central control processor synthesizes the information of these control sensory signals into a control response. For example, Gray proposed that the controller response was the algebraic sum of responses to arterial carbon dioxide partial pressure (Paco ) j arterial oxygen partial pressure (Pao ) and pH levels, while Lloyd and Cunningham assumed a multiplicative interaction of Paco and Pao levels and a net hyperbolic dependence of minute ventilation VE on Pao .

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Much of the early work on controller models considered the responses to step increases in CC>2 inputs and basic stability properties of the control system. Transport delays were considered a primary source for unstable breathing patterns such as Cheyne-Stokes respiration (CSR). The modeling process can be seen in the series of models presented by Horgan and Lange during the first half of the 1960s. Horgan and Lange [34] considered a control responding only to Paco (with CC>2 dynamics similar to the Gray model) and introduced a transport delay between controller and lung compartments. This two-compartment model could produce Cheyne-Stokes type respiratory oscillations. However, physiologically realistic CSR could not be produced when Po2 was included in the controller [47]. The model proposed by Horgan and Lange in 1963 [35] introduced a Gray-type controller involving both Paco an<^ ^ao • This model could produce Cheyne-Stokes respiration but was considered too unstable, and the transient response to a step CC>2 inhalation was nonphysiological without special assumptions. A further refinement by Horgan and Lange [36] included a more complex controller containing a peripheral sensory site responding to both Paco and Pao and containing three central sensory sections responding to CC-2 in the brain (PBCO )• Simulation predictions made with this model were never adequately compared with experimental human data, and the model had the drawback that it required the identification of several unknown (perhaps nonphysiological) time constants [47]. Milhorn et al. [62] included a Gray-type control which included a response to CC>2 in the brain (PBCO ) and arterial C>2 (Pao ) (Horgan and Lange in 1965 included both Pao and Paco in their peripheral control). The simulated response to step increases in inspired CC>2 was faster than what was demonstrated by experimental observation [47]. These early models had strong points and weaknesses reflecting the state of understanding of the control mechanism organization at the time. As more physiological detail was added, models more accurately predicted basic experimental results. Recent work incorporates features of both the early Gray and LloydCunningham controllers. Reflecting physiological research, control models today generally assume that ventilation (VE) responds additively to the central and peripheral sensory information, that the central sensor responds to brain CO2 levels (via PBCO and perhaps pH) and that the peripheral sensor responds nonlinearly to Paco anc^ ^"ao i^l- ^ee Cunningham, Robbins and Wolff [12] for an extensive discussion of the experimental data reflecting on these issues. For models of this type, see, for example, Khoo et al. [46, 45], Longobardo et al. [57, 56], Fincham and Tehrani [18] and Tehrani [82]. Issues remain as to whether cerebral spinal fluid pH levels should be modeled separately. Table 8.1 in the Appendix summarizes some of these developments, and further references and details are given in [12, 25, 47, 54]. Minimal Models

Minimal models have been devised to study the stability properties of the respiratory control system. Glass and Mackey [61, 21] and Carley and Shannon [6] considered a one-dimensional state-space model. Cleave et al. [10] studied a two-dimensional model. Elhefnawy, Saidel and Bruce [16] considered a three-dimensional model for

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simulations which they reduced to a one-dimensional model for stability analysis. These models, being simplifications of the overall system, were able to capture only some of the important properties of the respiratory system. Several features of the respiratory system in steady-state which models should exhibit are listed below: (i) Peripheral ventilatory control response is 10-25% of the total response. (ii) CC>2 sensitivity is normally 2 liters per min per mm Hg approximately, (iii) Total or minute ventilation is 7 liters per min approximately. (iv) Papo = 40 mm Hg and Pao = 95 to 100 mm Hg approximately. (v) Other factors being held fixed, VE varies directly with arterial CO2 and varies nonlinearly (for example, exponentially) with C>2. (vi) The central control responds to the CC>2 level in the brain which varies less than the arterial level of CC>2. For minimal models it is difficult to satisfy all of these criteria simultaneously. For example, Glass and Mackey matched items (iii) and (iv) above but CC>2 sensitivity could vary by as much as 100% during oscillatory behavior. It should be noted that the models of Glass and Mackey, Elhefnawy, Saidel and Bruce and Carley and Shannon considered only CC>2 control of ventilation and that there are trade-offs in steady-state values for Paco > VE and control gain. For example, if one considers only Paco control, then a control gain level sufficient to produce the correct steady-state values of PacO and VE might make the control hypersensitive to changing Paco levels- Cooke and Turi [11] considered a two-dimensional extension of the Glass-Mackey model that included a control responsive to both Paco and Pao (essentially a peripheral control) and included a single constant point delay. They acknowledged that the model would be more unstable than the actual physiological system as the peripheral control responds rapidly to arterial gas levels. Batzel and Tran [3, 4] considered a minimal model of two and three dimensions with one delay and a more realistic control equation than Cooke and Turi. It incorporated the effects of the central controller. This allowed for a closer match to the above criteria but was still amenable to an analytic analysis of stability. The analysis clearly showed that the central controller has an important role to play in enhancing the stability of the control system. The results were used to study the stability properties of the larger 5-dimensional model of Khoo et al. [46]. Comprehensive Models

A number of important comprehensive models of the respiratory system have been proposed. These models include such features as metabolic processes, multiple compartment volumes, breathing patterns, multiple transport delays, pH and acidbase factors, dissociation laws and associated Bohr and Haldane effects, as well as cerebral, lung and tissue blood flows.

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As mentioned above, a comprehensive model of the respiratory system focusing on control issues was given by Grodins et al. in 1954 [27]. This model provided an important reference point for all future modeling efforts. The Horgan-Lange model [35] introduced transport delays into a two-compartment model with a Gray-type controller involving both Paco and Pao . Their 1965 refinement [36] included a more complex controller containing both a peripheral controller responding to Paco and Pa0 and three central controller sensory sections in the brain (located in brain tissue, spinal fluid and ventrolateral areas) responding exclusively to PCO2. The Milhorn et al. model [62] included cerebral blood flow and a Gray-type controller responsive to CO2 in the brain (PBCO ) and arterial C-2 (Pao )• As observed above, the simulated response to a step increase in C02 was faster than that observed experimentally. In 1967, Grodins et al. [26] extended their earlier model to include variable delays, cerebral blood flow and a cerebral spinal fluid compartment which modeled the effects of pH on ventilation. The model was very successful in describing the lung-blood-tissue gas transport and exchange system, but issues remained concerning the organization of the control system. At about the same time, Longobardo et al. [55] included a Lloyd-Cunningham type controller [53] among other features and could produce Cheyne-Stokes respiration under various conditions such as congestive heart failure. However, a large disturbance in ventilation was necessary to precipitate these responses [47]. Other researchers also extended earlier models to include more physiological detail. These models could be applied to study more complex respiratory system phenomena and conditions. During the 1980s, for example, Longobardo et al. [57] introduced a model based on previous work that included a central controller responsive to PBCO and a peripheral control responsive to both Paco and Pao • The model included a mechanism for translating controller response into changes in depth and rate of breathing, thus modeling the breathing process. The effects of the transition to the sleep state were studied and a model for sleep apnea presented. Model simulation was compared to experimental data and was able to reproduce the transient effects of CC-2 inhalation on breathing and patterns of Cheyne-Stokes respiration. Beyond this, the model was used to predict more complicated phenomena. The model was used to predict relationships between sleep apnea duration, circulatory delay and respiratory muscle group sensitivity to hypercapnia and hypoxia. Khoo et al. [46] developed a model in the same family as the Longobardo model but did not include simulation of the breathing processes. The model was used to analytically study the stability properties of the respiratory control system. The study included analysis of a form of respiratory instability referred to as periodic breathing. The model was applied to a number of conditions in which periodic breathing is observed including high altitude exposure, sleep and cardiovascular dysfunction. Predictions were made regarding the influence on stability of such factors as prolonged lung-chemoreceptor delays, high controller sensitivity and lung and tissue CO2 and O2 storage volumes. The majority of models described above considered minute ventilation VE in terms of a continuous flow of air at varying rates through the lungs. They thus

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average out the ebb and flow of air in a breath, omitting the effects produced by the cycle of inspiration and expiration. They ignore also the question of how the variation in VE is divided between varying the rate and depth of breathing. Models describing how the central processor produces a change in VE by adjusting either the rate of breathing, the depth of a breath or a combination of both have been developed. The model developed by Saunders, Bali and Carson [77] is a well-known example as is the Longobardo model [57] mentioned above. The stability of the system can be affected by the degree to which VE is varied by changing the rate of breathing as opposed to accomplishing a change in VE by varying the depth of breathing, as can be seen in Batzel and Tran [3, 4]. A recent simulation model developed by Eldridge [15] incorporates a large number of factors and can be used to simulate a range of respiratory phenomena. Extensive reviews of the history of respiratory modeling can be found in Khoo and Yamashiro [47] (which has been very helpful in this report) and Swanson et al. [80]. The ventilatory response to hypoxia is quite complicated and, for a variety of reasons, not easy to isolate. The fundamental response to an hypoxic stimulus is via the highly perfused peripheral chemoreceptors, which are sensitive to reduced oxygen levels in the arterial blood. A number of additional secondary mechanisms interfere with this primary response. This is one source of the difficulty in analyzing respiratory behavior during hypoxia, resulting in much variability in experimental reports of the phenomenon in the extensive literature on this subject [89]. The biochemistry underlying the hypoxic response also remains poorly understood [52], hence further obscuring the mechanisms involved. A recent complex model by Ursino et al. [88, 89] of the human respiratory control system focuses on the complicated response to hypoxia (and hypercapnia). The model is applied to study the interactions of regulation mechanisms during both short- and long-term hypoxic periods. Models of Ventilation during Exercise

The ventilatory control mechanism involved in raising ventilation during exercise is different from the chemical control mechanism described above. The mechanism cannot depend on deviations in blood gas levels alone from steady-state levels. Indeed, the levels of CC>2 and C>2 concentrations in the blood deviate very little from normal steady-state levels during exercise. This is true even though the metabolic production of COa and utilization of C>2 are 3 to 4 times higher than normal and ventilation increases markedly. It appears that ventilation is not "proportional" to CO2 levels (for fixed O2 levels) but rather "proportional" to metabolic rates during exercise. Thus, some mechanism other than the chemical control system must be involved. For example, kinesthetic stimuli from the muscles in motion, or another feed-forward mechanism which anticipates the rise in metabolic rates, could be involved in setting ventilation to match metabolic rates during exercise. It is possible that the control response acts to minimize a total cost involving both chemical stimulation and work associated with breathing [72]. Thus the chemical control mechanism and CC>2 in particular would still have some effect on ventilation, but other mechanical features such as functional reserve capacity might also. Further discussion will be presented in the following section on cardiovascular-respiratory modeling.

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Current Issues

While the basic mechanisms surrounding the lung and tissue compartment processes are fairly well understood [47], processes involved in the control mechanism are still actively investigated, such as the relative importance of pH and PBCO to the central sensor stimulus. Stability issues in the control mechanism have important medical implications, especially involving the phenomena of periodic breathing, obstructive apnea and central apnea. Mechanisms connecting central and obstructive apnea are still being studied as well as the interrelation of breathing depth and breathing rate in periodic breathing. The phenomenon of "paradoxical breathing" in infants, and other phenomena related to the states of hypoxia and hypocapnia, are of great interest and are being actively studied. The mechanism by which a change in minute ventilation is resolved partly into alteration in rate of breathing and partly into change in depth of breathing is still not completely understood. Mechanisms relating heart rate to ventilation rate and the matching of ventilation and lung perfusion are still being investigated. The actual processes involved in setting ventilation rate during periods of exercise, which does not primarily depend on the chemical control system, is still not well understood, although it is clear it involves a number of factors including body movements, learned response and anticipatory reflexes. Chemical control may help to calibrate ventilation after exercise has begun and provide for optimal performance. The degree to which optimization functions as a design criterion is still open, though it clearly plays a role in the evolution of respiratory control organization.

8.2 8.2.1

Models of the Cardiovascular System Introduction

Blood flow to various regions of the body depends on a large number of factors including cardiac output, various blood pressures, cross section of arteries and veins and partial pressures of COa and 02 in the blood. Blood flow is regulated by both central mechanisms of control which can affect cardiac output, general vascular resistance and arterial pressure and by local mechanisms found in each tissue region which can alter the local vascular resistance. Furthermore, blood gas levels affect both vascular resistance and cardiac output in several ways. The interaction of the central and local controllers of blood flow, cardiac output and blood pressure allow for a very adaptive but very complex system. The sympathetic system is a part of the nervous system that acts to increase heart rate, constrict blood vessels and raise blood pressure. The sympathetic nervous system and the parasympathetic nervous system together constitute the autonomic nervous system, which controls involuntary functions. In general, the parasympathetic system acts to inhibit the action of the sympathetic system. It is through this intricate organization, with control centers located mainly in the brain stem, the spinal cord and the hypothalamus, that control of various important elements of cardiovascular function is implemented.

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Local control of resistance consists of changes in the cross section of the arterioles leading into the organs. A decrease in cross section, for example, increases resistance to blood flow and (with pressure held constant) reduces blood flow rate. If the metabolic rate increases or less oxygen is available to a tissue area, local vascular resistance will be reduced, allowing for increased blood flow which can then remove excess CC>2 and replenish 62. The most important factor in this local control appears to be oxygen concentration. In general, an increase in C>2 concentration causes local vasoconstriction while a decrease causes vasodilation [28]. This represents an important auto-regulatory mechanism responsive to local needs in a tissue or an organ. There is also a central mechanism controlling systemic resistance that responds to changes in system Paco , Pao , pH and blood pressure. The central mechanism which influences resistance is located in the medulla and is referred to as the vasomotor center [5]. This center affects general vasoconstriction and hence acts to regulate blood pressure. It is divided into two sites: the pressor or vasoconstrictor center and the depressor or vasodilator center. The pressor center acts via the sympathetic nervous system to increase vascular constriction and the depressor center acts via the parasympathetic system to inhibit the sympathetic signals, thereby resulting in vasodilation [5]. The ability to control short-term blood pressure and blood flow is important for the local distribution of blood flow. The vasomotor center reacts to information about arterial pressure (absolute change and derivative) sent from the baroreceptors found in both the aortic arch and the internal carotid arteries. To redirect blood flow while maintaining adequate blood pressure, the vasomotor system can increase vasoconstriction globally while local control lowers resistance where blood flow demands are increased. It is important to note that local control can override the central control. In this way, blood flow is restricted in regions where blood flow demands are less while blood flow is increased where it is needed. Certain areas such as the brain do not respond to the central vasoconstriction signal since adequate blood flow in these areas is necessary at all times. The heart's pumping activity or cardiac output Q depends on the interaction of a number of factors including heart rate, blood pressure, myocardial contractility (the force of contraction of the heart muscle), the degree of ventricular filling (Starling's law) and the blood gases PacO an
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nals have an inhibitory effect on the vasomotor center. When blood pressure falls and thus the baroreceptors are less stimulated, the vasomotor center increases the sympathetic tone and decreases the parasympathetic tone. An increase in sympathetic tone then raises blood pressure because sympathetic activity increases both the heart's pumping activity and vessel resistance. Sympathetic activity increases cardiac output by increasing both heart rate and contractility. Sympathetic activity increases resistance through constriction of the small arteries and large arterioles. When blood pressure rises and hence the baroreceptors are more stimulated, vasomotor center activity reacts with the opposite effects. This negative feedback baroreceptor reflex mechanism is referred to as the baroreflex and it plays a very important role in controlling the response of the cardiovascular system to many operating conditions. There are also chemoreceptors in the carotid and aortic bodies (near to and perhaps the same as the respiratory sites) whose afferent nerves mingle with the baroreceptor fibers and pass into the vasomotor center. The carotid and aortic bodies are provided with a richer supply of blood flow than is normally found in other tissues. As a result, they are quickly responsive to changes in arterial blood. The chemoreceptors are stimulated because of diminished oxygen and excess carbon dioxide, which may occur during asphyxia, for example, or when there is reduced blood flow (due to hypotension or hemorrhage, for example). The net result is that the vasomotor center is activated, resulting in vasoconstriction [63, 13, 5]. The central vasoconstriction helps to counteract local vasodilation that can occur when hypoxia occurs. The chemosensor role appears to be most significant as an emergency backup. However, the interaction of blood gas levels and vasomotor function is very complex and is interrelated to other cardiovascular functions. The respiratory chemoreceptor system plays a more active role in the normal control of respiration, while the chemoreceptor role in cardiovascular function plays a significant role in more extreme situations [63]. From the above discussion, it can be seen that the control mechanisms involved in adjusting blood flow are several in number and interlinked. The important elements are the vasomotor center, baroreflex and chemoreflex, which are linked via the sympathetic and parasympathetic systems, and local autonomous resistance control, which operates in the various tissues. All of these factors interact to modulate cardiac output, blood pressure and resistance in such a way as to maintain effective function in a variety of circumstances. The interrelation of these mechanisms has added to the challenge of developing models to study this system. In practice, due to these complex interrelationships, models often focus on one factor such as arterial blood pressure as a controlled quantity and view other factors, such as heart rate and resistance, as direct or indirect controllers. Models are also often restricted in scope, focusing on specific aspects of the system. For further details on the cardiovascular system see, e.g., [5] and [40], and for an overview of cardiovascular modeling approaches see [75] and [64].

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Survey of Modeling Efforts

Models of the cardiovascular system can be grouped in many ways, some of which are listed below: • • • •

models of the mechanical cardiovascular system; models of control of the cardiovascular system; pulsatile versus nonpulsatile models; comprehensive versus restricted models.

Modern mathematical analysis of the mechanical cardiovascular system began in 1899 with the work of Prank [20] and his students and was continued in the later work of Aperia [1] and MacDonald [60]. Modeling of the mechanical system concerns blood flow in arteries and veins, the heart cycle and other phenomena which do not involving active processes to control the system. Due to the complexity of the cardiovascular system, many models have focused on specific aspects of this system such as pressure-flow studies in arteries or veins (MacDonald [60]), left heart-artery studies (Kenner and Pfeiffer [44]), heart action and circulation (Palladino, Ribeiro and Noordergraaf [68]), the baroreflex loop (Ursino, Fiorenzi and Belardinelli [87] and Le Vey and Vermeiren [51]), local autoregulation of resistance (Peskin [69]), the dependence of resistance upon blood pressure and blood flow (Kenner and Ono [43]), baroreflex influence on heart rate variability (Cavalcanti [7]), contractility (Honda et al. [33]) and specific states of the system such as sleep (Timischl, Batzel and Kappel [84]), or for medical application (Tsuruta [85]). The interrelationship of the various mechanisms requires that models address the question of meaningful simplification of the global system. By the term comprehensive model we refer to models which take into account and integrate several systems of the cardiovascular system. This is contrasted with local models which attempt to isolate or limit the scope of study to a single significant feature. A number of comprehensive models of the cardiovascular system have been developed. These models can involve pulsatile blood flow (involving effects of the heart cycle) or nonpulsatile blood flow and generally incorporate one or more mechanisms of cardiovascular control. For reference purposes, a comprehensive model developed by Timischl [83] is given in the Appendix. Comprehensive Models

The majority of all comprehensive models for the cardiovascular system are extensions or modifications of Grodins' seminal four-compartment model presented in 1959 [23, 24]. This model consists of two circuits (systemic and pulmonary) arranged in series and two pumps (left and right ventricles) as depicted in Figure 8.2 in the Appendix. As in the Grodins respiratory model, the basic model structure consists of compartments representing volumes. In this case, the volumes are the artery and vein storage volumes of the systemic and pulmonary circuits. The compartment equations involve mass balance relations for blood flow in and out of the compartments.

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Control processes were explored by including relationships between resistance and metabolic factors which would affect arterial blood pressure Pas> and a baroreflex mechanism which acted to stabilize Pas during significant perturbations such as hemorrhage. These relationships were empirical and preliminary in nature but allowed for the study of the interrelationships of a number of cardiovascular quantities including Pas, heart rate H, systemic resistance Rs and cardiac output Q for both the left and right chambers of the heart. The frequencies of pulsations in human cardiovascular blood flow tend to be harmonics of the fundamental heart rate (approximately 1.2 beats per second). Pulsatility can play an important role in some cardiovascular phenomena. For example, pulsatile arterial blood flow tends to result in fairly equal distribution of blood to all tissues of the body. This phenomenon would not occur with a nonpulsatile steady flow. Pulsatility also plays an important role in the baroreflex response in certain situations; e.g., see Ursino, Fiorenzi and Belardinelli [87]. However, since these pulses are essentially superimposed on a slower general flow, models with nonpulsatile flow have been applied to areas such as pharmacokinetic studies as well as to other areas where the pulsatility does not play a substantial role. This would be the case, for example, when all the tissue compartments are lumped into one compartment or resistance site. The Grodins model described above is nonpulsatile as are later models presented in the work of Moller, Porovic and Thiele [65] and Rideout [75]. An example of a pulsatile model can be found in DeBoer, Karemaker and Strackee [14]. This model used difference equations to describe how the characteristics of each heartbeat are influenced by the features of previous beats. The model also includes important features of the cardiovascular system such as control of heart rate and peripheral resistance by the sympathetic system via the baroreflex, contractile properties, respiratory effects such as mechanical effects of respiration on blood pressure and Windkessel properties. The model was applied to study short-term changes in heart rate and blood pressure and systemic responses to blood pressure medication. One interesting result was the suggestion that the 10-second Meyer wave could be modeled via the delay in the baroreflex loop. Studies of bifurcation based on this model can be found in Eyal and Akselrod [17]. Kappel and Peer [38, 39], adapted the Grodins model to study transition from rest to exercise. The model included a relationship between systemic resistance and O2 concentration given by Peskin [69] and which was based on the work of Huntsman, Noordergraaf and Attinger [37]. In steady-state, heart rate H and contractility are related through the Bowditch effect which states that contractility is increased if the heart rate is increased. Studies conducted by Kappel and Peer [38] showed that for reasons of stability it is necessary to assume that the contractilities adapt to a higher heart rate with a certain delay. Based on this observation, a dynamical relationship between H and contractility was modeled by a second-order differential equation (see equations (8.15)-(8.18) in the Appendix). The most important feature of the model was the introduction of optimal control as the fundamental mechanism for controlling arterial blood pressure. In this approach, heart rate is the controller and optimality was defined in terms of minimizing deviations in blood pressure while also minimizing energy utilization by penalizing excessive variation

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in heart rate. The model control was used to move the system from the steady-state rest to steady-state exercise. Transient and steady-state results were compared favorably with experimental data obtained in bicycle ergometer tests. This approach provides a black box synthesis of the control mechanisms for the baroreceptor loop and makes it possible to examine the possible influence of optimality criteria for the operation of this system. Optimality Considerations

The concept that the heart and other cardiovascular mechanisms function in an optimal way, minimizing or maximizing certain quantities, has been applied in a number of settings. It is reasonable, for example, to study the possibility that respiration control or heart function control effectively act to minimize energy expenditure. Optimality criteria applied to regulation of respiration during exercise has been studied by Poon [72], while a study that looked at the optimal matching between the heart and arterial system was made in Kenner and Pfeiffer [44]. Pfeiffer and Kenner [70] studied the dynamic performance of the left ventricle during systole and developed an optimal strategy to eject a given volume of blood out of the left ventricle. As mentioned above, optimality criteria were applied to baroreflex function in Kappel and Peer [38, 39]. Optimality criteria have also been introduced into a study in Timischl [83] of cardiovascular-respiratory control during exercise. For a survey of applications of optimal control theory in biomedicine see, e.g., Swan [79]. Control Applications: Baroreflex Control Loop and Blood Pressure

Pressures and flows in the uncontrolled circulation (absent a strong disturbance) would tend toward stability due to the effect of the Frank-Starling mechanism (see Noordergraaf [66]) but the arterial pressure steady-state would be much higher than what we consider normal. The baroreflex control loop acts to stabilize arterial pressure at a lower level. A number of studies viewing the baroreceptor reflex from the point of view of control theory have been made. An example of early work utilizing computer simulation can be seen in Warner [91]. One of the important milestones in cardiovascular modeling is a comprehensive model developed by Guyton and Coleman [29] for the purpose of studying blood pressure and hypertension. Since the goal of the model was to study control of blood pressure and specifically hypertension, the model included the regulation of fluids and hence action of the kidneys. In addition to the inclusion of variable fluid volume and kidney action, the Guyton-Coleman model included effects due to load on the heart, autonomic reflexes (including baroreflex and chemoreflex function) and autoregulation of blood flow through individual tissues. The results from simulations with this model were a major factor in establishing the role that the kidneys play in long-term control of arterial pressure. In other applications, such as short-term blood pressure control, fluid level can be assumed constant (e.g., see Timischl [83]) and hence the kidneys need not be considered. A number of models, more limited in scope, have sought to isolate specific features of the baroreflex control loop or to study other control features. A model

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presented by Kenner [41] represented the autoregulatory pressure and flow controls in the peripheral arteries as well as the baroreflex mechanism by first-order transfer functions allowing for a linear analysis of the overall system stability. Ursino, Fiorenzi and Belardinelli [87] developed a mathematical model which described the influence of pulsatile blood flow on hemodynamic properties and cardiovascular function (which can be of importance in some contexts). Aside from the mass balance equations, the model also included representations for the cardiac pump as well as sympathetic control of heart rate, peripheral resistance and pressure-volume relations of the systemic venous system. Model simulations could reproduce experimental results which point to a role for pulsatility in the carotid baroreflex function. Lafer [48] studied the stimulation of heartbeat by the interaction of the sympathetic and parasympathetic nerve systems and also studied the dependency of stroke volume on ventricular elasticity. Cavalcanti and Belardinelli [8] and Ottesen [67] studied the effect of time delay on the baroreflex control of heart activity. They analyzed simple mathematical models of short-term blood pressure control and studied the change in stability properties with increasing time delay (in the neighborhood of one second) in the baroreceptor loop. A model recently presented by Ursino [86] included a range of important factors involved in controlling short-term arterial pressure. The mathematical model includes a six-compartment description of the vascular system, a model of the pulsating heart, two groups of baroreceptors and the activity of the sympathetic system. The influence of arterial pressure (baroreflex) and sympathetic action on system elements such as heart period, systemic peripheral resistance and heart contractility were also included in the model. The model was used to simulate medical situations involving the baroreflex response, including mild as well as severe acute hemorrhages. Results were consistent with experimental data. Bifurcation studies were made in a model of the baroreflex presented in Seidel and Herzel [78].

8.2.3

Current Issues

Among the many issues remaining to be resolved in cardiovascular research, we mention just a few, beginning with baroreceptor function and hypertension. Hypertension is an important risk factor for a number of diseases including congestive heart failure and stroke. Issues surrounding long- and short-term blood pressure control are of great interest. As described above, many studies have been made of various aspects of blood pressure control, and this topic remains an important current area of research. The interaction of the various factors which set cardiac output and vascular resistance just prior to and during onset of aerobic exercise is also an area of continuing research. Heart failure is common in the elderly population and too often fatal, with fewer than 30% of elderly persons surviving 6 years after their first hospitalization for this disease. Common risk factors leading to heart failure include coronary

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heart disease and systolic hypertension. The aging of the population worldwide will perpetuate the epidemic of heart failure well into the next century. Heart failure refers to the deterioration of the heart's pumping ability. There are four main forms: left-forward, left-backward, right-forward and right-backward heart failure. In left-forward heart failure, for example, the reduction in cardiac output is related to the reduction in contractility as the heart muscle is damaged and weakened. When contractility is reduced, stroke volume and cardiac output are decreased. Deterioration in contractility thus results in a reduction in blood pressure, leading to compensatory sympathetic system activation and vasoconstriction. The result is significant elevation of afterload, further reduction in stroke volume and damaging cardiac muscle compensatory changes (remodeling). As a result, the process of deterioration is exacerbated and becomes a self-perpetuating negative spiral. The kidneys may induce fluid retention to increase blood volume to compensate for the perceived reduced circulating blood volume. This acts to increase blood pressure but fluid retention also increases pre-load or filling pressure and symptoms of pulmonary congestion. Hence, the term "congestive heart failure." It is clear that the complex interrelation of factors which are involved in heart failure makes effective and long-term treatment very difficult. In the design of therapeutic strategies, models which can provide insight into these processes and quantify the interrelated factors are of great value. Areas of recent research include modeling contractility in heart failure (Honda et al. [33]) and the development of models for analyzing and diagnosing pathological states and drug treatment effects (Tsuruta et al. [85]).

8.3 8.3.1

Combined Cardiovascular-Respiratory Models Introduction

The cardiovascular and respiratory systems are linked in a number of ways and influence each other's performance. The cardiovascular control system affects the performance of the respiratory control system by determining the transport delay between the lungs, where Paco and Pao levels are adjusted by ventilation, and the sites where these levels are measured — Figure 8.1 in the Appendix. The rate of blood flow also influences the efficiency of gas exchange in the lungs and various tissue compartments. Vasomotor activity is related to respiratory center activity, and in general nearly any factor that increases vasomotor activity will also tend to (at least somewhat) increase respiration. The state of the respiratory system influences cardiovascular function in several ways. The chemosensors of the cardiovascular system (perhaps the same as or closely connected to the respiratory chemosensors in the carotid and aortic bodies) respond to Paco and Pao • As mentioned previously, activation of these sensors stimulates the vasomotor center. This stimulation in general acts to modulate heart rate, contractility and vasoconstriction in rather complicated ways and it is experimentally difficult to isolate specific interactions. It appears that, with ventilation maintained constant, chemoreceptor stimulation results in bradycardia, systemic

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vasoconstriction, reduced cardiac output and a negative left ventricular inotropic response [13]. Generally, however, stimulation of the peripheral chemoreceptors by Paco and Pao nas effects on both the respiratory and the cardiovascular centers. Under normal conditions, the respiratory effects on the heart via such mechanisms as the Lung Inflation reflex is more influential than the primary cardiovascular responses mentioned above, and thus heart rate tends to increase with peripheral chemosensor stimulation. Furthermore, the dilation of systemic arterioles which occurs as a direct consequence of hypoxic conditions locally (autoregulatory dilational response) also at least partially counteracts the central vasomotor vasoconstriction, with differing effects in different tissue regions. These responses are significant mainly when conditions of hypoxemia or low C>2 blood level occur. There can also be central control responses to local hypoxia and hypercapnia (high Pco2) conditions occurring in the central nervous system tissues which can come into play during asphyxia. The net effect of PacO an<^ P&O on carcuac output Q can be seen by holding other factors constant. A relation between Q, Paco and Pao is given by Richardson et al. [74] that was derived from experimental data. The effects of Paco and P^ are most important when deviations from normal levels in these blood gases are significant. Such conditions develop, for example, during hypoxic states which can occur with congestive heart failure, apnea, and during sleep. A basic curve fitting model describing the Pa<x> and Pao dependencies of cardiac output is given by Fincham and Tehrani [19], but the underlying interrelationships are not fully understood. Cerebral blood flow is also affected by Paco and data for this interaction can be found in Reivich [73] and Fincham and Tehrani [19]. The main cardiovascular responses to hypoxia involve several factors. Vasodilation occurs in the brain, heart and and any other organs where C>2 concentrations need to remain stable. There is also a central reflex for vasoconstriction in the other vascular beds as well as a moderate increase in systemic arterial pressure. As a net result of such changes, cardiac output increases. For further details on such effects see Heistad and Abboud [32]. The recent model by Ursino, Magosso and Avanzolini [88, 89] incorporates a number of these interacting cardio-respiratory features into their model of the human respiratory control system which focuses on the response to hypoxia and hypercapnia. Finally, a very important link between the two systems involves the level of oxygen concentration in the tissues. This concentration level influences the degree of vasoconstriction at the local vascular resistance level and is an important factor in directing blood flow to areas where it is needed. There are also well-noted processes which act to synchronize heart rhythms and respiratory rhythms as well as match ventilation rate and blood flow. These processes can include mechanical as well as neural influences.

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8.3.2

203

A Cardiovascular-Respiratory Model Application

One area where a comprehensive model involving both systems has been applied involves modeling ventilation and cardiovascular behavior during exercise. As mentioned above (Section 8.1.2) the ventilatory response to exercise can not depend on the chemical control mechanism. As the CC>2 and 02 concentrations do not deviate sufficiently to account for the observed increase in ventilation, other mechanisms or processes are involved. The model presented by Timischl [83] combined and extended the work of Kappel and Peer [38] and Khoo et al. [46]. The combined cardiovascular-respiratory model included an optimal control mechanism. The controlled quantities were arterial blood pressure Pas, arterial carbon dioxide PacO and arterial oxygen Pao . The two quantities used to control the system were the heart rate H and the ventilation rate VA- The control functions H and VA were adapted via the control loop such that Pas, PacO anc^ P&O were stabilized to their operating points when deviations occurred. These operating points were the steady-state values for a given condition defined by the choice of the parameters of the system. The optimal control characterized the operation of the baroreceptor loop and the respiratory control loop. The control drove the system from a set of initial conditions characterizing one physiological state to the steady-state of the system which characterized another physiological state. The transition between states was optimal in the sense that Pas, PacO and PaQ were stabilized such that deviations from their final steady-state values were as small as possible. Furthermore, heart rate and ventilation rate were prevented from changing too fast in order to provide a restraint on energy expenditure. The model was used to study the transient response predicted by the optimal control as the cardiovascular-respiratory system was transferred from the steady-state rest to the state of constant stable aerobic exercise. In this formulation, the ventilation rate of the exercise steadystate value depended on the metabolic rates. The optimality criteria defining the controls shaped the transient response with good results. The block diagram and model equations can be found in the Appendix. The respiratory and cardiovascular submodels were linked in the following ways. The respiratory mass balance equations included expressions for the blood flows Fs and Fp. The respiratory system in turn influenced the cardiovascular system via the local metabolic autoregulation. This was modeled by the assumption that systemic resistance Ra depends on venous oxygen concentration as described by Peskin [69]. This relation is given in equation (8.24) in the Appendix. It is based on a model for autoregulation first developed by Huntsman, Noordergraaf and Attinger [37]. Change in Rs due to the baroreflex or other sympathetic pathway and any delays in the process were not included. Heart rate H and ventilation rate VA acted on both systems through the control functions u\ and u%, while Pas, Pa-co anc^ Pao affected- the dynamical behavior through the cost functional. Recently, Timischl, Batzel and Kappel [84] extended the model to include transition to sleep and in [2] incorporated certain delays into the respiratory submodel. Also, in 1998, Wabel and Leonhardt [90] presented a model to simulate the cardiovascular and respiratory systems, based on the work of Coleman and co-workers

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and redesigned for simulation with the MATLAB toolbox Simulink. It included the heart and the peripheral circulation, the respiratory system, the kidneys and the major neural and hormonal control mechanisms. The model contains more than 30 blocks with over 200 physiological variables.

8.3.3

Current Issues

There are, of course, many open questions regarding the cardiovascular-respiratory control system. We mention here three areas: cardiovascular-respiratory entrainment, exercise physiology and congestive heart failure. While, the phenomenon of entrainment (correlation of certain respiratory and cardiovascular rhythms) is well known and has been extensively studied, a number of questions remain including how this phenomenon plays a role in the process of controlling and matching cardiovascular and respiratory functions. For example, it has recently been shown (Lorenzi-Filho et al. [58]) that the ventilatory oscillations of periodic breathing can amplify and entrain oscillations in blood pressure and heart rate. What are the stability implications of this process? Questions also remain on whether heartbeat triggers the onset of inspiration rather than modulation of cardiac rhythm being accomplished by ventilation or whether the coupling is by some phase relationship between the two systems. See, e.g., Larsen, Booth and Galletly [50]. In the second area, important questions remain as to how the cardiovascularrespiratory system anticipates, sets and matches responses in blood flow and ventilation rate to sudden and short-term aerobic exercise. In the area of exercise physiology, a topic of theoretical as well as practical interest concerns understanding where exercise performance limits are set within the cardiovascular-respiratory system and where the bottlenecks in the process of optimizing athletic performance lie. Finally, we note some issues related to congestive heart failure and CheyneStokes respiration (CSR). CSR is a form of periodic breathing instability in which periods of apnea are interspersed with periods of involuntary patterns of breathing. This can occur during sleep when voluntary control of breathing is suspended and the normal respiratory control response is muted. Congestive heart failure is an important medical condition associated with CSR. In congestive heart failure cardiac output is reduced due to deterioration of heart function. It is well known that delays in feedback control can create instability in a control system. The increased transport delay caused by reduced cardiac output in congestive heart failure will reduce the efficiency of the central and peripheral controllers of ventilation. This is due to the increased time it takes for blood to transfer blood gases from the site where these blood gas levels are adjusted (the lungs) to the sensory sites where these levels are measured. The actual mechanisms inducing CSR in congestive heart failure patients are still under active investigation. The increased feedback delay may be sufficient to contribute to the onset or persistence of CSR. See, e.g., Hall et al. [31], Pinna et al. [71] and Cherniack [9]. Reduced sensitivity to Paco or a reduction in Paco at the peripheral controller may also play a role. See, e.g., Lorenzi-Filho et al. [59]. Such factors would influence proper controller response.

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Conversely, recent epidemiological studies have indicated that sleep-related breathing disorders are an independent risk factor for hypertension. In particular, sleep apnea may lead to the development of cardiomyopathy and pulmonary hypertension (Roux, D'Ambrosio and Mohsenin [76]). Hence, modeling efforts which can elucidate some of the mechanisms surrounding sleep apnea (both central and obstructive) and its relation to hypertension would be of great value.

Appendix A.I Respiratory Model Proposed by Khoo et al. [46] Model equations for the block diagram of the Khoo model below represent basic mass balance equations. Equations (8.1) and (8.2) represent PacO an respectively, for the tissue compartment. The symbols that appear in the equations are denned in Table 8.2.

PICO refers to inspired CC>2 and PIO refers to inspired C>2. Note that delays (^ai TBI TT, TV) occur in the equations because the mass balance relations depend on the values of state variables which must be transported by blood flow between compartments. The delay depends upon cardiac output and the volume of the transporting arteries and veins. Because cardiac output in fact can vary with PacO and Pao , these delays are really variable and indeed distributed. However, the assumption of constant cardiac output is probably not a serious distortion, except in cases where cardiac output varies significantly. The peripheral sensory site and its contribution to the overall ventilation rate is referred to as the peripheral controller and is denoted by Vp, while the central sensor site and its contribution to ventilation rate is referred to as the central

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Figure 8.1. Block diagram of the respiratory system model.

controller and is denoted by Vc. The control equation represented below in (8.6) is a Gray-type of controller with the contributions from the peripheral controller Vp and central controller Vc additive. The peripheral controller describes a nonlinear relation between the effects of the delayed Paco and Pac>2 levels appearing at the carotid body sensor site. Thus the peripheral controller represents a LloydCunningham type control:

These equations essentially represent the response in ventilation rate (liters per minute) of the control processor to the state of the blood gas levels detected at the peripheral and central sensory sites. In the above equation GO and Gp represent control gains of the central and peripheral controllers while IP and IQ represent cutoff thresholds. The symbol [[Vp]] denotes that the formula for Vp will be set to zero if it becomes negative (similarly for Vc). Minute ventilation VE refers to the rate at which the volume of inspired air is passing into the lungs. Not all conducting airways and structures in the lungs can exchange gases with capillaries. Alveolar ventilation VA refers to that portion of inspired air which is in contact with gas-exchanging structures. The fraction of air contained in airways which cannot exchange gas is referred to as dead space ventilation VD- The relation between these quantities is

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A.2 Cardiovascular-Respiratory Model Proposed by Timischl and Kappel 1998 The symbols that appear in the equations are denned in Tables 8.2 and 8.3.

Equations (8.7)-(8.10) are basically the lung and tissue compartment equations found in the Khoo model but written in terms of concentrations Cac02, Cao2, etc. rather than partial pressures. Dissociation laws relate the two quantities. Equation (8.11) tracks brain COa concentration CBCO,- ^n equations (8.12)-(8.14) the dependencies of blood flow F on blood pressure are given by Ohm's law

where Pa is arterial blood pressure, Pv is venous pressure and R is vascular resistance. Equation (8.21) gives pulmonary arterial pressure. Details can be found in [38, 83]. Cardiac output Q is defined as the mean blood flow over the length of a pulse,

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where H is the heart rate and Vstr is the stroke volume. A relationship between stroke volume and blood pressure is given in Kappel and Peer [38]. It is based on the Prank-Starling mechanism and is described by

Resistance Rs depends on venous oxygen concentration CV02 via

Control functions HI and u2 are determined such that the cost functional

is minimized under the restriction

Figure 8.2. Block diagram of the cardiovascular-respiratory system model.

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Table 8.1. Timeline of respiratory modeling. Model and Year Gray (1946) Grodins et al. (1954) Horgan and Lange (1962)

Horgan and Lange (1963)

Lloyd and Cunningham (1963)

Horgan and Lange (1965)

Milhorn et al. (1965)

Longobardo et al. (1966)

Grodins et al. (1967)

Yamamoto and Hori (1971) Milhorn et al. (1972) Saunders et al. (1980)

Features Proposed algebraic sum of responses to Pa^, Pao and pH levels. Used Gray-type control and lung and tissue compartments to model step change in CO2. Used Gray-type control responding only to Pago and introduced transport delay between controller and lung compartment. This two-compartment model could produce CSR, but CSR could not be produced when Po2 was included in the controller. Used Gray-type control responding to both PacO2 and P^ . Could produce CSR but was too unstable. Postulated multiplicative interaction of response to Pacc,2 and Pa^ levels and a hyperbolic dependence of VE on Pa02 • Introduced three subcompartments to central controller including brain tissue, cerebral spinal fluid and medulla. Gray-type controller, but very complex with a number of unknown time constants. Used Gray-type control and several compartments. Controller responds to Paco2 m brain compartment and Pao2 m arteries. Cerebral blood flow depends on Pao2 and Pa^ • Designed to model CSR. First major model incorporating a multiplicative interaction between CO2 and O2 signals instead of the additive model of Gray. Postulated a single sensor for CO2 and O2 levels located in the arterial system. The controller is a Lloyd-Cunningham type as cited above. Complicated model with several compartments and incorporating acid-base buffering, variable cardiac output and cerebral blood flow and transport delay. Studied two types of Gray controllers incorporating response to pH, Paco and Pao • This was first major attempt to consider how changes in minute ventilation were translated into changes in tidal volume and breathing frequency. Cerebral spinal fluid compartment added to 1965 model. Sensors assumed responsive to pH. Major attempt to study the process of how ventilation was analyzed into frequency and tidal volume. Lloyd-Cunningham type controller was employed.

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Table 8.1 Continued. Timeline of respiratory modeling. Model and Year

Features

Longobardo et al. (1982)

Khoo et al. (1982) Fincham and Tehran! (1983)

Khoo et al. (1991) Tehrani (1993) Eldridge (1996)

Sleep apnea model. Central controller in brain responds to PBCO while peripheral controller responds to delayed PaCo an<^ P^o • Ventilation was analyzed into frequency and tidal volume. Designed to study PB. Similar to Longobardo (1982) model above but somewhat different controller. Developed a complex model incorporating variable cardiac output (hence variable transport delay) depending on Pac02 and Pao2 as well as a pH compartment and variable breath frequency and tidal volume mechanism. Adapted 1982 model to study sleep-induced PB. Adapted 1983 model to study infant respiratory phenomena. Complex computer model of respiratory control includes extensive sleep and obstructive apnea phenomena.

Table 8.2. Respiratory parameters. Symbol

Meaning

Unit

MRco2 MR02

metabolic COa production rate metabolic O? consumption rate partial pressure of CO? in arterial blood partial pressure of Oz in arterial blood partial pressure of COz in mixed venous blood partial pressure of O? in mixed venous blood alveolar ventilation effective COi storage volume of the lung compartment effective O% storage volume of the lung compartment effective tissue storage volume for COz effective tissue storage volume for O% effective brain tissue storage volume for CO?. dissociation constants relating concentration to partial pressure central controller gain factor peripheral controller gain factor central drive threshold value peripheral drive threshold value

ISTPD • rain ISTPD • min~ mmHg mmHg mmHg mmHg IBTPS • min 1 BTPS IBTPS 1 1 1

^"CO2 P

"02

Fvco2 Pvo2

VA VACO, V

AO, TC02

V

VT02 BCO, K,k,m Gc Gp V

Ic

Ip

l/(min -mmHg) l/(min -mrnHg) mmHg mmHg

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Table 8.3. Cardiovascular parameters. Symbol

Meaning

Unit

<*( ar

coefficient of Si in the differential equation for
min~ 2 min~ 2 mmHg • min-l" 1 mmHg • min""1 mmHg- min"1 1 • mmHg"1 1 • mmHg"1 1-mmHg- 1 I - mmHg-1 1- min"1 1- min"1 min"1 min"1 min"1 mmHg mmHg mmHg mmHg 1 • min~l 1 • min —1 mrnHg- min-1"1 mmHg • min-1"1 mmHg mmHg • min"1

•Apesk 01 0r

Cas Cap CVs Cvp

FP Fs

H ~il 7r

Pas *ap Pvs V

P

Qi

Qr

RP Rs S
1 1

Bibliography [1] A. Aperia, "Hemodynamical studies," Skandinavisches Archiv fur Physiologic, 83 (Supplement 16), 1940. [2] J. Batzel, S. Timischl and F. Kappel, Modeling the Human CardiovascularRespiratory Control System with Two State Delays: An Application to Congestive Heart Failure, Technical Report 191, Karl-Pranzens-Universitat, SFB Optimierung und Kontrolle, Graz, Austria, 2000. [3] J. J. Batzel and H.T. Tran, "Stability of the human respiratory control system. Part I: Analysis of a two dimensional delay state-space model," Journal of Mathematical Biology, 41(1), pp. 45-79, 2000.

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New England Journal of

[29] A.C. Guyton and T.G. Coleman, "Quantitative analysis of the pathophysiology of hypertension," Circulation Research, 24 (Suppl. 5), pp. 1-19, 1969. [30] J.S. Haldane and J.G. Priestley, "The regulation of lung ventilation," Journal of Physiology, 32, pp. 225-266, 1905. [31] M.J. Hall, A. Xie, R. Rutherford, S. Ando, J.S. Floras and T.D. Bradley, "Cycle length of periodic breathing in patients with and without heart failure," American Journal of Respiratory and Critical Care Medicine, 154, pp. 376-381, 1996.

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[32] D.D. Heistad and P.M. Abboud, "Circulatory adjustments to hypoxia," Circulation, 61, pp. 463-470, 1980. [33] H. Honda, K. Kinbara, J. Tani, T. Ogimura, Y. Koiwa and T. Takishima, "Simulated study of heart failure: Effects of contractility on cardiac function," Medical Engineering and Physics, 16, pp. 39-46, 1994. [34] J.D. Horgan and R.L. Lange, "Analog computer studies of periodic breathing," IRE Transactions on Bio-Medical Electronics, 9, pp. 221-228, 1962. [35] J.D. Horgan and R.L. Lange, "Digital computer simulation of the human respiratory system," IEEE International Conference Record, pp. 149-157, 1963. [36] J.D. Horgan and R.L. Lange, "Digital computer simulation of respiratory response to cerebrospinal fluid Pco2 m tne c&t>" Biophysical Journal, 5(6), pp. 935-945, 1965. [37] L.L. Huntsman, A. Noordergraaf and E.O. Attinger, "Metabolic autoregulation of blood flow in skeletal muscle: A model," in J. Baan, A. Noordergraaf and J. Raines, editors, Cardiovascular System Dynamics, pp. 400-414, MIT Press, Cambridge, MA, 1978. [38] F. Kappel and R.O. Peer, "A mathematical model for fundamental regulation processes in the cardiovascular system," Journal of Mathematical Biology, 31(6), pp. 611-631, 1993. [39] F. Kappel and R.O. Peer, Implementation of a Cardiovascular Model and Algorithms for Parameter Identification, Technical Report 26, Karl-FranzensUniversitat, SFB Optimierung und Kontrolle, Graz, Austria, 1995. [40] A.M. Katz, Physiology of the Heart, 2nd Edition, Raven Press, New York, 1992. [41] T. Kenner, "Dynamical control of flow and pressure in circulation," Kybernetik, 9, pp. 215-225, 1971. [42] T. Kenner, "Physical and mathematical modeling in cardiovascular systems," in N.H.C. Hwang, E.R. Gross and D.J. Patel, editors, Clinical and Research Applications of Engineering Principles, pp. 41-109, University Park Press, Baltimore, 1979. [43] T. Kenner and K. Ono, "The low frequency input impedance of the renal artery," Pflugers Archiv: European Journal of Physiology, 324, pp. 155-164, 1971. [44] T. Kenner and K.P. Pfeiffer, "Studies on the optimal matching between heart and arterial system," in J. Baan, A.C. Artnezenius, T. Yellin and M. Nijhoff, editors, Cardiovascular Dynamics, The Hague, 1980.

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[45] M.C.K. Khoo, A. Gottschalk and A.I. Pack, "Sleep-induced periodic breathing and apnea: A theoretical study," Journal of Applied Physiology, 70(5), pp. 2014-2024, 1991. [46] M.C.K. Khoo, R.E. Kronauer, K.P. Strohl and A.S. Slutsky, "Factors inducing periodic breathing in humans: A general model," Journal of Applied Physiology, 53(3), pp. 644-659, 1982. [47] M.C.K. Khoo and S.M. Yamashiro, "Models of control of breathing," in H.A. Chang and M. Paiva, editors, Respiratory Physiology: An Analytical Approach, pp. 799-829, Marcel Dekker, New York, 1989. [48] S. Lafer, Mathematical Modelling of the Baroreceptor Loop, Ph.D. thesis, Karl-Franzens-Universitat, Graz, Austria, 1996. [49] C. Lambertsen, "The chemical control of respiration at rest," in V.B. Mountcastle, editor, Medical Physiology, pp. 1447-1495, C.V. Mosby, St. Louis, MO, 1974. [50] P.D. Larsen, P. Booth and B.C. Galletly, "Cardioventilatory coupling in atrial fibrillation," British Journal of Anaesthesia, 82(5), pp. 685-690, 1999. [51] G. Le Vey and C. Vermeiren, "Short-term autonomic nervous control of the cardiovascular system: A system theoretic approach," Studies in Health Technology and Informatics, 71, pp. 163-178, 2000. [52] A.J. Lipton, M.A. Johnson, T. MacDonald, M.W. Lieberman, D. Gozal and B. Gaston, "S-nitrosothiols signal the ventilatory response to hypoxia," Nature, 413(6852), pp. 171-174, 2001. [53] B.B. Lloyd and D.J.C. Cunningham, "A quantitative approach to the regulation of human respiration," in D.J.C. Cunningham and B.B. Lloyd, editors, The Regulation of Human Respiration, pp. 331-349, Blackwell Scientific Publications, Oxford, UK, 1963. [54] G.S. Longobardo and N.S. Cherniack, "Abnormalities in respiratory rhythm," in A.P. Fishman, editor, Handbook of Physiology: Respiratory System Volume II, Control of Breathing Part 2, pp. 729-749. American Physiology Society, Bethesda, MD, 1986. [55] G.S. Longobardo, N.S. Cherniack and A.P. Fishman, "Cheyne-Stokes breathing produced by a model of the human respiratory system," Journal of Applied Physiology, 21(6), pp. 1839-1846, 1966. [56] G.S. Longobardo, N.S. Cherniack and B. Gothe, "Factors affecting respiratory system stability," Annals of Biomedical Engineering, 17(4), pp. 377-396, 1989. [57] G.S. Longobardo, B. Gothe, M.D. Goldman and N.S. Cherniack, "Sleep apnea considered as a control system instability," Respiratory Physiology, 50, pp. 311333, 1982.

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[58] G. Lorenzi-Filho, H.R. Dajani, R.S. Leung, J.S. Floras and T.D. Bradley, "Entrainment of blood pressure and heart rate oscillations by periodic breathing," American Journal of Respiratory and Critical Care Medicine, 159(4), pp. 11471154, 1999. [59] G. Lorenzi-Filho, F. Rankin, I. Bies and T.D. Bradley, "Effects of inhaled carbon dioxide and oxygen on Cheyne-Stokes respiration in patients with heart failure," American Journal of Respiratory and Critical Care Medicine, 159(5), pp. 1490-1498, 1999. [60] D.A. MacDonald Blood Flow in Arteries. Williams and Wilkins, Baltimore, 1960, 1974. [61] M.C. Mackey and L. Glass, "Oscillations and chaos in physiological control systems," Science, 197, pp. 287-289, 1977. [62] H.T. Milhorn, Jr., R. Benton, R. Rose and A.C. Guyton, "A mathematical model of the human respiratory control system," Biophysics, 5, pp. 27-46, 1965. [63] W.R. Milnor, "The cardiovascular control system," in V.B. Mountcastle, editor, Medical Physiology, pp. 958-983. C.V. Mosby, St. Louis, MO, 1974. [64] D.E. Mohrman and L.J. Heller, McGraw-Hill, New York, 1991.

Cardiovascular Physiology, 3rd Edition,

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[71] G.D. Pinna, R. Maestri, A. Mortara, M.T. La Rovere, F. Fanfulla and P. Sleight, "Periodic breathing in heart failure patients: Testing the hypothesis of instability of the chemoreflex loop" Journal of Applied Physiology, 89, pp. 21472157, 2000. [72] C.S. Poon, "Optimal control of ventilation in hypoxia, hypercapnia and exercise," in B.J. Whipp and D.M. Wiberg, editors, Modeling and Control of Breathing, pp. 189-196, Elsevier, New York, 1983. [73] M. Reivich, "Arterial Pco2 and cerebral hemodynamics," American Journal of Physiology, 206, pp. 25-35, 1964. [74] D.W. Richardson, A.J. Wasserman and J.L. Patterson, Jr., "General and regional circulatory response to change in blood pH and CO?, tension," Journal of Clinical Investigation, 40, pp. 31-43, 1961. [75] V.C. Rideout, "Linear analysis of the cardiovascular system" in J.S. Gravenstein et al., editors, Integrated approaches to Monitoring, Butterworths, Boston, 1983. [76] F. Roux, C. D'Ambrosio and V. Mohsenin, "Sleep-related breathing disorders and cardiovascular disease," American Journal of Medicine, 108(5), pp. 396402, 2000. [77] K.B. Saunders, H.N. Bali and E.R. Carson, "A breathing model of the respiratory system," Journal of Theoretical Biology, 84, pp. 135-161, 1980. [78] H. Seidel and H. Herzel, "Bifurcations in a nonlinear model of the baroreceptorcardiac reflex," Physica D, 115, pp. 145-160, 1998. [79] G.W. Swan, Applications of Optimal Control Theory in Biomedicine, Marcel Dekker, New York, 1984. [80] G.D. Swanson, F.S. Grodins and R.L. Hughson, editors, Respiratory ControlModeling Perspective, Plenum Press, New York, 1990. [81] M. Taylor "Optimality principles applied to the design and control mechanisms of the vascular system," in The Arterial System: Dynamics, Control Theory, and Regulation, R.D. Bauer and R. Busse, editors, pp. 181-194, Springer-Verlag, New York, 1978. [82] F.T. Tehrani, "Mathematical analysis and computer simulation of the respiratory system in the new born," IEEE Transactions on Biomedical Engineering, 40(5), pp. 475-481, 1993. [83] S. Timischl, A Global Model of the Cardiovascular and Respiratory System, Ph.D. thesis, Karl-Franzens-Universitat, Graz, Austria, 1998.

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[84] S. Timischl, J. Batzel and F. Kappel, Modeling the Human CardiovascularRespiratory Control System: An Optimal Control Application to the Transition to Non-Rem Sleep, Technical Report 190, Karl-Franzens-Universitat, SFB Optimierung und Kontrolle, Graz, Austria, 2000. [85] H. Tsuruta, T. Sato, M. Shirataka and N. Ikeda, "Mathematical model of cardiovascular mechanics for diagnostic analysis and treatment of heart failure; Part I: Model description and theoretical analysis," Medical and Biological Engineering and Computing, 32, pp. 3-11, 1994. [86] M. Ursino, "Modeling the interaction among several mechanisms in the shortterm arterial pressure control," Studies in Health Technology and Informatics, 71, pp. 139-161, 2000. [87] M. Ursino, A. Fiorenzi and E. Belardinelli, "The role of pressure pulsatility in the carotid baroreflex control: A computer simulation study," Computers in Biology and Medicine, 26(4), pp. 297-314, 1996. [88] M. Ursino, E. Magosso and G. Avanzolini, "An integrated model of the human ventilatory control system: The response to hypercapnia," Clinical Physiology, 21(4), pp. 447-464, 2001. [89] M. Ursino, E. Magosso and G. Avanzolini, "An integrated model of the human ventilatory control system: The response to hypoxia," Clinical Physiology, 21(4), pp. 465-477, 2001. [90] P. Wabel and S. Leonhardt, "A simulink model for the human circulatory system," Biomedical Technology, 43, pp. 314-315, 1998. [91] H. Warner, "Use of analogue computers in the study of control mechanisms in the circulation," Federation Proceedings, 21, pp. 87-96, 1962.

Chapter 9

Inverse Problems Related to Electromagnetic Nondestructive Evaluation

Fumio Kojima* Abstract This chapter is concerned with inverse analyses of electromagnetic measurements arising in quantitative nondestructive evaluation of material systems. First, problems on the identification of crack shape are considered. Second, our current efforts on the shape recoveries of natural cracks are discussed. The remainder of this chapter is devoted to challenging works on the characterization of material degradations for the purpose of aging and elongation of material systems.

9.1

Introduction

It is well known that quantitative nondestructive evaluation (QNDE) of material systems can be considered as a typical application of inverse problems in engineering sciences. Many efforts on inverse analyses have been directed to QNDE for the structural integrity of materials used in large-scale systems, such as airplanes, space shuttles, nuclear power plants, etc. The mathematical formulation of inverse problems have been effectively used in application to QNDE. Eddy current testing (ECT), a principal topic in this chapter, stands for a fundamental technique in QNDE of nuclear power plants. One of the inspection procedures is to detect output signal changes from an actuating magnetic probe. In this case, the input corresponds to an applied current to the probe, while the output is the signal of impedance change from the probe. Hence the relation between input and output of ECT can be described through the electromagnetic fields. Although there are many 'Graduate School of Science and Technology, Kobe University, 1-1, Rokkodai, Nada-ku, Kobe 657-8501 Japan. E-mail: kojimaQcs.kobe-u.ac.jp

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mathematical studies on inverse problems related to QNDE, practical implementations of those to real problems are limited. Prom a practical point of view, most real problems are well formulated in three spatial dimensions since material damages should be modeled on two-dimensional manifolds in engineering specifications. Moreover, recent visualization techniques are able to obtain observation data not on the line but on the surface. On the other hand, major mathematical discussions have focused on the two-dimensional case. In extending the results from 2D to 3D, the regularity properties of the inverse solutions are required by the Sobolev imbedding theorem. Those might be too restrictive of a constraint for setting the admissible class associated with material damages. Moreover, careful mathematical treatments on pointwise and/or distributed observations sometimes make no sense in engineering applications. In recent advanced sensor technologies, it is important that mathematical descriptions of observation mechanisms be strongly dependent upon the structure of measurement apparatus. We keep in mind these facts and our discussions in this chapter are concentrated on inverse problems from engineering perspectives. Our specific interest in this chapter is the discussion of electromagnetic inverse problems related to QNDE. In the past decade, demand has grown for QNDE of material systems used in nuclear power plants. Continuing efforts have been directed towards early discovery of any type of flaw in order to ensure a safe operation of the plants [1, 2, 3, 4, 5, 6, 7]. Periodic inspections and maintenance must be carried out for the reactor, turbine, steam generator and other components. Under such circumstances, it is essential to develop rapid and accurate inspection techniques for QNDE of the plants. Inverse problems in electromagnetic fields have attracted increasing attention in processing data from the advanced magnetic sensors. Useful computational algorithms have been developed that make it possible to image and recover a variety of defect information behind inspection data. Figure 9.1 illustrates the mechanism of a pressurized water-type nuclear reactor (PWR) plant. In this figure, steam generator (SG) tubes are the central parts for QNDE in the plant. The identification and the characterization of defect profiles from magnetic data are typical applications of the inverse problems arising in ECT. The QNDE for detecting the brittleness of reactor pressure vessels (RPVs) as illustrated in Figure 9.1 is the other critical issue for the structural integrity of the plants. It is also important to evaluate degradation of structural components in nuclear power plants for the purpose of the lifetime elongation. Inverse analyses with electromagnetic measurements play a key role in such inspection challenges. This chapter is organized as follows. In Section 9.2, the electromagnetic inverse problems are formulated for identifying crack shape. The mathematical model of the inspection is described by an "A-0" model derived from Maxwell's equations in three dimensions. A parameter estimation problem is considered for the inverse problem related to ECT. In Section 9.3, we discuss current efforts and achievements of electromagnetic inverse problems arising in crack profile identification of SG tubes. Research activities on ECT analysis done at the Japan Society of Applied Electromagnetics and Mechanics (JSAEM) are introduced and successful computational efforts are presented and discussed for recovering natural crack shapes of SG tubes. In Section 9.4, some possibilities of inverse formulations associated with

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Figure 9.1. Illustration of pressurized water reactor plants. QNDE are considered for the level of brittleness in RPVs . It is essentially required to understand the degradation mechanism from viewpoints of physics, magnetism and material science and to collect fundamental data from macroscopic and microscopic measurements, such as B-H curve measurement, Barkhausen noise measurement, superconducting quantum interference device (SQUID) measurement, etc. Those problems involve current challenging works on inverse problems in nonlinear electromagnetics. The final section of this chapter is devoted to future research directions on inverse problems related to aging material evaluation.

9.2 9.2.1

Parameter Estimation Arising in ECT Mathematical Model for Inspection Procedures

In this section, we consider the mathematical description of the inspection procedures for the ECT. Figure 9.2 illustrates the overall configuration of the ECT technique. The current source is applied to a coil probe as the input,

where J0 = (Jo>J%,J%) denotes the three-dimensional current density vector and where w denotes the angular frequency of the applied current. The applied frequency / = LJ/2ir must be determined in accordance with the so-called "skin depth" condition,

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Figure 9.2. Sample material with crack and measurement method for ECT. where /z denotes the magnetic permeability and where a denotes the electrical conductivity of the sample. Since a skin depth of sample material changes according to frequency of flowing current, it is possible to detect a configuration of deeply lying defects by changing the frequency. The eddy current inside the sample material is driven by magnetic flux generated by the coil probe. Conversely, the impedance of the coil probe is affected by eddy currents in the material sample. This output signal can detect the gain margin and the phase change of the voltage output of an LCZ meter

Namely, the output voltage has the representation

where the gain margin and the phase change are, respectively, given by

Consequently, it is natural that the system state can be described by an electromagnetic field using the complex phasor representations. Let H and E denote magnetic and electric fields in three dimensions and let H and J denote the associated magnetic flux density and current density vectors, respectively. Each component of the above vectors are defined on the three-dimensional Euclidean space, such as

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Thus the electromagnetic field in the neighborhood of the sample material as shown in Figure 9.2 is governed by Maxwell's equation in three spatial dimensions K3. Assuming that there is no electric charge density in the sample material, the phasor form of Maxwell's equations is represented by

From (9.4), we introduce the magnetic vector potential A = (Ai(x.), -A2(x), A3(x)), From the constitutive law,

and Ampere's law (9.7), we obtain ^

The current density vector J in (9.8) is represented by the summation of the source current vector (9.1) and the eddy current vector 3e given by where xs and Xc denote the characteristic function of the source region (coil probe) and the conducting sample domain, respectively. From Faraday's law (9.6), we obtain From this fact, the cross product of the above equation can be replaced by the gradient of a scalar electrical potential , Using this identity and applying Ohm's law, the eddy current vector can be rewritten by From (9.8) and (9.10), we obtain

Furthermore, from Gauss' law (9.5) and (9.9), it follows that The eddy current NDE system (9.11) and (9.12) is called the A-$ model.

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Mathematical Description of the Observation Mechanism

For the observation mechanism, the explicit forms of the mathematical model depend upon the sensor technologies. The conventional probe that is called the pancake-type coil as illustrated in Figure 9.2 consists of the single coil and is used for the input and the output. By virtue of the Biot-Savart law, the impedance change can be obtained bv

where Gc, N and / denote the inspection area of the tube, the number of coils per turn, and where u* is the Green's function in three dimensions. Thus the real output voltage given by (9.3) should be proportional to the impedance change,

In order to obtain model outputs by conventional probe, the partial differential equations (9.11) and (9.12) must be solved at each measurement point. Those procedures cause a considerable amount of computational volume in the inverse algorithm. To avoid this difficulty, the new EOT sensors have been developed, such as multiple coils probes, SQUIDs, etc. For instance, SQUIDs allow us direct measurements of magnetic flux density using a current injection method. Hence the remarkable reduction of computational costs can be desirable for SQUID-based NDE systems. The approach reported here is directly applicable to inverse problems using SQUIDs (e.g., see [8, 9, 10]).

9.2.3

Inverse Problems and the Associated Parameter Estimation

When a crack appears in a sample material, the output signal can detect the fluctuation of the impedance change AZ given by (9.13). This fluctuation is caused by the change of the skin depth (9.2) due to the existence of the material flaw. Suppose that the sample inspected here is a nonmagnetic conducting material. Then flows of electrical fields avoid a crack area, while magnetic flow fields can pass through the same area. This means that the existence of crack results in the fluctuation of the electrical conductivity a because the magnetic permeability becomes a constant value (j, = (J,Q = 4?r x 10~7 (H/m). Hence the inverse problem arising in EOT is to recover and visualize the distribution of the electrical conductivity,

using information on the input {J0, w} and the output {AZ} where C denotes the sample domain. Figure 9.3 shows the photograph of an artificial electric discharge machine (EDM) crack. Since the zone of the EDM crack is assumed to be zeroconductivity, the problem is reduced to the domain identification problem. In other words, the problem is to recover the shape function of the sample material based on information contained in the impedance trajectory AZ in (9.13).

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Figure 9.3. Photograph of EDM crack (Nuclear Engineering Ltd., 1997). Let us introduce the simple shape recovery problem in what follows. The crack region is assumed to be denned on

Assuming that the location, width and length of the crack can be initially determined, our focus is restricted to identifying the crack depth h(xi). To do this, let {Bi*}fci be the series of B-spline functions with knot sequence A^ (e.g., see [11]). To characterize the crack depth, the shape function h is approximated as

where q = {qf4}^ denotes the unknown parameter vector to be identified. Figure 9.4 illustrates an example of a sample specimen with inner crack. The parameter-to-output mapping is then obtained by

through (9.11) and (9.12) to (9.13) where Np denotes the location of the coil. The parameter estimation problem is then formulated as follows.

Figure 9.4. Admissible parameter class of EDM crack shape.

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Parameter Estimation Problem

Given an appropriate frequency ui and a set of current forces {Jj}j=i in accordance with the coil movement, and the measurement of the corresponding impedance trajectory Yd = {^j}jJi, find the optimal q = q* which minimizes the output least-square error

with respect to q € Qad subject to (9.11), (9.12) and (9.13). The precise numerical methods for solving the input-output relation play key roles in the practical implementation of numerical optimization routines. Although the finite element method using the nodal elements is a simple scheme and is easy to implement, it requires remeshing at each location of the pancake probe coil. This remeshing results in a considerable amount of computational effort for the problem considered here. Boundary element methods have an advantage for the problem considered here. However, serious computational efforts are necessary for implementing numerical integration of boundary element matrices. The hybrid scheme of the finite element method (FEM) and boundary element method (BEM) for the A- model is an effective method for the computational cost [12]. The edge finite element method for A takes advantage of a reduction in the required computational memory [13, 14]. To solve the optimization problem with the use of an appropriate numerical scheme, usual discrete optimization routines are applicable to the inverse problems treated here (see [15] for more details). In [16], the parameter estimation code was developed based on the hybrid use of FEM-BEM code that makes it possible to evaluate the probe impedance trajectory. Experimental data obtained at Nuclear Engineering Ltd., 1-3-7 Tosabori, Nishi-ku, Osaka 550-0001 Japan, were effectively tested using the proposed numerical code. Successful results of estimated shape and true shape are depicted in Figure 9.5.

Figure 9.5. True and estimated crack shapes in computational experiments [16].

Chapter 9. Electromagnetic Nondestructive Evaluation

9.3 9.3.1

227

Inverse Problems for Natural Crack JSAEM Benchmark Problems

Continuing efforts on the improvement of the advanced ECT have been accomplished by the Japan Society of Applied Electromagnetics (JSAEM). The purpose of this program is to evaluate the present status of ECT research, to develop forward and inverse numerical analysis methods for ECT and to develop advanced ECT technology for SG tubes with the help of experiments as well as numerical analysis from an academic point of view. Those results have been announced in technical books [1, 2, 3, 4, 5, 6, 7]. As described in the previous section, effective inverse algorithms were successfully tested using output least-square minimization problems [16], the Gaussian Process (GP)-based fuzzy inference system [17], etc. However there still exist critical problems for detecting natural cracks. The Research Committee on the Nondestructive Evaluation Technology by ECT has set up the international benchmark problem for the round-robin test. Major goals include the modeling of natural cracks and their characterization by advanced measurements and the development of inverse algorithms. To this end, benchmark problems with masked test specimens as shown in Figure 9.6 have been proposed. The tube samples were masked and the number and locations of natural flaws were unknown. Destructive tests were performed in April of 2000 and discussions and analysis were presented at the E'NDE Workshop which was held in Budapest in June of 2000 (see [6] for more details). Figure 9.7 is a photograph of the cross view of the sample with stress corrosion crack (SCC). The assessment of performance for inverse algorithms is listed in Table 9.1.

9.3.2

Inverse Analysis by Evolutionary Computation

Shape recoveries for natural cracks involve essential difficulties for their inverse procedures. The zero-conductivity model in the previous section has sometimes failed to predict the probe responses. Natural cracks such as the SCC have no distinct boundaries due to their nonzero conductivities. This implies that the inverse algorithm must take care of the estimation of conductivities (9.14) rather than the shape identification of the crack region. One possible solution is to invoke evolutionary

Figure 9.6. Masked tube sample for JSAEM benchmark problem (http://wwwsoc.hii.ac.jp/jsaem/html/ect-hl2/www-ect.html).

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Table 9.1. Performance criteria for inverse analysis. Priority for Evaluation 1

Evaluation Criteria

Specification

Orientation

Distinguishability of Circumferential Crack and Axial Crack Detectability of Number of Cracks 10% Accuracy for each Crack except less than 20% Depth Accuracy of Estimated Crack within l[mm] Error Time Required in On-site Inspection Conventional PC or Single Work Station Bench

2 3

Multiplicity Size of Depth

4 5 6

Size of Length Elapsed Time Computer System

programming (EP) for those inverse problems. The basic idea is summarized as follows (see [18] for more details): Suppose that a crack region is assumed to be characterized by

In the EP algorithm, each individual is directly set as the unknown vector q. At each generation cycle, the standard genetic operations performed and improved their populations. Since the shape parameters q of each crack can be represented as real numbers, we use the adaptive mutation mechanism

where f is the fitness value of the ith individual, that is, fi = Error(q t (t)), and where max f and min f are the maximum and minimum of fitness values at the current population. The parameters a and ß are the coefficient and offset, respectively.

Figure 9.7. Photograph of cross view of SCC sample from JSAEM.

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For the generation cycle, a steady-state genetic algorithm (SSGA) [19] is often used for computational savings. It should be noted that the electrical conductivity at each cell of the suspect region is determined by the genotype of each individual. If the cell is included in the crack region, then the conductivity is set as the estimated conductivity ac. Otherwise it is preassigned as the nominal value of the conductivity of the SG tube. Thus the fitness evaluation can be implemented by output least-square error (9.17) in the previous section. Many successful results have been achieved for a variety of sets of JSAEM benchmark problems. A successful result for one set of experiments is depicted in Figure 9.8.

9.3.3

Open Problems and Current Research Directions

Although subsequent tests using the EP algorithm worked properly, the identification of multiple cracks is a difficult problem to solve. Figure 9.9 shows the most difficult problem of the JSAEM benchmark problems. There exist four EDM cracks outside of the tube, and location, length and depth must be identified for each crack. It is almost impossible to determine those inverse solutions by applying the conventional output least-squares minimization problems. Co-evolutionary computation (CEC) [20] is a good candidate for solving the identification problems mentioned above. A CEC is generally composed of several species with different types of individuals (candidate solutions), while a standard EC has a single population of individuals. We are currently pursuing investigations to develop new algorithms based on CEC for solving this challenging identification problem (see [21, 22]).

9.4

Nondestructive Evaluation of Material Degradation

The evaluation of material degradation in pressure vessels in nuclear reactors is an important issue in lifetime elongation and safe operation. The Research Committee on the Electromagnetic Evaluation of Material Degradation of the JSAEM has been investigating new QNDE techniques for inspecting the degradation of materials in aged nuclear power plants using electromagnetic measurements before the initiation of macroscopic defects and flaws [23, 26]. Our current focus on the QNDE is directed to the low alloy steel (A533B) used in RPVs and a martensite stainless steel SUS410

Figure 9.8. Estimated results for a JSAEM benchmark problem (Sample No. NFI98-707).

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Figure 9.9. Measurement data and multiple EDM crack (JSAEM benchmark problem, Sample No. 98-25). as a reference. During the operation these structural materials suffer from high stress, heat and neutron radiation. The changes in microstructure cause changes in mechanical properties, such as hardness, yield stress, etc. Due to these effects, magnetic properties are affected and the assessments of the associated magnetic parameters provide valuable information on degradation levels before generating macroscopic crack. When facing difficulties arising in such QNDEs, it is essential to consider the fully nonlinear electromagnetic inverse problems. In the following, we introduce some challenging works on inverse analysis arising in material degradation.

9.4.1

Characterization of B-H Curve

Noting that the mechanism of material degradation is characterized by nano-scale flaws in the materials, electrical conductivities a are no longer sensitive parameters for QNDE. Magnetic materials are more concerned with the identification of magnetic permeabilities /x rather than that of electrical conductivities. JSAEM researchers have explored some interesting experimental findings for those parameters [23]. B-H curves as shown in Figure 9.10(a) reveal useful information with respect to degradation levels of magnetic materials. In those experiments, fatigue tests were applied to the sample A533B. For the case of low degradation level, we learned that the slope of the B-H curve at H = 0 increases and remnant flux density Br intends to decrease under the applied stress below yield. On the other hand, at the high degradation level, it was shown that the slope of the B-H curve at H = 0 saturates, while the coercive filed Hc increases. The fundamental technique for evaluating those magnetic parameters is to detect magnetic flux density B applying magnetic field H. Figure 9.10(b) depicts a prototype of measurement sensors for QNDE.

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Figure 9.10. (a) B-H curve and magnetic parameters associated with degradation., (b) Prototype of measurement system. Measurement apparatus is composed of magnetic yoke and pick-up coil. The yoke works as an input source in order to generate a magnetic field, and the pick-up coil can detect output signals by measurements of the magnetic flux density. By Ampere's law (9.7), the magnetic field of sample material A533B is governed by

where J denotes applied current to the magnetic yoke. For convenience, our concern in the inverse problem is restricted to the two-dimensional case. Namely, the magnetic vector potential A is approximated by (0, 0, A). Then it is well known that the B-H relation of the magnetic material is characterized by where M(H) is called a magnetic moment. As stated above, this moment function is characterized by the structural-sensitive magnetic parameters Hc and Br. Consequently, it yields that

The output signal from the pick-up coil is made by where O denotes the scanning area of measurements. Thus various kinds of inverse problems can be formulated for the identification of magnetic parameters associated with Hc and Br if the analytic model of the B-H curve is properly defined. 9.4.2

Stochastic Analysis of the Barkhausen Effect

Measurements of the Barkhausen effect (BE) in ferromagnetic materials have been used in the inspection of material degradation. The BE can be detected by means

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of a pickup coil as shown in Figure 9.10(b), when a ferromagnetic material is magnetized under the action of a varying external field. Suppose that magnetization is implemented by H = ct, where c is a constant parameter and t is time. Then, the B-H relation can be transformed into

This implies that BE measurements are proportional to the time differentiation of magnetic flux density. A comprehensive quantitative treatment of BE is connected with domain-wall (DW) dynamics in ferromagnetic systems. It is assumed that the induced voltage exhibits stochastic fluctuations and that this behavior reflects the stochastic nature of the interactions of magnetic DWs among themselves and with microscopic defects, such as impurities, dislocations, grain boundaries, etc. The stochastic DW analyses have been studied theoretically and experimentally in [24, 25]. Some experimental findings have shown that magnetic DW movement is dominant in the region of BE, while the rotation of magnetic moment M is dominant in the non-BE region [26]. Thus inverse analyses can be formulated as the detection and the characterization of BE signals using the stochastic dynamical model associated with DW motion.

9.4.3

Structure-Sensitive Magnetic Characterization by Nano-Scale Flaws

Studies of lifetime estimation and aging material evaluation will be essential works in the future. It is necessary to understand the mechanism of degradation from viewpoints of physics, material science, etc. The formulation of such inverse analyses is based on the connection between the magnetic sensitive parameters and the nanoscale flaws, such as impurities, dislocations and grain boundaries [27]. For the case of ferromagnetic materials, it was shown that the connection between a coercive force Hc and a parameter related to dislocation is modeled by

where F is the total area of DW in the unit volume, 4> is the angle between the normal of the Bloch wall, Ew is the DW energy, and Ms is the saturate value of magnetic moment, respectively. The other interesting evaluation index has been recently announced by [28]. That is, the relation between magnetic susceptibility and dislocations is given by

where 6 is the angle between H and Ms, and Ea is the magnetic anisotropy energy, respectively. Since those theoretical results are derived under the assumption of a single crystal, they are far from reality in current material systems. Thus, for the aging and lifetime estimates of material systems, there exist complete open inverse problems arising in QNDE.

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233

Concluding Remarks

Electromagnetic inverse problems arising in QNDE were discussed in this chapter. In the first part of this chapter, a feasible computational method related to EOT was represented for detecting and characterizing cracks in SG tubes of nuclear plants. The estimation package was tested with experimental data for EDM cracks and the computational results reported here are representative of the findings obtained in the experiments. The practical implementations of the inverse algorithm require computationally intensive iterative procedures in which the accurate forward problems must be solved numerous times. A rapid forward solver using a database has been developed and applied to the inverse algorithm for the purpose of computational savings (see [29, 30, 18]). The second part of this chapter was devoted to the applicability of inverse analysis to QNDE of material degradation. It can be expected that inverse analysis plays a central role in current and future efforts involving such QNDE techniques. In the course of inspection of material systems, the purpose of a lifetime estimate is to construct decision mechanisms, such as the exchange of the parts, the duration of inspection, etc. Roughly speaking, considerable inverse analyses for elongation and aging of material system might be recovering {da/dt, d^/dt} with the use of electromagnetic measurements. However, developments of such time marching analyses are not effective because both electrical conductivity and magnetic permeability provide less information on lifetime estimates of material systems. One possible approach is to look for appropriate sensitive parameters that are able to connect micro-cracks with nano-scale flaws. This is to sense dynamical changes in environmental conditions under operations, to explore physical parameters behind degradation mechanism and to construct new mathematical models for binding those factors. Useful prediction mechanisms based on new models will give accurate information and decision mechanisms for structural integrity of systems.

Acknowledgements This study was supported in part by the Research Committee on NDT by ECT of the Japan Society of Applied Electromagnetics and Mechanics (JSAEM) through a grant from 5 PWR utilities and NEL and by the Research Committee on NDE of Degradation of Steel Components in Nuclear Power Plants of the JSAEM through a grant from Japan Atomic Energy Research Institute and NFL. This research was also supported in part by the Grant-in-Aid for Scientific Research (Nos. 10680493, 07808055, 11895003, 10558075) by the Japan Society for the Promotion of Science.

Bibliography [1] R. Collins, W.D. Dover, J.R. Bowler and K. Miya, editors, Nondestructive Testing of Materials, Studies in Applied Electromagnetics and Mechanics 8, IOS Press, Amsterdam, 1995.

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[2] T. Takagi, J.R. Bowler and Y. Yoshida, editors, Electromagnetic Nondestructive Evaluation, Studies in Applied Electromagnetics and Mechanics 12, IOS Press, Amsterdam, 1997. [3] R. Albanese, G. Rubinacci, T. Takagi and S.S. Udpa, editors, Electromagnetic Nondestructive Evaluation (II), Studies in Applied Electromagnetics and Mechanics 14, IOS Press, Amsterdam, 1998. [4] D. Lesselier and A. Razek, editors, Electromagnetic Nondestructive Evaluation (III), Studies in Applied Electromagnetics and Mechanics 15, IOS Press, Amsterdam, 1999. [5] S.S. Udpa, T. Takagi, J. Pavo and R. Albanese, editors, Electromagnetic Nondestructive Evaluation (IV), Studies in Applied Electromagnetics and Mechanics 17, IOS Press, Amsterdam, 2000. [6] J. Pavo, C. Vertesy, T. Takagi and S.S. Udpa, editors, Electromagnetic Nondestructive Evaluation (V), Studies in Applied Electromagnetics and Mechanics 21, IOS Press, Amsterdam, 2001. [7] F. Kojima, T. Takagi, S.S. Udpa and J. Pavo, editors, Electromagnetic Nondestructive Evaluation (VI), Studies in Applied Electromagnetics and Mechanics, IOS Press, Amsterdam, to appear. [8] H.T. Banks and F. Kojima, "Boundary shape identification in two-dimensional electrostatic problems using SQUIDs," Journal of Inverse and El-posed problems, 8(5), pp. 487-504, 2000. [9] F. Kojima, R. Kawai, N. Kasai and Y. Hatsukade, Defect profiles identification of conducting materials using HTS-SQUID gradiometer with multiple frequencies, in Review of Progress in QNDE, AIP 20-A, pp. 377-384, 2001. [10] Y. Hatsukade, N. Kasai, H. Takashima, R. Kawai, F. Kojima and A. Ishiyama, "Development of NDE method using SQUID for reconstruction of defect shape," IEEE Transactions on Applied Superconductivity, 11, pp. 1311-1314, 2001. [11] C. de Boor, Practical Guide to Splines, Springer, New York, 1978. [12] F. Matsuoka, "Calculation of a three dimensional eddy current by the FEMBEM coupling method," in Proceedings of the IUTAM Conference on Electromagnetomechanical Interactions in Deformable Solids and Structures, NorthHolland, Amsterdam, pp. 169-174, 1987. [13] A. Bossavit, "A rational for edge-elements in 3-D fields computations," IEEE Transactions on Magnetics, MAG-24, pp. 74-79, 1988. [14] A. Kameari, Solution of axisymmetric conductor with a hole by FEM using edge-element, in COMPEL, 9, pp. 30-232, 1990.

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[15] F. Kojima, "Computational methods for inverse problems in engineering sciences," International Journal of Applied Electromagnetics and Mechanics, 7, pp. 1-16, 1996. [16] F. Kojima, Computational method for crack shape reconstruction using hybrid FEM-BEM scheme based on A-(j> method, in Electromagnetic Nondestructive Evaluation, Studies in Applied Electromagnetics and Mechanics 12, pp. 279286, IOS Press, Amsterdam, 1997. [17] F. Kojima, N. Kubota and S. Hashimoto, "Identification of crack profiles using genetic programming and fuzzy inference," Journal of Material Processing Technology, 108, pp. 263-267, 2001. [18] F. Kojima, N. Kubota, H. Kobayashi and T. Takagi, "Shape recovery of natural crack using evolutionary programming related to eddy current testing," International Journal of Applied Electromagnetics and Mechanics, to appear. [19] G. Syswerda, A study on reproduction in generational and steady-state genetic algorithms, in Foundations of Genetic Algorithms, Morgan-Kaufmann, San Mateo, CA. [20] D.B. Fogel, Evolutionary Computation, IEEE Press, Piscataway, NJ, 1995. [21] N. Kubota and F. Kojima, "Coevolutionary optimization in uncertain environments," in Energy and Information in Non-linear Systems, Proceedings of the 4th Japan-Central Europe Joint Workshop on Energy and Information in Non-linear Systems, CSAEM, pp. 20-23, 2001. [22] F. Kojima and N. Kubota, "Electromagnetic inverse analysis using coevolutionary algorithm and its application to crack profiles identification," in Proceedings of the 7th International MENDEL Conference on Soft Computing, pp. 75-80, 2001. [23] A. Gilanyi, K. Kitsuta, M. Uesaka and K. Miya, "Magnetic property assessment as basis for nondestructive evaluation for steel components in nuclear engineering," in Nonlinear Electromagnetic Systems, Studies in Applied Electromagnetics and Mechanics 10, IOS Press, Amsterdam, 1996. [24] G. Bertotti, "Langevin and Fokker-Planck equation with nonconventional boundary conditions for the description of domain-wall dynamics in ferromagnetic systems," Physical Review B, 39, pp. 6737-6743, 1989. [25] K. Yamada and T. Saito, "Observation of Barkhausen effect in ferromagnetic amorphous ribbon by sensitive pulsed magnetometer," Journal of Magnetism and Magnetic Materials, 104, pp. 341-342, 1992. [26] M. Uesaka et al., "NDE-based life science of Japanese nuclear reactors," in Proceedings of the 5th International Workshop on Electromagnetic Nondestructive Evaluation, IOS Press, Amsterdam, pp. 57-58, 1999.

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[27] H. Trauble, "The influence of crystal defects on magnetization processes in ferromagnetic single crystals," in Magnetism and Metallurgy, Academic Press, New York, 1969. [28] S. Takahashi, J. Echigoya and Z. Motoki, "Magnetization curves of plastically deformed Fe metals and alloys," Journal of Applied Physics, 87, pp. 805-813, 2000. [29] Z. Badics, Y. Matsumoto, K. Aoki, F. Nakayama, M. Uesaka and K. Miya, "Accurate probe-response calculation in eddy current NDE by finite element method," Journal of Nondestructive Evaluation, 14, pp. 181-192, 1995. [30] Z. Chen and K. Miya, "ECT inversion using a knowledge-based forward solver," Journal of Nondestructive Evaluation, 17, pp. 167-175, 1998.

Chapter 10

Some Suboptimal Strategies for Numerical Realization of Large-Scale Optimal Control Problems Karl Kunisch* Abstract A brief introduction to selected topics of suboptimal strategies for numerical realization of large-scale optimal control problems is given. Receding horizon strategies, reduced-order modeling methods as well as suboptimal methods to solve certain Hamilton-Jacobi-Bellman equations are discussed.

10.1

Introduction

We survey some of the techniques developed and modified within the last decade on numerical realization of optimal control problems governed by large-scale partial differential equations. Large scale is a vague term, of course, depending on the available resources in manpower, hardware and software. What may appear to be large-scale at a certain instance of time can become quite tractable soon thereafter. The study of suboptimal techniques, nevertheless, is a viable one. First, because as resources increase, the models become increasingly more complex. Second, the interest in suboptimal strategies is motivated not only by making large-scale problems feasible but also by reducing computing time for smaller problems and by systemtheoretic questions which go beyond optimal control, be it open- or closed-loop control. In Section 10.6 we state a model problem from fluid dynamics that serves both as motivation and as reference in the following sections. Section 10.2 is devoted to "Institut fur Mathematik, Karl-Franzens-Universitat Graz, A-8010 Graz, Austria. This research was supported in part by the Fonds zur Forderung der wissenschaftlichen Forschung under SFB 03, Optimierung und Kontrolle.

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the instantaneous control technique which can be considered as a special case of a receding horizon strategy. Reduced-order techniques are the subject of Section 10.3. We address both order reduction by proper orthogonal decomposition and by the reduced basis method. In Section 10.4 we give a very brief account of methods that can be utilized to obtain suboptimal solutions to the Hamilton-Jacobi-Bellman equation. The fields of suboptimal strategies and of optimal control of fluids are growing rapidly. We have not made an attempt to give complete lists of methods let alone authors who contributed to these topics. We hope, however, that the interested reader will find many relevant issues and that the references serve as an adequate introduction to the topics addressed.

10.2

A Model Problem

To explain some of the concepts for suboptimal control we shall repeatedly return to a model problem in fluid mechanics. For this purpose let fi be a bounded domain in R2, let T > 0 and set Q = (0,T] x fi, £ = [0,T] x dft, where dtt denotes the boundary of fi. We consider the controlled un-stationary Navier-Stokes equations

where Re > 0, g E W1'2(0,T;Ho/2(dQ)) and y0 E L2(ty, div y0 = 0 are given, B E £(U, L 2 (fi)) and u E L2(0, T; U). Here U denotes the control space, which is assumed to be a Hilbert space with inner product (-, -)t/- Further, y(t) — y(t, x) E R 2 ,

where n denotes the unit outer normal to d£l. For a function space treatment of (10.1) we refer to [10, 15, 36], for example. We consider the following optimal control problem associated with (10.1):

where F is a smooth real-valued functional that is bounded from below. Typical

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choices for F are

and

where z is a fixed control target. Let us assume that (P) admits an optimal control u* e U with associated velocities and pressure (y*,p*) = (y(u*),p(u*)). To formally derive the optimality condition for (P) it is convenient to introduce the Lagrangian

Here y(0, •) = yo in fi is kept as an explicit constraint. Setting the partial derivatives of L with respect to y,p, and u equal to zero we obtain the equations for (£, TT):

Equation (10.3) is referred to as the adjoint equation and (10.4) is the optimality condition. Combined, (10.1), (10.3) and (10.4) are called the optimality system. Due to the forward-backward nature of the primal equation (10.1) and the adjoint equation (10.3), as well as the strong coupling of the primal variables (y,p, u) and the adjoint variables (£, TT), the efficient numerical solution of the optimality system is a challenging task. If (P) is posed in 3-D or if it involves further coupling e.g., with thermal, chemical or mechanical processes [29, 34, 35], then it may become an almost impossible task to efficiently solve the optimality system directly. This is one of the reasons why suboptimal strategies are important. Another motivation is the reduction of computing time. We shall address suboptimal schemes in Sections 10.2-10.4. It will be useful to characterize the gradient of J at a control u in direction 6u. This can be achieved efficiently by means of the Lagrangian. Let us proceed formally by expressing (P) as

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where x = (y,p, u) and F(y) — JQ F(y)dx dt. Here e(x) = 0 represents the equality constraint given by (10.1). We have where A satisfies Here eu denotes the derivative of e with respect to u, e*u stands for the adjoint of eu and analogous notation is used for e*. The Lagrangian associated with (10.5) is given by

with (•, •) the inner product in the range space of e. The condition Ly = 0 is squivalent to which is (10.7), and further Hence by (10.6) we have Applying (10.8) to (10.2) we find for the Riesz representation of the gradient of J in (P}. The functional analytical framework for optimality systems related to optimal control of flow phenomena such as (P) has been investigated in several papers; we refer to [3, 13, 14] and the references given there. If the distributed control term in (10.1) is replaced by boundary control then (10.4) becomes

and

The correct functional analytic setting of boundary control problems requires timederivative bounds for the controls; see [14], for example. In our case this could be realized by choosing | /0 (|w(i)|^ + J^ u(i)|^)di as control cost. This would result in the extra term —/3(-j^)2u(t) plus boundary conditions in (10.10). An alternative approach, approximating Dirichlet by penalized Neumann boundary conditions, was pursued in [22].

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10.3

241

Instantaneous Control-Receding Horizon Control

To explain the approach let m > 1 be fixed and set 6t = T/m, ti = i 6t, for i = 0,... , m. As a first step in the presentation let us consider the case where the Navier-Stokes equations (10.1) are approximated by a Crank-Nicolson scheme. At the ith level of the instantaneous control method one solves the following stationary optimal control problem, where the variables (y,p,u) correspond to (l/(*i),p(*i), «(*<)) =

where

is a known forcing term. Let Ui = u(ti) denote a solution to (10.11) and set (ViiPi) = (y(ui),p(ui)). Then (yi,pi,Ui) satisfy the optimality system for (10.11) consisting of the equality constraints in (10.11), together with the adjoint equations:

and the optimality condition

Note that yi enters (10.12) through F as well as through the linearization terms. Even though F may depend on y only in a subdomain fi C fi corresponding to locations where observations are available, (10.12) requires yi throughout fi due to the contribution of the linearized convection terms. If instead of the semi-implicit

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scheme an explicit Euler scheme is used, we have

where R = ^ y»_i — (t/j_i • V)y,_i + /(£»). In this case information from time level i,_i is passed to ti solely through the inhomogeneity R. Let Ui denote a solution to (10.14) and as before set (yi,pi) = (y(ui),p(ui)). Then the optimality system for (10.14) consists of the equality constraints in (10.14) together with

together with

In the optimality system for (10.14) the coupling between primal and adjoint equations occurs only through F. The instantaneous control strategy considered as open-loop control consists in iteratively solving (10.14) (or (10.11) or some variation thereof) for optimal controls Uj and associated states (yi,pi). The feedback use and interpretation of instantaneous control consists in evaluating F in (10.15) on the basis of data available from the real system, solving (10.15) for (£i,7Tj), evaluating the optimal control at time level ti via (10.16), applying it to the system to obtain the new observation and then continue to the next time level. The instantaneous control approach replaces (P) by a sequence of stationary optimal control problems. Clearly it cannot be claimed that a solution to (P) is obtained by this technique and its justification therefore needs to be addressed. In the context of control of fluids the instantaneous control technique was probably first discussed in [12] and utilized and refined in several papers thereafter; e.g., see [11] and the references given there. One justification for the instantaneous control strategy is its success in achieving the control objective in numerical tests for diverse control problems in fluid mechanics; we refer to [5, 11, 12] and the references given there. Further frameworks for the analysis of the instantaneous control strategy are discussed next. (i) In [26] the connection between instantaneous control and receding horizon control was pointed out. Receding horizon control is a well-known technique

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in optimal control of nonlinear ordinary differential equations; see e.g. [9] and the references given there. To briefly explain some concepts we follow [26] and consider the infinite-horizon optimal control problem in finite dimensions:

where f°: Rn x Rm -> R and /: R" x RTO -> R™. Next (10.17) is replaced by a sequence of finite horizon problems. Let T > 0 denote the so-called prediction horizon, let G denote a continuous mapping from Rn to R, and consider

where Xk denotes the solution on \(k — 1)T, kT] which is assumed to exist. Let x denote the function defined on [0, oo) which arises from concatenation of the solutions Xk, k = 2 , . . . , to (10.18) which are assumed to exist. So far, the replacement of (10.17) by the sequence of problems (10.18) has almost exclusively been justified by means of the asymptotic stabilization property for (10.7), which can be guaranteed under appropriate assumptions on / and G and/or additional explicit constraints on the state x in (10.17). The framework that best fits the application to the discretized Navier-Stokes equations uses the concept of closed-loop dissipativity. Definition 10.1. Problem (10.17) is called closed-loop dissipative if there exist a feedback law u = —K(x) and a > 0 such that

Examples for closed-loop dissipative systems are given in [26]. Assume henceforth that (10.17) is closed-loop dissipative, and set G(x) = f |x|^n. We define for T > 0

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and analogously V: R™ —> R denotes the minimal value functional for (10.17). Then for 0 < f < T it can be shown that

Therefore, a longer prediction horizon results in a better estimate of V(x). Moreover, if

then

and hence x(kT) —> 0 for k —> oo. Estimate (10.20) holds, for example, for f°(x,u) = /3i|x|2 +/3-2\u\2. If (10.17) is not necessarily closed-loop dissipative, then similar results can be obtained if G is chosen as a control-Lyapunov function. Related results are also available for discrete-time systems. The instantaneous control strategy is an extreme case with only one discrete-time predication horizon step. (ii) Another analysis for the receding horizon optimal control concept applied to the Navier-Stokes equations was proposed in [24]. To briefly explain the approach we consider

subject to (10.1) with B = I and Y, U given. The infinite horizon problem (10.22) is replaced by a sequence of finite horizon problems

subject to (10.1) with initial condition y ( k T , - ) = yk(kT,-), where yk is the solution to (10.23) on \(k - l)T,kT\. Let y denote the function constructed from {yk}'j*L1 by concatenation. The main assumptions for the analysis in T9A1

QT-O

According to (Cl) the optimal control body forces u in (10.22) should be "close" to the body forces U corresponding to the desired flow field Y. With

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(Cl), (C2) and appropriate technical assumptions holding, it can be shown that there exist constants M > 0 and « > 0 such that

(iii) In [23] the authors analyze an instantaneous control strategy based on an implicit time stepping scheme. They consider optimal control of Burgers' equation

subject to u € L 2 (0,T;L 2 (ft)) and

where f3 > 0, v > 0 and z are given, and ft = (0,1). Let A denote the negative Laplacian b(y) = yyx and let h > 0 stand for the step size. Given {u^}kL1 and setting Zk = z(tk) we consider the following algorithm: (a) fc = 0, t0 = 0. (b) Solve for (y, A):

, (c) Set VJ(i/(u£),u£) - /3u°k - B*\. (d) Given p > 0 set uk+i -u°k- pVJ(u°k). (e) Solve (/ + hA)yk+i - yk + hb(yk) + Buk+i. (f) Set ifc+i = tk + h, k = k + 1 and return to step (b). The two equations in step (b) are readily seen as primal and adjoint systems of a discrete-time linear-quadratic optimal control problem. Accordingly, an optimal step length for p in step (d) can easily be computed. Under the assumption that B = I and u^ = 0, for all fc, the determination of yk+i in the above algorithm is equivalent to

where Sh is the solution operator to v — v h v" = /, with homogeneous Dirichlet boundary conditions. If h is sufficiently small and appropriate technical assumptions are satisfied, then there exists K 6 (0,1) such that and hence \yk - zk\L*(tt) for k -> oo.

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(iv) In a recent paper [17] the close connection between instantaneous control, the multiple shooting approach, well known as a numerical method in optimal control for ordinary differential equations, and the Gauss-Seidel method applied to a discrete-time reformulation of continuous time optimal control problems is pointed out. It is shown that certain instantaneous control techniques coincide with the first step of a forward Gauss-Seidel iteration applied to the discrete-time problems. The analysis in [17], which is carried out for linear quadratic problems, can be extended to certain nonlinear problems and will be instrumental in improving numerical aspects of instantaneous control and receding horizon strategies.

10.4

Reduced-Order Methods

A powerful and structurally completely different possibility for solving optimal control problems for complex systems is the use of reduced-order methods. The underlying idea consists of projecting the partial differential equation onto some lowerdimensional state-space, to project the cost-functional accordingly and to solve the resulting lower-dimensional problem. A popular method for obtaining reducedorder methods is based on proper orthogonal decomposition (POD). An alternative is given by reduced basis methods. We shall explain these two techniques and turn to POD first. Let X denote a Hilbert space with inner product (•, -)x- In the case of (10.1) it could be the closure of {v € C$°(ty2: div v = 0} in L 2 (n) 2 or H1^)2 so that elements of X are divergence-free. For given n € N let denote a grid in the interval [0, Tj. Let {T/J}"=O denote the velocity components of the solution (yj,pj) to (10.1) at the grid points {tj} corresponding to some fixed reference control. POD does not address the question of how these solutions, which are referred to as snapshots, are obtained. They must be available from an independent numerical technique or from experimental data. Let us set V = span {yj}"=0 and d = dim V. If {^i}f=l denotes an orthonormal basis for V, then each member of V can be expressed as

The method of POD consists of choosing an orthonormal basis such that for every t 6 {!,-•• , d} the mean square error between y j , j = Q,... ,n and the corresponding Ith partial sum in (10.26) is minimized on average as follows:

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The solution {^t}f=1 to (10.27) is called the POD basis of rank i. It is characterized by a necessary optimality condition. We introduce the bounded linear operator ;y : l"+i_>Xby

Its adjoint y*: X -» Rn+1 is given by

It follows that n - yy* and 1C = y*y are given by

respectively. Using a Lagrangian framework the optimality condition for (10.27) is given by

Note that 71 is bounded, self-adjoint, nonnegative and, since it has a finite-dimensional range, it is also compact. By Hilbert-Schmidt theory there exists an orthonormal basis {V'iheN for X and a sequence {Aj}i€N of nonnegative real numbers so that

and V = span {V'ij'iLi- Setting

we find Ttvi = A» Vi and (vi,ttj) R n+i = 5ij. Thus {u,}f=1 is an orthonormal basis of eigenvectors of 72. for the image of 7£. Conversely, {^i}f=i can be obtained from {u«}jLi by means of ^i = -4- yvt, i = 1,... ,d. The sequence {V)i}f=i solves (10.27). Note also that due to orthonormality of {^1)^=1 the "min-expression" in (10.27) can be replaced by ££=o£?=i K^V^xl 2 . If {ijji]l=l is the POD basis of rank £ < d, then we have the following error formula:

For computations the spatial variable must be discretized as well. Then both 72- and K, are matrices and the computation of the POD basis will be carried out by whichever matrix has smaller dimension. In finite dimensions, moreover, the close connection between POD and singular value analysis becomes quite obvious.

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The question about the choice of i is certainly a critical one. It is commonly resolved by defining the relative information content

If a basis is required that contains 6% of the total information, then t is determined according to

The reduced dynamical system is obtained by a Galerkin approximation applied to (10.1); i.e., one makes an Ansatz

and the coefficients oti(t) are determined from

where Pe denotes the projection onto span {ipi}i=\- In (10.29) the inner products are in L2(Q) and for simplicity we assumed g = 0 in (10.1). Note that the divergence-free condition is incorporated in the basis elements and hence it does not explicitly enter into (10.29). Making a further Ansatz for the controls,

(10.30) can be expressed as

with A and B matrices and n a nonlinear mapping. Inserting the expressions for y* and ue into J ( y , u), the model problem (P) can be expressed as a finite-dimensional control problem of the form

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Different from a generic approximation, the POD-based system reduction leading to (Pf) has the property that its basis elements are related to the structure of the dynamical system (10.1). The basis elements, however, are computed for a reference control, which does not represent the optimal control for (Pf). Hence the problem of unmodeled system dynamics occurs. It can partially be compensated by repeatedly adapting the POD basis leading to the following algorithm. Algorithm 1. Initialize the snapshot set {j/°}™_0 and set i = 0. 2. Compute t according to (10.29) with n replaced by n(i + 1). 3. Compute the POD basis and solve (P^) for 7*. 4. Compute the state yl according to ul(t) = E 7J(t)V>j, add resulting snapshots {j/j}j==o to existing snapshots. 5. Check stopping criterion, set i = i + 1, goto 2. The above algorithm was suggested in [1]. An alternative adaptive POD-based strategy combined with a trust-region approach was proposed in [4]. For a general treatment of POD application to dynamical systems we refer to [6] and the references there. The use of POD-based system reduction in optimal control recently attracted a significant amount of attention; we refer to [2, 21, 29, 30, 34, 35], for example. In some of these references the complexity of the system is such that without system reduction (or some alternative suboptimal technique) the problems could not be solved within reasonable computing and/or manpower time. Closing our discussion on POD approximation, we mention a recent result in [38] on the relation between POD and balanced truncation for linear systems, and an error estimate for the POD approximation in the Woo-norm. Convergence rate estimates for GalerkinPOD approximation of nonlinear dynamical systems are given in [31, 32]. Let us now turn to the reduced basis method, which was first proposed for system reduction in structural problems and which was utilized for optimal control of fluids in [27, 28]. Consider a stationary parameter-dependent equation formally expressed as

where X stands for the state-space of the differential equation and F for the parameter space. In applications of the reduced basis method to control problems, A denotes the control variable. Order reduction by means of the reduced basis method proceeds in two steps. In the first one the reduced basis subspace XR C X is determined. In the second step linear combinations of XR called reduced basis functions are determined which properly accommodate the boundary conditions of the differential equation. Three general reduced basis techniques are formulated using Taylor, Lagrange and Hermite subspaces.

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1. Taylor Subspace: In this case the reduced basis functions are linear combinations of Taylor basis functions generated by computing the Taylor expansion of x(X) at a reference value A*. The reduced basis subspace is

where M € N. Equations for |^f, j = 1,... ,M, are obtained from the implicit function theorem applied to (10.32), e.g.,

2. Lagrange Subspace: Here the reduced basis functions are linear combinations of basis functions generated by solving the nonlinear system (10.32) at various parameters Xj. The reduced basis subspace is

3. Hermite Subspace: Here XR is a combination of the Taylor and Lagrange subspaces. Let us illustrate one possibility of obtaining Lagrange subspace reduced basis functions by a procedure which suggests itself for boundary velocity control on Fc C dfl. We define the reduced subspace as

where j/j satisfies

with {ui}££i given boundary velocities, 6 fixed and T the unit tangent vector to dtl. Let 2/0 denote the solution to (10.33) with Vi = 0. Then reduced basis functions {tpi}iLi are defined as

where the constants a, and Cj are chosen such that homogeneous boundary conditions are enforced on dfl. A reduced-order solution

to the time-dependent version of (10.33) can be obtained by means of a Galerkin procedure with test functions {
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S»=i Pi(t}viTi $ < M, as boundary control, and {^(t)}^ as control parameters, an appropriate Ansatz would be

with test functions {Vi}££f+i f°r the Galerkin scheme. The details for obtaining the finite-dimensional optimal control problems are given in [28].

10.5

Comments on Suboptimal Closed-Loop Methods

So far we focused on open-loop methods. In this final section we briefly survey some of the concepts developed for suboptimal feedback computations and partially follow [7]. To explain the ideas it will be convenient to consider a variant of (10.17):

where g: Mn -+ Rn is a nonlinear function and B is an n x m matrix. The optimal feedback control for (10.34) is known to be of the form

where V is the solution to the Harnilton-Jacobi-Bellman equation

In case g is linear and g(x) = Ax with A an n x n matrix, Vx(x) in (10.35) is replaced by II x with II the positive definite solution to The necessity of resorting to suboptimal strategies stems from the fact that it is rather difficult to numerically realize the Hamilton-Jacobi-Bellman equation unless n is small. One possibility of obtaining a suboptimal feedback solution is to linearize, g1 at a nominal solution ~x and to utilize a Riccati feedback controller based on A = gx(x). An alternative is to use a power series expansion for the value function, i.e., V(x) = £^0 Vn(x), where Vn(x) = O(xn+2) and the associated expansion

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The resulting equation for n = 0 is the Riccati equation (10.37). A third possibility is to use "state-dependent Riccati equations." The idea is to write the nonlinear term in (10.34) as g(x) = A(x)x, to consider

and to proceed by using a power series expansion for II(a;). We mention three further possibilities of obtaining suboptimal feedback solutions which, unlike the previous ones, do not utilize the Riccati equation. As already described in Section 10.3, the instantaneous control method with explicit time stepping has a natural interpretation as a feedback control method. We now describe how interpolation of two-point boundary value solutions provides a feedback mechanism. For this purpose we recall that the optimality condition for (10.34) is given by

The relation to the open-loop control is given by

For numerical realization the terminal condition for the adjoint equation in (10.38) is replaced by p(T) = 0 for large T. For the following arguments it is convenient to indicate the dependence of p on x so that p(t) = p(t, x ( t ) ) . Assuming that T is large with respect to t, (10.39) is approximated by

A practical interpretation of (10.40) is the following: When the system has reached the state x(t), the corresponding feedback control is set to — ^R~1BTp(Q,x(t)), where p(0,~x(t)) is the second component of the solution to (10.38) evaluated at t = 0, and the initial condition in (10.38) is set to x(0) = x(t). This feedback strategy can be realized by pre-computing the solutions to (10.38) with initial conditions x(0) = Xj, with {xj}^ chosen in a neighborhood of the expected optimal trajectory. The feedback solution at state x(t) is then obtained by proper interpolation of the values {p(0, Xj)}^=l. The last suboptimal strategy that we describe here is especially well suited for control synthesis of systems arising in fluid mechanics. We consider the controlled system

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where A is a nonnegative self-adjoint operator in a Hilbert space, F is a locally Lipschitz nonlinear operator satisfying

where ze is an equilibrium solution to (10.41), and B is of the form

Note that (10.42) is satisfied, e.g., for the convection term arising in the NavierStokes equation. For Q a nonnegative self-adjoint operator we consider

and seek feedback solutions of the form

Under some technical assumptions it can be shown that the closed-loop system (10.41), (10.44) is closed-loop dissipative. The optimal feedback control (i.e., the optimal choices for 7^) can again be characterized by a Hamilton-Jacobi-Bellman equation which, however, differs somewhat from (10.36) due to the constraints 7, > 0. The structure of this equation suggests that c(x)(x — xe), with c a real-valued function, is an appropriate Ansatz for Vx. The value of c(x) can be obtained from the Hamilton-Jacobi-Bellman equation. The details of this approach and numerical examples are given in [25].

10.6

Conclusions

We addressed selected suboptimal strategies for optimal control of partial differential equations with emphasis on examples in fluid dynamics. The impact of such methods, we hope, will be a significant one, since they provide a means of solving practical problems which may otherwise be quite intractable. The reader will have noticed that many interesting questions for the methods we presented still need to be answered. Also many additional system-theoretical aspects may become numerically feasible for large-scale problems by suboptimal techniques. We mention robust control, the theory of dynamical observers, estimators and compensators. While we focused on suboptimal strategies here, this is not to indicate that exact methods would not be of equal importance and require further research. Secondorder methods [16, 20, 33] and numerical methods for constrained problems, for example, present interesting challenges.

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Bibliography [1] K. Afanasiev and M. Hinze, "Adaptive control of a wake flow using proper orthogonal decomposition," Lecture Notes in Pure and Applied Mathematics, 216, Marcel Dekker, New York, pp. 317-332, 2001. [2] J. A. Atwell and B. King, "Proper orthogonal decomposition for reduced basis feedback controllers for parabolic systems," Mathematical and Computer Modelling, 33, pp. 1-19, 2001. [3] F. Abergel and R. Temam, "On some control problems in fluid mechanics," Theoretical and Computational Fluid Dynamics, 1, pp. 303-325, 1990. [4] E. Arian, M. Fahl and E. W. Sachs, Trust-Region Proper Orthogonal Decomposition for Flow Control, ICASE Report 2000-25, available online at http:/www.icase.edu/library/reports/rdp/2000.html. [5] M. Berggren, "Numerical solution of a flow-control problem: Vorticity reduction by dynamic boundary action," SIAM Journal on Scientific Computing, 19, pp. 829-860, 1998. [6] G. Berkooz, P. Holmes and J. L. Lumley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, UK, 1996. [7] S. C. Beeler, H. T. Tran and H. T. Banks, "Feedback control methodologies for nonlinear systems," Journal of Optimization Theory and Applications, 107, pp. 1-33, 2000. [8] T. R. Bewley, R. Temam and M. Ziane, "A general framework for robust control in fluid mechanics," Physica D, 138, pp. 360-392, 2000. [9] H. Chen and F. Allgower, "A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability," Automatica, 34, pp. 12051217, 1998. [10] P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press, Chicago, 1988. [11] H. Choi, M. Hinze and K. Kunisch, "Instantaneous control of backward facing step flow," Applied Numerical Mathematics, 31, pp. 133-158, 1999. [12] H. Choi, R. Temam, P. Moin and J. Kim, "Feedback control for unsteady flow and its application to the stochastic Burgers equation," Journal of Fluid Mechanics, 253, pp. 509-543, 1993. [13] M. D. Gunzburger and S. Mansversisi, "Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control," SIAM Journal on Numerical Analysis, 37, pp. 1481-1512, 2000.

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[14] M. D. Gunzburger and S. Mansversisi, "The velocity tracking problem for Navier-Stokes flows with bounded distributed controls," SIAM Journal on Control and Optimization, 37, pp. 1913-1945, 1999. [15] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. [16] M. Heinkenschloss, "Formulation and analysis of a sequential quadratic programming method for the Dirichlet boundary control of Navier-Stokes flow," in Optimal Control: Theory, Algorithms and Applications, W.W. Hager and P.M. Paralos, editors, Kluwer Academic Publishers B.V., Dordrect, The Netherlands, pp. 178-203, 1998. [17] M. Heinkenschloss, Time-Domain Decomposition Iterative Methods for the Solution of Distributed Linear Quadratic Optimal Control Problems, Preprint, Rice University, Houston, TX, 2000. [18] J. W. He, R. Glowinski, R. Metcalfe, A. Nordlander and J. Periaux, "Active control and drag optimization for flow past a circular cylinder," Journal of Computational Physics, 163, pp. 83-117, 2000. [19] M. Hinze and K. Kunisch, "On suboptimal control strategies for the NavierStokes equations," ESAIM: Proceedings, 4, pp. 181-198, 1998. [20] M. Hinze and K. Kunisch, "Second order methods for optimal control of time-dependent fluid flow," SIAM Journal on Control and Optimization, 40, pp. 925-946, 2001. [21] M. Hinze and K. Kunisch, "Three control methods for time-dependent fluid flow," Journal of Flow, Control and Combustion, 65, pp. 273-298, 2000. [22] L. S. Hou and S. S. Ravindran, "A penalized Neumann control approach for solving an optimal Dirichlet control problem for the Navier-Stokes equations," SIAM Journal on Control and Optimization, 36, pp. 1795-1814, 1998. [23] M. Hinze and S. Volkwein, "Analysis of instantaneous control for the Burgers equation," Nonlinear Analysis, 50, pp. 1-26, 2002. [24] L. S. Hou and Y. Yan, "Dynamics and approximations of a velocity tracking problem for the Navier-Stokes flows with piecewise distributed controls," SIAM Journal on Control and Optimization, 35, pp. 1847-1885, 1997. [25] K. Ito and S. Kang, "A dissipative feedback control synthesis for systems arising in fluid dynamics," SIAM Journal on Control and Optimization, 32, pp. 831-854, 1994. [26] K. Ito and K. Kunisch, "Asymptotic properties of receding horizon optimal control problems," SIAM Journal on Control and Optimization, 40, pp. 1585-1610, 2002.

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[27] K. Ito and S. S. Ravindran, Reduced basis method for control problems governed by partial differential equations, in W. Desch et al., editors. Control and Estimation of Distributed Parameter Systems, Internat. Conference in Vorau, Austria, International Series on Numerical Mathematics 126, pp. 153-168, Birkhauser, Basel, 1998. [28] K. Ito and J. D. Schroeter, Reduced Order Feedback Synthesis for Viscous Incompressible Flows, Preprint, North Carolina State University, Raleigh, NC. [29] G. M. Kepler, H. T. Iran and H. T. Banks, "Compensator control for chemical vapor deposition film growth using reduced order design models," IEEE Transactions on Semiconductor Manufacturing, 14, pp. 231-241, 2001. [30] K. Kunisch and S. Volkwein, "Control of the Burgers equation by a reducedorder approach using proper orthogonal decomposition," Journal of Optimization Theory and Applications, 102, pp. 345-371, 1999. [31] K. Kunisch and S. Volkwein, "Galerkin proper orthogonal decomposition methods for parabolic systems," Numerische Mathematik, 90, pp. 117-148, 2001. [32] K. Kunisch and S. Volkwein, "Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics," SIAM Journal on Numerical Analysis, 40, pp. 492-515, 2002. [33] M. Laumen, "Newton's mesh independence principle for a class of optimal shape design problems," SIAM Journal on Control and Optimization, 37, pp. 1070-1088, 1999. [34] H. V. Ly and H. T. Iran, "Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor," Quarterly of Applied Mathematics, 60(4), pp. 631-656, 2002. [35] C. H. Lee and H. T. Tran, Reduced Order Feedback Control for Liquid Film Growth, Preprint, North Carolina State University, Raleigh, NC. [36] R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1979. [37] S. Volkwein, "Optimal control of a phase-field model using the proper orthogonal decomposition," Zeitschrift fur Angewandte Mathematik und Mechanik, 81, pp. 83-97, 2001. [38] H. Zwart, P.O.D. for Linear Systems, Preprint.

Chapter 11

Results and Conjectures for the Control of Navier-Stokes Equations

J. L. Lions* Abstract We consider a flow governed by the Navier-Stokes equations. We assume that we can act on the flow, through a control vector-function. We assume the control to be distributed in order to simplify the exposition, but what is presented here applies to the physically more relevant situation of boundary control. Approximate controllability means that we can drive in a given finite time the system from an initial state to an arbitrarily small neighborhood (in a suitable topology) of another given state (the target). We conjectured in 1990 that, under very mild conditions on the control, there is approximate controllability for the Navier-Stokes system (and for "all" distributed systems of an unstable or turbulent or chaotic nature). We recall in Section 11.2 what has already been obtained on this question, namely by J.M. Coron, A. Pursikov and Y. Imanuvilov. We have also conjectured that the "energy" needed to achieve approximate controllability will remain bounded (and may even decrease) as the viscosity tends to 0 (i.e., the Reynolds number goes to infinity). Results along these lines due to E. Zuazua and the author are also recalled in Section 11.2. But as the viscosity tends to zero, an "optimal control" for achieving approximate controllability may become very sensitive (cf. Section 11.3), a situation which calls for suboptimal control. This idea is developed in Section 11.4, where we use duality and penalty arguments.

'Deceased. Formerly of the Academic des Sciences and Dassault Aviation. 257

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11.1

Introduction

Let fi be a bounded open set of Md, d = 2 or d = 3. Inside fJ we consider the flow of a perfect incompressible viscous fluid. The state (velocity) of the flow is denoted by y, and the control is denoted by v. We assume that the state equations are the Navier-Stokes equations, i.e.,

In (11.1), O denotes an open set contained in fi which can be arbitrarily small, and 10 is the characteristic function of O. In (11.1) p denotes the pressure. Roughly speaking (this will be made precise in what follows), we want to choose v, if it is possible, so that the flow behaves according to our wishes. Remark 11.1.1. The control v which appears in (11.1) is a distributed control. Physically, in particular in the aerospace industry, it is much more important to consider boundary control; i.e., the control appears on the boundary and it is applied on parts of the boundary dtl of fi. We do not study this situation here, giving only bibliographical citations. Essentially, all that we are going to present is valid, up to technical difficulties, for the case of boundary control. A notable exception is indicated in the following remark. Remark 11.1.2. In what follows, neither the size nor the geometrical position of O plays any role. The "best strategy" for choosing the location of O seems to be a completely open question. But, after all, it is not as fundamental as for the boundary control. In that case it is important that there be a piece of the surface where one can apply control on each connected component of the boundary o/fi; cf. 0. Glass [8J. It is now necessary to introduce some function spaces. We define

where

and H = closure of V in (L2((l))d, i.e., H = {
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The space V (resp., H) is a Hilbert space with the norm

We assume that v satisfies where T is the fixed time horizon. Given v with (11.4), there exists a state y (weak) solution of (11.1), (11.3) such that

and, after possible modification on a set of measure 0, t —» y ( t ) , is continuous from [0, T] -> V(resp., H) if d = 3 (resp., d = 2). (In this respect, cf. also Remark 11.4.1 later on.) In (11.6), V denotes the dual of V when H is identified with its dual. Remark 11.1.3. The existence of y satisfying (1.5) goes back to J. Leray [14]Uniqueness has been proven, for the above spaces, only if d = 2 (in G. Prodi and the author [17]). If d = 3, uniqueness is still an open question. Taking into account the possibility of nonuniqueness, we denote by any solution of (11.1), (11.2), (11.3) which satisfies (11.5) and (11.6). This is the state of our system that we wish to control. The control problem we are interested in here is the controllability problem. Given T, we consider the set of all functions when v spans (L2(fi x (0, T)))d and where y denotes the set of all possible solutions (cf. (11.7)). One says that the problem is approximately controllable when the set given by all functions (11.8) is dense in V' (or in If). In what follows we recall conjectures and results on this question (Section 11.2). We then proceed by showing that the "best" choice of the control v (in a sense to be made precise below) is very sensitive to /u when p, —> 0 (i.e., when the Reynolds number gets very large); this is presented in Section 11.3. We then introduce some apparently new ideas related to "robust control" based on some mixture of duality and penalty; these are presented in Section 11.4. Some extensions and further problems are indicated in Section 11.5.

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The First Two Conjectures Relative to Approximate Controllability

The first conjecture, made by the author [14], states that there is approximate controllability. In case d = 2, the conjecture has been proven by J. M. Coron [2] (who also proved it for the case of boundary control with special boundary conditions by [3]). In case d = 3, results are not yet "clear cut," but they all seem to support the conjecture; cf. 0. Yu Imanuvilov [12, 13], A. Fursikov and O. Yu Imanuvilov [7], A. Fursikov [6] and the bibliography therein. Remark 11.2.1. For the case p, = 0 (Euler's equation), J. M. Coron (loc. cit.) has proven the exact controllability (in suitable function spaces), arguing that there exists a control which drives the system exactly to a given state yT at time T. (In approximate controllability, one can only drive the system as close as one wishes to Remark 11.2.2. The situation is, of course, much simpler in the linear case (i.e., without the term yVy in (ll.l)j. Then there is approximate controllability for any dimension d, in the space H. The result is still true if, for d = 3,

i.e., one uses only a 2-component control. If one uses a one-component control,

This situation has been studied by E. Zuazua and the author [18]. For a special geometrical situation, it is shown there that there is approximate controllability generically with respect to f2; i.e., approximate controllability may be untrue, but it becomes valid after an arbitrary C°° small variation of fi. We conjecture that a similar result is true for the nonlinear Navier-Stokes system. Remark 11.2.3. The conjecture has been proven by E. Zuazua and the author [21] for the Galerkin's approximation of the Navier-Stokes equations. Remark 11.2.4. The approximate controllability conjecture has been presented here for the Navier-Stokes equations but it appears likely it is more general. We conjecture approximate controllability is true for all distributed systems which contain some turbulence, chaos and strong mixing of scales. For instance, all models for meteorology or climatology are probably approximately controllable with perhaps the notable exception of ice caps. It seems likely that similar results will eventually be obtained for compressible fluids, based on the results of P. L. Lions [23].

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We now proceed with the cost of approximate controllability. Let us assume that there is approximate controllability. Then one introduces the quantity

where B = unit ball of V (resp., H) if d = 3 (resp., d = 2), /3 > 0 is arbitrarily small. Let us denote by

the inf in (11.11), where we retain only the dependence in p,, with the other elements yT and /? being fixed. One conjectures that

i.e., turbulent flows are not more "expensive" to control than the nonturbulent ones. Remark 11.2.5. More generally, it is conjectured that the more unstable (turbulent, chaotic) a system is, the cheaper it is to control (at least to drive it to a "stable" state). Results along this line are proven for the Galerkin's approximation of Navier-Stokes equations and for some linear systems in [21, 19, 20]. One can "replace" the functional (11.11) by

where a is positive (and quite "large"), || || denotes the norm in V' (resp., H) and where y(T; v) denotes (if d = 3) the set of all possible solutions at time T (this set being possibly reduced to one element). Remark 11.2.6. The advantages o/(11.14) with respect to (11.11), (11.12) are the following: (i) Equation (11.14) makes sense even without approximate controllability. (ii) Formulation (11.14) is more convenient for numerical computations (cf. R. Glouiinski and J.L. Lions [9]; one will find there some formulas for choosing a such that if v achieves the inf in (11.14), then y(T;v) — yr is likely to be in J3B). A variant of conjecture (11.13) is the following conjecture:

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Remark 11.2.7. Conjecture (11.15) has been verified experimentally by J.W. He et al. [11] for the Kuramoto-Shivashinski equation. In conclusion, the conjectures stated above are that unstable, turbulent, chaotic distributed systems are (i) approximately controllable, (ii) at a cost which does not increase (and even may decrease) as the system becomes increasingly unstable. There are more and more "pieces of evidence" in favor of these conjectures. But the above remarks are (essentially) theoretical. An important question remains: Can one hope to practically achieve control of Navier-Stokes equations?

11.3

Sensitivity to Reynolds Number

It is beyond the scope of this chapter to even attempt to present a survey of industrial applications of the control of Navier-Stokes equations or of related systems. The amount of related work is enormous since control of flows is fundamental in the aerospace industry as well as in many other fields. We consider here a kind of preliminary theoretical question. It will illustrate that controlling Navier-Stokes equations in real situations is difficult (hardly a new remark). We will then introduce in Section 11.4 a new method which could be of some usefulness. Let us consider the linearized Navier-Stokes equations

subject to the initial and boundary conditions (11.2) and (11.3). In (11.16), g is given such that Problem (11.16), (11.2), (11.3) admits a unique solution which depends on the control v and the function g. Remark 11.3.1. Of course one recovers the Navier-Stokes equations if one finds a fixed voint of the mavvina

One then introduces

using the same notation as in (11.14).

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The problem

with g fixed, admits a unique solution that we denote by v(n) (all parameters are fixed except /x). A very natural step then is to study the sensitivity ofv(u.) to u,. Let us define

We have proven (cf. [16]) that the mapping

and we have obtained the estimate

Remark 11.3.2. We think (but we have not proven) that the estimate (11.23) can not be significantly improved. In other words the choice of the optimal control v(fj,) is very sensitive to (i~l = Reynolds number. Remark 11.3.3. We conjecture that for the nonlinear Navier-Stokes system, one can find an optimal control v(p) such that (11.22) holds with an estimate similar to (11.23). Remark 11.3.4. One has similar estimates and conjectures for the case of boundary control. The next step is then to try to avoid the situation leading to (11.23), i.e., to try to find "sub-optimal controls" which are more robust. This problem has, again, given rise to a huge amount of work. No attempt is made to give a bibliography on this topic. A new method towards this goal is presented in the following section.

11.4

An Attempt to Nonlinear Duality

Let us start from the state equation (11.16) and consider problem (11.20), where g is fixed. One can apply Fenchel-Rockafellar duality [4, 5] in this linear situation. One obtains that

where £(f,g) is given in the following fashion. One defines the adjoint state ip =

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U ( f , g) as the solution of

and

In (11.25), (gV)* denotes the adjoint of gV. One then introduces

where we take ||/|| — norm of / in V (resp., H) if d — 3 (resp., d = 2). Remark 11.4.1. In the linearized case one could take in (11.19) and (11.28) the norm of H. But this would not make sense for the full nonlinear NavierStokes equations if d = 3, where one can show the existence of a solution which is continuous from [0,T] -> (H–1/4(O))3 (cf. [22], Corollary 6.2, Chapter 1). It is in order to avoid the use of H~ 1 / 4 that we have introduced V1. But, of course, inf J(v, g) or — inf L(f, g) does not solve the problem for Navier-Stokes equations. We introduce now a penalty argument. We define

where e > 0 is "small" and where || any solution of

|| denotes the norm in V. We conjecture that

is "robust" with respect to variations of u when u —> 0 and gives an approximation of the solution to (11.14). Let us first give some estimates for (11.30). Let w be a given control and let z be a solution (or the solution if d = 2) of

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265

with the initial and boundary conditions analogous to (11.2), (11.3). We choose then

The corresponding solution of (11.16) is then y = z so that

where

Therefore

where

where £ = C(/> w ) is a solution of (11.34). It follows from the estimate (11.35) that i n f M e ( v , g , f) admits a solution (Vei9ei fe) which is such that, as e —> 0,

Moreover, if ys = ys(v£,g£), then

One can then extract a subsequence, still denoted by (v£, ge, fs), such that

and we can verify that y is a solution of (11.1), (11-2), (11.3).

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Remark 11.4.2. Starting from (11.16) one could introduce the simpler functional

but it seems to give less flexibility than (11.29) (and thus probably leads to a less robust solution). Actually the inf of functional (11.41) converges, when e —> 0, towards the inf \ /QT fo v2 dxdt+% \\y(T; v) - yT ||2. Remark 11.4.3. There are other ways to use penalty techniques for unstable or singular systems. Let us consider for instance (cf. [22], Chapter 1, Section 3) the state equation

where y is subject to y\t=o = 0 and y = 0 on d£t. Because of the —y3 term in (11.42), there is, in general, blow up infinite time: given v, there is, in general, no solution in the time interval (0,T). Then one thinks of (11.42) as a constraint, not as a state equation, and one adds to the cost function the penalty term

(cf. [21]). The same technique can be applied to the Navier-Stokes equations, with some supplementary technical difficulties due to the pressure term. The idea of "penalizing" the state equation (and the boundary conditions and the initial conditions as well) has been introduced by the author in [15], Chapter 5, Section 3.

11.5

Further Remarks

We have not considered here numerical algorithms for controllability. A rather systematic study of all types of linear problems has been made by R. Glowinski and the author [10]. In [16] we have introduced a time decomposition method which is aimed at finding completely parallel algorithms; numerical experiments are in progress. Of course, finally, everything will rest on the actuators, where smart materials are playing an ever increasing role. We refer for this point to H.T. Banks, R.C. Smith and Y. Wang [1].

Dedication Dedicated to H.T. Banks on his sixtieth anniversary.

Chapter 11. Control of the Navier-Stokes Equations

267

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[4] I. Ekeland and R. Temam, Analyse Converge et Problemes Variationnels, Dunod, Gauthier-Villars, Paris, 1973. [5] W. Fenchel, "On conjugate convex functions," Canadian Journal of Mathematics, 1, pp. 73-77, 1973. [6] A. Fursikov, Optimal Control of Distributed Systems: Theory and Applications, Translation of Mathematical Monographs, Vol. 187, AMS, Providence, RI, 2000. [7] A. Fursikov and O. Y. Imanuvilov, "On the exact boundary zero controllability of the two dimensional Navier Stokes equations," Ada Applicandae Mathematicae, 36, pp. 1-10, 1994. [8] O. Glass, "An addendum to a J.M. Coron theorem concerning the controllability of the Euler system for 2D incompressible inviscid fluids," Journal de Mathematiques Pures et Appliquees, 80, pp. 845-877, 2001. [9] R. Glowinski and J. Lions, "Exact and approximate controllability for distributed parameter systems (i)," Acta Numerica, pp. 269-378, 1994. [10]

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J. L. Lions

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Index anisotropic grids, 82

frequency response function, 171

balanced truncations, 101 Barkhausen effect, 231 boundary element methods, 226 Burgers' equation abstract strong form, 26 abstract weak form, 27 Dirichlet boundary control, 30 distributed control, 25 instantaneous control, 245 Neumann boundary control, 32

gap topology, 96, 107 convergence of transfer functions, 108 Ghidaglia and Temam framework, 28 graph topology, 96 Hamilton-Jacobi-Bellman (HJB) equations, 123, 125, 238, 253 Hankel operator, 100 health monitoring, see structural health monitoring Hilbert-Schmidt operator, 36, 247 homogenization methods anisotropic grids, 82 multiple-scale, 63 perforated domains, 73 periodic unfolding, 68, 71, 86 rubber-like materials, 88 Tartar's oscillating test functions, 65 truss-like structures, 76, 85 two-scale convergence, 67 hysteresis, 183, 230

cardiovascular-respiratory, 201 closed-loop dissipativity, 243 congestive heart failure, 201 controllability gramian, 100 coprime factorizations left, 96, 105 right, 96 corrector functions, 61 damage detection, 170 damping delayed, 160 polarization, 6 structural, 20 viscous, 20 denaturation rate, 8 Dirichlet map, 31

input-output stability, 112 Khoo model, 205 Lame coefficients, 83 left-coprime factorization, 96, 105 Lorenz equation, 38 LQG-balanced realization, 103 LQG-balanced truncations, 98 numerical computation, 108 robust control design, 112

eddy current testing (ECT), 219 e-pseudospectrum, 38, 42 evolutionary programming, 228 feedback gain operators, 23 Fenchel-Rockafellar duality, 263 Frank-Starling mechanism, 208 269

270

magnetic field, 4 max-plus basis expansions, 133 linearity, 125 martingales, 131 operations, 123 probability, 130 solutions, 129 stochastic calculus, 130 Maxwell's equations, 3, 223 medical risk, 2 medical therapy, 12 membrane kinetics, 12 method of snapshots, 97, 246 models cardiovascular system, 194 Grodin's four-compartment model, 197 polarization, 3 reactor pressure vessels, 221 respiratory system, 188 ventilation, 193 multiple-scale method, 63 Navier- Stokes equations, 238, 258 approximate controllability, 260 boundary control, 258 instantaneous control, 242 linearized, 262 penalty argument, 264 receding horizon control, 242 neo-Hookian materials, 89 nondestructive evaluation, 170 crack regions, 224 material degradation, 229 nano-scale flaws, 232 quantitative nondestructive evaluation (QNDE), 219 nonlinear elastomers, 88 nonlinear filtering, 134 nuclear power plants, 219 nuclear systems, 98, 102 observability gramian, 100 Ohm's law, 3 periodic material, 57

Index

periodic unfolding method, 68, 71, 86 perforated domains, 73 periodic unfolding operator, 68 piezoelectric materials damage detection, 172, 173 homogenization issues, 90 POD, see proper orthogonal decomposition polarization model, 3 positive position feedback, 175 principle component analysis, 97 Pritchard-Salamon (PS) realization, 100 system, 99 proper orthogonal decomposition (POD) 21, 97, 246 protein denaturation, 8 reduced basis method, 249 reduced-order models, 97, 246 reference cell, 56 regulator equations, 144, 145, 152 regulator problem bounded inputs and outputs, 142 error feedback, 147 examples, 152 state feedback, 143 regular systems, 151 well-posedness, 144 systems with delays, 155 unbounded inputs and outputs, 150 respiratory system control mechanisms, 189 models, 188 ventilatory control mechanisms, 193 Riccati equations state-dependent, 252 right-coprime factorization, 96 robust control design LQG-balanced truncation, 112 rubber-like materials, 88 self-healing, 179 semiconvex solutions, 125

Index sensors

fiber optic, 175 interferometric strain, 175 structural health monitoring, 170 crack-type damage, 179 genetic algorithms, 179 superconducting quantum interference device (SQUID), 221, 224 system sensitivity, 40 Tartar's oscillating test functions method, 65 thermal convection loop abstract model, 37 control, 34 thermoacoustic effect, 10 tissue damage, 8 topologies gap, 96, 107 convergence of transfer functions, 108 graph, 96 transmission zero, 149 Trotter-Kato conditions, 23 trust region approach, 249 two-scale convergence method, 67 vanishing viscosity, 127 vibration suppression, 174 viscosity solutions, 124

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