PI and PID controller tuning rules for time delay processes: a summary Technical Report AOD-00-01, Edition 1 A. O’Dwyer, School of Control Systems and Electrical Engineering, Dublin Institute of Technology, Kevin St., Dublin 8, Ireland. 15 May 2000 Phone: 353-1-4024875 Fax: 353-1-4024992 e-mail:
[email protected] Abstract: The ability of proportional integral (PI) and proportional integral derivative (PID) controllers to compensate many practical industrial processes has led to their wide acceptance in industrial applications. The requirement to choose either two or three controller parameters is perhaps most easily done using tuning rules. A summary of tuning rules for the PI and PID control of single input, single output (SISO) processes with time delay are provided in this report. Inevitably, this report is a work in progress and will be added to and extended regularly. Keywords: PI, PID, tuning rules, time delay. 1. Introduction The ability of PI and PID controllers to compensate most practical industrial processes has led to their wide acceptance in industrial applications. Koivo and Tanttu [1], for example, suggest that there are perhaps 510% of control loops that cannot be controlled by SISO PI or PID controllers; in particular, these controllers perform well for processes with benign dynamics and modest performance requirements [2, 3]. It has been stated that 98% of control loops in the pulp and paper industries are controlled by SISO PI controllers [4] and that, in process control applications, more than 95% of the controllers are of PID type [3]. The PI or PID controller implementation has been recommended for the control of processes of low to medium order, with small time delays, when parameter setting must be done using tuning rules and when controller synthesis is performed either once or more often [5]. However, Ender [6] states that, in his testing of thousands of control loops in hundreds of plants, it has been found that more than 30% of installed controllers are operating in manual mode and 65% of loops operating in automatic mode produce less variance in manual than in automatic (i.e. the automatic controllers are poorly tuned); this is rather sobering, considering the wealth of information available in the literature for determining controller parameters automatically. It is true that this information is scattered throughout papers and books; the purpose of this paper is to bring together in summary form the tuning rules for PI and PID controllers that have been developed to compensate SISO processes with time delay. Tuning rules for the variations that have been proposed in the ‘ideal’ PI and PID controller structure are included. Considerable variations in the ideal PID controller structure, in particular, are encountered; these variations are explored in more detail in Section 2. 2. PID controller structures The ideal continuous time domain PID controller for a SISO process is expressed in the Laplace domain as follows: U( s) = G c (s) E (s) (1) with
G c (s) = Kc (1 +
1 + Tds) Ts i
(2)
and with Kc = proportional gain, Ti = integral time constant and Td = derivative time constant. If Ti = ∞ and Td = 0 (i.e. P control), then it is clear that the closed loop measured value, y, will always be less than the desired value, r (for processes without an integrator term, as a positive error is necessary to keep the measured value constant, and less than the desired value). The introduction of integral action facilitates the achievement of equality between the measured value and the desired value, as a constant error produces an increasing controller output. The introduction of derivative action means that changes in the desired value may be anticipated, and
thus an appropriate correction may be added prior to the actual change. Thus, in simplified terms, the PID controller allows contributions from present controller inputs, past controller inputs and future controller inputs. Many tuning rules have been defined for the ideal PI and PID structures. Tuning rules have also been defined for other PI and PID structures, as detailed in Section 4. 3. Process modelling Processes with time delay may be modelled in a variety of ways. The modelling strategy used will influence the value of the model parameters, which will in turn affect the controller values determined from the tuning rules. The modelling strategy used in association with each tuning rule, as described in the original papers, is indicated in the tables. Of course, it is possible to use the tuning rules proposed by the authors with a different modelling strategy than that proposed by the authors; applications where this occurs are not indicated (to date). The modelling strategies are referenced as indicated. The full details of these modelling strategies are provided in Appendix 2. K e − sτ m A. First order lag plus delay (FOLPD) model ( G m ( s) = m ): 1 + sTm Method 1: Parameters obtained using the tangent and point method (Ziegler and Nichols [8], Hazebroek and Van den Waerden [9]); Appendix 2. Method 2: Km , τ m assumed known; Tm estimated from the open loop step response (Wolfe [12]); Appendix 2. Method 3: Parameters obtained using an alternative tangent and point method (Murrill [13]); Appendix 2. Method 4: Parameters obtained using the method of moments (Astrom and Hagglund [3]); Appendix 2. Method 5: Parameters obtained from the closed loop transient response to a step input under proportional control (Sain and Ozgen [94]); Appendix 2. Method 6: Km , Tm , τ m assumed known. Method 7: Parameters obtained using a least squares method in the time domain (Cheng and Hung [95]); Appendix 2. Method 8: Parameters obtained in the frequency domain from the ultimate gain, phase and frequency determined using a relay in series with the closed loop system in a master feedback loop. The model gain is obtained by the ratio of the integrals (over one period) of the process output to the controller output. The delay and time constant are obtained from the frequency domain data (Hwang [160]). Method 9: Parameters obtained from the closed loop transient response to a step input under proportional control (Hwang [2]); Appendix 2. Method 10: Parameters obtained from two points estimated on process frequency response using a relay and a relay in series with a delay (Tan et al. [39]); Appendix 2. Method 11: Tm and τ m are determined from the ultimate gain and period estimated using a relay in series with the process in closed loop; Km assumed known (Hang and Cao [112]); Appendix 2. Method 12: Parameters are estimated using a tangent and point method (Davydov et al. [31]); Appendix 2. Method 13: Parameters estimated from the open loop step response and its first time derivative (Tsang and Rad [109]); Appendix 2. Method 14: Tm and τ m estimated from Ku , Tu determined using Ziegler-Nichols ultimate cycle method;
Km estimated from the process step response (Hang et al. [35]); Appendix 2. Method 15: Tm and τ m estimated from Ku , Tu determined using a relay autotuning method; Km estimated from the process step response (Hang et al. [35]); Appendix 2. Method 16: G p ( jω135 ) , ω135 and Km are determined from an experiment using a relay in series with the 0
0
process in closed loop; estimates for Tm and τ m are subsequently calculated. (Voda and Landau [40]); Appendix 2. Method 17: Parameter estimates back-calculated from discrete time identification method (Ferretti et al. [161]); Appendix 2. * Method 18: Parameter estimates calculated from process reaction curve using numerical integration procedures (Nishikawa et al. [162]). * Method 19: Parameter estimates determined graphically from a known higher order process (McMillan [58] … also McMillan (1983), pp. 34-40.
* Method 20: Km estimated from the open loop step response. T90% and τ m estimated from the closed loop step response under proportional control (Astrom and Hagglund [93]?) Method 21: Parameters estimated from linear regression equations in the time domain (Bi et al. [46]); Appendix 2. Method 22: Tm and τ m estimated from relay autotuning method (Lee and Sung [163]); Km estimated from the closed loop process step response under proportional control (Chun et al. [57]); Appendix 2. * Method 23: Parameters are estimated from a step response autotuning experiment – Honeywell UDC 6000 controller (Astrom et al. [30]). Method 24: Parameters are estimated from the closed loop step response when process is in series with a PID controller (Morilla et al. [104a]); Appendix 2. Method 25: τm and Tm obtained from an open loop step test as follows: Tm = 1.4( t 67% − t 33% ) ,
τ m = t 67% − 1.1Tm . K m assumed known (Chen and Yang [23a]). Method 26: τm and Tm obtained from an open loop step test as follows: Tm = 1.245 ( t 70% − t 33% ) , τ m = 1. 498 t 33% − 0. 498 t 70% . K m assumed known (Miluse et al. [27b]). * Method 27: Data at the ultimate period is deduced from an open loop impulse response (Pi-Mira et al. [97a]). B. Non-model specific Method 1: Parameters K u , K m , ωu are estimated from data obtained using a relay in series with the process in closed loop and from the process step response (Kristiansson and Lennartsson [157]). need to check how the other methods define these parameters –
C. Integral plus time delay (IPD) model ( G m ( s) =
K me− sτ m s
)
Method 1: τ m assumed known; Km determined from the slope at start of the open loop step response (Ziegler and Nichols [8]); Appendix 2. Method 2: Km , τ m assumed known. Method 3: Parameters estimated from the ultimate gain and frequency values determined from an experiment using a relay in series with the process in closed loop (Tyreus and Luyben [75]); Appendix 2. Method 4: Parameters are estimated from the servo or regulator closed loop transient response, under PI control (Rotach [77]); Appendix 2. Method 5: Parameters are estimated from the servo closed loop transient response under proportional control (Srividya and Chidambaram [80]); Appendix 2. Method 6: K u and Tu are estimated from estimates of the ultimate and crossover frequencies. The ultimate frequency estimate is obtained by placing an amplitude dependent gain in series with the process in closed loop; the crossover frequency estimate is obtained by also using an amplitude dependent gain (Pecharroman and Pagola [165]); Appendix 2. D. First order lag plus integral plus time delay (FOLIPD) model ( G m (s) =
Km e− sτ ) s(1 + sTm ) m
* Method 1: Method of moments (Astrom and Hagglund [3]). Method 2: Km , Tm , τ m assumed known. Method 3: Parameters estimated from the open loop step response and its first and second time derivatives (Tsang and Rad [109]); Appendix 2. Method 4: K u and Tu are estimated from estimates of the ultimate and crossover frequencies (Pecharroman and Pagola [165]) – as in Method 6, IPD model. E.
Second order system plus time delay (SOSPD) model ( G m (s) =
Km e− sτ ) s + 2ξ m Tm1s + 1 (1 + Tm1s)(1 + Tm2 s) K m e− sτ
Tm1
2 2
m
m
,
Method 1: Km , Tm1 , Tm2 , τ m or Km , Tm1 , ξ m , τ m assumed known. Method 2: Parameters estimated using a two-stage identification procedure involving (a) placing a relay in series with the process in closed loop and (b) placing a proportional controller in series with the process in closed loop (Sung et al. [139]); Appendix 2. * Method 3: Parameters obtained in the frequency domain from the ultimate gain, phase and frequency determined using a relay in series with the closed loop system in a master feedback loop. The model gain is obtained by the ratio of the integrals (over one period) of the process output to the controller output. The other parameters are obtained from the frequency domain data (Hwang [160]). Method 4: Tm and τ m estimated from Ku , Tu determined using a relay autotuning method; Km estimated from the process step response (Hang et al. [35]); Appendix 2. * Method 5: Parameter estimates back-calculated from discrete time identification method (Ferretti et al. [161]). Method 6: Parameteres estimated from the underdamped or overdamped transient response in open loop to a step input (Jahanmiri and Fallahi [149]); Appendix 2. * Method 7: Parameters estimated from a least squares time domain method (Lopez et al. [84]). Method 8: Parameters estimated from data obtained when the process phase lag is − 900 and − 180 0 , respectively (Wang et al. [143]); Appendix 2. * Method 9: Parameter estimates back-calculated from discrete time identification method (Wang and Clements [147]). Method 10: Km , Tm1 and τm are determined from the open loop time domain Ziegler-Nichols response (Shinskey [16], page 151); Tm 2 assumed known. Method 11: Parameters estimated from two points determined on process frequency response using a relay and a relay in series with a delay (Tan et al. [39]); Appendix 2. * Method 12: Parameter estimated back-calculated from discrete time identification method (Lopez et al. [84]). * Method 13: Parameters estimated from a step response autotuning experiment – Honeywell UDC 6000 controller (Astrom et al. [30]). Method 14: Tm1 = T m2 . τm and T m1 obtained from an open loop step test as follows:
Tm1 = 0. 794 ( t 70% − t 33% ) , τ m = 1.937 t 33% − 0.937 t 70% . K m assumed known (Miluse et al. [27b]). Method 15: K u and Tu are estimated from estimates of the ultimate and crossover frequencies (Pecharroman and Pagola [165]) – as in Method 6, IPD model. F. Integral squared plus time delay ( I 2PD ) model ( G m (s) =
K me − sτ m ) s2
Method 1: Km , Tm , τ m assumed known. G. Second order system (repeated pole) plus integral plus time delay (SOSIPD) model ( G m (s) =
Km e −s τ
m
s (1 + sTm )
2
)
Method 1: K u and Tu are estimated from estimates of the ultimate and crossover frequencies (Pecharroman and Pagola [165]) – as in Method 6, IPD model. Method 2: Km , Tm , τ m assumed known.
H. Third order system plus time delay (TOLPD) model ( G m (s) =
Km e − sτ ). (1 + sTm1 )(1 + sTm2 )(1 + sTm 3 ) m
Method 1: Km , Tm1 , Tm2 , Tm3 , τ m known.
I. Unstable first order lag plus time delay model ( G m (s) =
K me − sτ 1 − sTm
m
)
Method 1: Km , Tm , τ m known. Method 2: The model parameters are obtained by least squares fitting from the open loop frequency response of the unstable process; this is done by determining the closed loop magnitude and phase values of the (stable) closed loop system and using the Nichols chart to determine the open loop response (Huang and Lin [154], Deshpande [164]).
J. Unstable second order system plus time delay model ( G m (s) =
K m e− s τ ) (1 − sTm1 )(1 + sTm 2 ) m
Method 1: Km , Tm1 , Tm2 , τ m known. Method 2: The model parameters are obtained by least squares fitting from the open loop frequency response of the unstable process; this is done by determining the closed loop magnitude and phase values of the (stable) closed loop system and using the Nichols chart to determine the open loop response (Huang and Lin [154], Deshpande [164]).
K. Second order system plus time delay model with a positive zero ( G m (s) =
K m (1 − sTm3 ) e− s τ
m
(1 + sTm1 )(1 + sTm2 )
)
Method 1: Km , Tm1 , Tm2 , Tm3 , τ m known.
L. Second order system plus time delay model with a negative zero ( G m (s) =
K m (1 + sTm3 )e − sτ
m
(1 + sTm1 )(1 + sTm 2 )
)
Method 1: Km , Tm1 , Tm2 , Tm3 , τ m known. M. Fifth order system plus delay model ( G m ( s) =
K m (1 + b1s + b2 s2 + b3s3 + b4 s4 + b ss5 ) e− s τ
(1 + a s + a s
2
1
2
Method 1: Km , b1 , b 2 , b 3 , b 4 , b5 , a1 , a2 , a3 , a4 , a5 , τ m known. N. General model with a repeated pole ( G m ( s) =
+ a3s 3 + a 4 s4 + a 5s5
)
m
)
K m e − sτ ) (1 + sTm ) n m
* Method 1: Strejc’s method O. General stable non-oscillating model with a time delay P.
Delay model ( G m ( s) = e − sτ m )
Note: * means that the procedure has not been fully described to date. 4. Organisation of the report The tuning rules are organised in tabular form, as is indicated in the list of tables below. Within each table, the tuning rules are classified further; the main subdivisions made are as follows: (i) Tuning rules based on a measured step response (also called process reaction curve methods). (ii) Tuning rules based on minimising an appropriate performance criterion, either for optimum regulator or optimum servo action. (iii) Tuning rules that gives a specified closed loop response (direct synthesis tuning rules). Such rules may be defined by specifying the desired poles of the closed loop response, for instance, though more generally, the desired closed loop transfer function may be specified. The definition may be expanded to cover techniques that allow the achievement of a specified gain margin and/or phase margin. (iv) Robust tuning rules, with an explicit robust stability and robust performance criterion built in to the design process. (v) Tuning rules based on recording appropriate parameters at the ultimate frequency (also called ultimate cycling methods).
(vi) Other tuning rules, such as tuning rules that depend on the proportional gain required to achieve a quarter decay ratio or magnitude and frequency information at a particular phase lag. Some tuning rules could be considered to belong to more than one subdivision, so the subdivisions cannot be considered to be mutually exclusive; nevertheless, they provide a convenient way to classify the rules. Tuning rules for the variations that have been proposed in the ‘ideal’ PI and PID controller structure are included in the appropriate table. In all cases, one column in the tables summarise the conditions under which the tuning rules are designed to operate, if appropriate ( Y ( s) = closed loop system output, R( s) = closed loop system input). Tables 1-3: PI tuning rules – FOLPD model - G m ( s) =
K me − sτ 1 + sTm
m
1 Table 1: Ideal controller G c ( s) = Kc 1 + . Eighty-five such tuning rules are defined; the references are Ts i (a) Process reaction methods: Ziegler and Nichols [8], Hazebroek and Van der Waerden [9], Astrom and Hagglund [3], Chien et al. [10], Cohen and Coon [11], Wolfe [12], Murrill [13] – page 356, McMillan [14] – page 25, St. Clair [15] – page 22 and Shinskey [15a]. Twelve tuning rules are defined. (b) Performance index minimisation (regulator tuning): Minimum IAE - Murrill [13] – pages 358-363, Shinskey [16] – page 123, ** Shinskey [17], Huang et al. [18], Yu [19]. Minimum ISE - Hazebroek and Van der Waerden [9], Murrill [13]– pages 358-363, Zhuang and Atherton [20], Yu [19]. Minimum ITAE - Murrill [13] – pages 358-363, Yu [19]. Minimum ISTSE - Zhuang and Atherton [20]. Minimum ISTES – Zhuang and Atherton [20]. Thirteen tuning rules are defined. (c) Performance index minimisation (servo tuning): Minimum IAE – Rovira et al. [21], Huang et al. [18]. Minimum ISE - Zhuang and Atherton [20], han and Lehman [22]. Minimum ITAE - Rovira et al. [21]. Minimum ISTSE - Zhuang and Atherton [20]. Minimum ISTES – Zhuang and Atherton [20]. Seven tuning rules are defined. (d) Direct synthesis: Haalman [23], Chen and Yang [23a], Pemberton [24], Smith and Corripio [25], Smith et al. [26], Hang et al. [27], Miluse et al. [27a], Gorecki et al. [28], Chiu et al. [29], Astrom et al. [30], Davydov et al. [31], Schneider [32], McAnany [33], Leva et al. [34], Khan and Lehman [22], Hang et al. [35, 36], Ho et al. [37], Ho et al [104], Tan et al. [39], Voda and Landau [40], Friman and Waller [41], Smith [42], Cox et al. [43], Cluett and Wang [44], Abbas [45], Bi et al. [46], Wang and Shao [47]. Thirty-one tuning rules are defined. (e) Robust: Brambilla et al. [48], Rivera et al. [49], Chien [50], Thomasson [51], Fruehauf et al. [52], Chen et al. [53], Ogawa [54], Lee et al. [55], Isaksson and Graebe [56], Chun et al. [57]. Ten tuning rules are defined. (f) Ultimate cycle: McMillan [58], Shinskey [59] – page 167, **Shinskey [17], Shinskey [16] – page 148, Hwang [60], Hang et al. [65], Zhuang and Atherton [20], Hwang and Fang [61]. Twelve tuning rules are defined. 1 Table 2: Controller G c ( s) = Kc b + Two direct synthesis tuning rules are defined by Astrom and Hagglund T i s [3] - page 205-208.
1 E (s ) − α Kc R( s) . One performance index Table 3: Two degree of freedom controller: U(s) = K c 1 + Tis minimisation tuning rule is defined by Taguchi and Araki [61a]. Tables 4-7: PI tuning rules - non-model specific 1 Table 4: Controller G c ( s) = Kc 1 + . Nineteen such tuning rules are defined; the references are: Ts i (a) Ultimate cycle: Ziegler and Nichols [8], Hwang and Chang [62], ** Hang et al. [36], McMillan [14] – page 90, Pessen [63], Astrom and Hagglund [3] – page 142, Parr [64] – page 191, Yu [122] – page 11. Seven tuning rules are defined. (b) Other tuning rules: Parr [64] – page 191, McMillan [14] – pages 42-43, Parr [64] – page 192, Hagglund and Astrom [66], Leva [67], Astrom [68], Calcev and Gorez [69], Cox et al. [70]. Eight tuning rules are defined. (c) Direct synthesis: Vrancic et al. [71], Vrancic [72], Friman and Waller [41], Kristiansson and Lennartson [158a]. Four tuning rules are defined.
1 Table 5: Controller G c ( s) = K c b + . One direct synthesis tuning rule is defined by Astrom and Hagglund Ti s [3] – page 215.
1 Table 6: Controller U( s) = Kc 1 + E( s) + Kc ( b − 1)R (s) . One direct synthesis tuning rule is defined by T is Vrancic [72]. Table 7: Controller U( s) = Kc Y(s) −
Kc E ( s) . One direct synthesis tuning rule is defined by Chien et al. [74]. Ti s K me− sτ m s
Tables 8-11: PI tuning rules – IPD model G m ( s) =
1 Table 8: Controller G c ( s) = Kc 1 + . Twenty such tuning rules are defined; the references are: Ts i (a) Process reaction methods: Ziegler and Nichols [8], Wolfe [12], Tyreus and Luyben [75], Astrom and Hagglund [3] – page 138. Four tuning rules are defined. (b) Regulator tuning – performance index minimisation: Minimum ISE – Hazebroek and Van der Waerden [9]. Minimum IAE - Shinskey [59] – page 74. Minimum ITAE - Poulin and Pomerleau [82]. Four tuning rules are defined. (c) Ultimate cycle: Tyreus and Luyben [75], ** Shinskey [17]. Two tuning rules are defined. (d) Robust: Fruehauf et al. [52], Chien [50], Ogawa [54]. Three tuning rules are defined. (e) Direct synthesis: Wang and Cluett [76], Cluett and Wang [44], Rotach [77], Poulin and Pomerleau [78], Kookos et al. [38]. Five tuning rules are defined. (f) Other methods: Penner [79], Srividya and Chidambaram [80]. Two tuning rules are defined. 1 1 Table 9: Controller G c ( s) = Kc 1 + . One robust tuning rule is defined by Tan et al. [81]. Ts i 1 + Tf s Table 10: Controller U( s) = Kc Y(s) −
Kc E ( s) . One direct synthesis tuning rule is defined by Chien et al. [74]. Ti s
1 E (s ) − α Kc R( s) . Two performance index Table 11: Two degree of freedom controller: U(s) = K c 1 + Tis minimisation - servo/regulator tuning rules are defined by Taguchi and Araki [61a] and Pecharroman and Pagola [134b]. Tables 12-14: PI tuning rules – FOLIPD model G m (s) =
Km e− sτ s(1 + sTm ) m
1 Table 12: Ideal controller G c ( s) = Kc 1 + . Six such tuning rules are defined; the references are: Ts i (a) Ultimate cycle: McMillan [58]. One tuning rule is defined. (b) Regulator tuning – minimum performance index: Minimum IAE – Shinskey [59] – page 75. Shinskey [59] – page 158. Minimum ITAE - Poulin and Pomerleau [82]. Four tuning rules are defined. (c) Direct synthesis - Poulin and Pomerleau [78]. One tuning rule is defined. 1 Table 13: Controller G c ( s) = Kc b + . One direct synthesis tuning rule is defined by Astrom and Hagglund Ti s [3] – pages 210-212.
1 E (s ) − α Kc R( s) . Two performance index Table 14: Two degree of freedom controller: U(s) = K c 1 + Tis minimisation tuning rules are defined by Taguchi and Araki [61a] and Pecharroman and Pagola [134b]. Tables 15-16: PI tuning rules – SOSPD model
K m e− sτ
Tm1 s2 + 2ξ m Tm1s + 1 2
Km e− sτ (1 + Tm1s)(1 + Tm2 s) m
m
or
1 Table 15: Ideal controller G c ( s) = Kc 1 + . Ten tuning rules are defined; the references are: Ts i (a) Robust: Brambilla et al. [48]. One tuning rule is defined. (b) Direct synthesis: Tan et al. [39]. One tuning rule is defined. (c) Regulator tuning – minimum performance index: Minimum IAE - Shinskey [59] – page 158, ** Shinskey [17], Huang et al. [18], Minimum ISE – McAvoy and Johnson [83], Minimum ITAE – Lopez et al. [84]. Five tuning rules are defined. (d) Servo tuning – minimum performance index: Minimum IAE - Huang et al. [18]. One tuning rule is defined. (e) Ultimate cycle: Hwang [60], ** Shinskey [17]. Two tuning rules are defined. 1 E (s ) − α Kc R( s) . Three performance index Table 16: Two degree of freedom controller: U(s) = K c 1 + T is minimisation tuning rules are defined by Taguchi and Araki [61a] and Pecharroman and Pagola [134a], [134b]. Table 17: PI tuning rules – SOSIPD model (repeated pole) G m (s) =
Km e −s τ
m
s (1 + sTm )
2
1 E (s ) − α Kc R( s) . Two performance index Table 17: Two degree of freedom controller: U(s) = K c 1 + Tis minimisation tuning rules are defined by Taguchi and Araki [61a] and Pecharroman and Pagola [134b].
Km e − sτ (1 + sTm1 )(1 + sTm2 )(1 + sTm 3 ) m
Tables 18-19: PI tuning rules – third order lag plus delay (TOLPD) model
1 Table 18: Ideal controller G c ( s) = Kc 1 + . One *** tuning rule is defined. The reference is Hougen [85]. Ts i 1 E (s ) − α Kc R( s) . One performance index Table 19: Two degree of freedom controller: U(s) = K c 1 + Tis minimisation tuning rule is defined by Taguchi and Araki [61a]. K me − sτ 1 − sTm
m
Table 20: PI tuning rules - unstable FOLPD model
1 Table 20: Ideal controller G c ( s) = Kc 1 + . Six tuning rules are defined; the references are: Ts i (a) Direct synthesis: De Paor and O’Malley [86], Venkatashankar and Chidambaram [87], Chidambaram [88], Ho and Xu [90]. Four tuning rules are defined. (b) Robust: Rotstein and Lewin [89]. One tuning rule is defined. (c) Ultimate cycle: Luyben [91]. One tuning rule is defined.
K m e− s τ (1 − sTm1 )(1 + sTm 2 ) m
Table 21: PI tuning rules - unstable SOSPD model
1 Table 21: Ideal controller G c ( s) = Kc 1 + . Three tuning rules are defined; the references are: Ts i (a) Ultimate cycle: McMillan [58]. One tuning rule is defined. (b) Minimum performance index – regulator tuning: Minimum ITAE – Poulin and Pomerleau [82]. Two tuning rules are defined. Table 22: PI tuning rules – delay model e −s τ m
1 Table 22: Ideal controller G c ( s) = Kc 1 + . Two tuning rules are defined; the references are: Ts i
(a) Direct synthesis: Hansen [91a]. (b) Minimum performance index – regulator tuning: Shinskey [57].
K m e− sτ 1 + sTm
m
Tables 23-40: PID tuning rules - FOLPD model
1 Table 23: Ideal controller G c ( s) = Kc 1 + + Td s . Fifty-seven tuning rules are defined; the references are: Ti s (a) Process reaction: Ziegler and Nichols [8], Astrom and Hagglund [3] – page 139, Parr [64] – page 194, Chien et al. [10], Murrill [13]- page 356, Cohen and Coon [11], Astrom and Hagglund [93]pages 120-126, Sain and Ozgen [94]. Eight tuning rules are defined. (b) Minimum performance index – regulator tuning: Minimum IAE – Murrill [13] – pages 358-363, Cheng and Hung [95]. Minimum ISE - Murrill [13] – pages 358-363, Zhuang and Atherton [20]. Minimum ITAE - Murrill [13] – pages 358-363. Minimum ISTSE - Zhuang and Atherton [20]. Minimum ISTES - Zhuang and Atherton [20]. Minimum error - step load change - Gerry [96]. Eight tuning rules are defined. (c) Minimum performance index – servo tuning: Minimum IAE - Rovira et al. [21], Wang et al. [97]. Minimum ISE - Wang et al. [97], Zhuang and Atherton [20]. Minimum ITAE - Rovira et al. [21], Cheng and Hung [95], Wang et al. [97]. Minimum ISTSE – Zhuang and Atherton [20]. Minimum ISTES – Zhuang and Atherton [20]. Nine tuning rules are defined. (d) Ultimate cycle: Pessen [63], Zhuang and Atherton [20], Pi-Mira et al. [97a], Hwang [60], Hwang and Fang [61], McMillan [58], Astrom and Hagglund [98], Li et al. [99], Tan et al. [39], Friman and Waller [41]. Fourteen tuning rules are defined. (e) Direct synthesis: Gorecki et al. [28], Smith and Corripio [25], Suyama [100], Juang and Wang [101], Cluett and Wang [44], Zhuang and Atherton [20], Abbas [45], Camacho et al.[102, Ho et al. [103], Ho et al [104], Morilla et al. [104a]. Fourteen tuning rules are defined. (f) Robust: Brambilla et al. [48], Rivera et al. [49], Fruehauf et al. [52], Lee et al. [55]. Four tuning rules.
1 1 Table 24: Ideal controller with first order filter G c ( s) = Kc 1 + + Td s . Three robust tuning rules are Ti s Tf s + 1 defined by ** Morari and Zafiriou [105], Horn et al. [106] and Tan et al. [81]. 1 1 + b1s Table 25: Ideal controller with second order filter G c ( s) = Kc 1 + + Td s . One robust tuning 2 Ti s 1 + a 1s + a 2s rule is defined by Horn et al. [106].
1 Table 26: Ideal controller with set-point weighting G c ( s) = Kc b + + Td s . One direct synthesis tuning rule Ti s is defined by Astrom and Hagglund [3] – pages 208-210. Table 27. Ideal controller with first order filter and set-point weighting: 1 1 1 + 0.4Trs U(s) = K c 1 + + Td s − Y( s) . One direct synthesis tuning rule is defined R (s ) 1 + sTr Tis Tf s + 1 by Normey-Rico et al. [106a]. 1 1 + sTd Table 28: Classical controller G c ( s) = K c 1 + . Twenty tuning rules are defined; the references are: T Ti s 1+ s d N (a) Process reaction: Hang et al. [36] – page 76, Witt and Waggoner [107], St. Clair [15] – page 21, Shinskey [15a]. Five tuning rules are defined. (b) Minimum performance index – regulator tuning: Minimum IAE - Kaya and Scheib [108], Witt and Waggoner [107]. Minimum ISE - Kaya and Scheib [108]. Minimum ITAE - Kaya and Scheib [108], Witt and Waggoner [107]. Five tuning rules are defined. (c) Minimum performance index – servo tuning: Minimum IAE - Kaya and Scheib [108], Witt and Waggoner [107]. Minimum ISE - Kaya and Scheib [108]. Minimum ITAE - Kaya and Scheib [108], Witt and Waggoner [107]. Five tuning rules are defined. (d) Direct synthesis: Tsang and Rad [109], Tsang et al. [111]. Two tuning rules are defined. (e) Robust: Chien [50]. One tuning rule is defined.
(f) Ultimate cycle: Shinskey [59] – page 167, Shinskey [16] – page 143. Two tuning rules are defined. 1 Td s . Two tuning rules are defined; the E( s) − Table 29: Non-interacting controller U (s) = K c 1 + Y ( s ) Td s Tis 1+ N references are: (a) Minimum performance index – regulator tuning: Minimum IAE - Huang et al. [18]. (b) Minimum performance index – servo tuning: Minimum IAE - Huang et al. [18]. 1 Td s Table 30: Non-interacting controller U( s) = Kc 1 + Y(s) . Five tuning rules are defined; the E ( s) − sT Ti s 1+ d N references are: (a) Minimum performance index – servo tuning: Minimum ISE - Zhuang and Atherton [20], Minimum ISTSE - Zhuang and Atherton [20]. Minimum ISTES - Zhuang and Atherton [20]. Three tuning rules are defined. (b) Ultimate cycle: Zhuang and Atherton [20], Shinskey [16] – page 148. Two tuning rules are defined. Td s 1 E (s) − Table 31: Non-interacting controller U (s) = K c + Y ( s) . Six tuning rules are defined; the sTd T s i 1+ N references are: (a) Minimum performance index – regulator tuning: Minimum IAE – Kaya and Scheib [108]. Minimum ISE – Kaya and Scheib [108]. Minimum ITAE – Kaya and Scheib [108]. (b) Minimum performance index – servo tuning: Minimum IAE – Kaya and Scheib [108]. Minimum ISE – Kaya and Scheib [108]. Minimum ITAE – Kaya and Scheib [108]. Table 32: Non-interacting controller with setpoint weighting: 1 Kc Td s U( s) = Kc b + Y( s) + Kc ( b − 1)Y( s) . Three ultimate cycle tuning rules are E (s) − Ti s 1 + Td s N defined by Hang and Astrom [111], Hang et al. [65] and Hang and Cao [112]. 1 1 + Td s Table 33: Industrial controller U( s) = Kc 1 + Y(s) . Six tuning rules are defined: the R( s) − Ts Ti s 1+ d N reference are: (a) Minimum performance index – regulator tuning: Minimum IAE - Kaya and Scheib [108]. Minimum ISE - Kaya and Scheib [108]. Minimum ITAE - Kaya and Scheib [108]. Three tuning rules are defined. (b) Minimum performance index – servo tuning: Minimum IAE - Kaya and Scheib [108]. Minimum ISE Kaya and Scheib [108]. Minimum ITAE - Kaya and Scheib [108]. Three tuning rules are defined. 1 Table 34: Series controller Gc ( s) = Kc 1 + (1 + sTd ) . Three tuning rules are defined; the references are: Ti s (a) ******: Astrom and Hagglund [3] – page 246. (b) Ultimate cycle: Pessen [63]. (c) Direct synthesis: Tsang et al. [110].
1 sTd . One robust tuning rule is Table 35: Series controller with filtered derivative Gc ( s) = Kc 1 + 1 + sT Ti s 1 + d N defined by Chien [50]. 1 Td s . Three tuning rules are defined; the Table 36: Controller with filtered derivative G c ( s) = Kc 1 + + Ti s 1 + s Td N references are:
(a) Robust: Chien [50], Gong et al. [113]. Two tuning rules are defined. (b) Direct synthesis: Davydov et al. [31]. One tuning rule is defined. 1 Table 37: Alternative non-interacting controller 1 - U( s) = Kc 1 + E( s) − Kc Td sY ( s) . Six ultimate cycle Ti s tuning rules are defined; the references are: Shinskey [59] – page 167, ** Shinskey [17], Shinskey [16] – page 143, VanDoren [114]. 2 1 1 + 05 . τm s + 0.0833τ m s 2 Table 38: Alternative filtered derivative controller - G c ( s) = Kc 1 + . One direct 2 Ti s [1 + 01. τ ms] synthesis tuning rule is defined by Tsang et al. [110]. Table 39: I-PD controller U(s) =
Kc E (s) − K c (1 + Td s)Y(s) . Two direct synthesis tuning rules are defined by Chien Ti s
et al. [74] and Argelaguet et al. [114a].
1 T s β T s d d E( s) − K c α + R (s) . One Table 40: Two degree of freedom controller: U (s) = K c 1 + + Tis Td Td 1+ s 1+ s N N performance index minimisation tuning rule is defined by Taguchi and Araki [61a]. Tables 41-48: PID tuning rules - non-model specific 1 Table 41: Ideal controller G c ( s) = Kc 1 + + Td s . Twenty five tuning rules are defined; the references are Ti s (a) Ultimate cycle: Ziegler and Nichols [8], Blickley [115], Parr [64] – pages 190-191, De Paor [116], Corripio [117] – page 27, Mantz andTacconi [118], Astrom and Hagglund [3] – page 142, Astrom and Hagglund [93], Atkinson and Davey [119], ** Perry and Chilton [120], Yu [122] – page 11, Luo et al. [121], McMillan [14] – page 90, McAvoy and Johnson [83], Karaboga and Kalinli [123], Hang and Astrom [124], Astrom et al. [30], St. Clair [15] - page 17, Shin et al. [125]. Nineteen tuning rules are defined. (b) Other tuning: Harriott [126], Parr [64] – pages 191, 193, McMillan [14] - page 43, Calcev and Gorez [69], Zhang et al. [127], Garcia and Castelo [127a]. Six tuning rules are defined. 1 Td s . Eight tuning rules are defined; the Table 42: Controller with filtered derivative Gc ( s) = Kc 1 + + Ti s 1 + Td s N references are: (a) Direct synthesis: Vrancic [72], Vrancic [73], Lennartson and Kristiansson [157], Kristiansson and Lennartson [158], Kristiansson and Lennartson [158a]. Six tuning rules are defined. (b) Other tuning: Leva [67], Astrom [68]. Two tuning rules are defined. Table 43: Ideal controller with set-point weighting: 1 U( s) = Kc Fp R(s) − Y(s) + ( Fi R(s) − Y(s)) + Td s( Fd R(s) − Y(s)) . One ultimate cycle tuning rule is Ts i
(
)
defined by Mantz and Tacconi [118].
1 Table 42: Ideal controller with proportional weighting G c ( s) = Kc b + + Td s . One direct synthesis tuning Ti s rule is defined by Astrom and Hagglund [3] – page 217. 1 K Ts Table 44: Non-interacting controller U( s) = Kc 1 + E( s) − c d Y( s) . One ultimate cycle tuning rule is sT Ti s 1+ d N defined by Fu et al. [128].
1 Table 45: Series controller U( s) = Kc 1 + (1 + sTd ) . Three ultimate cycle tuning rules are defined by Pessen Ti s [131], Pessen [129] and Grabbe et al. [130].
1 sTd . One ultimate cycle tuning Table 46: Series controller with filtered derivative U( s) = Kc 1 + 1 + sT Ti s 1 + d N rule is defined by Hang et al. [36] - page 58. 1 1 + sTd Table 47: Classical controller U( s) = Kc 1 + . One ultimate cycle tuning rule is defined by Corripio T Ti s 1+ s d N [117]. 1 Table 48: Non-interacting controller U( s) = Kc 1 + E ( s) − Kc Td sY (s) . One ultimate cycle tuning rule is Ti s defined by VanDoren [114]. K m e− sτ m s 1 Table 49: Ideal controller Gc ( s) = K c 1 + + Td s . Five tuning rules are defined; the references are: T s i Tables 49-58: PID tuning rules - IPD model
(a) Process reaction: Ford [132], Astrom and Hagglund [3] – page 139. Two tuning rules are defined. (b) Direct synthesis: Wang and Cluett [76], Cluett and Wang [44], Rotach [77]. Three tuning rules are defined. Table 50: Ideal controller with first order filter, set-point weighting and output feedback: 1 1 1 + 0.4Trs U(s ) = K c 1 + + Tds − Y(s ) − K 0 Y(s) . One direct synthesis tuning rule R( s) Ti s 1 + sTr Tf s + 1 has been defined by Normey-Rico et al. [106a]. 1 Td s . One robust tuning rule has Table 51: Ideal controller with filtered derivative G c ( s) = Kc 1 + + sTd Ts i 1+ N been defined by Chien [50]. 1 Td s . One robust tuning rule has Table 52: Series controller with filtered derivative G c ( s) = Kc 1 + 1 + Ts Ts i 1 + d N been defined by Chien [50].
1 1 + Td s . Five tuning rules have been defined; the references Table 53: Classical controller G c ( s) = Kc 1 + Ti s 1 + Td s N are: (a) Ultimate cycle: Luyben [133], Belanger and Luyben [134]. Two tuning rules have been defined. (b) Robust: Chien [50]. One tuning rule has been defined. (c) Performance index minimisation – regulator tuning: ** Minimum IAE - Shinskey [17], Shinskey [59] – page 74. Two tuning rules have been defined. 1 Table 54: Alternative non-interacting controller 1: U( s) = Kc 1 + E( s) − Kc Td sY ( s) . Two performance index T is minimisation rules – minimum IAE regulator tuning have been defined by Shinskey [59] – page 74 and ** Shinskey [17].
Table 55: I-PD controller U(s) =
Kc E (s) − K c (1 + Td s)Y(s) . One direct synthesis tuning rule has been defined by Ti s
Chien et al. [74].
1 ) E( s) + Kc ( b − 1) R (s) − Kc TdsY( s) . One direct synthesis tuning rule has Tis been defined by Hansen [91a]. 1 T s β T s d d E( s) − K c α + R (s) . Two Table 57: Two degree of freedom controller: U (s) = K c 1 + + Tis Td Td 1 + s 1 + s N N minimum performance index – servo/regulator tuning have been defined by Taguchi and Araki [61a] and Pecharroman and Pagola [134a]. Table 56: Controller U(s) = Kc (1 +
Km e − sτ s(1 + sTm ) m
Tables 58-67: PID tuning rules - FOLIPD model
1 Table 58: Ideal controller Gc ( s) = K c 1 + + Td s . One ultimate cycle tuning rule has been defined by Millan Ti s [58].
1 1 Table 59: Ideal controller with filter G c ( s) = Kc 1 + + Td s . Three robust tuning rules have been T s 1 + Tf s i defined by Tan et al. [81], Zhang et al. [135] and Tan et al. [136]. 1 Table 60: Ideal controller with set-point weighting G c ( s) = Kc b + + Td s . One direct synthesis tuning rule Ti s has been defined by Astrom and Hagglund [3] - pages 212-213. 1 1 + Td s . Five tuning rules have been defined; the references Table 61: Classical controller G c ( s) = Kc 1 + Ti s 1 + Td s N are as follows: (a) Robust: Chien [50]. One tuning rule is defined. (b) Minimum performance index – regulator tuning: Minimum IAE – Shinskey [59] – page 75, Shinskey [59] – pages 158-159, Minimum ITAE - Poulin and Pomerleau [82], [92]. Four tuning rules are defined. 1 Td s . One robust tuning rule has Table 62: Series controller with derivative filtering G c ( s) = Kc 1 + 1 + Ts Ts i 1 + d N been defined by Chien [50].
1 U(s) = Kc 1 + E (s) − K c Td sY ( s) . Two minimum Ti s performance index (minimum IAE) – regulator tuning rules have been defined by Shinskey [59] – page 75, page 159. 1 Td s . One robust tuning rule has Table 64: Ideal controller with filtered derivative: G c ( s) = Kc 1 + + Ti s 1 + s Td N been defined by Chien [50]. Table 63: Alternative non-interacting controller 1:
Table 65: Ideal controller with set-point weighting:
(
)
Kc [ Fi R(s) − Y(s)] + Kc Td s[ Fd R (s) − Y(s)] . One ultimate cycle tuning rule Ti s has been defined by Oubrahim and Leonard [138]. 1+ T s 1 + T s i d . One direct synthesis tuning rule has Table 66: Alternative classical controller: G c ( s) = Kc 1 + Td s 1 + Td s N N been defined by Tsang and Rad [109]. U(s) = Kc Fp R ( s) − Y( s) +
Table 67: Two degree of freedom controller: 1 T s β T s d d E( s) − K c α + R (s) . Two minimum performance index – U (s) = K c 1 + + Tis Td Td 1+ s 1+ s N N servo/regulator tuning have been defined by Taguchi and Araki [61a] and Pecharroman and Pagola [134a].
Km e− sτ K m e− sτ or 2 2 (1 + sTm1 )(1 + sTm 2 ) Tm1 s + 2ξ m Tm1s + 1 m
Tables 68-79: PID tuning rules - SOSPD model
m
1 Table 68: Ideal controller Gc ( s) = K c 1 + + Td s . Twenty seven tuning rules have been defined; the Ti s references are: (a) Minimum performance index – servo tuning: Minimum ITAE – Sung et al. [139]. One tuning rule is defined. (b) Minimum performance index – regulator tuning: Minimum ITAE – Sung et al. [139], Lopez et al. [84]. One tuning rule is defined. (c) Ultimate cycle: Hwang [60], Shinskey [16] – page 151. Three tuning rules are defined. (d) Direct synthesis: Hang et al. [35], Ho et al. [140], Ho et al. [141], Ho et al. [142], Wang et al. [143], Leva et al. [34], Wang and Shao [144], Pemberton [145], Pemberton [24], Suyama [100], Smith et al. [146], Chiu et al. [29], Wang and Clemens [147], Gorez and Klan [147a], Miluse et al. [27a], Miluse et al. [27b], Seki et al. [147b], Landau and Voda [148]. Nineteen tuning rules are defined. (e) Robust: Brambilla et al. [48], Chen et al. [53], Lee et al. [55]. Three tuning rules are defined.
1 1 Table 69: Filtered controller G c ( s) = Kc 1 + + Td s . One robust tuning rule has been defined by Hang Ti s Tf s + 1 et al. [35].
b s+1 1 Table 70: Filtered controller G c ( s) = Kc 1 + + Td s 1 . One robust tuning rule has been defined by T s a 1s + 1 i Jahanmiri and Fallahi [149].
1 1 + Td s . Seven tuning rules have been defined; the Table 71: Classical controller G c ( s) = Kc 1 + Ti s 1 + Td s N references are: (a) Minimum performance index – regulator tuning: Minimum IAE - Shinskey [59] – page 159, ** Shinskey [59], ** Shinskey [17], ** Shinskey [17]. Minimum ISE – McAvoy and Johnson [83]. Five tuning rules are defined. (b) Direct synthesis: Astrom et al. [30], Smith et al. [26]. Two tuning rules are defined. 1 1 + NTd s Table 72: Alternative classical controller G c ( s) = Kc 1 + . One ***** tuning rule has been Ti s 1 + Td s defined by Hougen [85].
1 Table 73: Alternative non-interacting controller 1: U( s) = K c 1 + E ( s) − Kc Td sY ( s) . Three minimum Ti s performance index (minimum IAE) – regulator tuning rules have been defined by Shinskey [59] – page 158, ** Shinskey [17], ** Shinskey [17]. 1 Table 74: Series controller G c ( s) = K c 1 + (1 + Td s) . One minimum performance index - regulator tuning rule Ti s has been defined by Haalman [23].
1 Td s . Two tuning rules have been E( s) − Table 75: Non-interacting controller U (s) = K c 1 + Y ( s ) Td s Tis 1+ N defined. The references are: (a) Minimum performance index – regulator tuning: Minimum IAE - Huang et al. [18]. (b) Minimum performance index – servo tuning: Minimum IAE - Huang et al. [18]. Table 76: Ideal controller with set-point weighting: K U(s) = Kc Fp R ( s) − Y( s) + c [ Fi R( s) − Y(s) ] + Kc Td s[ Fd R ( s) − Y( s)] . One ultimate cycle tuning rule Ti s
(
)
has been defined by Oubrahim and Leonard [138]. 1 Table 77: Non-interacting controller U( s) = Kc b + [ R (s) − Y(s) ] − ( c + Tds)Y( s) . One direct synthesis tuning Ts i rule has been defined by Hansen [150].
1 Td s . Two tuning rules are defined; Table 78: Ideal controller with filtered derivative G c ( s) = Kc 1 + + sT Ts i 1 + d N the references are: (a) Direct synthesis: Hang et al. [151]. (b) Robust: Hang et al. [151]. 1 T s β T s d d E( s) − K c α + R (s) . Three Table 79: Two degree of freedom controller: U (s) = K c 1 + + Tis Td Td 1 + s 1 + s N N minimum performance index – servo/regulator tuning rules have been defined by Taguchi and Araki [61a] and Pecharroman and Pagola [134a], [134b]. Table 80: PID tuning rules - I 2PD model G m (s) =
K me − sτ m s2
1 ) E( s) + Kc ( b − 1) R (s) − Kc TdsY( s) . One direct synthesis tuning rule has Tis been defined by Hansen [91a].
Table 80: Controller U(s) = Kc (1 +
Table 81: PID tuning rules – SOSIPD model (repeated pole) G m (s) =
Km e −s τ
m
s (1 + sTm )
2
1 T s β T s d d E( s) − K c α + R (s) . Two Table 81: Two degree of freedom controller: U (s) = K c 1 + + Tis Td Td 1+ s 1+ s N N minimum performance index – servo/regulator tuning rules have been defined by Taguchi and Araki [61a] and Pecharroman and Pagola [134a].
Tables 82-84: PID tuning rules - SOSPD model with a positive zero
K m (1 − sTm3 ) e− s τ
m
(1 + sTm1 )(1 + sTm2 )
1 Td s . One robust tuning rule has been Table 82: Controller with filtered derivative G c ( s) = Kc 1 + + Ti s 1 + Td s N defined by Chien [50]. 1 1 + Td s . One robust tuning rule has been defined by Chien Table 83: Classical controller G c ( s) = Kc 1 + Ti s 1 + Td s N [50]. 1 Td s . One robust tuning rule has Table 84: Series controller with filtered derivative G c ( s) = Kc 1 + 1 + Ts Ts i 1 + d N been defined by Chien [50]. Tables 85-88: PID tuning rules - SOSPD model with a negative zero
K m (1 + sTm 3 ) e− sτ
m
(1 + sTm1 )(1 + sTm 2 )
1 Table 85: Ideal controller Gc ( s) = K c 1 + + Td s . One minimum performance index tuning rule has been Ti s defined by Wang et al. [97].
1 Td s . One robust tuning rule has been Table 86: Ideal controller with filtered derivative G c ( s) = Kc 1 + + Ti s 1 + Td s N defined by Chien [50]. 1 1 + Td s . One robust tuning rule has been defined by Chien Table 87: Classical controller G c ( s) = Kc 1 + Ti s 1 + Td s N [50]. 1 Td s . One robust tuning rule has Table 88: Series controller with filtered derivative G c ( s) = Kc 1 + 1 + Ts Ts i 1 + d N been defined by Chien [50].
Km e− sτ (1 + sTm1 )(1 + sTm 2 )(1 + sTm 3 ) m
Table 89-90: PID tuning rules - TOLPD model
1 Table 89: Ideal controller Gc ( s) = K c 1 + + Td s . Two minimum performance index tuning rules have been Ti s defined by Polonyi [153].
1 T s β T s d d E( s) − K c α + R (s) .One Table 90: Two degree of freedom controller: U (s) = K c 1 + + Tis Td Td 1+ s 1+ s N N minimum performance index tuning rule has been defined by Taguchi and Araki [61a].
K m e− sτ 1 − sTm
m
Tables 91-93: PID tuning rules - unstable FOLPD model
1 Table 91: Ideal controller Gc ( s) = K c 1 + + Td s . Three direct synthesis tuning rules are defined by De Paor Ti s and O’Malley [86], Chidambaram [88] and Valentine and Chidambaram [154]. 1 KTs Table 92: Non-interacting controller U( s) = Kc 1 + E ( s) − c d Y(s) . Two tuning rules have been sT Ti s 1+ d N defined; the references are: (a) Minimum performance index – servo tuning: Minimum IAE - Huang and Lin [155] (b) Minimum performance index – servo tuning: Minimum IAE - Huang and Lin [155] 1 1 + Td s . One performance index minimisation – regulator Table 93: Classical controller G c ( s) = Kc 1 + Ti s 1 + Td s N tuning rule has been defined by Shinskey [16]– page 381. Tables 94-97: PID tuning rules - unstable SOSPD model G m ( s) =
Km e − sτ (1 − sTm1 )(1 + sTm 2 ) m
1 Table 94: Ideal controller Gc ( s) = K c 1 + + Td s . Two tuning rules have been defined; the references are Ti s (a) Ultimate cycle: McMillan [58] (b) Robust: Rotstein and Lewin [89].
1 1 + Tds . Two minimum performance index tuning rules Table 95: Classical controller G c ( s) = Kc 1 + Ti s 1 + Td s N (regulator - minimum ITAE) have been defined by Poulin and Pomerleau [82], [92]. 1 Table 96: Series controller Gc ( s) = Kc 1 + (1 + Td s) . One direct synthesis tuning rule has been defined by Ti s Ho and Xu [90]. Table 97: Non-interacting controller
1 KTs U( s) = Kc 1 + E ( s) − c d Y(s) . Two tuning rules have been sT Ti s 1+ d N
defined; the references are (a) Minimum performance index – servo tuning: Minimum IAE - Huang and Lin [155] (b) Minimum performance index – regulator tuning: Minimum IAE - Huang and Lin [155] Table 98: PID tuning rules – general model with a repeated pole G m ( s) =
K m e − sτ (1 + sTm ) n m
1 Table 98: Ideal controller Gc ( s) = K c 1 + + Td s . One direct synthesis tuning rule has been defined by Ti s Skoczowski and Tarasiejski [156]
Table 99: PID tuning rules – general stable non-oscillating model with a time delay 1 Table 99: Ideal controller Gc ( s) = K c 1 + + Td s . One direct synthesis tuning rule has been defined by Gorez Ti s and Klan [147a].
Tables 100-101: PID tuning rules – fifth order model with delay K (1 + b1s + b2 s2 + b3s3 + b4 s4 + b ss5 ) e− s τ m G m ( s) = m 1 + a1s + a2 s2 + a3s 3 + a 4 s4 + a 5s5
(
)
1 Table 100: Ideal controller Gc ( s) = K c 1 + + Td s . One direct synthesis tuning rule is defined by Vrancic et Ti s al. [159].
1 Tds . One direct synthesis tuning rule is Table 101: Controller with filtered derivative G c ( s) = Kc 1 + + Td s Ts i 1+ N defined by Vrancic et al. [159].
** some more information needed. The number of tuning rules in each table is included in the data. Servo and regulator tuning rules are counted separately; otherwise, rules in which different tuning parameters are provided for a number of variations in process parameters or desired response parameters (such as desired gain margin, phase margin or closed loop response time constant) are counted as one tuning rule. Tabular summaries are provided below.
Table A: Model structure and tuning rules – a summary for PI controllers
Model Stable FOLPD Non-model specific IPD FOLIPD SOSPD SOSIPD TOLPD Unstable FOLPD Unstable SOSPD Delay model TOTAL
Process reaction
Direct Synthesis
Ultimate cycle
Robust tuning
Other rules
Total
12
Minimise Performanc e index 21
33
10
12
0
88 (53%)
0
0
7
7
0
8
23 (14%)
4 0 0 0 0 0
6 6 9 2 1 0
6 2 1 0 0 4
2 1 2 0 0 1
4 0 1 0 0 1
2 0 0 0 1 0
24 (14%) 9 (5%) 13 (7%) 2 (1%) 2 (1%) 6 (4%)
0
2
0
1
0
0
3 (2%)
0
1
1
0
0
0
2 (1%)
16
48
54
24
18
11
171
Table B: Model structure and tuning rules – a summary for PID controllers
Model Stable FOLPD Non-model specific IPD FOLIPD SOSPD
I 2PD SOSIPD SOSPD – pos. zero SOSPD – neg. zero TOLPD Unstable FOLPD Unstable SOSPD Higher order TOTAL
Process reaction
Direct Synthesis
Ultimate cycle
Robust tuning
Other rules
Total
13
Minimise Performanc e index 45
24
28
12
1
123 (44%)
0
0
7
27
0
8
42 (15%)
2 0 0 0
6 8 16 0
6 2 23 1
2 2 4 0
3 6 6 0
0 0 1 0
19 (7%) 18 (6%) 50 (17%) 1 (0%)
0 0
2 0
0 0
0 0
0 3
0 0
2 (1%) 3 (1%)
0
1
0
0
3
0
4 (1%)
0 0
3 3
0 3
0 0
0 0
0 0
3 (1%) 6 (2%)
0
4
1
1
1
0
7 (2%)
0
0
4
0
0
0
4 (1%)
15
88
71
64
34
10
282
Table C: Model structure and tuning rules – a summary for PI/PID controllers Process reaction
Model Stable FOLPD Non-model specific IPD FOLIPD SOSPD
I 2PD SOSIPD SOSPD – pos. zero SOSPD – neg. zero TOLPD Unstable FOLPD Unstable SOSPD Delay model Higher order TOTAL
Direct Synthesis
Ultimate cycle
Robust tuning
Other rules
Total
25
Minimise Performanc e index 66
57
38
24
1
211 (47%)
0
0
14
34
0
16
65 (15%)
6 0 0 0
12 14 25 0
12 4 24 1
4 3 6 0
7 6 7 0
2 0 1 0
43 (9%) 27 (6%) 63 (14%) 1 (0%)
0 0
4 0
0 0
0 0
0 3
0 0
4 (1%) 3 (1%)
0
1
0
0
3
0
4 (1%)
0 0
4 3
0 7
0 0
0 1
1 0
5 (1%) 12 (3%)
0
6
1
2
1
0
10 (2%)
0
1
1
0
0
0
2 (0%)
0
0
4
0
0
0
4 (1%)
31
136
125
88
52
21
453
Table D: PI controller structure and tuning rules – a summary Stable FOLPD
Nonmodel specific
IPD
1 G c ( s) = Kc 1 + Ts i
85
19
20
1 G c ( s) = Kc b + T i s
2
1
1 E (s ) − α Kc R( s) U(s) = K c 1 + Tis
1
Controller structure
FOLIP D
SOSPD
Other
Total
6
10
12
152 (92%)
0
1
2
0
6 (4%)
1
2
2
1
3
10 (4%)
0
1
1
0
0
0
2 (1%)
1 1 G c ( s) = Kc 1 + Ts i 1 + Tf s
0
0
1
0
0
0
Total
88
21
23
8
12
15
1 (0%) 167
U( s) = Kc Y(s) −
Kc E ( s) Ti s
Table E: PID controller structure and tuning rules – a summary Stable FOLPD
Nonmodel specific
IPD
57
25
5
3
8
1 1 G c ( s) = Kc 1 + + Td s Ti s Tf s + 1
3
b s+1 1 G c ( s) = Kc 1 + + Td s 1 Ti s a 1s + 1
Controller structure
FOLIP D
SOSPD
Other
Total
1
27
11
126 (45%)
1
1
2
3
0
0
3
1
0
7 (3%)
0
0
0
0
1
0
1 (0%)
1 1 + b1s G c ( s) = Kc 1 + + Td s T s 1 + a 1s + a 2s 2 i
3
0
0
0
0
0
3 (1%)
1 G c ( s) = Kc b + + Td s Ti s
1
1
0
1
0
0
3 (1%)
Subtotal
67
34
6
6
31
14
158 (56%)
1 1 + sTd G c ( s) = K c 1 + T Ti s 1+ s d N
20
1
5
5
7
5
43 (15%)
1+ T s 1 + T s i d G c ( s) = Kc 1 + Td s 1 + Td s N N
0
0
0
1
0
0
1 (0%)
1 1 + NTd s G c ( s) = Kc 1 + T i s 1 + Td s
0
0
0
0
1
0
1 (0%)
1 Gc ( s) = Kc 1 + (1 + sTd ) Ti s
3
3
0
0
1
1
8 (3%)
1
1
1
1
0
2
6 (2%)
2 1 1 + 05 . τ m s + 0.0833τ m s 2 G c (s) = K c 1 + 2 Ti s [1 + 01. τ ms]
1
0
0
0
0
0
1 (0%)
Subtotal
25
5
6
7
9
8
60 (22%)
5
0
0
0
0
0
5 (2%)
2
0
0
0
4
0
6 (2%)
1 G c ( s) = Kc 1 + + Td s T s i 1 Td s G c ( s) = Kc 1 + + Ti s 1 + s Td N
1 sTd Gc ( s) = Kc 1 + 1 + sT Ti s 1+ d N
1 Td s U( s) = Kc 1 + E ( s) − Y(s) sTd T s i 1+ N 1 Td s E( s) − U (s) = K c 1 + Y ( s ) Td s Tis 1+ N
18 (6%)
Stable FOLPD
Nonmodel specific
IPD
Td s 1 E (s) − U (s) = K c + Y ( s) sTd T s i 1+ N
6
0
0
1 K Ts U( s) = Kc 1 + E( s) − c d Y( s) sT T s i 1+ d N
0
1
3
Controller structure
FOLIP D
SOSPD
Other
Total
0
0
0
6 (2%)
0
0
1
0
6 (2%)
0
0
0
0
0
6
0
0
0
0
0
6
1
2
2
3
0
14 (5%)
0
0
0
0
1
0
1 (0%)
2
0
1
0
0
0
3 (1%)
E(s ) − K α + βTds R (s ) c T s 1+ d s N
1
0
1
1
3
3
9 (3%)
1 ( Fi R(s) − Y(s)) + Td s( Fd R(s) − Y(s)) Ti s
0
1
0
1
1
0
1 ) E (s ) + K c ( b − 1)R ( s) − K cTd sY(s ) Tis
0
0
1
0
0
1
1 1 1 + 0.4Trs U( s) = K c 1 + + Td s R (s) − Y( s) T s +1 Ti s 1 + sTr f
1
0
0
0
0
0
1 1 1 + 0.4Tr s U(s) = Kc 1 + + Td s − Y(s) − K0 Y (s ) R (s ) Ti s 1 + sTr Tf s + 1
0
0
1
0
0
0
Subtotal
32
3
6
4
8
8
3 (1%) 2 (1%) 1 (0%) 1 (0%) 61 (22%)
Total
124
42
18
17
49
30
1 Kc Td s U(s) = Kc b + Y(s) + Kc ( b − 1) Y(s) E( s) − Ti s 1 + Td s N
1 1 + Td s U( s) = Kc 1 + Y(s) R( s) − Ts Ti s 1+ d N 1 U( s) = Kc 1 + E( s) − Kc Td sY ( s) Ti s 1 U(s) = Kc b + [R (s) − Y(s) ] − ( c + Td s) Y(s) Ts i U(s) =
Kc E (s) − K c (1 + Td s)Y(s) Ti s
1 Td s U(s ) = Kc 1 + + T Ti s 1+ d N
(
)
U(s) = K c Fp R(s) − Y(s) +
U( s) = K c (1 +
3 (1%) 6 (2%)
280
3. Tuning rules for PI and PID controllers
Table 1: PI tuning rules - FOLPD model - G m ( s) =
K me − sτ 1 – ideal controller G c ( s) = Kc 1 + . 84 tuning 1 + sTm Ts i m
rules Rule Process reaction Ziegler and Nichols [8] Model: Method 1. Hazebroek and Van der Waerden [9] Model: Method 1
τm Tm 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Kc
Ti
09 . Tm K mτ m
3.33τm
α Tm K mτ m α
β
0.68 7.14 0.70 4.76 0.72 3.70 0.74 3.03 0.76 2.50 0.79 2.17 0.81 1.92 0.84 1.75 0.87 1.61 τ α = 0.5 m + 01 . Tm
τm Tm
α
0.90 1.49 0.93 1.41 0.96 1.32 0.99 1.25 1.02 1.19 1.06 1.14 1.09 1.10 1.13 1.06 1.17 1.03 τm β= 16 . τ m − 1.2Tm
3.2τm
Chien et al. [10] - regulator
0.6Tm K mτ m
4τ m
Chien et al. [10] - servo Model: Method 1
Cohen and Coon [11] process reaction Model: Method 1.
07 . Tm K mτ m
β
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
063 . Tm Km τ m
Astrom and Hagglund [3] – regulator – page 150 Model: Method 1
Quarter decay ratio.
τm ≤1 Tm
βτ m
Astrom and Hagglund [3] – page 138 Model: Not relevant
Model: Method 1
Comment
2.33
τm Km
τm Tm
α
β
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
1.20 1.28 1.36 1.45 1.53 1.62 1.71 1.81
1.00 0.95 0.91 0.88 0.85 0.83 0.81 0.80
τm > 35 . Tm Ultimate cycle ZieglerNichols equivalent 0% overshoot τ 011 . < m < 10 . Tm 20% overshoot τ 011 . < m < 10 . Tm
07 . Tm K mτ m
2.3τm
20% overshoot
035 . Tm Km τ m
117 . Tm
0% overshoot τ 011 . < m < 10 . Tm
0.6Tm K mτ m
Tm
20% overshootτ 011 . < m < 10 . Tm
1 Tm + 0083 . 0.9 K m τm
2 3.33 τ m + 0.31 τm Tm Tm Tm τ 1 + 2.22 m Tm
Quarter decay ratio
Rule
Kc
Ti
τm Tm
α Tm K mτ m α
βτ m
Two constraints criterion Wolfe [12]
β
τm Tm
α
β
0.2 0.5
4.4 1.8
3.23 2.27
1.0 5.0
0.78 0.30
1.28 0.53
Model: Method 2.
Two constraints criterion Murrill [13] – page 356
Tm 0928 . K m τm
0.946
Tm τ m 1078 . Tm
Comment Decay ratio = 0.4; minimum error integral (regulator mode).
0 .583
Quarter decay ratio; minimum error integral (servo mode). τ 01 . ≤ m ≤ 10 . Tm
Model: Method 3 McMillan [14] – page 25 Model: Method 3 St. Clair [15] – page 22 Model: Method 3 Shinskey [15a] Model: Method 1 Regulator tuning Minimum IAE - Murrill [13] – pages 358-363 Model: Method 3
τm
Time delay dominant processes
Tm
Tm ≤ 30 . τm
3.78τ m
Tm = 0.167 τm
Km 3 0333 . Tm K mτ m 0. 667 Tm K mτ m
Performance index minimisation
Tm 0984 . Km τm
0.986
Tm τ m 0608 . Tm
0. 707
01 . ≤
100 . Tm Km τ m
30 . τm
τm Tm = 0.2
Minimum IAE - Shinskey [16] – page 123 Model: Method 6
104 . Tm K mτ m
2.25τ m
τm Tm = 0.5
111 . Tm Km τ m
145 . τm
τm Tm = 1
139 . Tm K mτ m
τm
τm Tm = 2
Minimum IAE - Shinskey [17] – page 38 Model: Method 6 Minimum IAE – Huang et al. [18] Model: Method 6
0.95Tm K mτ m
34 . τm
τm Tm = 01 .
0.95Tm K mτ m
2.9τ m
τm Tm = 0.2
Minimum IAE – Yu [19] K e− sτ L (Load model = L ) 1 + sTL Model: Method 6
Kc 0.685 TL K m Tm
( 1) 1
Ti τm
0.214 − 0 .346
Tm
τm Tm
− 1.256
Tm TL 0. 214 Tm
01 . ≤
( 1)
τm − 055 . Tm
1977 .
τm Tm
1123 .
TL Tm
τm Tm
TL 0874 . Km Tm
− 0099 . + 0159 .
τm Tm
τm Tm
− 1041 .
Tm TL 0.415 Tm
−4 .515
τm + 0. 067 Tm
τm Tm
0 .876
2. 641
Kc
( 1)
1 = Km
τm ≤1 Tm
≤ 2 .641
τm Tm
τm Tm
+ 016 . ;
≤ 0.35 TL
+ 0.16 ≤
τm Tm
1
τm ≤ 10 . Tm
Tm
≤1;
≤ 0.35
−0.9077 −0.063 0.5961 τ τ τm τm τ 6.4884 + 4.6198 m + 0.8196 m − 52132 . − 7.2712 − 0.7241e T Tm Tm Tm Tm
m
m
2 3 4 5 6 τm τm τm τm τm τ Ti (1) = Tm 00064 . + 3.9574 m − 64789 . + 9 . 4348 − 10 . 7619 + 7 . 5146 − 2 . 2236 Tm Tm Tm Tm Tm Tm
Kc
Rule Minimum IAE – Yu [19] K e− sτ L (Load model = L ) 1 + sTL (continued) Model: Method 6 Minimum ISE - Hazebroek and Van der Waerden [9]
Ti
τ − 0015 . + 0 .384 m Tm
TL 0871 . K m Tm
TL 0513 . K m Tm
0 .218
τm Tm
τm Tm
τm Tm
− 1055 .
− 1451 .
Tm TL 0. 444 Tm
Comment
τ − 0. 217 m − 0. 213 Tm
Tm TL 0. 670 Tm
Tm τ . + 0.3 m 074 K mτ m Tm
− 0. 003
τm − 0. 084 Tm
τm Tm
0867 .
τm Tm
0 .56
τm Tm
α
β
τm Tm
0.2 0.3 0.5
0.80 0.83 0.89
7.14 5.00 3.23
0.7 1.0 1.5
0.96 1.07 1.26
2.44 1.85 1.41
2.0 3.0 5.0
0.959
Tm τ m 0492 . Tm
0.739
0.945
Tm τ m 0535 . Tm
0.586
Minimum ISE – Zhuang and Atherton [20]
Tm 1279 . K m τm Tm 1346 . Km τm
0 .675
Tm τ m 0552 . Tm
0. 438
τm Tm
0 .181− 0 .205
τm Tm
− 0. 045 + 0. 344
TL 1289 . K m Tm
Minimum ITAE - Murrill [13] – pages 358-363 Model: Method 3 Minimum ITAE – Yu [19] K e− sτ L (Load model = L ) 1 + sTL Model: Method 6
0. 04 + 0. 067
τm Tm
τm Tm
Tm 0859 . Km τm TL 0598 . K m Tm
−1. 214
τm Tm
Tm TL 0. 430 Tm
τm − 0 .49 Tm
0. 954
τm Tm
0 .639
τm Tm
−1. 014
Tm TL 0. 359 Tm
τm − 0 .292 Tm
− 2. 532
τm ≤ 10 . Tm
11 . ≤
τm ≤ 2.0 Tm
TL
≤ 2.310
Tm
τm
τm Tm
0. 899
− 1047 .
τm Tm
− 0889 .
0. 977
τm Tm
0. 272 − 0. 254
τm Tm
τm Tm
Tm TL 0.347 Tm
τm − 0. 094 Tm
− 1112 .
Tm TL 0596 . Tm
τm − 0. 44 Tm
0 .372
Tm τ m 0674 . Tm − 1341 .
Tm TL 0805 . Tm
0 .304
τm Tm
2 .310
τm Tm
τm Tm
τm Tm
Tm
Tm
01 . ≤ 0. 196
TL Tm
τm Tm
−0 .011− 1945 .
τm Tm
−1. 055
Tm TL 0. 425 Tm
−5. 809
τm + 0 .241 Tm
τm Tm
2 .385
τm Tm
τm
0 .084 + 0154 .
τm Tm
τm Tm
− 1. 042
Tm TL 0. 431 Tm
− 0148 .
τm − 0. 365 Tm
τm Tm
0. 901
1<
TL Tm
TL Tm
≤1;
≤ 0.35 τm Tm
≤ 0.35
> 035 .
τm ≤ 10 . Tm τm Tm
+ 0.112 ;
≤ 0.35
+ 0112 . ≤
Tm
0.787 TL Km Tm
+ 0. 077 ;
≤ 0.35
≤ 2.385 τm
0 .901
Tm
≤3; τm
0. 46
0.680
τm − 0. 112 Tm
TL
1<
τm
+ 0.077 ≤ τm
0 .898
β
01 . ≤
Tm
0. 735 TL K m Tm
> 035 .
1.46 1.18 1.89 0.95 2.75 0.81 τ 01 . ≤ m ≤ 10 . Tm
Tm −0 .065+ 0 .234
≤ 0.35
α
Tm
107 . TL K m Tm
Tm
βτ m β
TL 1157 . Km Tm
τm
τm Tm < 0.2
Tm 1305 . K m τm
Model: Method 6
Tm
τm Tm
0. 921 TL K m Tm
≤3; τm
143 . Tm
Minimum ISE - Murrill [13] – pages 358-363 Model: Method 3
Minimum ISE – Yu [19] K e− sτ L (Load model = L ) 1 + sTL
Tm
α Tm K mτ m α
Model: Method 1
Model: Method 6
TL
1<
TL Tm
≤1;
≤ 0.35
≤3;
τm Tm
≤ 0.35
Kc
Rule Minimum ITAE – Yu [19] K e− sτ L (Load model = L ) 1 + sTL
TL 0878 . K m Tm
0172 . − 0. 057
Ti τm Tm
τm Tm
−0 .909
Tm TL 0. 794 Tm
0 .228
Comment
τm − 0257 . Tm
τm Tm
0. 489
τm
> 035 .
Tm
(continued) Model: Method 6 Minimum ISTSE - Zhuang and Atherton [20] Model: Method 6
Minimum ISTES – Zhuang and Atherton [20] Model: Method 6 Servo tuning Minimum IAE – Rovira et al. [21] Model: Method 3 Minimum IAE - Huang et al. [18] Model: Method 6 Minimum ISE - Zhuang and Atherton [20] Model: Method 6
Minimum ISE – Khan and Lehman [22] Model: Method 6
Minimum ITAE – Rovira et al. [21] Model: Method 3
Tm 1015 . K m τm
0.957
Tm τ m 0667 . Tm
0.552
Tm 1065 . K m τm
0.673
Tm τ m 0687 . Tm
0.427
Tm 1021 . Km τ m
0.953
Tm τ m 0629 . Tm
0.546
Tm 1076 . Km τm
0 .648
Tm τ m 0650 . Tm
0. 442
Tm 0758 . K m τm
0.861
Kc( 2) 2
Performance index minimisation Tm τ 1.020 − 0.323 m Tm
Ti (2 )
Tm 0980 . Km τm
0.892
Tm 1072 . K m τm
0.560
τm ≤ 2.0 Tm
01 . ≤
τm ≤ 10 . Tm
11 . ≤
τm ≤ 2.0 Tm
01 . ≤
τm ≤ 10 . Tm
01 . ≤
τm ≤1 Tm
τm ≤ 10 . Tm
Tm
11 . ≤
τm ≤ 2.0 Tm
Kc Km
0.01 ≤
τm ≤ 0.2 Tm
0.2 ≤
τm ≤ 20 Tm
01 . ≤
τm ≤ 10 . Tm
τ 0.648 − 0.114 m Tm
0 .916
11 . ≤
01 . ≤
0.5291 00003082 . − 2 τm Tm τ m
Kc Km 0.808 0.511 0.255 T m + − τm Tm Tm τ m K m 0.095 + 0.846 − 0.381 τ 2 τ m Tm τ m Tm τ m m
Tm 0586 . Km τ m
τm ≤ 10 . Tm
Tm
τ 0.690 − 0.155 m Tm
0.7388 03185 Tm . + Tm Km τm
01 . ≤
Tm
τm 1.030 − 0165 . Tm
−1.0169 3.5959 3.6843 τ τm τm τm 1 τm T − 130454 Kc = . − 9.0916 + 0.3053 + 11075 . − 2.2927 + 4.8259e Km Tm Tm Tm Tm 2 3 4 5 6 τm τm τm τm τm τm (2 ) Ti = Tm 0.9771 − 0.2492 + 3.4651 − 7.4538 + 8.2567 − 4.7536 + 11496 . Tm Tm Tm Tm Tm Tm m
2
( 2)
m
Rule Minimum ISTSE - Zhuang and Atherton [20] Model: Method 6
Minimum ISTES – Zhuang and Atherton [20] Model: Method 6
Direct synthesis Haalman [23] Model: Method 6 Chen and Yang [23a] Model: Method 25 Minimum IAE – regulator Pemberton [24], Smith and Corripio [25] – page 343346. Model: Method 6 Minimum IAE – servo Smith and Corripio [25] – page 343-346. Model: Method 6 5% overshoot – servo – Smith et al. [26] – deduced from graph. Model: Method not stated 1% overshoot – servo – Smith et al. [26] – deduced from graph Model: Method not stated 5% overshoot - servo Smith and Corripio [25] – page 343-346. Model: Method 6 5% overshoot - servo Hang et al. [27] Model: Method 1 Miluse et al. [27a] Model: Method not stated
Kc
Ti
Tm 0712 . Km τm
0.921
Tm 0786 . Km τm
0.559
Tm 0569 . K m τm
0.951
Tm 0628 . Km τ m
0.583
2Tm 3K m τ m 0. 7T m Km τ m
Comment
Tm
01 . ≤
τm ≤ 10 . Tm
Tm
11 . ≤
τm ≤ 2.0 Tm
Tm
01 . ≤
τm ≤ 10 . Tm
Tm
11 . ≤
τm ≤ 2.0 Tm
τ 0.968 − 0.247 m Tm τ 0.883 − 0.158 m Tm τ 1.023 − 0.179 m Tm τ 1.007 − 0.167 m Tm
Tm
Closed loop sensitivity M s = 19 . . (Astrom and Hagglund [3]) Ms = 1. 26 ; A m = 2. 24 ;
Tm
φ m = 500 01 . ≤
τm ≤ 0.5 Tm
Tm
01 . ≤
τm ≤ 05 . Tm
052 . Tm K mτ m
Tm
0.04 ≤
τm ≤ 14 . Tm
044T . m K mτ m
Tm
0.04 ≤
τm ≤ 14 . Tm
Tm Km τ m
3Tm 5Km τ m
Tm
Tm 2K m τ m
Tm
13Tm 25K m τ m
Tm
0. 368Tm K m τm
Tm
0. 514Tm K m τm
Tm
Closed loop response overshoot = 0% (Model: Method 26 – Miluse et al. [27b]) Closed loop response overshoot = 5%
0. 581Tm K m τm
Tm
Closed loop response overshoot = 10%
Rule
Kc
Ti
Comment
Miluse et al. [27a] continued
0. 641Tm K m τm
Tm
Closed loop response overshoot = 15%
0. 696Tm K m τm
Tm
Closed loop response overshoot = 20%
0. 748Tm K m τm
Tm
Closed loop response overshoot = 25%
0. 801Tm K m τm
Tm
Closed loop response overshoot = 30%
0. 853Tm K m τm
Tm
Closed loop response overshoot = 35%
0. 906Tm K m τm
Tm
Closed loop response overshoot = 40%
0. 957 Tm K mτ m
Tm
Closed loop response overshoot = 45%
1.008 Tm Km τ m
Tm
Closed loop response overshoot = 50%
Model: Method not stated
Regulator – Gorecki et al. [28] (considered as 2 rules) Model: Method 6 Chiu et al. [29] Model: Method 6
Kc
Astrom et al. [30]
Kc
Ti
Ti ( 4)
Low freq. part of magnitude Bode diagram is flat
λ Tm K m (1 + λτm )
Tm
λ variable; suggested values: 0.2, 0.4, 0.6, 1.0.
1
Model: Method 12
τ K m 1.905 m + 0.826 Tm
2 τ 2 Tm = 2 + m − 1 e K m τm 2Tm
= τm
Tm
Honeywell UDC 6000 controller
τm . + 0.362 Tm 0153 Tm
Closed loop response damping factor = 0.9; 02 . ≤ τ m Tm ≤ 1 .
2
τm τm 2 + −2 − 2Tm 2 Tm
τ 1+ m 2Tm
2
2 3 2 2 τ τ τ τ τ 3 + m + m + m − 2 + m 2 + m 2Tm Tm 2Tm 2Tm 2Tm 2
Kc ( 4) =
Pole is real and has max. attainable multiplicity
( 3)
Kc( 4) 3
Davydov et al. [31]
( 3)
( 3)
Ti
3 3τ m K m1 + Tm
Model: Method 22
3
( 3)
1 Km
T T Tm + 6 m + 6 m τm τm τm 2 Tm Tm 4 1+ 3 + 3 τm τm
3
2
1+3
T T Tm + 6 m + 6 m τm τm τm 2 Tm Tm 3 1+ 2 + 2 τm τm
1+ 3
, Ti ( 4) = τ m
3
Kc
Rule
Ti
Comment
Schneider [32]
0.368
Tm K mτ m
Tm
Closed loop response damping factor = 1
Model: Method 6
0.403
Tm K mτ m
Tm
Closed loop response damping factor = 0.6
McAnany [33]
(1.44Tm + 0.72Tm τ m − 0.43τ m − 2.14)
(
K m 12 . τ m + 0.36 τ m 2 + 2
Model: Method 5 Leva et al. [34]
ω cn Tm Km
Model: Method 16 Khan and Lehman [22]
Tm Km
Tm Km
Model: Method 11 Or Model: Method 14
Gain and phase margin – Ho et al. [37]
)
Closed loop time constant = 167 . Tm .
π tanφ m − + τ m ω cn + tan −1 (ω cn Tm ) 2 ω cn
1 + ω cn 2 Tm 2 1 + ω cn 2 Ti 2
0.404 0.256 01275 . + − τm T τ m m Tm
ω cn
Tm π 2.82 − φ m − tan −1 − φm τ m 2 = τm
KcKm 0.4104 0.00024 0.525 − − τ m2 Tm 2 τ m Tm KcKm 0.719 0.0808 0.324 + − 2 τ m Tm τ τ m m τ mTm
0.01 ≤
τm ≤ 0.2 Tm
0.2 ≤
τm ≤ 20 Tm
1048 . Tm Km τ m
Tm
Gain Margin = 1.5 Phase Margin = 30 0
07854 . Tm K mτ m
Tm
Gain Margin = 2 Phase Margin = 45 0
0524 . Tm K mτ m
Tm
Gain Margin = 3 Phase Margin = 60 0
0.393Tm K mτ m
Tm
Gain Margin = 4 Phase Margin = 67 .5 0
0314T . m Km τ m
Tm
Gain Margin = 5 Phase Margin = 72 0
ω p Tm
1
K m Am
4ω p τ m
ωp =
2
2ω p −
(considered as 2 rules) Model: Method 6
2
4Tm + 128 . τ m − 2.4
0. 3852 0.723 τ + − 0.404 m2 Tm τm Tm
Model: Method 6
Hang et al. [35, 36]
)
(
Km 556 . + 2τ m + τ m
π
+
A m φ m + 0.5πA m (A m − 1)
(A
1
2 m
)
− 1 τm
Tm
Given A m , ISE is minimised when φ m = 688884 . − 34.3534 A m + 91606 .
τm for servo Tm
tuning (Ho et al [104]). Given A m , ISE is minimised when φ m = 45.9848A m 0.2677 ( τ m Tm )
0. 2755
for regulator
tuning (Ho et al [104]).
(
Tan et al. [39]
β Ti ω φ 1 + βTm ω φ
Model: Method 10
A m 1 + βTi ω φ
Symmetrical optimum principle - Voda and Landau [40]
(
)
)
2
[
ωφ < ω u
2
1
4.6 ω135
35 . G p ( jω135 ) 0
1 Model: Method not relevant
1
βω φ tan − tan − 1 βT mω φ − βτ m ω φ − φ
2.828 G p ( jω135 ) 1
115 . G p ( jω 135 0 ) + 0.75K m
0
4.6 G p ( jω135 ) − 0.6K m 0
[
β = 08 . ,
τm < 05 . ; Tm
β = 0.5,
τm > 05 . Tm
τm ≤ 01 . Tm
0
4 ω135
]
01 . < 0
ω 135 0 2.3 G p ( jω 1350 ) − 0.3K m
]
τm ≤ 015 . Tm
0.15 <
τm ≤1 Tm
Rule
Kc
Ti
Comment
Friman and Waller [41] Model: Method 6
0.2333
1
τ m > 2Tm . Gain margin = 3;
G p ( jω135 )
ω135
Model: Method 6
Tm 2K mτ m
Tm
Smith [42] Model: Method not specified
0.35 Km
042 . τm
Kc( 5)
Ti (5)
0.5Tm Km τ m
Tm
0
0
Voda and Landau [40]
Modulus optimum principle - Cox et al. [43]4 Model: Method 17
Cluett and Wang [44] Model: Method 6
Bi et al. [46] Model: Method 20
4
Kc
( 5)
=
0.5 Km
6 dB gain margin - dominant delay process
τm ≤1 Tm τm >1 Tm
0.099508τ m + 0.99956Tm Closed loop time constant = τm 4τm 0.99747τ m − 8742510 . . − 5 Tm
0.05548 τ m + 0.33639Tm K mτm
016440 . τ m + 0.99558Tm Closed loop time constant = τm 2τm 0.98607τ m − 15032 . .10− 4 Tm
0.092654τ m + 0.43620Tm K mτm
0.20926τ m + 098518 . Tm Closed loop time constant = τm 133 . τm 0.96515τ m + 4.255010 . −3 Tm
012786 . τ m + 051235 . Tm Km τ m
0.24145τ m + 0.96751Tm Closed loop time constant = τ 0.93566 τ m + 2.298810 . −2 Tm m τm
016051 . τ m + 0.57109Tm Km τ m
0.26502τ m + 0.94291Tm Closed loop time constant = τ 089868 . τ m + 6.935510 . −2 Tm m 08 . τm 0.28242τ m + 0.91231Tm τ 0.85491τ m + 015937 . Tm m
−1.045
Model: Method 6
Phase margin = 60 0 ; τ 0.25 ≤ m ≤ 1 Tm
0.019952τ m + 0.20042Tm K mτm
019067 . τ m + 0.61593Tm Km τ m
Abbas [45]
Phase margin = 45 0
τ 0148 . + 0186 . m Tm Km (0.497 − 0.464 V0.590 ) 05064 . Tm Km τ m
Tm + 0.5τ m
Closed loop time constant = 067 . τm V = fractional overshoot, 0 ≤ V ≤ 0.2 τ 01 . ≤ m ≤ 5.0 Tm
Tm
Tm 3 + Tm2 τ m + 0.5Tm τ m 2 + 0.167τ m 3 Tm3 + Tm 2τ m + 0.5Tmτ m 2 + 0.167τ m 3 ( 5) , T = i 2 2 3 2 2 Tm τm + Tm τ m + 0.667τ m Tm + Tm τ m + 0.5τ m
Kc
Rule Wang and Shao [47]
Kc
Ti
( 5a ) 5
Ti
Comment
λ = inverse of the maximum of the absolute real part of the open loop transfer function; λ = [1.5, 2.5]
(5 a )
Model: Method 6 Robust
Tm + 0.5τ m K m( λ + τ m )
Brambilla et al. [48] -
Closed loop response has less than 5% overshoot with no model uncertainty: τ τ τ λ = 1 , 01 . ≤ m ≤ 1 ; λ = 1 − 05 . log 10 m , 1 < m ≤ 10 Tm Tm Tm
Model: Method 6
Rivera et al. [49]
Tm λKm
Tm
λ ≥ 17 . τ m , λ > 01 . Tm .
Model: Method 6
2Tm + τ m 2λK m
Tm + 0.5τ m
λ ≥ 17 . τ m , λ > 01 . Tm .
Tm K m (τ m + λ)
Tm
τm 2Km ( τ m + λ )
05 . τm
Fruehauf et al. [52]
5Tm 9τ m Km
5τ m
τm < 0.33 Tm
Model: Method 1
Tm 2τ m Km
Tm
τm ≥ 0.33 Tm
Chen et al. [53]
0. 50Tm τm K m
Tm
A m = 3.14 , φ m = 61. 40 , Ms = 1. 00
Tm
A m = 2.58 , φ m = 55. 00 , Ms = 1. 10
0. 67Tm τ mK m
Tm
A m = 2. 34 , φ m = 51.6 0 , Ms = 1. 20
0. 70Tm τm K m
Tm
A m = 2. 24 , φ m = 50.0 0 , M s = 1. 26
0. 72Tm τm K m
Tm
A m = 2.18 , φ m = 48. 70 , Ms = 1. 30
Chien [50] Model: Method 6 Thomasson [51] Model: Method not defined
0. 61Tm τ m Km
Model: Method 6
5
Kc
( 5a )
f 1 (ω90
0
0
( 5a )
=
λ = Tm [50]; λ > Tm + τ m , τ m << Tm (Thomasson [51]) τ m >> Tm ; λ = desired closed loop time constant
1 1 f 2 ( ω900 ) − , λf 1 (ω90 0 ) ω90 0 Km 2 2 2 )= ( Tm + {1 + ω90 Tm }τ m ) sin(−ω90 τm − tan −1 ω90 Tm ) − ω90 Tm cos(− ω90 τ m − tan− 1 ω90 Tm ) 2 2 1 .5 1 + ω90 Tm =
(
)
0
f 2 ( ω90 ) = − Ti
Tm + 0.5τ m
1 1 + ω 90 Tm 2
0
[
2
[
0
[(T
m
0
0
0
+ {1 + ω 90 Tm }τ m ) cot(−ω90 τm − tan −1 ω90 Tm ) + ω90 Tm 2
2
0
0
0
0
0
2
]
]
ω900 Tm + (1 + ω90 0 2Tm 2 ) τ m cos(− ω90 0 τm − tan− 1 ω90 0 Tm ) + (1 + 2ω 900 2 Tm 2 ) sin(− ω90 0 τm − tan −1 ω900 Tm ) − ω900 Tm cos(−ω90 0 τm − tan −1 ω90 0 Tm ) + ω90 0 [Tm + (1 + ω900 Tm ) τm ] sin(− ω900 τ m − tan− 1 ω90 0 Tm ) 3
2
2
2
2
0
]
Rule
Kc
Chen et al. [53] - continued Model: Method 6
Ogawa [54] – deduced from graphs Model: Method 6
Tm
A m = 2. 07 , φ m = 46. 50 , Ms = 1. 40
0. 80Tm τm K m
Tm
A m = 1.96 , φ m = 44.10 , Ms = 1. 50
βTm β
τm Tm
α
β
0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0
0.9 0.6 0.7 0.47 0.47 0.36 0.4 0.33
1.3 1.6 1.3 1.7 1.3 1.8 1.3 1.8
2.0 10.0 2.0 10.0 2.0 10.0 2.0 10.0
0.45 0.4 0.4 0.35 0.32 0.3 0.3 0.29
2.0 7.0 2.2 7.5 2.4 8.5 2.4 9.0
Ti K m ( λ + τm )
Tm +
Km (τ m + λ )
Model: Method 21 Ultimate cycle
τm 2( λ + τ m ) 2
λ = 0.333 τm Tm > τ m
Tm ( τ m + 2λ ) − λ2
Tm ( τ m + 2λ ) − λ2
Chun et al. [57]
20% uncertainty in the process parameters 33% uncertainty in the process parameters 50% uncertainty in the process parameters 60% uncertainty in the process parameters Desired closed loop e− τ m s response = , ( λ s + 1)
Tm + 025 . τm
Tm + 0.25τ m Km λ
Isaksson and Graebe [56] Model: Method 6
λ = 0.4Tm
τ m + Tm
2
0.65 T 1881 . Tm 1 m 166 . τ m 1 + 0.65 Km τ m T Tm + τ m m 1 + T + τ m m
Regulator – minimum IAE – Shinskey [59] – page 167. Model: Method not specified Regulator – minimum IAE – Shinskey [17] – page 121. Model: Method not specified
T 3.05 − 0.35 u τm
µ1 =
(
Tuning rules developed from Ku , Tu
2 T T Tu 0.87 − 0.855 u + 0.172 u τm τ m
Ku
Regulator – minimum IAE – Shinskey [16] – page 148 Model: Method 6 Regulator – nearly minimum IAE, ISE, ITAE – Hwang [60] Model: Method 8
0. 76Tm τm K m
τm Tm
Model: Method 6
Model: Method 1 or Method 18
Comment
α Km α
Lee et al. [55]
McMillan [58]
Ti
055 . Ku
0.78Tu
τ m Tm = 0.2
0. 48Ku
0.47 Tu
τ m Tm = 1
0.5848Ku
0.81Tu
τ m Tm = 0.2
0.5405K u
0.66Tu
τ m Tm = 0.5
0.4762K u
0.47Tu
τ m Tm = 1
0.4608 K u
0.37Tu
τ m Tm = 2
( 1 − µ1 ) K u ,
114 . 1 − 0.482ω u τ m + 0.068ω 2u τ 2m Ku K m 1 + K u K m
Kc µ 2 Ku ω u ,
)
µ2 =
(
0.0694 −1 + 2.1ω u τ m −
0.367ω 2u τ m2
Ku Km 1 + Ku Km
)
Decay ratio = 0.15 τ 01 . ≤ m ≤ 2.0 Tm
Kc
Rule Regulator – nearly minimum IAE, ISE, ITAE – Hwang [60] (continued) Model: Method 8
Servo – small IAE – Hwang [60]
µ1 =
Model: Method not specified
0.0724ω 2u τ 2m
K u K m 1 + K u Km
( 1 − µ1 ) K u ,
µ1 =
µ1 =
µ1 =
µ1 =
Hang et al. [65]
(
1.09 1 − 0.497ω u τ m +
Model: Method 8 r = parameter related to the position of the dominant real pole.
( 1 − µ1 ) K u ,
(
1.03 1 − 0.51ωu τm +
(
0.0759ω u2τ m2
Ku Km 1 + Ku Km
)
1.07 1 − 0.466ω uτ m + 0.0667ω u2τ m2 Ku Km 1 + Ku Km
(
(
−1 + 2.54ω u τm − 0.457ωu2 τm2 K u Km 1 + K u Km
)
( 1 − µ1 ) K u , 0.0647ω 2u τ m2
(
0.0386 −1 + 3.26ω uτ m − 0.6ω 2u τ m2 K u Km 1 + K u Km
µ2 =
(
0.0328 −1 + 2.21ω u τm −
Decay ratio = 0.25 τ 01 . ≤ m ≤ 2.0 Tm
)
0.338ω 2u τ m2
K u Km 1 + K u Km
)
µ2 =
(
0.0477 − 1 + 2.07ω u τ m −
0.333ω 2u τ 2m
µ2 =
)
Ku K m 1 + K u K m
Kc µ 2 Ku ω u ,
)
Decay ratio = 0.1, r = 0.5 τ 01 . ≤ m ≤ 2.0 Tm
)
Kc µ 2 Ku ω u ,
0.0657ω 2u τ 2m
Ku Km 1 + Ku Km
Decay ratio = 0.2 τ 01 . ≤ m ≤ 2.0 Tm
0.054
µ2 =
µ2 =
)
Ku Km 1 + K u K m
114 . 1 − 0.466ωu τ m +
Kc µ 2 Ku ω u ,
Kc µ 2 Ku ω u ,
( 1 − µ1 ) K u ,
(
Comment
Kc µ 2 Ku ω u ,
( 1 − µ1 ) K u ,
111 . 1 − 0.467ω u τ m +
)
Ti
(
0.0609 −1 + 197 . ω uτ m − 0.323ωu2 τm2
τm 11 T + 13 12 + 2 m τ 37 m − 4 4 0.2 5 15 Tm Ku 6 τm 11 T + 13 m 15 + 14 τm 37 − 4 Tm
Ku Km 1 + Ku Km
)
Decay ratio = 0.1, r = 0.75 τ 01 . ≤ m ≤ 2.0 Tm Decay ratio = 0.1, r = 1.0 τ 01 . ≤ m ≤ 2.0 Tm
τm τ 11 T + 13 016 . ≤ m < 096 . ; m + 1Tu Tm 37 τm − 4 Servo response: 10% T m overshoot, 3% undershoot
Servo – minimum ISTSE – Zhuang and Atherton [20] 0.361K u 0.083 (1.935 K m K u + 1)Tu Model: Method not relevant Regulator – minimum ISTSE 1892 . K m K u + 0244 . 0.706Km K u − 0.227 Ku Ku – Zhuang and Atherton [20] 3249 . K K + 2 . 097 . 0.7229Km K u + 12736 m u Model: Method not relevant 2 ( Kc / K uω u ) Servo – nearly minimum IAE τm τ m 2 . + 0.0376 Ku τ m and ITAE – Hwang and 0.438 − 0110 τm Tm T m 0.0388 + 0108 . − 0.0154 Tm T m Fang [61] Model: Method 9 2 ( Kc / K uω u ) Regulator – nearly minimum τ m τm 2 τm IAE and ITAE – Hwang and 0.515 − 0.0521 Tm + 0.0254 Tm Ku τm 0.0877 + 0.0918 − 0.0141 Tm Tm Fang [61] Model: Method 9 2 ( Kc / K u ω u ) Simultaneous 0.46 − 0.0835 τ m + 0.0305 τ m K 2 u Servo/regulator – Hwang Tm Tm 0.0644 + 0 .0759 τ m − 0.0111 τ m Tm and Fang [61] Tm Model: Method 9
01 . ≤
τm ≤ 2.0 Tm
01 . ≤
τm ≤ 2.0 Tm
τm ≤ 2.0 ; Tm decay ratio = 0.03 01 . ≤
τm ≤ 2.0 ; Tm decay ratio = 0.12 01 . ≤
01 . ≤
τm ≤ 2.0 Tm
K me − sτ 1 – Controller G c ( s) = Kc b + . 2 tuning rules 1 + sTm Ti s m
Table 2: PI tuning rules - FOLPD model - G m ( s) =
Kc
Rule
Ti
Direct synthesis
(Maximum sensitivity)
029 . e Astrom and Hagglund [3] dominant pole design – page 205-208
Comment
− 2.7 τ + 3.7τ 2
Tm
K mτ m
8.9τ m e −6.6 τ + 3.0 τ or 2
,
τ = τ m ( τ m + Tm )
0.79Tm e −1.4 τ + 2.4 τ
2
Model: Method 3 or 4
078 . e− 4.1τ + 5.7 τ Tm Km τ m
8.9τ m e −6.6 τ + 3.0 τ or
Astrom and Hagglund [3] modified Ziegler-Nichols – page 208 Model: Method 3 or 4
04 . Tm K mτ m
0.7Tm
2
2
0.79Tm e −1.4 τ + 2.4 τ
2
b = 0.81e0.73 τ + 1.9 τ τ Ms = 1.4 , 014 . ≤ m ≤ 55 . Tm 2
b = 0.44e0.78 τ − 0.45τ ; τ Ms = 2.0, 014 . ≤ m ≤ 55 . Tm 2
b = 0.5; 01 . ≤
τm ≤2 Tm
Table 3: PI tuning rules - FOLPD model - G m ( s) =
K me − sτ – Two degree of freedom controller: 1 + sTm m
1 E (s ) − α Kc R( s) . 1 tuning rule U(s) = K c 1 + Tis Rule Minimum servo/regulator Taguchi and Araki [61a] Model: ideal process
Kc
Ti Performance index minimisation
1 0.7382 0 . 1098 + τm K m − 0.002434 Tm
Ti
( 5b ) 6
τ τ α = 0. 6830 − 0.4242 m + 0.06568 m Tm Tm
6
Ti
( 5b )
Comment
2 3 τ τ τ = Tm 0.06216 + 3.171 m − 3.058 m + 1. 205 m Tm Tm Tm
2
τm ≤ 1.0 . Tm Overshoot (servo step) ≤ 20% ; settling time ≤ settling time of tuning rules of Chien et al. [10]
1 Table 4: PI tuning rules - non-model specific – controller G c ( s) = Kc 1 + . 20 tuning rules Ts i Kc
Ti
Comment
0.45Ku
0.83Tu
Quarter decay ratio
0.45Ku
1 5.22 . − 522 p1 T1
p1 , T1 = decay rate, period measured under proportional control when Kc = 05 . Ku
** Hang et al. [36]
0.25Ku
0.2546Tu
dominant time delay process
McMillan [14] – page 90
0.3571K u
Tu
Pessen [63]
0.25Ku
0.042 K uTu
dominant time delay process
0.4698 K u
0.4373Tu
01988 . Ku
0.0882Tu
0.2015 Ku
01537 . Tu
Parr [64] – page 191
05 . Ku
043 . Tu
Gain margin = 2, phase margin = 20 0 Gain margin = 2.44, phase margin = 61 0 Gain margin = 3.45, phase margin = 46 0 Quarter decay ratio
Yu [122] – page 11
0.33Ku
2Tu
Other tuning rules Parr [64] – page 191
0667 . K25%
T25%
Quarter decay ratio
042 . K25%
T25%
‘Fast’ tuning
0.33K25%
T25%
‘Slow’ tuning
0333 . G p ( jω u )
2Tu
Bang-bang oscillation test
0.5
4
Alfa-Laval Automation ECA400 controller
Rule Ultimate cycle Ziegler and Nichols [8]
Hwang and Chang [62]
Astrom and Hagglund [3] – page 142
McMillan [14] – pages 4243 Parr [64] – page 192
Hagglund and Astrom [66]
G p ( jω 135 )
ω135
0.25
1.6 ω135
0
G p ( jω 135 ) 0
0
0
Alfa-Laval Automation ECA400 controller - process has a long delay
Kc
Rule
(
)
tan φ m − φ p ω − 0 .5 π
Leva [67]
(
)
Ti
(
)
tan φ m − φ pω − 05 . π
G p ( jω ) 1 + tan φ m − φ p ω − 05 .π
ω
sin φ m
tanφ m ω 90
2
Astrom [68]
G p ( jω 90 )
Comment
φ m > φ pω + 0.5π , φ pω = process phase at frequency ω
0
0
1 ωu
φ m = 450 ,‘small’ τ m
016 . Tu tan φ m
V = relay amplitude, A = limit cycle amplitude.
A3 A2
A m ≥ 2 , φ m ≥ 60 0
1 Calcev and Gorez [69]
2 2 G p ( jω u )
Cox et al. [70]
020 . VTu sin φ m A
φ m = 15 0 , ‘large’ τ m
Direct synthesis Vrancic et al. [71]
05 . A3 A1A2 − Km A 3
Vrancic [72]
05 . A3 A1A2 − Km A 3
1
333 . τm 3.7321 ω150
Gain margin = 2, Phase margin = 30 0
1.18K u K m − 1.72 (0.33K u K m − 0.17) ωu
0.1 ≤ Ku Km ≤ 0.5
0. 50K u K m − 0. 36 (0.33K u K m − 0.17) ωu
K u K m > 0.5
0.4830 Friman and Waller [41]
G p ( jω150 )
0
0
1.18 K u K m − 1. 72 Kristiansson and Lennartson [158a]
K uK m
2
0. 50K u K m − 0. 36 Ku Km
2
20K135 Km − 160
20K135 K m − 160
0
0
K135 K m
2
5.4K135 K m − 13. 6 0
K135 K m
2
0
1
(0.315 K135 Km − 0.175 )ω u 5. 4K135 K m − 13. 6
K u K m < 0. 1 ; K 135 K m > 0 .1
0
(1.32K135 K m − 3.2)ω u 0
A1 = y1( ∞) , A 2 = y 2 ( ∞) , A 3 = y3 ( ∞)
Alternatively, if the process model is G m (s) = K m
(
1 + b1 s + b2 s2 + b 3s3 − s τ e , then 1 + a1s + a 2 s3 + a 3s3 m
)
A1 = Km ( a1 − b1 + τ m ) , A2 = K m b 2 − a2 + A1a1 − b1 τm + 05 . τ m2 ,
(
K u K m < 0. 1 ; K 135 0 K m ≤ 0 .1
0
0
Modified Ziegler-Nichols process reaction method
A3 = Km a 3 − b3 + A 2 a1 − A1a 2 + b2 τ m − 05 . b1τ m 2 + 0167 . τ m3
)
0
1 Table 5: PI tuning rules - non-model specific – controller G c ( s) = K c b + . 1 tuning rule Ti s Rule
Kc
Ti
Direct synthesis
(Maximum sensitivity) 2.9 κ − 2.6κ 2
Astrom and Hagglund [3] Ms = 1.4 – page 215
Comment
0.053 Ku e κ = 1 Km Ku
,
b = 11 . e −0.0061κ + 1.8 κ ; 0 < Km K u < ∞ . maximum sensitivity Ms =1.4 2
0.90Tu e− 4.4κ + 2. 7κ
2
b = 048 . e 0.40κ − 0.17 κ ; 0 < Km K u < ∞ . Ms = 2.0 2
013 . K ue
1.9κ − 1.3κ 2
0.90Tu e
− 4.4κ + 2. 7κ 2
1 Table 6: PI tuning rules - non-model specific – controller U( s) = Kc 1 + E( s) + Kc ( b − 1)R (s) . 1 tuning rule Ti s Kc
Rule
Ti
Comment
Direct synthesis
b = [0.5,08 . ] - good servo
A1
Vrancic [72]
2
Kc
Kc
( 6)
( 6)
=
=
A 1A 2 − K m A 3 −
A 1A 2 − K m A 3 +
2
(1 − b )( K
0
2
2
2 K c (6 )
+
K c( 6) K m 2 2
(1 − b ) 2
m
A 3 + A 1 − 2K mA 1A 2 3
)
( K m A3 − A 1A 2 ) 2 − (1 − b2 )A 3 ( Km 2 A3 + A13 − 2K m A1A2 )
(1 − b )( K
y( τ ) y1( t) = Km − dτ , y2 ( t) = ∆u
∫
Km +
1
and regulator response
( K mA 3 − A 1A 2 ) 2 − (1 − b2 )A 3 ( Km 2 A3 + A13 − 2K mA 1A2 )
2
2
t
Kc
( 6)
m
A 3 + A 1 − 2K mA 1A 2 3
t
∫ (A 1 − y1 (τ))dτ , 0
)
t
y3 ( t) =
∫(A 0
2
− y2 ( τ) )dτ
, Km A 3 − A1A 2 < 0
, Km A 3 − A1A 2 > 0
Table 7: PI tuning rules - non-model specific – controller U( s) = Kc Y(s) −
Kc E ( s) . 1 tuning rule Ti s
Kc
Ti
Comment
− TCL + 1414 . TC LTm + τ mTm
− T CL + 1414 . T C L Tm + Tm τ m
Underdamped system response - ξ = 0.707 .
Rule Direct synthesis 2
Chien et al. [74]
(
K m TC L2
+ 1414 . TCL τ m + τ m
2
2
)
Tm + τ m
τm > 0.2Tm
Table 8: PI tuning rules - IPD model G m ( s) =
K me− sτ m s
- controller G c ( s) = Kc 1 + 1 . 21 tuning rules
Rule
Kc
Ti
Process reaction Ziegler and Nichols [8] Model: Method 1.
0.9 Km τ m
333 . τm
0.6 Km τ m
2.78τ m
Two constraints method – Wolfe [12] Model: Method 1
087 . Km τ m
4.35τm
0.487 Km τ m
8.75τ m
Tyreus and Luyben [75] Model: Method 2 or 3 Astrom and Hagglund [3] – page 138 Model: Not relevant Regulator tuning Minimum ISE – Hazebroek and Van der Waerden [9] Model: Method 1 Shinskey [59] – minimum IAE regulator – page 74. Model: Method not specified Poulin and Pomerleau [82] – minimum ITAE (process output step load disturbance) Model: Method 2 Poulin and Pomerleau [82] – minimum ITAE (process input step load disturbance) Model: Method 2 Ultimate cycle Tyreus and Luyben [75]
0.63 Km τ m
Comment Quarter decay ratio Decay ratio = 0.4; minimum error integral (regulator mode). Decay ratio is as small as possible; minimum error integral (regulator mode). Maximum closed loop log modulus = 2dB ; closed loop time constant = τ m 10 Ultimate cycle ZieglerNichols equivalent
3.2τm Performance index minimisation
15 . K mτ m
556 . τm
09259 . Km τ m
4τ m
05264 . K mτ m
4.5804 τ m
05327 . K mτ m
38853 . τm
0.31Ku
2.2Tu
061 . Ku
Tu
05 . Km τ m
5τ m
Model: Method 2 or 3. Regulator – minimum IAE – Shinskey [17] – page 121. Model: method not specified Robust Fruehauf et al. [52] Model: Method 5
Ts i
Maximum closed loop log modulus = 2dB ; closed loop time constant = τ m 10
Rule
Kc
Chien [50]
1 2λ + τ m Km [λ + τ m ]2
Model: Method 2
λ 15 . τm
Ogawa [54] – deduced from graph Model: Method 5
Overshoot 58%
TS 6τ m
Ti
Comment
2λ + τm
1 λ= , τm [50]; Km λ > τ m + Tm (Thomasson [51])
PP 17 . Km τm
RT 7Km τm
2.5τ m
35%
11τ m
2.0K m τm
16K m τm
35 . τm
26%
16τ m
2.2K m τm
23Km τm
45 . τm
22%
20τ m
25 . Km τm
30K m τm
Zhang et al. [135] λ = [15 . τ m ,4.5τ m ] - values deduced from graphs
11τ m
20% uncertainty in the process parameters
Kmτm
12τ m
30% uncertainty in the process parameters
0.34 K mτ m
13τ m
40% uncertainty in the process parameters
0.30 Km τ m
14τ m
50% uncertainty in the process parameters
0.27 K mτ m
15τ m
60% uncertainty in the process parameters
0.45 K mτ m 0.39
Rule
Kc
Ti
α K mτ m
βτ m
Comment
Direct synthesis
Wang and Cluett [76] – deduced from graph Model: Method 2
Cluett and Wang [44] Model: Method 2
Rotach [77] Model: Method 4 Poulin and Pomerleau [78] Model: Method 2
Closed Damp. Gain Phase loop Factor margin margin time ξ Am φm const. [deg.]
τm
0.707
1.3
11
Closed Damp. Gain Phase loop Factor margin margin α β time ξ Am φm const. [deg.] 0.9056 2.6096 τ m 1.0 1.3 14 0.8859 3.212
2τm
0.707
2.5
33
0.5501 4.0116
2τm
1.0
2.3
37
0.6109 5.2005
3τ m
0.707
3.6
42
0.3950 5.4136
3τ m
1.0
3.0
46
0.4662 7.1890
α
β
4τm
0.707
4.7
47
0.3081 6.8156
4τm
1.0
4.0
52
0.3770 9.1775
5τ m
0.707
5.9
50
0.2526 8.2176
5τ m
1.0
4.8
56
0.3164 11.166
6τ m
0.707
7.1
52
0.2140 9.6196
6τ m
1.0
5.6
59
0.2726 13.155
7τm
0.707
8.2
54
0.1856 11.022
7τm
1.0
6.3
61
0.2394 15.143
8τ m
0.707
9.2
55
0.1639 12.424
8τ m
1.0
7.2
62
0.2135 17.132
9τm
0.707
10.4
56
0.1467 13.826
9τm
1.0
8.0
64
0.1926 19.120
10τ m 0.707
11.5
57
0.1328 15.228 10τ m
1.0
8.7
65
0.1754 21.109
11τm 0.707
12.7
58
0.1213 16.630 11τm
1.0
9.6
66
0.1611 23.097
12τ m 0.707
13.8
59
0.1117 18.032 12τ m
1.0
10.4
67
0.1489 25.086
13τ m 0.707
14.9
59
0.1034 19.434 13τ m
1.0
11.2
67
0.1384 27.074
14τ m 0.707
16.0
60
0.0963 20.836 14τ m
1.0
12.0
68
0.1293 29.063
15τ m 0.707
17.0
60
0.0901 22.238 15τ m
1.0
12.7
68
0.1213 31.051
16τ m 0.707
18.2
60
0.0847 23.640 16τ m
1.0
13.6
69
0.1143 33.040
0.9588 Km τ m
30425 . τm
Closed loop time constant = τm
0.6232 Km τ m
52586 . τm
Closed loop time constant = 2τ m
0.4668 Km τ m
7.2291τ m
Closed loop time constant = 3τ m
0.3752 Km τ m
91925 . τm
Closed loop time constant = 4τ m
0.3144 Km τ m
111637 . τm
Closed loop time constant = 5τ m
0.2709 Km τ m
131416 . τm
Closed loop time constant = 6τ m
0.75 K mτ m
241 . τm
034 . K u or
2.13 Km Tu
104 . Tu
Damping factor for oscillations to a disturbance input = 0.75. Maximum sensitivity = 5 dB
Rule Gain and phase margin – Kookos et al. [38] Model: Method 2 Representative results
Kc
Ti
ωp
1 ω p 05 . π − ωpτ m
A m Km
(
Comment
)
ωp =
A m φ m + 0.5πA m (A m − 1)
(A
2 m
)
− 1 τm
0.942 K mτ m
4510 . τm
Gain Margin = 1.5 Phase Margin = 22 .5 0
0.698 K mτ m
4.098τ m
Gain Margin = 2 Phase Margin = 30 0
0.491 K mτ m
6942 . τm
Gain Margin = 3 Phase Margin = 45 0
0384 . K mτ m
18.710τm
Gain Margin = 4 Phase Margin = 60 0
Other methods Penner [79] Model: Method 2
Srividya and Chidambaram [80] Model: Method 5
0.58 K mτ m
10τ m
Maximum closed loop gain = 1.26
0.8 K mτ m
59 . τm
Maximum closed loop gain = 2.0
0.67075 K mτ m
36547 . τm
Table 9: PI tuning rules - IPD model G m ( s) =
Rule
K me− sτ m s
- controller G c ( s) = Kc 1 + 1
Kc
Ti
0463 . λ + 0.277 K mτ m
τm 0.238λ + 0.123
1 . 1 tuning rule Ts 1 + Tf s i Comment
Robust Tan et al. [81] Model: Method 2
Tf =
τm , 5.750λ + 0.590 λ = 0.5
Table 10: PI tuning rules - IPD model G m ( s) =
Kc
Rule Direct synthesis Chien et al. [74] Model: Method 1
K me− sτ m s
(
1.414TCL + τ m
K m TCL + 1.414TCL τ m + τ m 2
- controller U( s) = Kc Y(s) −
Kc E ( s) . 1 tuning rule Ti s
Ti
Comment
1414 . TCL + τ m
Underdamped system response - ξ = 0.707 . τ m ≤ 0.2Tm
Table 11: PI tuning rules - IPD model G m ( s) =
K me− sτ m s
- Two degree of freedom controller: 1 E (s ) − α Kc R( s) . 1 tuning rule U(s) = K c 1 + Tis
Rule
Kc
Servo/regulator tuning Taguchi and Araki [61a] Model: ideal process
Ti Performance index minimisation
0.049 Ku
2. 826 Tu
τm ≤ 1.0 . Tm Overshoot (servo step) ≤ 20% ; settling time ≤ settling time of tuning rules of Chien et al. [10] α = 0.506 , φ c = −1640
0.066 Ku
2. 402 Tu
α = 0.512 , φ c = −1600
0.099 Ku
1.962 Tu
α = 0.522 , φ c = −1550
0.129 Ku
1.716 Tu
α = 0.532 , φ c = −1500
0.159 Ku
1.506 Tu
α = 0.544 , φ c = −1450
0.189 Ku
1.392 Tu
α = 0.555 , φ c = −1400
0.218 K u
1.279 Tu
α = 0.564 , φ c = −1350
0.250 Ku
1.216 Tu
α = 0.573 , φ c = −1300
0.286 Ku
1.127 Tu
α = 0.578 , φ c = −1250
0.330 Ku
1.114 Tu
α = 0.579 , φ c = −1200
0.351K u
1.093Tu
α = 0.577 , φ c = −1180
0. 7662 K m τm
4.091 τm
α = 0. 6810
Minimum ITAE Pecharroman and Pagola [134b]
Km =1 Model: Method 6
Comment
Table 12: PI tuning rules – FOLIPD model G m (s) =
Km e− sτ - controller G c ( s) = Kc 1 + 1 . 6 tuning rules s(1 + sTm ) Ts i m
Kc
Rule
Ti
Comment
Ultimate cycle 2
T 0.65 1477 . Tm 1 332 . τ m 1 + m Tuning rules developed τ m Model: Method not relevant K m τ 2 0.65 from Ku , Tu m Tm 1 + τm Regulator tuning Minimum performance index Minimum IAE – Shinskey [59] – page 75. 0.556 3.7( τ m + Tm ) Model: Method not K m ( τ m + Tm ) specified Minimum IAE – Shinskey [59] – page 158 0.952 4(Tm + τ m ) Model: Open loop method Km (Tm + τ m ) not specified 2 Minimum ITAE – Poulin and b Tm Pomerleau [82] – deduced K m ( τ m + Tm ) a τ + T 2 + 1 a( τ m + Tm ) ( m m) from graph McMillan [58]
Model: Method 2
τm Tm
a
b
τm Tm
a
b
τm Tm
a
b
Output step load disturbance
0.2 0.4 0.6 0.8
5.0728 4.9688 4.8983 4.8218
0.5231 0.5237 0.5241 0.5245
1.0 1.2 1.4 1.6
4.7839 4.7565 4.7293 4.7107
0.5249 0.5250 0.5252 0.5254
1.8 2.0
4.6837 4.6669
0.5256 0.5257
0.2 0.4 0.6 0.8
3.9465 3.9981 4.0397 4.0397
0.5320 0.5315 0.5311 0.5311
1.0 1.2 1.4 1.6
4.0397 4.0337 4.0278 4.0278
0.5311 0.5312 0.5312 0.5312
1.8 2.0
4.0218 4.0099
0.5313 0.5314
(2 tuning rules) Input step load disturbance Direct synthesis Poulin and Pomerleau [78] Model: Method 2
034 . K u or
2.13 Km Tu
104 . Tu
Maximum sensitivity = 5 dB
Table 13: PI tuning rules – FOLIPD model G m (s) =
Rule
Kc
Km e− sτ - controller G c ( s) = Kc b + 1 . 1 tuning rule s(1 + sTm ) Ti s m
Ti
Comment
Direct synthesis
041 . e− 0.23τ + 0.019τ , Km ( Tm + τ m )
b = 0.33e2.5τ −1.9 τ . τ Ms = 1.4; 014 . ≤ m ≤ 55 . Tm
2
Astrom and Hagglund [3] maximum sensitivity – pages 210-212 Model: Method 1.
2
57 . τ m e1.7 τ − 0.69τ
2
τ = τ m ( τ m + Tm ) 081 . e −1.1τ + 0.76 τ Km (Tm + τ m )
2
b = 078 . e−1.9 τ + 1. 2τ . τ Ms = 2.0; 014 . ≤ m ≤ 55 . Tm 2
3.4τ me 0.28τ − 0.0089 τ
2
Table 14: PI tuning rules - FOLIPD model G m (s) =
Km e− sτ - Two degree of freedom controller: s(1 + sTm ) m
1 E (s ) − α Kc R( s) . 1 tuning rule. U(s) = K c 1 + Tis Rule
Kc
Ti
Servo/regulator tuning Taguchi and Araki [61a] Model: ideal process
Minimum performance index
0.049 Ku
2. 826 Tu
τm ≤ 1.0 . Tm Overshoot (servo step) ≤ 20% ; settling time ≤ settling time of tuning rules of Chien et al. [10] α = 0.506 , φ c = −1640
0.066 Ku
2. 402 Tu
α = 0.512 , φ c = −1600
0.099 Ku
1.962 Tu
α = 0.522 , φ c = −1550
0.129 Ku
1.716 Tu
α = 0.532 , φ c = −1500
0.159 Ku
1.506 Tu
α = 0.544 , φ c = −1450
0.189 Ku
1.392 Tu
α = 0.555 , φ c = −1400
0.218 K u
1.279 Tu
α = 0.564 , φ c = −1350
0.250 Ku
1.216 Tu
α = 0.573 , φ c = −1300
0.286 Ku
1.127 Tu
α = 0.578 , φ c = −1250
0.330 Ku
1.114 Tu
α = 0.579 , φ c = −1200
0.351K u
1.093Tu
α = 0.577 , φ c = −1180
1 0 .2839 0 .1787 + τm Km + 0. 001723 Tm
τ τ 4. 296 + 3. 794 m + 0. 2591 m Tm Tm
τ α = 0. 6551+ 0. 01877 m Tm
Minimum ITAE Pecharroman and Pagola [134b]
K m = 1 ; Tm = 1 Model: Method 4
Comment
2
Table 15: PI tuning rules - SOSPD model
K m e− sτ
Km e− sτ (1 + Tm1s)(1 + Tm2 s) m
m
Tm1 s2 + 2ξ m Tm1s + 1 2
or
1 - controller G c ( s) = Kc 1 + . 11 tuning rules. Ts i Kc
Ti
Comment
Tm1 + Tm 2 + 0.5τ m K m τ m ( 2λ + 1)
Tm1 + Tm 2 + 05 . τm
λ varies graphically with τm (Tm1 + Tm 2 ) -
Rule Robust
Brambilla et al. [48]
τ m (Tm1 + Tm 2 )
Model: Method 1
0.1 0.2 0.5
Direct synthesis Gain and phase margin - Tan et al. [39] – repeated pole Model: Method 11
Regulator tuning Minimum IAE - Shinskey [59] – page 158 Model: Open loop method not specified
Model: Method 7
Minimum IAE – Shinskey [17] – page 48. Model: method not specified.
τ m (Tm1 + Tm 2 )
λ 3.0 1.8 1.0
(
β Ti ω φ 1 + βTm ω φ
(
A m 1 + βTi ω φ
)
)
2
τ m (Tm1 + Tm 2 )
λ 0.6 0.4 0.2
1.0 2.0 5.0 1
[
]
β = 0.8,
ωφ < ω u
2
λ 0.2
10.0
βω φ tan − 2 tan −1 βTm ω φ − βτ m ω φ − φ m
τm < 0.33 ; Tm
β = 0.5,
τm > 0.33 Tm
Minimum performance index 100Tm1 K m (τ m
Minimum ISE – McAvoy and Johnson [83] – deduced ξm from graph 1 1 Model: Method 1 1 Minimum ITAE – Lopez et al. [84] – deduced from graph
01 . ≤ τ m ( Tm1 + Tm 2 ) ≤ 10
T m1 − + Tm2 ) 50 + 551 − e τ m + T m2
3 Tm1 − τ +T τ m 0.5 + 35 . 1 − e ( m m2 )
α Km
βτ m
τ m Tm 1
α
β
ξm
τ m Tm 1
α
β
ξm
τ m Tm 1
α
β
0.5 4.0 10.0
0.8 5.7 13.6
1.82 12.5 25.0
4 4
0.5 4.0
4.3 27.1
3.45 6.67
7 7
0.5 4.0
7.8 51.2
3.85 5.88
α
β
α Km ξm 0.5 0.5 0.5
βTm α
τ m Tm 1
0.1 3.0 1.0 0.2 10.0 0.3 0.77 Tm1 K m τm
070 . Tm1 Km τ m 080 . Tm1 Km τ m 080 . Tm1 Km τ m
β
ξm
τ m Tm 1
α
β
ξm
τ m Tm 1
2.86 0.83 4.0
1 1 1
0.1 1.0 10.0
7.0 0.95 0.35
2.00 2.22 5.00
4 4 4
0.1 1.0 4.0
283 . ( τ m + Tm 2 )
2.65( τ m + Tm 2 ) 2.29( τ m + Tm 2 ) 167 . ( τ m + Tm2 )
τm Tm1
40.0 0.83 6.0 3.33 0.75 10.0 T = 02 . , m2 = 01 . Tm 1
τm T = 02 . , m2 = 0.2 Tm1 Tm1 τm T = 02 . , m2 = 05 . Tm1 Tm 1 τm T = 02 . , m2 = 10 . Tm1 Tm 1
Rule
Kc
Minimum IAE - Huang et al. [18]
K c ( 31)
Ti 1
Comment
0<
Ti ( 31)
Tm 2 τ ≤ 1 ; 01 . ≤ m ≤1 Tm1 Tm1
Model: Method 1
1
Kc (31) =
− 0.9077 −0.063 τm τm 1 τ Tm2 τ mTm2 6.4884 + 4.6198 m − 3491 . − 253143 . + 0 . 8196 − 52132 . 2 Km Tm 1 Tm1 Tm1 Tm1 Tm1
0.5961 0.7204 1.0049 1.005 τm Tm2 Tm2 Tm 2 1 T − 7.2712 + − 180448 . + 5.3263 + 139108 . + 0.4937 m2 Km τm Tm1 Tm1 Tm1 Tm1
1 + Km
0.8529 0.5613 0.557 1.1818 Tm 2 τ m Tm 2 τ m τm Tm2 τ m Tm 2 191783 . + 12.2494 + 8.4355 − 17.6781 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1
1 + Km Ti
( 31)
τ − 0.7241eT
τ m Tm 2
Tm 2
m
m1
− 2.2525e
Tm 1
+ 54959 . e
Tm1 2
2 2 3 τm Tm2 τm τm Tm2 τ m Tm2 = Tm1 0.0064 + 3.9574 + 4.4087 − 6.4789 . − 15083 . − 128702 + 9.4348 2 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1
T + Tm 1 17.0736 m2 Tm1
2
τm τ + 15.9816 m T T m1 m1
2
2
Tm2 T τ . m 2 − 10.7619 m − 3909 T T m1 m1 Tm1
3 2 2 τ T T τ τ + Tm 1 − 10.684 m2 m − 22.3194 m m 2 − 6.6602 m Tm1 Tm1 Tm1 Tm1 Tm1
3
4
Tm2 T + 6.8122 m 2 Tm 1 Tm1
4
5 4 2 3 2 3 τm Tm2 τ m τ m Tm 2 Tm2 τ m + Tm1 75146 . . . . + 28724 + 114666 + 111207 Tm1 Tm1 Tm1 Tm1 Tm 1 Tm1 Tm1 4 5 6 5 6 τm Tm 2 Tm 2 τm Tm2 Tm2 τm + Tm 1 − 12174 . . . − 4.3675 − 2.2236 − 0112 + 10308 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 4 2 3 3 2 4 τ T τm Tm2 τ m Tm2 τm + Tm1 − 1. 9136 m m2 − 34994 . − 15777 . . + 11408 T T T T T T T m1 m1 m1 m1 m1 m1 m1
Tm 2 Tm1
5
Rule
Kc
Minimum IAE - Huang et al. [18]
K c ( 32)
Ti
Comment
0.4 ≤ ξ m ≤ 1 ;
2
Ti ( 32)
0.05 ≤
Model: Method 1
2
Kc
( 32)
1 = Km
1.4439 0.1456 τm τm τm τm − 10.4183 − 209497 . − 55175 . ξ m − 265149 . ξm + 42.7745 + 105069 . Tm1 Tm1 Tm1 Tm1
+
0.3157 −0.0541 τ τ 1 15.4103 m + 34.3236ξ m 3.7057 − 17.8860ξ m 4 .5359 − 54.0584 ξ m 1.9593 + 22.4263ξ m m Km Tm1 Tm1
+
4. 7426 τ τ τm T 1 2.7497ξ m m + 50.2197ξ m 1.8288 m − 171968 . ξ m 2.7227 + 10293 . ξ m m1 Km Tm1 Tm1 Tm1 τm
+
τ ξ 1 − 167667 . e T + 14.5737eξ − 7.3025 e Km m
Ti
( 32)
τm ≤1 Tm1
m1
m
m
τm Tm1
2 3 τm τm τm τm 2 = Tm1 11447 . + 45128 . − 75.2486ξ m − 110807 . + 345.3228ξ m + 191.9539 − 12.282ξ m Tm1 Tm1 Tm1 Tm1 2 4 τ τ τ + Tm1 359 .3345ξ m m − 158.7611 m ξ m 2 − 770.2897ξ m2 − 153633 . m Tm1 Tm1 Tm1 3 2 τ τ τm + Tm1 − 412 .5409 ξ m m − 414 .7786ξ m 2 m + 4850976 . ξ m 3 + 864.5195ξ m 4 Tm1 Tm1 Tm1 5 4 3 2 τ τ τ τm 3 + Tm 1 55.4366 m + 222 .2865 ξ m m + 275166 . ξ m 2 m + 2052493 . ξm Tm1 Tm1 Tm1 Tm1 6 5 τm 4 τm τm 5 6 + Tm 1 − 479 .5627 . ξ m − 6.547 . ξ m ξ m − 4731346 − 432822 + 99.8717ξ m Tm1 Tm1 Tm1 4 3 2 τm τm τm τ 2 3 4 5 + Tm1 − 735666 . ξ − 56 . 4418 ξ − 37 . 497 m m ξ m + 160.7714 m ξ m Tm1 Tm1 Tm1 Tm1
Rule Servo tuning Minimum IAE - Huang et al. [18] Model: Method 1
3
Kc (33) =
Kc
Ti
Comment
Minimum performance index K c ( 33)
3
0<
Ti ( 33)
Tm 2 τ ≤ 1 ; 01 . ≤ m ≤1 Tm1 Tm1
− 1.0169 3.5959 τ τm 1 τ T τ T − 130454 . − 9.0916 m + 2.6647 m2 + 9.162 m m2 2 + 0.3053 m + 11075 . Km Tm1 Tm 1 Tm1 Tm1 T m 1
+
3.6843 0.8476 2.6083 2.9049 τ Tm2 T T 1 T − 2.2927 m − 310306 . − 13.0155 m2 + 9.6899 m 2 − 0.6418 m 2 Km τm Tm1 Tm1 Tm1 Tm1
−0.2016 1.3293 0.801 1 Tm2 τ m Tm 2 τ m τ m Tm 2 189643 + . − 39.7340 + 28155 . Km Tm1 Tm1 Tm1 Tm1 Tm 1 Tm1
Ti
( 33)
τ T 3.956 τ T 1 τ m Tm2 T T − 2.0067 + + 4.8259e + 2.1137e + 84511 . eT Km Tm1 Tm1 2 2 3 τm Tm 2 τm τm Tm2 τ m Tm2 = Tm1 0.9771− 0.2492 + 0.8753 + 3.4651 − 38516 . + 7.5106 − 7 .4538 2 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 2 2 3 4 Tm2 τ m τ m Tm 2 Tm 2 τm + Tm1 116768 . . . − 10.9909 − 161461 + 82567 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 m
m2
m1
m1
m
m2 2
m1
3 2 2 3 4 τ T Tm 2 Tm2 τ m τ m Tm 2 + Tm 1 − 181011 . . . + 6.2208 m m2 + 219893 + 158538 Tm1 Tm1 Tm 1 Tm1 Tm1 Tm1 Tm1 5 4 2 3 2 3 τ T τ T τ τ T + Tm1 − 4.7536 m + 14.5405 m 2 m − 2.2691 m2 m − 8.387 m m 2 Tm 1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 4 5 6 5 6 τ m Tm2 Tm 2 τm Tm 2 Tm2 τ m + Tm 1 − 16.651 . . . − 71990 + 11496 − 4.728 + 11395 Tm 1 Tm1 Tm1 Tm 1 Tm1 Tm1 Tm1 4 2 3 3 2 4 τ T τ m Tm 2 τ m Tm2 τ + Tm 1 0.6385 m m 2 + 10885 . + 31615 . + 4.5398 m T T T T T T T m1 m1 m1 m1 m1 m1 m1
Tm 2 Tm1
5
Rule
Kc
Minimum IAE - Huang et al. [18]
K c ( 34 )
Ti
Comment
0.4 ≤ ξ m ≤ 1 ;
4
Ti ( 34)
0.05 ≤
Model: Method 1
4
Kc
Ti
( 34)
( 34)
1 = Km
τm ≤1 Tm1
3 2 τm τm τm τm − 10.95 − 18845 . − 3.4123ξ m + 4.5954ξ m − 17002 . . ξ m − 21324 Tm1 Tm1 Tm1 Tm 1
+
0.421 0.1984 1.8033 τ τ τ τm 1 − 14.4149ξ m 2 m − 0.7683ξ m 3 + 7.5142 m + 3.7291 m + 53444 . Km Tm1 Tm1 Tm1 Tm1
+
−0.6753 −0.1642 τ τ 1 − 0.0819 ξ m 19 .5419 − 3603 . ξ m 1.0749 + 71163 . ξ m 1.1006 + 3206 . ξm m − 7.8480ξ m m Km Tm1 Tm1
+
τ τ 1 113222 . ξ m1.9948 m + 2.4239e T Km Tm1
m
m1
+ 34137 . eξ + 10251 . e m
ξ m τm Tm1
− 05593 . ξm
Tm1 τm
2 3 τm τm τm τm 2 = Tm1 2.4866 − 233234 . + 53662 . ξ m + 656053 . − 24.1648ξ m − 83.6796 + 29.0062ξ m Tm1 Tm1 Tm1 Tm1 2 4 τ τm τ + Tm1 − 1359699 . ξ m m + 431477 . ξ m 2 + 519749 . ξ m 3 + 86.0228 m Tm1 Tm 1 Tm1 3 2 τ τ τm + Tm 1 704553 . ξ m m + 1534877 . ξ m 2 m − 1250112 . ξ m 3 − 685893 . ξ m4 Tm1 Tm1 Tm1 5 4 3 2 τ τ τ τ + Tm 1 − 62.7517 m + 27.6178 ξ m m − 152 .7422 ξ m 2 m + 20.8705 m ξ m 3 Tm1 Tm1 Tm1 Tm 1 6 5 τm 4 τm τm 5 + Tm1 54.0012 . . ξm . ξm 6 ξ m + 58.7376ξ m + 131193 + 202645 − 232064 Tm1 Tm1 Tm1 4 3 2 τm τm τm τm 2 3 4 + Tm 1 − 616742 . ξ + 136 . 2439 ξ − 954092 . . ξm 5 m m ξ m + 204168 Tm1 Tm1 Tm1 Tm1
Kc
Rule
Ti
Comment
Ultimate cycle Decay ratio = 0.15 - ε < 2.4 , Regulator - nearly minimum IAE, ISE, ITAE – Hwang [60]
Kc
( 7)
Ti
( 7)
0.2 ≤
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.15 -
Kc( 8)
Model: Method 3
Ti (8)
2.4 ≤ ε < 3 , 0.2 ≤
τm Tm1
≤ 2.0 ,
0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.15 -
Kc ( 9 ) 5
Ti ( 9)
3 ≤ ε < 20 , 0.2 ≤
τm Tm1
≤ 2.0 ,
0.6 ≤ ξ m ≤ 4.2
6Tm1 + 4ξ m Tm1 τm + K H Km τ m 9 , KH = 2 2 2 Tm1 τ mω H 2τ m K m 2
5
ε=
ωH = Tm12
Kc
( 7)
Kc
(8 )
Kc
( 9)
2
1 + KH K m 2T τ ξ K K τ 2 + m1 m m + H m m 3 6
[
2 0.674 1 − 0.447ω H τ m + 0.0607 (ω H τ m ) = 1 − K H K m (1 + K H K m )
[
2 0.778 1 − 0.467ω H τ m + 0.0609(ω H τ m ) = 1 − K H K m (1 + K H K m )
[
]
2 τm 2 ξ m Tm1τ m 2 − T − , m1 18 18
] K
H
, Ti ( 7) =
H
, Ti (8 ) =
] K
(
K c (1 + K H K m )
0.0607ω H K m 1 + 1.05ω H τ m − 0.233ω H τ m
(
2
2
K c (1 + K H K m )
0.0309ω H K m 1 + 2.84ω H τ m − 0.532 ω H τ m 2
2
ω τ 131 . ( 0519 . ) 1 − 103 . ε + 0514 . ε 2 K c (1 + KH K m ) ( 9) = 1 − K H , Ti = KH Km 1 + KH K m 0.0603 1 + 0.929 ln[ ω H τ m ] 1 + 2.01 ε − 12 . ε2 H
m
(
)
(
)(
) ) )
Kc
Rule
Kc
Ti
( 10 ) 6
Ti
Comment Decay ratio = 0.15 - ε > 20 ,
( 10)
0.2 ≤
Regulator – nearly minimum IAE, ISE, ITAE - Hwang [60]
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.2 - ε < 2.4 ,
Kc
Model: Method 3
( 11)
Ti
( 11)
Ti
( 12)
0.2 ≤
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.2 -
Kc
( 12 )
2.4 ≤ ε < 3 , 0.2 ≤
τm
≤ 2.0 ,
Tm1
0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.2 -
Kc(13 )
Ti (13)
3 ≤ ε < 20 , 0.2 ≤
τm
≤ 2.0 ,
Tm1
0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.2 - ε > 20 ,
Kc(14 )
Ti (14)
Kc(15)
Ti (15)
(16 )
( 16)
0.2 ≤
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.25 - ε < 2.4 , 0.2 ≤
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.25 -
Kc
Ti
2.4 ≤ ε < 3 , 0.2 ≤
τm
≤ 2.0 ,
Tm1
0.6 ≤ ξ m ≤ 4.2
6
Kc
( 10)
Kc
( 11)
Kc
( 12 )
[
2 114 . 1 − 0.482ω H τ m + 0.068( ω H τ m ) = 1 − K H K m (1 + K H K m )
] K
[
2 0.622 1 − 0.435ω Hτ m + 0.052(ω H τm ) = 1 − K H K m (1 + K H K m )
H
H
] K
[
2 0.724 1 − 0.469ω H τ m + 0.0609 (ω H τ m ) = 1 − K H K m (1 + K H K m )
[
, Ti (10) =
, Ti (11) =
] K
]
H
(
K c (1 + K H K m ) 2
)
0.0697 ω H K m 1 + 0.752ω H τ m − 0.145ω H τ m
2
)
2
)
0.0694ω H K m − 1 + 2.1ω H τ m − 0.367ω H τm
(
, Ti (12 ) =
2
K c (1 + K H K m ) 2
(
K c (1 + K H K m )
0.0405ω H K m 1 + 1.93ω H τ m − 0.363ω H τ m 2
ω τ 126 . (0.506) 1 − 107 . ε + 0.616 ε 2 K c (1 + K H K m ) Kc (13) = 1 − K H , Ti (13) = K H Km 1 + KH K m 0.0661(1 + 0.824 ln[ω H τ m ])(1 + 171 . ε − 117 . ε2 ) H
Kc
( 14 )
Kc
( 15)
Kc
( 16)
[
m
(
)
2 109 . 1 − 0.497ω H τ m + 0.0724(ω Hτ m ) = 1 − K H K m (1 + K H K m )
[
] K
2 0.584 1 − 0.439ω H τ m + 0.0514 ( ω H τ m ) = 1 − K H K m (1 + K H K m )
[
2 0.675 1 − 0.472ω H τ m + 0.061( ω H τ m ) = 1 − K H K m (1 + K H K m )
H
] K
] K
H
, Ti (14 ) =
H
(
0.054 ω H K m − 1 + 2.54ω H τ m − 0.457ω H τ m
, Ti (15) =
, Ti (16) =
K c (1 + K H K m ) 2
(
K c (1 + K H K m )
0.0714ω H K m 1 + 0.685ω H τ m − 0.131ω H 2 τ m 2
(
)
K c (1 + K H K m )
0.0484ω H K m 1 + 1.43ω H τ m − 0.273ω H τ m 2
2
)
2
)
Rule
Kc
Ti
Regulator - nearly minimum IAE, ISE, ITAE - Hwang [60]
Kc(17 ) 7
Ti (17)
Comment Decay ratio = 0.25 τm
3 ≤ ε < 20 , 0.2 ≤
≤ 2.0 ,
Tm1
0.6 ≤ ξ m ≤ 4.2
Model: Method 3
Decay ratio = 0.25 - ε > 20 ,
Kc (18 )
Ti (18)
0.2 ≤
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.1 - ε < 2.4 , Servo - nearly minimum IAE, ISE, ITAE - Hwang [60]
Kc
( 19 )
Ti
0.2 ≤
( 19)
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
ξ m ≤ 0.776 + 0.0568
Model: Method 3
τ τm + 018 . m Tm1 T m1
2
Decay ratio = 0.1 - 2.4 ≤ ε < 3 ,
Kc
( 20)
Ti
0.2 ≤
( 20)
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
ξ m ≤ 0.776 + 0.0568
τ τm + 018 . m Tm1 T m1
2
Decay ratio = 0.1 - 3 ≤ ε < 20 ,
Kc
( 21)
Ti
0.2 ≤
( 21)
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
ξ m ≤ 0.776 + 0.0568
τ τm + 018 . m Tm1 T m1
2
Decay ratio = 0.1 - ε > 20 ,
Kc
( 22 )
Ti
0.2 ≤
( 22 )
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
ξ m ≤ 0.776 + 0.0568
Kc
( 17)
Kc
( 18)
Kc
( 19 )
Kc
( 20)
[
]
ω τ 2 12 . ( 0495 . ) 1 − 11 . ε + 0.698 ε Kc (1 + KH K m ) K , T (17 ) = = 1− i H 0.0702 1 + 0.734 ln[ ω H τ m ] 1 + 148 . ε − 11 . ε2 KH Km (1 + K H Km ) H
7
τ τm + 018 . m Tm1 T m1
m
[
2 1.03 1 − 0.51ω H τ m + 0.0759( ω H τ m ) = 1 − K H K m (1 + K H K m )
[
] K
2 0.822 1 − 0.549ω H τ m + 0.112(ω Hτ m ) = 1 − K H K m (1 + K H K m )
[
H
] K
2 0.786 1 − 0.441ω H τ m + 0.0569(ω H τ m ) = 1 − K H K m (1 + K H K m )
[
)(
(
, Ti (18) =
H
] K
]
0.0386ω H K m − 1 + 3.26ω H τ m − 0.6ω H τm
, Ti (19 ) =
H
(
K c (1 + K H K m ) 2
(
2
, Ti ( 20) =
)
K c (1 + K H K m )
0.0142ω H K m 1 + 6.96ω H τ m − 177 . ωH τm 2
(
2
)
K c (1 + K H K m )
0.0172 ω H K m 1 + 4.62ω H τ m − 0.823ω H τ m 2
ω τ 128 . ( 0542 . ) 1 − 0.986 ε + 0558 . ε 2 K c (1 + K H K m ) ( 21) Kc ( 21) = 1 − KH , Ti = 0.0476 1 + 0.996 ln ω τ 1 + 213 K K 1 + K K . ε2 ) ( [ H m ])( . ε − 113 H m H m H
Kc
( 22)
[
m
(
)
2 114 . 1 − 0.466 ω H τ m + 0.0647 ( ω H τ m ) = 1 − K H K m (1 + K H K m )
] K
H
, Ti (22) =
)
(
K c (1 + K H K m )
0.0609ω H K m − 1 + 1.97ω H τ m − 0.323ω H 2 τ m2
)
2
)
2
Kc
Rule
Ti
Comment Decay ratio = 0.1 - ε < 2.4 ,
Servo - nearly minimum IAE, ISE, ITAE - Hwang [60]
8
K c ( 23)
0.2 ≤
Ti ( 23)
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
ξ > 0889 . + 0.496
Model: Method 3
τ τm + 0.26 m Tm1 Tm1
2
Decay ratio = 0.1 - 2.4 ≤ ε < 3 , K c ( 24 )
0.2 ≤
Ti ( 24)
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
ξ > 0889 . + 0.496
τ τm + 0.26 m Tm1 Tm1
2
Decay ratio = 0.1 - 3 ≤ ε < 20 , Kc
( 25)
Ti
0.2 ≤
( 25)
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
ξ > 0889 . + 0.496
τ τm + 0.26 m Tm1 Tm1
2
Decay ratio = 0.1 - ε > 20 , K c (26)
0.2 ≤
Ti (26)
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
ξ > 0889 . + 0.496
τ τm + 0.26 m Tm1 Tm1
2
Decay ratio = 0.1 - ε < 2.4 , K c ( 27 )
0.2 ≤
Ti ( 27)
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2 τ τm + 018 . m Tm1 T m1
ξ m > 0.776 + 0.0568 ξ m ≤ 0.889 + 0.496
8
Kc
( 23)
Kc
( 24)
Kc
( 25)
[
2 0.794 1 − 0541 . ω H τ m + 0.126( ω H τ m ) = 1 − K H K m (1 + KH K m )
[
0.738 1 − 0.415ω τ + 0.0575 ω τ 2 ( H m) H m = 1 − KH K m (1 + KH K m )
[
[
Kc
H
, Ti (23) =
] K
H
, Ti (24) =
]
K c (1 + K H K m )
(
0.0078ω H K m 1 + 8.38ω H τ m − 197 . ω H2 τ m 2
)
K c (1 + K H K m )
(
0.0124ω H K m 1 + 4.05ω H τ m − 0.63ω H 2 τ m 2
)
ω τ 2 115 . (0.564) H m 1 − 0.959 ε + 0.773 ε K c (1 + K H K m ) K , T (25) = = 1 − H i K K ( 1 + K K ) 0 . 0355 1 + 0 . 947 ln[ω H τ m ] 1 + 19 . ε − 107 . ε2 H m H m
107 . 1 − 0.466ω H τ m + 0.0667( ω H τ m ) Kc ( 26) = 1 − K H Km (1 + K H Km ) ( 27)
] K
[
0.789 1 − 0.527ω τ + 0.11 ω τ 2 ( H m) H m = 1 − K H K m (1 + K H K m )
2
] K
] K
H
)(
(
H
, Ti (26) =
, Ti (27) =
(
τ τm + 0.26 m Tm1 Tm1
)
K c (1 + K H K m )
0.0328ω H K m − 1 + 2.21ω H τ m − 0338 . ω H2 τ m 2 K c (1 + K H K m )
(
0.009ω H K m 1 + 9.7ω H τ m − 2.4ω H 2 τ m 2
)
)
2
2
Kc
Rule
Ti
Comment Decay ratio = 0.1 - 2.4 ≤ ε < 3 ,
Servo - nearly minimum IAE, ISE, ITAE - Hwang [60]
9
K c (28)
0.2 ≤
Ti (28)
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2 τ τm + 018 . m Tm1 T m1
ξ m > 0.776 + 0.0568
Model: Method 3
ξ m ≤ 0.889 + 0.496
τ τm + 0.26 m Tm1 Tm1
2
2
Decay ratio = 0.1 - 3 ≤ ε < 20 , K c ( 29 )
0.2 ≤
Ti ( 29)
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2 τ τm + 018 . m Tm1 T m1
ξ m > 0.776 + 0.0568 ξ m ≤ 0.889 + 0.496
τ τm + 0.26 m Tm1 Tm1
2
2
Decay ratio = 0.1 - ε > 20 , K c ( 30)
0.2 ≤
Ti (30)
τm Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2 τ τm + 018 . m Tm1 T m1
ξ m > 0.776 + 0.0568 ξ m ≤ 0.889 + 0.496
Regulator - minimum IAE Shinskey [17] – page 121. Model: method not specified
9
Kc
( 28)
048 . Ku
[
[
τm T = 02 . , m2 = 0.2 Tm1 Tm1
083 . Tu
2 0.76 1 − 0426 . ω H τ m + 0.0551( ω H τ m ) = 1 − K H Km ( 1 + K H Km )
] K
H
, Ti (28) =
]
(
τ τm + 0.26 m Tm1 Tm1
K c (1 + K H K m )
0.0153ω H K m 1 + 4.37ω H τ m − 0.743ω H 2 τ m 2
)
Kc
( 29)
ω τ 2 122 . (0.55) H m 1 − 0.978 ε + 0.659 ε K c (1 + K H K m ) K , T (29) = = 1− H i K K ( 1 + K K ) 0.0421 1 + 0.969 ln[ω H τ m ] 1 + 2.02 ε − 111 . ε2 H m H m
Kc
( 30)
2 111 . 1 − 0.467ω H τ m + 0.0657( ω H τ m ) = 1 − KH K m ( 1 + K H Km )
[
] K
)(
(
H
, Ti (30) =
(
)
K c (1 + K H K m )
0.0477ω H K m − 1 + 2.07ω H τ m − 0.333ω H2 τ m 2
)
2
2
Table 16: PI tuning rules - SOSPD model -
K m e− sτ
m
Tm1 s2 + 2ξ m Tm1s + 1 2
- Two degree of freedom controller: U(s) = K c 1 + 1 E (s ) − α Kc R( s) . 3 tuning rules.
Kc
Rule
Ti
Servo/regulator tuning Taguchi and Araki [61a] Model: ideal process
Tis
Minimum performance index 1 0.5613 0 . 3717 + τm K m + 0 .0003414 Tm
Ti
1 Km
τ τ τm + 0. 3087 m − 0. 1201 m Tm Tm Tm
0 . 05627 0 . 1000 + τm 2 [ + 0.06041] T m
Ti
2
τm ≤ 1.0 ; ξ m = 1 Tm Overshoot (servo step) ≤ 20% ; settling time ≤ settling time of tuning rules of Chien et al. [10]
( 30a ) 10
2
α = 0. 6438− 0. 5056
τ τ τ α = 0. 6178− 0. 4439 m − 7. 575 m + 9. 317 m Tm T m Tm
Minimum ITAE Pecharroman and Pagola [134a]
Comment
3
( 30b )
3
τ − 3. 182 m T m
4
τm ≤ 1.0 ; ξ m = 0.5 Tm Overshoot (servo step) ≤ 20% ; settling time ≤ settling time of tuning rules of Chien et al. [10] α = 0. 4002 , φ c = − 139.65 0 K m = 1 ; Tm = 1 ; ξ m = 1
0.1713 Ku
1.0059 Tu
0.147 K u
1.150 Tu
α = 0. 411 , φ c = −1460
0.170 K u
1.013Tu
α = 0. 401 , φ c = −1400
0.195 K u
0.880 Tu
α = 0.386 , φ c = −1330
0.210 K u
0.720 Tu
α = 0.342 , φ c = −1250
0.234 K u
0.672 Tu
α = 0.345 , φ c = −1150
0.249 K u
0.610 Tu
α = 0.323 , φ c = −1050
0.262 K u
0.568 Tu
α = 0.308 , φ c = −940
0.274 K u
0.545 Tu
α = 0. 291 , φ c = −840
0.280 K u
0.512 Tu
α = 0. 281 , φ c = −730
0.291K u
0.503 Tu
α = 0.270 , φ c = −630
0.297 K u
0.483 Tu
α = 0.260 , φ c = −520
0.303 K u
0.462 Tu
α = 0.246 , φ c = −410
0.307 K u
0.431Tu
α = 0.229 , φ c = −300
Model: Method 15 Minimum ITAE Pecharroman and Pagola [134b]
K m = 1 ; Tm = 1 ; ξ m = 1 Model: Method 15
2 3 τm τ m τm = Tm 2.069 − 0.3692 + 1.081 − 0.5524 Tm Tm Tm 2 3 4 τ τ τ τ = Tm 4. 340 − 16. 39 m + 30. 04 m − 25 .85 m + 8.567 m Tm Tm Tm Tm
( 30a )
10
Ti
Ti
( 30b )
Rule
Kc
Ti
Comment
Minimum ITAE Pecharroman and Pagola [134b] - continued
0.317 K u
0.386 Tu
α = 0. 171 , φ c = −190
0.324 K u
0.302 Tu
α = 0.004 , φ c = −100
0.320 K u
0.223 Tu
α = −0.204 , φ c = − 60
Table 17: PI tuning rules - SOSIPD model (repeated pole) -
K m e −sτ
m
s (1 + Tm1s) 2
- Two degree of freedom controller: U(s) = K c 1 + 1 E (s ) − α Kc R( s) . 1 tuning rule.
Rule
Kc
Servo/regulator tuning Taguchi and Araki [61a] Model: ideal process
Ti
K m = 1 ; Tm = 1 Model: Method 1
Comment
Minimum performance index
τm Tm
τm ≤ 1.0 ; Tm Overshoot (servo step) ≤ 20% ; settling time ≤ settling time of tuning rules of Chien et al. [10]
0.049 Ku
2. 826 Tu
α = 0.506 , φ c = −1640
0.066 Ku
2. 402 Tu
α = 0.512 , φ c = −1600
0.099 Ku
1.962 Tu
α = 0.522 , φ c = −1550
0.129 Ku
1.716 Tu
α = 0.532 , φ c = −1500
0.159 Ku
1.506 Tu
α = 0.544 , φ c = −1450
0.189 Ku
1.392 Tu
α = 0.555 , φ c = −1400
0.218 K u
1.279 Tu
α = 0.564 , φ c = −1350
0.250 Ku
1.216 Tu
α = 0.573 , φ c = −1300
0.286 Ku
1.127 Tu
α = 0.578 , φ c = −1250
0.330 Ku
1.114 Tu
α = 0.579 , φ c = −1200
0.351K u
1.093Tu
α = 0.577 , φ c = −1180
1 0 .3840 0 . 07368 + τm K m + 0.7640 Tm
τ 8.549 + 4.029 m Tm
α = 0. 6691 + 0.006606
Minimum ITAE Pecharroman and Pagola [134b]
Tis
Km e − sτ - controller G c ( s) = Kc 1 + 1 . 1 Ts (1 + sTm1 )(1 + sTm2 )(1 + sTm 3 ) i tuning rule m
Table 19: PI tuning rules - TOLPD model
Rule Hougen [85]
Kc
0.7 Tm1 Km τm
Ti 15 . τm
0.08
Tm1( Tm 2 + Tm3 )
15 . τm
0.08
Tm1( Tm 2 + Tm3 )
τm ≤ 0.04 ; Tm1 ≥ Tm 2 ≥ Tm3 Tm1
Model: Method 1 0. 333 1 Tm 1 T + Tm2 + Tm 3 0.7 + 0.8 m1 0. 333 2K m τ m ( Tm 1Tm 2 Tm 3 )
Comment
τm > 0.04 ; Tm1 ≥ Tm 2 ≥ Tm3 Tm1
0.333
Table 20: PI tuning rules - TOLPD model (repeated pole) -
K m e −sτ
m
(1 + Tm1s ) 3
- Two degree of freedom controller: U(s) = K c 1 + 1 E (s ) − α Kc R( s) . 1 tuning rule.
Rule
Kc
Servo/regulator tuning Taguchi and Araki [61a] Model: ideal process
Ti
Ti
( 30c )
Comment
Minimum performance index 1 0.7399 0 . 2713 + τm K m + 0.5009 Tm
Ti
( 30 c ) 1
τ τ α = 0. 4908 − 0. 2648 m + 0. 05159 m Tm Tm
1
Tis
2 τ τ = Tm 2.759 − 0. 003899 m + 0.1354 m Tm Tm
2
τm ≤ 1.0 ; Tm Overshoot (servo step) ≤ 20% ; settling time ≤ settling time of tuning rules of Chien et al. [10]
K me − sτ 1 − sTm
m
Table 21: PI tuning rules - unstable FOLPD model
Kc
Rule
- controller G c ( s) = Kc 1 + 1 . 6 tuning rules
Ts i
Ti
Comment
Direct synthesis
1 De Paor and O’Malley [86]
Kc
( 35)
Model: Method 1 Venkatashankar and Chidambaram [87] Model: Method 1 Chidambaram [88] Model: Method 1
1 − Tm τm Tm tan( 0.5φm ) Tm τ m
Kc( 36)
25( Tm − τ m )
1 T 1 + 0.26 m Km τm
25Tm − 27τ m
ω p Tm
Ho and Xu [90]
A m Km
Model: Method 1
τm <1 Tm
gain margin = 2; φ m = tan − 1
1 − Tm τ m
(
− T m τ m 1 − Tm τ m
Tm τ m
τm < 0.67 Tm
τm < 06 . Tm ωp =
1 1.57ω p − ω p τ m − 2
1 Tm
A m φ m + 157 . A m (A m − 1)
(A
2 m
)
−1 τm
,
τm < 0.62 Tm
Robust Rotstein and Lewin [89]
λ Tm λ + 2 Tm 2 λ Km
λ λ + 2 Tm
Km uncertainty = 50%
τm Tm = 0.2
λ = [0.6Tm ,1.9Tm ]
τm Tm = 0.2
λ = [ 0.5Tm , 45 . Tm ]
τm Tm = 0.4
λ = [15 . Tm ,4.5Tm ]
τm Tm = 0.6
λ = [ 39 . Tm ,41 . Tm ]
Model: Method 1
Km uncertainty = 30%
λ obtained graphically – sample values below
Ultimate cycle Luyben [91] Model: Method not relevant
031 . Ku
Maximum closed loop log modulus = 2 dB; closed loop time constant = 316 . τm
2.2Tu
2
2
Kc
Kc
( 35)
( 36)
=
Tm 1 − Tm τ m sin cos (1 − Tm τ m )Tm τ m + K m Tm τ m
1 = Km
2 0.98 1 + 0.04Tm 2 ( Tm − τ m )
(1 − Tm τ m )Tm τ m
β 2 Tm2 (Tm − τ m )
)
2
1+ (Tm − τ m ) 2 τ 2m : τm 25 β( T − τ ) < 025 . β = 1373 . , m m τm 625 2 Tm 1+β2 T − τ ( ) m m 2 τm β = 0.953, 0.25 ≤
τm < 0.67 Tm
K m e− s τ - controller G c ( s) = Kc 1 + 1 . 3 Ts (1 − sTm1 )(1 + sTm 2 ) i tuning rules. m
Table 22: PI tuning rules - unstable SOSPD model
Rule
Kc
Ti
Comment
Ultimate cycle McMillan [58] Model: Method not relevant
Kc( 37) 3
Ti (37 )
Tuning rules developed from Ku , Tu
Regulator tuning
Minimum performance index
Minimum ITAE – Poulin and Pomerleau [82] – deduced from graph Model: Method 1 Output step load disturbance (considered as 2 tuning rules)
Input step load disturbance
bTm1 1 +
(
4Tm1 ( τ m + Tm2 )
aTm2 2
4( τm + Tm 2 )
aTm1 − 4( τ m + Tm2 )
2
K m aTm1 − 4[ τ m + Tm2 ]
)
τm Tm
a
b
τm Tm
a
b
τm Tm
a
b
0.05 0.10 0.15 0.20
0.9479 1.0799 1.2013 1.3485
2.3546 2.4111 2.4646 2.5318
0.25 0.30 0.35 0.40
1.4905 1.6163 1.7650 1.9139
2.5992 2.6612 2.7368 2.8161
0.45 0.50
2.0658 2.2080
2.9004 2.9826
0.05 0.10 0.15 0.20
1.1075 1.2013 1.3132 1.4384
2.4230 2.4646 2.5154 2.5742
0.25 0.30 0.35 0.40
1.5698 1.6943 1.8161 1.9658
2.6381 2.7007 2.7637 2.8445
0.45 0.50
2.1022 2.2379
2.9210 3.0003
2
3
Kc ( 37) =
1477 . Tm1Tm 2 K m τ m2
Ti A max Km
(
)
Tm1 Tm2 ω max + Tm1 + Tm 2 ω max + ω max 2
Kc ( 38) =
0.65 (T + T )T T 1 m1 m2 m1 m 2 ( 37) , T = 332 . τ 1 + i m 0.65 ( Tm1 + Tm 2 )Tm1Tm 2 (Tm1 − Tm 2 )(Tm1 − τ m )τ m 1+ (Tm1 − Tm2 )(Tm1 − τ m )τ m 2
6
2
1 + Ti ω max 2
2
2
4
2
1 Table 23: PI tuning rules – delay model e − sτ m - controller G c ( s) = Kc 1 + . 2 tuning rules Ts i Rule Direct synthesis Hansen [91a] Model: Method not specified Regulator tuning Shinskey [57] – minimum IAE – regulator - page 67. Model: method not specified
Kc
Ti
0.2
0.3τm Minimum performance index
0.4 Km
0.5τ m
Comment
K m e− sτ 1 - ideal controller G c ( s) = Kc 1 + + Td s . 56. 1 + sTm Ti s m
Table 25: PID tuning rules - FOLPD model
Kc
Rule Process reaction Ziegler and Nichols [8] Model: Method 1 Astrom and Hagglund [3] – page 139 Model: Method 6 Parr [64] – page 194 Model: Method 1 Chien et al. [10] – regulator Model: Method 1
Chien et al. [10] servo Model: Method 1
Three constraints method - Murrill [13]page 356
[
Ti
Td
Comment
2τ m
05 . τm
Quarter decay ratio
0.94Tm K mτ m
2τ m
05 . τm
Ultimate cycle ZieglerNichols equivalent
125 . Tm K mτ m
2.5τ m
0. 4τ m
095 . Tm Km τ m
2.38τ m
0.42τ m
0% overshoot; τ 011 . < m <1 Tm
12 . Tm K mτ m
2τ m
0.42τ m
20% overshoot; τ 011 . < m <1 Tm
0.6Tm K mτ m
Tm
0.5τ m
0% overshoot; τ 011 . < m <1 Tm
095 . Tm Km τ m
1.36Tm
0.47τ m
20% overshoot; τ 011 . < m <1 Tm
12 . Tm 2Tm , ] K m τ m K m τm
Tm 1370 . Km τm
0.950
Tm τm 1351 . Tm
0.738
τ 0.365Tm m Tm
0.950
Model: Method 3
Quarter decay ratio; minimum integral error (servo mode); Kc K m Td = 05 . ; Tm
01 . ≤
Cohen and Coon [11] Model: Method 1
Astrom and Hagglund [93]pages 120-126
2 2.5 τ m + 0.46 τ m 1 Tm Tm . + 025 . T Tm 135 Km τm m τm 1 + 0.61 Tm
3 Km
Model: Method 19 Sain and Ozgen [94] 1 T . 0. 6939 m + 01814 Model: Method 5 K m τm Regulator tuning Minimum IAE – Murrill [13] – pages 358-363 Model: Method 3
0.37 τ m τ 1 + 0.2 m Tm
τm ≤1 Tm
Quarter decay ratio
T90% = 90% closed loop step response time. Leeds and Northrup Electromax V.
T90%
05 . τm
0.8647Tm + 0.226τ m Tm + 0.8647 τm
0.0565Tm T 0.8647 m + 0.226 τm
Minimum performance index
Tm 1435 . Km τ m
0.921
Tm τm 0878 . Tm
0.749
τ 0.482 Tm m Tm
1.137
01 . <
τm ≤1 Tm
Rule Modified minimum IAE - Cheng and Hung [95] Model: Method 7 Minimum ISE Murrill [13] – pages 358-363 Model: Method 3 Minimum ISE Zhuang and Atherton [20]
Kc
3 Tm K m τm
Ti 0 .921
Td
Tm τm 0878 . Tm
Comment
0.749
τ 0.482 Tm m Tm
1.137
Tm 1495 . Km τ m
0.945
Tm τm 1101 . Tm
0.771
τ 0.56Tm m Tm
1. 006
Tm 1473 . Km τ m
0.970
Tm τm 1115 . Tm
0.753
τ 0.55Tm m Tm
0.948
Tm 1524 . Km τm
0.735
Tm τ m 1130 . Tm
0.641
τ 0.552Tm m Tm
0.851
Tm 1357 . Km τm
0 .947
Tm τ m 0842 . Tm
0.738
τ 0.381Tm m Tm
0.995
Tm 1468 . Km τm
0.970
Tm τm 0942 . Tm
0.725
τ 0.443Tm m Tm
0.939
Tm 1515 . Km τ m
0.730
Tm τm 0957 . Tm
0.598
τ 0.444 Tm m Tm
0 .847
Model: Method 6 Minimum ISTES Zhuang and Atherton [20]
Tm 1531 . Km τm
0.960
Tm τ m 0971 . Tm
0.746
τ 0.413Tm m Tm
0.933
Tm 1592 . Km τm
0.705
Tm τm 0957 . Tm
0.597
τ 0.414 Tm m Tm
0.850
Model: Method 6
Model: Method 6 Minimum ITAE Murrill [13] – pages 358-363 Model: Method 3 Minimum ISTSE Zhuang and Atherton [20]
Minimum error - step load change - Gerry [96] Model: Method 6 Servo tuning Minimum IAE Rovira et al. [21] Model: Method 3
03 . Km
Tm 1086 . Km τm
0.5τm
0.869
0.6032 0.7645 + (Tm + 05 . τm) τ m Tm
Tm + 0.5τ m
Model: Method 6 Minimum ISE - Wang et al. [97]
0.7524 0.9155 + (Tm + 0 .5τ m ) τ m Tm
Tm + 0.5τ m
K m ( Tm + τ m )
K m (Tm + τ m )
Model: Method 6 Minimum ISE Zhuang and Atherton [20] Model: Method 6
Tm 1048 . Km τ m
0.897
Tm 1154 . K m τm
0.567
0.5Tm τ m Tm + 05 . τm
0.5Tm τ m Tm + 05 . τm
Tm
τ 0.489 Tm m Tm
0.888
Tm
τ 0.490 Tm m Tm
0.708
τ 1195 . − 0.368 m Tm τ 1.047 − 0.220 m Tm
τm ≤1 Tm
01 . ≤
τm ≤ 10 . Tm
11 . ≤
τm ≤ 2.0 Tm
01 . <
τm ≤1 Tm
01 . ≤
τm ≤ 10 . Tm
11 . ≤
τm ≤ 2.0 Tm
01 . ≤
τm ≤ 10 . Tm
11 . ≤
τm ≤ 2.0 Tm
τm >5 Tm
Tm
Minimum performance index 0.914 Tm τm 0.348 Tm τ 0.740 − 0.13 m Tm Tm
Minimum IAE Wang et al. [97]
01 . <
01 . <
τm ≤1 Tm
0.05 <
τm <6 Tm
0.05 <
τm <6 Tm
01 . ≤
τm ≤ 10 . Tm
11 . ≤
τm ≤ 2.0 Tm
Rule
Kc
Ti
Minimum ITAE Rovira et al. [21] Model: Method 3
Tm 0965 . K m τm
Modified minimum ITAE - Cheng and Hung [95] Model: Method 7 Minimum ITAE – Wang et al. [97]
12 . Tm Km τm
0.85
0.855
Td
Tm
τ 0.308 Tm m Tm
Tm
τ 0.308 Tm m Tm
0.929
τ 0.796 − 0.1465 m Tm τ 0.796 − 0.147 m Tm
05307 . 0.7303 + (Tm + 0.5τ m ) τ m Tm K m (Tm + τm )
Tm + 0.5τ m
0.5Tm τ m Tm + 05 . τm
Model: Method 6 Minimum ISTSE – Zhuang and Atherton [20] Model: Method 6
Minimum ISTES – Zhuang and Atherton [20] Model: Method 6
Tm 1042 . Km τm
0 .897
Tm 1142 . Km τ m
0.579
Tm 0968 . Km τm
0.904
Tm 1061 . Km τ m
0 .583
Comment 0.929
Tm
τ 0.385Tm m Tm
0.906
Tm
τ 0.384 Tm m Tm
0 .839
Tm
τ 0.316 Tm m Tm
0.892
Tm
τ 0.315Tm m Tm
0.832
τ 0.987 − 0.238 m Tm 0.919 − 0.172
τm Tm
τ 0.977 − 0.253 m Tm τ 0.892 − 0.165 m Tm
Ultimate cycle Regulator – minimum IAE – Pessen [63] 0.7K u 0.4Tu Model: Method 6 Servo – minimum 0.051( 3. 302 K m Ku + 1) Tu ISTSE – Zhuang and 0.509Ku Atherton [20] Model: Method not relevant Servo – minimum ISTSE – Pi-Mira et 0.604 K u 0.04 (4. 972 K m K u + 1)T al. [97a] Model: Method 27 Regulator - minimum ISTSE - Zhuang and 4.434 K m K u − 0.966 1751 . K m K u − 0.612 Ku Tu Atherton [20] 512 . K m K u + 1734 . 3776 . K m K u + 1388 . Model: Method not relevant
0149 . Tu
0125 . Tu
01 . ≤
τm ≤1 Tm
Damping factor of closed loop system = 0.707.
0.05 <
τm <6 Tm
01 . ≤
τm ≤ 10 . Tm
11 . ≤
τm ≤ 2.0 Tm
01 . ≤
τm ≤ 10 . Tm
11 . ≤
τm ≤ 2.0 Tm
01 . ≤
τm ≤ 10 . Tm
01 . ≤
τm ≤ 2.0 Tm
01 . ≤
τm ≤ 2.0 Tm
0.130 Tu
0144 . Tu
Kc
Rule
Ti
Td
Ti (38 )
0471 . Ku K mω u
Comment
1
Regulator - nearly minimum IAE, ISE, ITAE - Hwang [60]
Kc( 38)
Model: Method 8 Decay ratio = 0.15 0.1 ≤
τm ≤ 2.0 ; ε < 2.4 Tm
Decay ratio = 0.15 -
Model: Method 8
Kc
( 39)
Ti
( 39 )
0471 . Ku K mω u
0.1 ≤
τm ≤ 2.0 ; Tm
2.4 ≤ ε < 3
Decay ratio = 0.15 -
Kc( 40)
Ti (40 )
0471 . Ku K mω u
0.1 ≤
τm ≤ 2.0 ; Tm
3 ≤ ε < 20
Decay ratio = 0.15 -
Kc( 41)
Ti (41)
0471 . Ku K mω u
( 42)
( 42 )
0471 . Ku K mω u
Ti (43)
0471 . Ku K mω u
0.1 ≤
τm ≤ 2.0 ; ε ≥ 20 Tm
Decay ratio = 0.2 -
Kc
Ti
0.1 ≤
τm ≤ 2.0 ; ε < 2.4 Tm
Decay ratio = 0.2 -
Kc( 43)
0.1 ≤
τm ≤ 2.0 ; Tm
2.4 ≤ ε < 3
2
Kc
Rule
1
KH =
ωH =
Kc
( 38)
Kc
( 39)
2
Ti
2 9 . Km Kc Ku τ m τm − τm Tm + 1884 + 2 18 9ωu 2 K mτ m 18
Td
τ m4 49 τm 2Tm 2 7Tm τm 3 0.471Ku K c + + − 324 324 162 ωu
1+ KH Km τ m Tm K H K m τ m 0.942 K c K u τ m + − 3 6 3ω u
[
2 2 0.674 1 − 0.447ω H τ m + 0.0607ω H τ m = KH − K m (1 + K H K m )
[
2 2 0.778 1 − 0.467ω H τ m + 0.0609ω H τ m = KH − K m (1 + KH K m )
[
10τ m3 16 Tm τ m2 1775 . Kc 2 Ku 2 τ m2 + − 2 81 81ω u 81
2Tm ω u + K H K m τ m ω u − 1884 . Ku Kc 0471 . K c K uω H τ m
,ε=
2
Comment
] , T
i
=
] , T
=
( 38)
( 39)
i
Kc
( 38)
(
(1 + KH K m )
ω H Km 0.0607 1 + 105 . ω Hτ m − 0233 . ωH 2τ m2 Kc
(
( 39)
(1 + K H Km )
ω H K m 0.0309 1 + 2.84ω H τ m − 0532 . ω H2 τ m 2
]
ω τ ( 40) 131 . ( 0519 . ) H m 1 − 103 . ε + 0.514 ε 2 K c (1 + K H K m ) Kc ( 40) = K H − , Ti (40) = K m (1 + K H Km ) ω H K m 00603 . 1 + 0.929 ln[ ω Hτ m ] 1 + 2.01 ε − 12 . ε2
[
114 . 1 − 0.482ω H τ m + 0.068ω H 2 τ m 2 Kc ( 41) = KH − K m (1 + K H Km )
[
] , T
0622 . 1 − 0.435ω H τ m + 0.052ω H τ m Kc ( 42) = K H − K m (1 + K H K m )
[
2
2
)(
(
( 41)
i
] , T
0.724 1 − 0469 . ω H τ m + 00609 . ωH 2 τm2 Kc ( 43) = KH − K m (1 + K H K m )
( 42)
i
] , T
=
i
Kc
( 41)
(
(1 + KH K m )
ω H K m 0.0694 − 1 + 21 . ω H τ m − 0.367ω H 2 τ m2 =
( 43)
Kc
( 42)
(
)
(1 + KH K m )
ω H K m 00697 . 1 + 0752 . ω H τ m − 0.145ω H 2 τ m 2 =
Kc
(
( 43)
(1 + K H K m )
ω H K m 0.0405 1 + 193 . ω H τ m − 0.363ω H 2 τ m 2
) )
)
) )
3
Regulator – nearly minimum IAE, ISE, ITAE - Hwang [60] (continued)
Kc
Model: Method 8
Decay ratio = 0.2 -
( 44)
Ti
( 44 )
Kc( 45)
Ti (45)
Kc
( 46)
Ti
( 46 )
Kc
( 47)
Ti
( 47 )
0471 . Ku K mω u
0.1 ≤
τm ≤ 2.0 ; Tm
3 ≤ ε < 20
Decay ratio = 0.2 -
0471 . Ku K mω u
τm ≤ 2.0 ; ε ≥ 20 Tm
0.1 ≤
Decay ratio = 0.25 -
0471 . Ku K mω u
0.1 ≤
τm ≤ 2.0 ; ε < 2.4 Tm
Decay ratio = 0.25 -
0471 . Ku K mω u
0.1 ≤
τm ≤ 2.0 ; Tm
2.4 ≤ ε < 3
Decay ratio = 0.25 -
Kc( 48)
Servo - nearly min. IAE, ISE, ITAE Hwang [60]
Kc
( 49)
Kc
( 5 0)
Ti (48 )
0471 . Ku K mω u
Ti
( 49 )
0471 . Ku K mω u
Ti
( 50)
0471 . Ku K mω u
0.1 ≤
τm ≤ 2.0 ; Tm
3 ≤ ε < 20
Decay ratio = 0.25 τm ≤ 2.0 ; ε ≥ 20 Tm
0.1 ≤
Decay ratio = 0.1 01 . ≤
τm ≤ 2.0 ; ε < 2.4 Tm
Model: Method 8
3
[
]
ω τ ( 44) 126 . ( 0.506) H m 1 − 107 . ε + 0616 . ε 2 K c (1 + K H K m ) Kc ( 44) = K H − , Ti (44) = K m (1 + K H K m ) ω H K m 00661 . 1 + 0.824 ln[ ω H τ m ] 1 + 171 . ε − 117 . ε2
Kc
( 45)
[
109 . 1 − 0.497ω H τ m + 0.0724ω H2 τ m 2 = KH − K m (1 + K H Km )
] , T
[
0.584 1 − 0439 . ω H τ m + 00514 . ωH 2 τm2 Kc ( 46) = K H − K m (1 + K H Km )
[
0675 . 1 − 0.472ω H τ m + 0061 . ω H 2 τm2 Kc ( 47) = K H − K m (1 + KH K m )
[
)(
(
( 45)
i
] , T
=
] , T
( 47)
i
(
ω H K m 0.054 − 1 + 2.54ω H τ m − 0.457 ω H τ m
=
( 46)
i
=
]
K c (45) (1 + K H K m ) 2
Kc
( 46)
(
(1 + K H K m )
ω H K m 0.0714 1 + 0.685ω H τ m − 0131 . ωH 2 τm2 Kc
( 47)
(
(1 + K H K m )
ω H K m 00484 . 1 + 143 . ω H τ m − 0.273ω H 2 τ m 2
)
)
ω τ ( 48) 12 . ( 0.495) H m 1 − 11 . ε + 0.698 ε 2 K c (1 + K H K m ) Kc ( 48) = K H − , Ti (48) = K m (1 + K H K m ) ω H K m 0.0702 1 + 0.734 ln[ ω H τ m ] 1 + 148 . ε − 11 . ε2
Kc
( 49)
[
103 . 1 − 051 . ω H τ m + 0.0759ω H 2 τ m 2 = KH − K m (1 + KH K m )
[
)(
(
] , T
0.822 1 − 0.549ω H τ m + 0112 . ω H2 τ m 2 Kc (50) = KH − K m (1 + K H K m )
( 49)
i
] , T
i
=
( 50)
Kc
( 49)
(
(1 + K H K m )
ω H K m 0.0386 −1 + 3.26ω H τ m − 0.6ω H 2 τ m2 =
Kc
(
( 50)
)
(1 + K H K m )
ω H K m 0.0142 1 + 6.96ω H τ m − 177 . ω H 2 τm2
)
2
)
)
)
Kc
Rule
Ti
Td
Ti (51)
0471 . Ku K mω u
Comment
4
Servo - nearly min. IAE, ISE, ITAE Hwang [60] (continued)
Decay ratio = 0.1 -
Kc(51)
2.4 ≤ ε < 3
Decay ratio = 0.1 -
Kc(5 2)
Model: Method 8
Ti (52)
Kc(53) Servo – nearly minimum IAE, ITAE – Hwang and Fang [61] Model: Method 9
2 c1 + c2 τ m + c3 τ m K u Tm Tm
c 1 = 0.537, c 2 = −0.0165 c 3 = 0.00173
0471 . Ku K mω u
3 ≤ ε < 20
Decay ratio = 0.1 -
0471 . Ku K mω u
Ti (53)
01 . ≤
2 Kc c + c τ m + c 9 τ m K u 2 τ m 7 8 Tm τm Tm ω u Kc K uω u c 4 + c 5 + c6 Tm Tm c = 0.350, c = − 00344 .
c 4 = 0.0503 , c 5 = 0.163
7
8
c 6 = −0.0389
Model: Method 1 or Method 18
Ku Am
arbitrary
Ku cosφ m
αTd
Astrom and Hagglund [98] Model: Method not relevant
T m τ m 1 + T + τ m m
[
0786 . 1 − 0441 . ω H τ m + 00569 . ω H 2 τm2 Kc (51) = K H − Km (1 + KH K m )
[
] , T
( 51)
i
0. 65
T m 0 .25 τ m 1 + T + τ m m
0 .65
τm ≤ 2.0 Tm decay ratio = 0.03 01 . ≤
Tu
Kc
( 51)
(
τm ≤ 2.0 Tm decay ratio = 0.12 01 . ≤
01 . ≤
Tuning rules developed from Ku , Tu
4 + tan 2 φ m α 4π
Specify phase margin φ m ; α = 4 (Astrom et al. [30])
(1 + K H K m )
ω H K m 0.0172 1 + 4.62ω H τ m − 0.823ω H2 τ m 2
]
τm ≤ 2.0 Tm
Specify gain margin Am
Tu 4π 2 Ti tan φ m +
=
τm ≤ 2.0 ; ε ≥ 20 Tm
c 9 = 0.00644
2 2 Kc Simultaneous τ τ c 1 + c 2 m + c 3 m K u c 7 + c 8 τ m + c 9 τ m K u 2 τ τ Servo/regulator Tm Tm Tm Tm ω u K c K u ω u c 4 + c 5 m + c 6 m Tm Tm nearly minimum IAE, c 7 = 0.371, c 8 = −00274 . c 1 = 0.713, c 2 = −0176 . ITAE - Hwang and c 4 = 0.149, c 5 = 0.0556 c 9 = 0.00557 c 3 = 0.0513 Fang [61] c 6 = −0.00566 Model: Method 9 McMillan [58]
1.415 Tm 1 0 .65 K m τm Tm 1+ Tm + τ m
τm ≤ 2.0 ; Tm
01 . ≤
2 2 Kc Regulator – nearly τm τm c + c τ m + c τ m K u 2 minimum IAE, ITAE c 1 + c 2 T + c 3 T K K uω u c 4 + c 5 τ m + c 6 τ m 7 8 Tm 9 T m ω u Kc m m Tm Tm – Hwang and Fang c 7 = 0.421, c 8 = 0.00915 c = 0802 . , c = − 0154 . 1 2 [61] c 4 = 0.190 , c 5 = 0.0532 c 9 = −000152 . c 3 = 0.0460 Model: Method 9 c 6 = −0.00509
4
τm ≤ 2.0 ; Tm
0.1 ≤
)
( 52) 128 . ( 0.542) 1 − 0.986 ε + 0.558 ε 2 K c (1 + K H K m ) Kc (52 ) = K H − , Ti (52) = K m (1 + K H K m ) ω H K m 0.0476 1 + 0.996 ln[ ω H τ m ] 1 + 2.131 ε − 113 . ε2
ωH τm
[
114 . 1 − 0.466ω H τ m + 0.0647ω H 2 τ m 2 Kc (53) = KH − K m (1 + KH K m )
] , T
)(
(
i
( 53)
=
Kc
(
( 53)
(1 + K H K m )
ω H K m 0.0609 − 1 + 197 . ω H τ m − 0.323ω H 2 τ m2
)
)
Rule
Kc
Ti
Li et al. [99]
Kc(5 4) 5
εTd
Model: Method not relevant
φm 0
15 20 0 25 0
η
Am 2 1.67 1.67 4h πAA m
2.8 3.2 3.5
Tan et al. [39] Model: Method 10
Kφ
1.67 1.43 1.43
30 35 0 40 0
0
3.8 4.0 4.2
45 50 0 55 0
Tu
ω uωφ
2
2
2
)
K u − r 2 Kφ 2
4 + tan 2 φ m α 4π
ω φ Ti 2
2
01443 . ωu
04830 .
3.7321 ω150
0.9330 ω 150
)
0
(
0
η
60 65 0
1.11 1.11
5.4 5.5
simplified algorithm
Am = 2 , φ m = 450 ; α chosen arbitrarily ωφ ; r = 01 . + 09 . (K u K φ )
05774 . ωu
(
Am
φm
Arbitrary A m , φ m at
1
0.25 G p ( jω u )
G p jω 150 0
4.6 4.9 5.2
00796 . T
αTd
(
1.25 1.25 1.25
tan φ m +
rK φ ω u − ω φ
Am
Model: Method not relevant
η
Am
0
Comment
T 4 2 η + η + 4π η η φm Am
03183 . T
Ku cosφ m Am
Friman and Waller [41]
φm
Td
0
)
τm < 0.25 , Am = 2 , Tm φ m = 60 0
0.25 ≤
τm ≤ 2.0 Tm
Am = 2 , φ m = 450
tan φ − b A 2 − b2 m 4h cos φm Kc = , η= with ±h = amplitude of relay, ±b = 2πb 2 1 + b A2 − b 2 2 πA m A − b + (Ti − Td ) T deadband of relay, A, T = limit cycle amplitude and period, respectively. 5
( 54 )
(
)
Kc
Rule
Ti
Td
Comment
Direct synthesis Regulator - Gorecki et al. [28]
Kc
( 55)
Ti
( 55)
Td
Model: Method 6
Regulator - minimum IAE - Smith and Corripio [25] – page 343-346 Model: Method 6 Servo – minimum IAE – Smith and Corripio [25] – page 343-346 Model: Method 6 Servo – 5% overshoot – Smith and Corripio [25] – page 343-346 Model: Method 6
6
Kc
( 55)
Ti (56)
Td (5 6)
Tm K mτ m
Tm
0.5τ m
5Tm 6Km τ m
Tm
0.5τ m
Tm 2K m τ m
Tm
0.5τ m
Kc
(2 tuning rules)
( 56) 6
2 2 τm τm τm 2 Tm τm = 6+ 3+ −9− − e Km τ m 2Tm 2Tm 2Tm 2Tm 2
Ti
( 55)
Pole is real and has maximum attainable τ multiplicity; m < 2 Tm
( 5 5)
Low frequency part of magnitude Bode diagram is flat.
2
τ τ 3+ m − 3− m 2 Tm 2Tm
2
τm τ τm τ −9− m − m 6 + 3+ 2Tm 2Tm 2Tm 2Tm = τm , 2 2 2 3 τm τm τ m τm τm τm 21 + 3 + − 6 3+ − 36 − 4.5 − Tm 2Tm Tm 2Tm 2Tm 2Tm 2
Td
( 55)
= 0.5τ m
τ 3+ m − 1 2Tm 2
τm τ τm τ −9− m − m 6 + 3+ 2Tm 2Tm 2Tm 2Tm
2
2
Kc ( 56) =
1 Km
1 , Ti (56 ) = τ m τ m Tm 2 (56) + 1 − 2 Ti τ m 2
1+ 7 Td
( 56 )
= τm
3
T T T Tm + 135 m + 240 m + 180 m τm τ τ m m τm 2 Tm Tm Tm 152 + 1 1 + 3 + 6 τm τm τm
3
Tm T T T + 27 m + 60 m + 60 m τm τm τm τm 2
7 + 42
7 + 42
3
4
T T T Tm + 135 m + 240 m + 180 m τm τm τm τm
4
4
01 . ≤
τm ≤ 15 . Tm
01 . ≤
τm ≤ 15 . Tm
Suyama [100] Model: Method 6
1 Tm + 0.2236 0.7236 Km τm
Tm + 0.309 τ m
2.236Tm τ m 7.236Tm + 2.236τ m
Kc
Rule Juang and Wang [101] Model: Method 6 Cluett and Wang [44] Model: Method 6
Gain and phase margin - Zhuang and Atherton [20] Model: Method not relevant
α+
Ti
τ τm + 05 . m Tm Tm
τ Km α + m Tm
Model: Method 6
α+ Tm
2
Td
τ τm + 05 . m Tm Tm
2
Comment
2
τ τ τ 0.5 m Tm α + m − 0.5α m Tm Tm Tm 2 τ m τ m τm α + α + + 0.5 Tm Tm Tm
τ α + m Tm
Closed loop time constant = αTm , 0<α <1
0.019952 τ m + 0. 20042Tm Km τm
0.099508τ m + 0.99956Tm τm 0.99747 τ m − 8 .7425.10 − 5 Tm
−0.0069905τ m + 0.029480Tm τm 0.029773τ m + 0.29907 Tm
Closed loop time constant = 4τm
0.055548 τ m + 0. 33639Tm Km τm
0 .16440τ m + 0.99558 Tm τm 0.98607τ m − 15032 . .10 − 4 T m
−0.016651τ m + 0.093641Tm τm 0.093905τm + 0.56867 Tm
Closed loop time constant = 2τm
0.092654 τ m + 0.43620Tm Km τm
0.20926 τ m + 0 .98518Tm τm 0.96515τ m + 4255010 . − 3 Tm
−0.024442τ m + 0.17669Tm τm 0.17150τm + 080740 . Tm
Closed loop time constant = 133 . τm
012786 . τ m + 0.51235Tm K mτ m
0.24145τ m + 0 .96751Tm τm 0.93566 τ m + 2 .298810 . − 2Tm
−0.030407 τ m + 0 .27480Tm τm 0.25285 τ m + 1.0132Tm
Closed loop time constant = τ m
016051 . τ m + 057109 . Tm K mτ m
0 .26502τ m + 0 .94291Tm τm 0.89868τ m + 6.935510 . − 2 Tm
−0.035204τ m + 0.38823Tm τm 0.33303 τm + 11849 . Tm
Closed loop time constant = 08 . τm
019067 . τ m + 0.61593Tm K mτ m
0.28242τ m + 0.91231T m τm 0.85491τ m + 0.15937 Tm
−0.039589τ m + 0.51941Tm τm 0.40950τ m + 13228 . Tm
Closed loop time constant = 067 . τm
mK u cos(φ m ), m = 0 .614(1 − 0.233 e 0
(
−0 .347 K m K u
− 0. 45K m Ku
φ m = 338 . 1 − 0.97e
τ 0177 . + 0.348 m Tm
Abbas [45]
2
)
) α
tan(φ m ) +
4 + tan2 (φ m ) α 2ω u
,
tan(φm ) +
α = 0.413( 3.302K mK u + 1)
− 1.002
Tm + 05 . τm
1 Tm + τ m Km Tm τm
0.0821 Servo – minimum ISE 18578 . φm - Ho et al. [103] K A 0.9087 m
m
τm Tm
4Tm τ m Tm + τ m − 0.9471
Ti
( 57)
V = fractional overshoot
Tm τ m 2Tm + τ m
K m ( 0531 . − 0.359V 0.713 )
Camacho et al.[102] Model: Method 6
Gain margin = 2, phase margin = 60 0 τ 01 . ≤ m ≤ 2.0 Tm
4 + tan2 ( φm ) α 2ω u
0 ≤ V ≤ 0.2 τ 01 . ≤ m ≤ 5.0 Tm
Tm τ m Tm + τ m 0.4899Tm φ m 0.1457 τ m 0.0845 Tm Am
7
1.0264
Am ∈[ 2 ,5] ,
[
]
φ m ∈ 30 0 ,60 0 ,
τm ≤ 10 . . Tm
01 . ≤
Model: Method 6
621189 . 403182 . τ m 76.2833τ m + − Am Tm A m Tm (Ho et al [104]) Am ∈[ 2 ,5] ,
Given A m , ISE is minimised when φ m = 29.7985 +
− 0.908 0.3678 1.0317 Regulator - minimum 10722 . φ m −0.116 τ m 12497 . Tm φ m 1.0082 τ m 0.4763Tm φ m −0.328 τ m 0.2099 0.0961 ISE – Ho et al. [103] Km A m 0.8432 Tm Tm Tm Am Am
Model: Method 6 Given A m , ISE is minimised when φ m = 46.5489A m0.2035 ( τ m Tm )
Kc
Rule
7
Ti
( 57 )
=
[
Ti
0.0211Tm 1 + 0.3289 A m + 6.4572φ m + 251914 . ( τ m Tm ) 1 + 0.0625A m − 0.8079φ m + 0.347( τm Tm )
Td
]
0.3693
[
]
01 . ≤
τm ≤ 10 . Tm
φ m ∈ 30 0 ,60 0 ,
(Ho et al [104]) Comment
Morilla et al. [104a] Model: Method 24
Kc
Robust Robust - Brambilla et al. [48] Model: Method 6
τm
( 57a ) 8
α Ti
1 + 1 − 4α
Tm + 0.5τ m
1 Tm + 05 . τm Km λτ m
α = 0. 1; δ 0 = [ 0. 2, 0. 5]
( 57 a )
τm ≤ 10 and no Tm model uncertainty -
01 . ≤
Tm τ m 2Tm + τ m
λ ≈ 0.35
1 Tm + 05 . τm Km λ + 05 . τm
Tm + 0.5τ m
Tm τ m 2Tm + τ m
λ > 01 . Tm , λ ≥ 0.8τ m .
Fruehauf et al. [52]
5Tm 9τ m Km
5τ m
≤ 0.5τ m
τm < 033 . Tm
Model: Method 1
Tm 2τ m Km
Tm
≤ 0.5τ m
τm ≥ 0.33 Tm
Lee et al. [55]
Ti K m ( λ + τm )
Rivera et al. [49] Model: Method 6
Tm +
Model: Method 6
2 τm τ 1 − m 2( λ + τ m ) 3Ti
τm 2( λ + τ m ) 2
λ + ατ m − 05 . τm 2λ + τ m − α 2
τm 3
3 2 Two degrees of τm ατ − m freedom controller; 2 Tmα − 6 α = T − m 2λ + τ m − α τ 2 λ −T ; Ti Tm 1 − e Tm 2 λ2 + ατ m − 05 . τm Desired response = 2λ + τ m − α e− τ m s
Tm + α − Ti K m ( 2λ + τm − α )
λ=
2
m m
1 + λs
8
Kc
( 57a )
=
with δ 0 =
Ti
( 57 a )
(
K m [ 2δ 0 + ω n 0 τ m − Ti 1 2π 1 + log e [b a ]
, 2
( 57 a )
)] , ω
n0
=
(
− δ 0 Tm + δ 0 Tm + Tm τ m − Ti 2
ωn0
(
2
Tm τ m − Ti
( 57 a )
)− αT
b = desired closed loop response decay ratio a
i
( 57a )
)− αT
i
( 57 a ) 2
( 57a ) 2
K m e− sτ - ideal controller with first order filter 1 + sTm m
Table 26: PID tuning rules - FOLPD model
1 1 G c ( s) = Kc 1 + + Td s . 3 tuning rules. Ti s Tf s + 1 Rule
Kc
Ti
Td
1 Tm + 05 . τm K m λ + τm
Tm + 0.5τ m
Tm τ m 2Tm + τ m
TF =
τm Tm τ m + Tm
Tf =
Comment
Robust ** Morari and Zafiriou [105]
Horn et al. [106] Model: Method 6
H∞ optimal – Tan et al. [81] Model: Method 6
2Tm + τ m 2 (λ + τ m ) K m
Tm + 0.5τ m
0.265λ + 0.307 Tm + 05 . Km τm
Tm + 0.5τ m
τ m Tm τ m + 2Tm
λτ m , 2( Tm + τ m )
λ > 0.25τ m , λ > 0.2Tm . λτ m ; 2( λ + τ m )
λ > τ m , λ < Tm . τm 5.314 λ + 0.951 λ = 2 - ‘fast’ response λ = 1 - ‘robust’ tuning λ = 1.5 - recommended Tf =
K m e− sτ - ideal controller with second order filter 1 + sTm m
Table 27: PID tuning rules - FOLPD model
1 1 + b1s G c ( s) = Kc 1 + + Td s . 1 tuning rule. 2 Ti s 1 + a 1s + a 2s Kc
Ti
Td
2Tm + τ m , 2( 2λ + τm − b1) Km
Tm + 05 . τm
τ m Tm τ m + 2Tm
Rule
Comment
Robust Horn et al. [106]
[
λ τ m + 2 Tm τ m ( τ m − λ ) 2
Model: Method 6
b1 =
+
[
Tm (τ m + 2 λ)
2 λ (2T m − λ)
( τ m + 2λ)
]
]
Filter 1 + b1 s 1+
2λτ m + 2 λ + b1 τ m
a2 =
2
2( 2λ + τ m − b 1 ) λ 2τ m
2( 2λ + τ m − b1 )
λ < Tm
.
s + a 2s 2
; λ > τm ,
K m e− sτ - ideal controller with set-point weighting 1 + sTm m
Table 28: PID tuning rules - FOLPD model
1 G c ( s) = Kc b + + Td s . 1 tuning rule. Ti s Kc
Rule
Ti
Direct synthesis
Comment
(Maximum sensitivity)
38 . e Astrom and Hagglund [3] – pages 208-210
Td
−8.4 τ + 7.3τ 2
Tm
K mτ m
,
τ = τ m ( τ m + Tm )
52 . τ me
− 2.5τ − 1.4 τ 2
0.46Tme 2.8τ − 2.1τ
or 2
0.89 τm e −0.37 τ − 4.1τ or 2
0.077 Tm e5.0τ − 4 .8 τ
2
b = 0.40e 0.18 τ + 2 .8 τ ; M s = 1.4 ; 2
014 . ≤
Model: Method 3
τm ≤ 55 . Tm
b = 0.22e0.65τ + 0.051τ ; Ms = 2.0 ; 2
8.4e− 9.6 τ + 9.8τ Tm Km τ m 2
32 . τ m e− 1.5τ − 0.93τ or 2
0.28Tm e3.8τ − 1.6 τ
2
0.86τ m e− 1.9 τ − 0.44 τ or 2
0.076 Tm e3.4 τ − 1.1τ
2
014 . ≤
τm ≤ 55 . Tm
K m e− sτ - ideal controller with first order filter and set-point 1 + sTm m
Table 29: PID tuning rules - FOLPD model
1 1 1 + 0.4Trs weighting U(s) = K c 1 + + Td s − Y( s) . 1 tuning rule. R (s ) 1 + sTr Tis Tf s + 1 Rule Direct synthesis Normey-Rico et al. [106a] Model: Method not specified
Kc
Ti
Td
Comment
0. 375(τ m + 2Tm ) K m τm
Tm + 0. 5τ m
Tm τ m 2Tm + τ m
Tf = 0.13τ m Tr = 0.5τ m
K m e− sτ 1 1 + Td s . 20. Table 30: PID tuning rules - FOLPD model - classical controller G c ( s) = Kc 1 + 1 + sTm Ti s 1 + Td s N m
Kc
Ti
Td
083 . Tm Km τ m
15 . τm
0.25τ m
τm
τm
Rule Process reaction Hang et al. [36] – page 76 Model: Method 1 Witt and Waggoner [107] Model: Method 1
[
0.6Tm Tm , ] K mτ m K mτ m
Witt and Waggoner [107] Model: Method 2
Kc
Regulator tuning Minimum IAE - Kaya and Scheib [108] Model: Method 3 Minimum IAE – Witt and Waggoner [107] Model: Method 1
1
Kc
( 58 )
=
Kc
Ti
( 59 )
‘aggressive’ tuning;
05 . Tm K mτ m
5τ m
0.5τ m
‘conservative’ tuning;
0. 889Tm K m τm
1.75τ m
0.70 τm
τm = 0.167 Tm
( 59 )
=
Minimum performance index
Tm 098089 . Km τ m
0.76167
Tm τm 091032 . Tm
Kc(5 9) 2
Ti (59)
Td (59 )
τ τ 0.7425 + 0.0150 m + 0.0625 m Tm Tm
τ Tm τ + 0.25 m 0.7425 + 0.0150 m + 0.0625 m τm Tm Tm
2
Tm τ Tm τ 1350 . + 025 . ± 07425 . + 0.0150 m + 0.0625 m τm Tm Tm −0.921
τ 0.964 Tm m Tm
2
− 1.886 τm 1 ± 1 − 1693 , . Tm
1.137
τ 1 − 1 m 1693 . m Tm
1.886
, Td
τ 0.59974 Tm m Tm
1.05221
Tm
τm 0718 . = Km Tm
Equivalent to Cohen and Coon [11]; N = [10,20]
0.5τ m
2 Km
1350 .
2
Td
( 5 8)
5τ m
T T 1.350 m + 0.25 ± m τm τm
Ti (58 ) =
Td (58 ) =
Ti
( 58)
Foxboro EXACT controller ‘pretune’; N=10 Equivalent to Ziegler and Nichols [8]; N = [10,20]
Tm Km τ m
St. Clair [15] – page 21 Model: Method 1 Shinskey [15a] Model: Method 1
( 5 8) 1
Comment
( 59 )
=
τ 0.964 Tm m Tm
1.137
τ 1 + 1 ± 1.693 m Tm
1.886
2
0.89819
0<
0.1 <
τm ≤ 1 ; N=10 Tm
τm < 0.258 Tm
; N = [10,20]
Kc
Rule Minimum ISE - Kaya and Scheib [108] Model: Method 3 Minimum ITAE Kaya and Scheib [108] Model: Method 3 Minimum ITAE Witt and Waggoner [107] Model: Method 1 Servo tuning Minimum IAE - Kaya and Scheib [108] Model: Method 3 Minimum IAE - Witt and Waggoner [107] Model: Method 1 Minimum ISE - Kaya and Scheib [108] Model: Method 3 Minimum ITAE Kaya and Scheib [108] Model: Method 3
3
Kc
Td
( 60 )
Kc
Ti
=
( 60)
( 61)
( 61)
=
=
Tm 111907 . Km τ m
1.06401
Tm 077902 . K m τm
065 . Tm Km τm
Tm τ m 07987 . Tm
τ 0.54766 Tm m Tm
Tm τ m 114311 . Tm
0.70949
τ 0.57137 Tm m Tm
1.03826
Ti( 60) 3
1.04432
τ 0.762Tm m Tm
1.03092
Td ( 61)
0.80368
0.86411
τ 0.54568 Tm m Tm
Tm
τ 0.42844Tm m Tm
τ 0.99783 + 0.02860 m Tm
τm ≤ 1 ; N=10 Tm
0<
τm ≤ 1 ; N=10 Tm
0<
1.0081
τm ≤ 1 ; N=10 Tm
0.995
0.914 τm τm 1 ± 1 − 1392 . . − 013 . 074 , Tm Tm
τ 1 m 1 − 1.392 m Tm
0.869
τm 0.74 − 0.13 Tm
, Td
( 61)
=
τ 0.696Tm m Tm τ 1 ± 1 − 1.392 m Tm
0.869
τm ≤ 1 ; N = [10,20] Tm
0<
.
0.914
τm ≤ 1 ; N=10 Tm
τm ≤ 1 ; N=10 Tm
0.995
τ 0.696 Tm m Tm
; N = [10,20]
0<
−1.733 1 ± 1 − 1.283 τ m T ( 60) = , 1.733 i Tm τ 1 − 1 m 1283 . m Tm
1.733
τm < 0.379 Tm
0 .1 ≤
Tm
τ 112666 . − 0.18145 m Tm
0<
0 .1 <
Minimum performance index 1.08433 Tm τm 0 . 50814 T τ m 0.9895 + 0.09539 m Tm Tm
τ 0.762Tm m Tm
τ 1 + 1 ± 1.283 m Tm − 0.869
Td ( 60)
Ti (61)
Tm 071959 . Km τ m
Comment 0.87798
Kc( 61)
0.947
Td 0.9548
Kc( 60)
Tm 112762 . Km τ m
0.679 τ m K m Tm
τm 1086 . = Km Tm
Ti 0.89711
0.914
τm 0.74 − 0.13 Tm
Kc
Ti
Td
Kc( 62) 4
Ti (62 )
Td ( 62)
0809 . Tm Km τ m
Tm
05 . τm
Overshoot = 16%; N=5
Tm
025 . τm
N = 2.5
Rule Minimum ITAE Witt and Waggoner [107] Model: Method 1 Direct synthesis Tsang and Rad [109] Model: Method 13
aTm Km τ m
Tsang et al. [111]
ξ
a
Model: Method 6
1.681 8 1.382 9
ξ
a
0.0 1.161 0 0.1 0.991 6
a
0.2 0.859 4 0.3 0.754 2
ξ
ξ
a
0.4 0.669 3 0.5 0.600 0
ξ
a
0.6 0.542 9 0.7 0.495 7
Comment
a
0.8 0.456 9 0.9
0 .1 ≤
τm ≤ 1 ; N = [10,20] Tm
ξ 1.0
Robust Chien [50] Model: Method 6
1 Tm K m λ + 05 . τm
Tm
05 . τm
λ = [ τ m , Tm ] ; N=10
1 05 . τm K m λ + 05 . τm
05 . τm
Tm
λ = [ τ m , Tm ] ; N=10
K mτ m 3τ m − 0.32 Tu
Tu Tu 015 . − 0.05 τm
014 . Tu
095 . Tm Km τ m
1.43τ m
052 . τm
τ m Tm = 0.2
095 . Tm Km τ m
117 . τm
048 . τm
τ m Tm = 0.5
114 . Tm Km τ m
1.03τ m
0.40τ m
τ m Tm = 1
139 . Tm Km τ m
077 . τm
0.35τ m
τ m Tm = 2
Ultimate cycle Minimum IAE regulator – Shinskey [59] – page 167. Model: Method not specified Minimum IAE regulator - Shinskey [16] – page 143. Model: Method 6
4
Kc
Ti
( 62)
( 62)
=
0.965 τ m = K m Tm
− 0.85
0.929 τm τm 1 ± 1 − 1232 . . 0.796 − 01465 , Tm Tm
τ 0.616 Tm m Tm τ 1 m 1 − 1.232 m Tm
0.85
0.929
τm 0.796 − 0.1465 Tm
, Td
( 62)
=
τ 0.616Tm m Tm τ 1 ± 1 − 1.232 m Tm
0.85
0.929
τm 0.796 − 0.1465 Tm
K m e− sτ - non-interacting controller 1 + sTm m
Table 31: PID tuning rules - FOLPD model
Rule Regulator tuning Minimum IAE – Huang et al. [18] Model: Method 6 Servo tuning Minimum IAE Huang et al. [18] Model: Method 6
1 Td s . 2 tuning rules. E( s) − U (s) = K c 1 + Y ( s ) Td s Tis 1+ N Kc Ti Td
Comment
Minimum performance index
Kc( 63) 5
Ti (63)
Td ( 63)
01 . ≤
τm ≤ 1 ; N =10 Tm1
01 . ≤
τm ≤ 1 ; N =10 Tm
Minimum performance index
Kc( 64) 6
Ti (64 )
Td ( 64)
−0.8058 0.6642 2.1482 τ τm τm τm 1 τ T 01098 . − 8.6290 m + 11863 . + 231098 . + 20 . 3519 − 191463 . e Km Tm Tm Tm Tm
m
5
Kc ( 63) =
Ti
( 63)
Td
( 63)
m
2 3 4 5 τ τ τ τ τ = Tm − 0.0145 + 2 .0555 m − 4.4805 m + 7 .7916 m − 7 .0263 m + 2.4311 m Tm Tm Tm Tm Tm 2 3 4 5 6 τm τm τm τm τm Tm τm −0.0206 + 0.9385 = − 2 .3820 + 7.2774 − 111018 . + 8.0849 − 2.274 ( 63) Tm K c Tm Tm Tm Tm Tm
0.0865 −0.4062 2 .6405 τ τm τ τm 1 τ 7.0636 + 66.6512 m + 261928 . + 7.3453 m + 336578 . − 57.937e T Km Tm Tm Tm Tm
m
6
Kc ( 64) =
Ti
( 64 )
Td
( 64 )
m
2 3 4 5 τm τ τm τ τ = Tm 0.9923 + 0.2819 m − 14510 . . + 2.504 m − 18759 + 0.5862 m Tm Tm Tm Tm Tm 2 3 4 5 6 τm τm τm τm τm Tm τm = ( 64) 0.0075 + 0.3449 − 0.0793 + 0.8089 − 1.0884 + 0.352 + 0.0471 Tm K c Tm Tm Tm Tm Tm
K m e− sτ - non-interacting controller 1 + sTm m
Table 32: PID tuning rules - FOLPD model
Rule
Tds 1 E (s ) − U (s) = K c 1 + Y (s) . 5 tuning rules. sT Tis 1+ d N Kc Ti Td
Servo tuning Minimum ISE Zhuang and Atherton [20] Model: Method 6
Minimum ISTSE Zhuang and Atherton [20] Model: Method 6
Minimum ISTES Zhuang and Atherton [20] Model: Method 6
Tm 1260 . Km τm
0.887
Tm 1295 . Km τ m
0.619
Tm 1053 . Km τ m
0.930
Tm 1120 . K m τm
0.625
Tm 1001 . Km τ m
Minimum performance index 0.886 Tm τm 0 . 375 T τ m 0.701 − 0.147 m Tm Tm
τ 0.378 Tm m Tm
0.756
τ 0.349 Tm m Tm
0 .907
Tm
τ 0.350 Tm m Tm
0.811
Tm
τ 0.308 Tm m Tm
0.897
Tm
τ 0.308 Tm m Tm
0.813
Tm
τ 0.661 − 0.110 m Tm Tm 0.736 − 0.126
τm Tm
τ 0.720 − 0.114 m Tm
Tm 0942 . Km τm
0.933
0 .624
Comment
τ 0.770 − 0.130 m Tm τ 0.754 − 0.116 m Tm
Ultimate cycle Servo – minimum . ISTSE – Zhuang and 4.437K m K u − 1587 0.037(589 . K m K u + 1)Tu Ku Atherton [20] 8.024K m Ku − 1435 . Model: Method not relevant Regulator - minimum 0.5556Ku 039 . Tu IAE - Shinskey [16] – 0.4926Ku 0.34Tu page 148. 05051 . Ku 0.33Tu Model: Method 6 0.4608Ku 0.28Tu
0112 . Tu
01 . ≤
τm ≤ 10 . ; N=10 Tm
11 . ≤
τm ≤ 2.0 ; N=10 Tm
01 . ≤
τm ≤ 10 . ; N=10 Tm
11 . ≤
τm ≤ 2.0 ; N=10 Tm
01 . ≤
τm ≤ 10 . ; N=10 Tm
11 . ≤
τm ≤ 2.0 ; N=10 Tm
01 . ≤
τm ≤ 2.0 ; N=10 Tm
014 . Tu
τ m Tm = 0.2
014 . Tu
τ m Tm = 0.5
013 . Tu
τ m Tm = 1
013 . Tu
τ m Tm = 2
K m e− sτ - non-interacting controller 1 + sTm m
Table 33: PID tuning rules - FOLPD model
Rule
Td s 1 E (s) − U (s) = K c + Y ( s) . 6 tuning rules. sT Ti s 1+ d N Kc Ti Td
Regulator tuning Minimum IAE - Kaya and Scheib [108] Model: Method 3 Minimum ISE - Kaya and Scheib [108] Model: Method 3 Minimum ITAE Kaya and Scheib [108] Model: Method 3 Servo tuning Minimum IAE - Kaya and Scheib [108] Model: Method 3
Tm 113031 . Km τ m
Minimum ISE - Kaya and Scheib [108] Model: Method 3
Tm 126239 . Km τm
0.8388
Minimum ITAE Kaya and Scheib [108] Model: Method 3
Tm 098384 . Km τ m
0.49851
Comment
Minimum performance index
Tm 131509 . Km τm
0.8826
1.3756
τ 0.5655Tm m Tm
1.25738
τ 0.79715Tm m Tm
0.41941
τ 0.49547Tm m Tm
0.41932
Tm τm 12587 . Tm
Tm 13466 . Km τm
0.9308
Tm τ m 16585 . Tm
Tm 13176 . Km τm
0.7937
Tm τ m 112499 . Tm
0.81314
1.42603
0.4576
Minimum performance index 0.17707 Tm τm 0.32175Tm τ 5.7527 − 5.7241 m Tm Tm
Tm
τ 0.47617Tm m Tm
0.24572
Tm
τ 0.21443Tm m Tm
0.16768
τ 6.0356 − 6.0191 m Tm τ 2.71348 − 2.29778 m Tm
0<
τm ≤ 1 ; N=10 Tm
0<
τm ≤ 1 ; N=10 Tm
0<
τm ≤ 1 ; N=10 Tm
0<
τm ≤ 1 ; N=10 Tm
0<
τm ≤ 1 ; N=10 Tm
0<
τm ≤ 1 ; N=10 Tm
K m e− sτ - non-interacting controller with set-point weighting 1 + sTm m
Table 32: PID tuning rules - FOLPD model
Rule
1 K Ts U( s) = Kc b + E( s) − c d Y( s) + Kc ( b − 1)Y(s) . 3 tuning rules. sT Ti s 1+ d N Kc Ti Td
Comment
Ultimate cycle Hang and Astrom [111]
0.6Ku
0.5Tu
0125 . Tu
τm < 0.3 . Tm b = 2( x − 01 . )+
N=10 Model: Method 1
0.6Ku
0.5Tu
0.6Ku
τm 0.5 15 . −.083 . Tu Tm
0125 . Tu
0.6Ku
τm 0.5 15 . −.083 . Tu Tm
0125 . Tu
0.6Ku
0.335Tu
0125 . Tu
0.5Tu
0125 . Tu
0.6Ku Hang et al. [65] Model: Method 1
36 15 − K b= , , ' 27 + 5K ' 15 + K 10% overshoot - servo 20% overshoot - servo '
b=
0.6Ku
Hang and Cao [112] Model: Method 11
K =
11[τ m Tm ] + 13 2 37[ τ m Tm ] − 4
0.6Ku
τ . − 0.22 m Tu 053 Tm
'
x = overshoot. τ 0.3 ≤ m < 0.6 . Tm
b = 2x + 0.6 ≤
K = '
08 . ≤
0125 . Tu
τ T . − 0.22 m u 053 Tm 4
τm Tm
τm < 10 . ; b=0.8 Tm
10 . < 016 . ≤
0.57 ≤
01 . ≤
τm Tm
τm < 0.8 . Tm
b = 16 . −
11[τ m Tm ] + 13 2 37[ τ m Tm ] − 4
0.222 K ' Tu
8 4 ' K + 1 , 17 9 20% overshoot, 10% undershoot; servo
b=
0125 . Tu
166 . τm , Tm
τm ; b=0.8 Tm τm < 0.57 ; Tm N=10
τm < 0.96 ; Tm N=10
τm < 0.5 ; N=10 Tm
K m e− sτ - industrial controller 1 + sTm m
Table 33: PID tuning rules - FOLPD model
1 1 + Td s U( s) = Kc 1 + Y(s) . 6 tuning rules. R( s) − Ts Ti s 1+ d N Rule
Kc
Ti
Regulator tuning Minimum IAE - Kaya and Scheib [108] Model: Method 3 Minimum ISE - Kaya and Scheib [108] Model: Method 3 Minimum ITAE Kaya and Scheib [108] Model: Method 3 Servo tuning Minimum IAE - Kaya and Scheib [108] Model: Method 3
Tm 081699 . Km τ m
Minimum ISE - Kaya and Scheib [108] Model: Method 3
Tm 11427 . Km τm
0.9365
Minimum ITAE Kaya and Scheib [108] Model: Method 3
Tm 08326 . Km τm
0.7607
Td
Comment
Minimum performance index
091 . Tm Km τm
0 .7938
Tm τ m 101495 . Tm
Tm 11147 . Km τm
0.8992
Tm τ m 09324 . Tm
Tm 07058 . Km τm
0.8872
Tm τ m 103326 . Tm
1.004
1.00403
0.8753
0.99138
τ 0.5414Tm m Tm τ 0.56508 Tm m Tm
0.7848
0.91107
τ 0.60006Tm m Tm
0.971
Minimum performance index 0.97186 Tm τm τ 0.44278Tm 1.09112 − 0.22387 m Tm Tm
Tm
τ 0.35308Tm m Tm
0.78088
Tm
τ 0.44243Tm m Tm
1.11499
τ 0.99223 − 0.35269 m Tm τ 1.00268 + 0.00854 m Tm
0<
τm ≤ 1 ; N=10 Tm
0<
τm ≤ 1 ; N=10 Tm
0<
τm ≤ 1 ; N=10 Tm
0<
τm ≤ 1 ; N=10 Tm
0<
τm ≤ 1 ; N=10 Tm
0<
τm ≤ 1 ; N=10 Tm
K m e− sτ 1 - series controller Gc ( s) = Kc 1 + (1 + sTd ) . 3 tuning 1 + sTm Ti s m
Table 34: PID tuning rules - FOLPD model
rules.
Kc
Ti
Td
Comment
5Tm 6Km τ m
15 . τm
0.25τ m
Foxboro EXACT controller
035 . Ku
0.25 Tu
0.25 Tu
Tm
025 . τm
Rule Autotuning ** Astrom and Hagglund [3] – page 246 Model: Not specified Ultimate cycle Pessen [63] Model: Method 6
01 . ≤
Direct synthesis
aTm Km τ m
Tsang et al. [110] Model: Method 13
a 1.819 4 1.503 9
ξ
a
0.0 1.269 0 0.1 1.089 4
ξ
a
0.2 0.949 2 0.3 0.837 8
ξ
a
0.4 0.748 2 0.5 0.675 6
ξ
a
0.6 0.617 0 0.7 0.570 9
ξ
a
0.8 0.541 3 0.9
ξ 1.0
τm ≤1 Tm
K m e− sτ - series controller with filtered derivative 1 + sTm m
Table 35: PID tuning rules – FOLPD model
1 sTd Gc ( s) = Kc 1 + 1 + sT Ti s 1+ d N
. 1 tuning rule.
Kc
Ti
Td
Comment
Chien [50]
Tm K m (λ + 0.5τm )
Tm
05 . τm
λ = [ τ m , Tm ] ; N=10
Model: Method 6
0.5τ m K m ( λ + 0.5τm )
05 . τm
Tm
λ = [ τ m , Tm ] ; N=10
Rule Robust
K m e− sτ - controller with filtered derivative 1 + sTm m
Table 36: PID tuning rules – FOLPD model
1 Td s G c ( s) = Kc 1 + + Ti s 1 + s Td N Rule Robust Chien [50] Model: Method 6 Gong et al. [113]
. 3 tuning rules.
Kc
Ti
Td
Comment
Tm + 0.5τ m K m ( λ + 0.5τm )
Tm + 05 . τm
Tm τ m 2Tm + τ m
λ = [ τ m , Tm ] ; N=10
Tm + 0.3866τ m K m (λ + 1.0009τm )
Tm + 0.3866τ m
03866 . Tm τm Tm + 0.3866τ m
N = [3,10]
Model: Method 6
λ=
( 0.1388 + 0.1247N )Tm + 0.0482 Nτ m τ 0.3866 ( N − 1)Tm + 0.1495 Nτ m
m
Direct synthesis Davydov et al. [31] Model: Method 12
1 τm Km 1552 . + 0078 . Tm
τm 0186 . + 0532 . Tm T m
1 τm K m 1209 . + 0103 . Tm
τ 0.382 m + 0.338 Tm Tm
Closed loop response τm 0.25 0186 . + 0532 . Tm damping factor = T m 0.9; 02 . ≤ τ m Tm ≤ 1 ; N = Km
Closed loop response τ 0.4 0.382 m + 0.338 Tm damping factor = Tm 0.9; 02 . ≤ τ m Tm ≤ 1 ; N = Km
K m e− sτ - alternative non-interacting controller 1 1 + sTm m
Table 37: PID tuning rules – FOLPD model
1 U( s) = Kc 1 + E( s) − Kc Td sY ( s) . 6 tuning rules. Ti s Rule
Kc
Ti
Td
Comment
012 . Tu
Tu < 2 .7 τm
012 . Tu
Tu ≥ 2.7 τm
Ultimate cycle 2
Ku Regulator - minimum IAE - Shinskey [59] – page 167. Model: method not specified Minimum IAE Shinskey [17] – page 117. Model: Method 6 Minimum IAE Shinskey [16] – page 143. Model: Method 6 Regulator - minimum IAE - Shinskey [16] – page 148. Model: Method 6 Regulator – minimum IAE - Shinskey [17] – page 121. Model: method not specified Process reaction – VanDoren [114] Model: Method 1
3.73 − 0.69
Tu
0125 .
Tu τm
0125 .
Tu τm
τm
Ku T 2.62 − 0.35 u τm
2
132 . Tm K m τ m
180 . τm
0.44τm
τ m Tm = 01 .
132 . Tm K m τ m
1.77τm
0.41τ m
τ m Tm = 0.2
135 . Tm Km τ m
1.43τ m
0.41τ m
τ m Tm = 0.5
149 . Tm K m τ m
117 . τm
0.37τm
τ m Tm = 1
182 . Tm Km τ m
0.92τ m
0.32τ m
τ m Tm = 2
0.7692 Ku
0.48Tu
011 . Tu
τ m Tm = 0.2
0.6993Ku
0.42Tu
012 . Tu
τ m Tm = 0.5
0.6623Ku
0.38Tu
012T . u
τ m Tm = 1
0.6024K u
0.34Tu
012 . Tu
τ m Tm = 2
0.7576 Ku
0.48Tu
011 . Tu
τ m Tm = 0.2
15 . Tm Km τ m
25 . τm
04 . τm
K m e− sτ - Alternative filtered derivative controller 1 + sTm m
Table 38: PID tuning rules – FOLPD model
2 1 1 + 05 . τm s + 0.0833τ m s 2 G c ( s) = Kc 1 + . 1 tuning rule. 2 Ti s [1 + 01. τ ms]
Kc
Ti
Td
aTm Km τ m
Tm
025 . Tm
Rule
Comment
Direct synthesis Tsang et al. [110] Model: Method 13
a 1.851 2 1.552 0
ξ
a
0.0 1.329 3 0.1 1.159 5
ξ
a
0.2 1.028 0 0.3 0.924 6
ξ
a
0.4 0.841 1 0.5 0.768 0
ξ
a
0.6 0.695 3 0.7 0.621 9
ξ
a
0.8 0.552 7 0.9
ξ 1.0
K m e− sτ K - I-PD controller U(s) = c E (s) − K c (1 + Td s)Y(s) . 2 tuning 1 + sTm Ti s m
Table 39: PID tuning rules – FOLPD model
rules.
Kc
Rule
Ti
Td
Comment
Ti ( 64a )
Td ( 64 a )
Underdamped system response - ξ = 0.707 .
Direct synthesis Chien et al. [74] K c ( 64a )
1
τ m ≤ 0.2Tm
Model: Method 6 Minimum ISE – Argelaguet et al. [114a]. Model: Method not defined
1
Kc (64a ) =
Td (64 a ) =
Tm + 0. 5τ m
2Tm + τ m 2K m τ m
1.414TCLTm + τ m Tm + 0.25τ m 2 − TCL2
(
Km TCL2
+ 0.707TCL τ m + 0.25τ m
2
)
, Ti (64 a ) =
0707 . TmTC Lτ m + 0.25Tm τ m2 − 05 . τ m TCL2 2 Tm τ m + 025 . τ m + 1414 . TCLTm − TC L2
Tm τ m 2Tm + τ m
1414 . TCL Tm + τ mTm + 0.25τ m2 − TCL2 Tm + 05 . τm
First order Pade approximation for τm
Table 40: PID tuning rules - FOLPD model - G m ( s) =
K me − sτ – Two degree of freedom controller: 1 + sTm m
1 Td s U (s) = K c 1 + + Tis T 1+ d N
Kc
Rule
Ti
Servo/regulator tuning
Kc
( 64b ) 2
Ti
Model: ideal process
Kc
Td
( 64 b)
( 64 b)
Td
Comment
Minimum performance index
Taguchi and Araki [61a]
2
β T s d E( s) − K c α + R (s) . 1 tuning rule. Td s 1+ s N
1 1. 224 = 0.1415 + τm Kc − 0.001582 Tm
( 64 b)
Td
( 64b )
τm ≤ 1.0 ; Tm Overshoot (servo step) ≤ 20% ; settling time ≤ settling time of tuning rules of Chien et al. [10]
2 3 ( 64 b) 0.01353 + 2.200 τm − 1. 452 τm + 0. 4824 τm , T = T m i Tm Tm Tm
2 2 τm τm τm τm = Tm 0.0002783 + 0. 4119 − 0. 04943 , α = 0. 6656 − 0. 2786 + 0. 03966 , Tm Tm Tm Tm
β = 0. 6816 − 0.2054
τm τ + 0.03936 m Tm Tm
2
1 Table 41: PID tuning rules - non-model specific – ideal controller G c ( s) = Kc 1 + + Td s . 25 tuning rules. Ti s Rule
Kc
Ti
Td
Comment
Ultimate cycle Ziegler and Nichols [8]
[ 0.6Ku , Ku ]
0.5Tu
0125 . Tu
Quarter decay ratio
Blickley [115]
0.5K u
Tu
. Tu , 0167 . Tu ] [ 0125
Quarter decay ratio
0.5K u
Tu
0.2Tu
Parr [64] – pages 190-191
05 . Ku
034 . Tu
008 . Tu
Overshoot to servo response ≈ 20% Quarter decay ratio
De Paor [116]
0.866Ku
0.5Tu
0125 . Tu
phase margin = 30 0
Corripio [117] – page 27 Mantz andTacconi [118]
0. 75Ku
0. 63Tu
01 . Tu
Quarter decay ratio
0.906K u
0.5Tu
0125 . Tu
phase margin = 25 0
0.4698 K u
0.4546Tu
01136 . Tu
01988 . Ku
12308 . Tu
0.3077Tu
Gain margin = 2, phase margin = 20 0 Gain margin = 2.44, phase margin = 61 0
0.2015K u
0.7878Tu
01970 . Tu
Gain margin = 3.45, phase margin = 46 0
0.35Ku
0.77Tu
0.19Tu
Atkinson and Davey [119] ** Perry and Chilton [120]
0. 25Ku
0.75Tu
0. 25Tu
033 . Ku
05 . Tu
033 . Tu
Gain margin ≥ 2 , phase margin ≥ 450 20% overshoot - servo response ‘Some’ overshoot
02 . Ku
05 . Tu
033 . Tu
‘No’ overshoot
Yu [122] – page 11
033 . Ku
05 . Tu
0125 . Tu
‘Some’ overshoot
02 . Ku
05 . Tu
0125 . Tu
‘No’ overshoot
Luo et al. [121]
0.48Ku
0.5Tu
0125 . Tu
McMillan [14] – page 90 McAvoy and Johnson [83] Karaboga and Kalinli [123]
0.5K u
0.5Tu
0125 . Tu
054 . Ku
Tu
0.2Tu
Astrom and Hagglund [3] – page 142 Astrom and Hagglund [93]
. Ku ,0.6Ku ] [ 032
04267 . K u 15 . Ku , Tu Tu Tu (1 − cos φ m ) π sin φm
. Ku Tu ,015 . Ku Tu ] [ 008 Tu (1 − cos φ m ) 4π sin φ m
φ m = phase margin
Tu 2 tan φ m + 1 + tan φ m 4π
Tu 2 tan φ m + 1 + tan φ m 16π
‘small’ time delay
05 . Ku
12 . Tu
0125 . Tu
‘aggressive’ tuning
0.25Ku
12 . Tu
0125 . Tu
‘conservative’ tuning
Hang and Astrom [124]
Ku sinφ m
Astrom et al. [30]
Ku cos φ m
St. Clair [15] page 17
Kc
Rule Pole placement - Shin et al. [125]
Kc
Other rules Harriott [126] – pages 179-180
Ti
Td
αTi
( 65)
Comment Typical α : 0.1 Typical ξ : [0.3,0.7]
( 65)
K25%
0167 . T25%
0.667T25%
Quarter decay ratio Quarter decay ratio
K25%
0.67T25%
017 . T25%
Parr [64] – pages 191, 193
05 . G p ( jω u )
Tu
0.25Tu
McMillan [14] Page 43
083 . K25%
0.5T25%
01 . T25%
‘Fast’ tuning
0.67K25%
0.5T25%
01 . T25%
‘Slow’ tuning
1
Ti 4
φ m = 450 ,‘small’ τ m
2 2 G p ( jω u )
1 ωu
Kc( 66) 2
Ti (66 )
Td ( 66)
Calcev and Gorez [69] Zhang et al. [127]
1
Ti
( 65) 1
K c ( 65) =
Ti ( 65) =
ρ 1 1 ( 65) ( 65) a 1 αω uTi − − a 2 αω 1Ti − − b2 − ρ a 2 ( 65) ( 65) ω u Ti ω 1Ti
φ m = 15 0 , ‘large’ τ m
,
1
( a2 − a1 ) + (a 1 − a 2 ) 2 + 4(αρa1ω u + αb2 ω 1 ) ρωa1 + ωb 2 2(αρa1ω u + αb2 ω 1 )
u
1 ( ω − ω1 ) 1 − ξ2 , b = 1 sin ∠G ( jω ) , a = − 1 cos ∠G p ( jω 1 ) , ρ = u 2 p 1 1 K1 ωu ξ K1 Ku K1 , ω1 = modified ultimate gain and corresponding angular frequency
(
a2 =
2
)
(
1
Kc ( 66) =
(
1 + tan φm − φ p
(
π 2 A2 − ε 2
)
) + sin
)
, d, ε = relay amp. and deadband, A = limit cycle amp.
φm 16d 2 Crossing point of the Nyquist curve and relay with hysteresis is outside the unit circle: 2
2
2
Ti ( 66) =
ωu −1 ωc ω ω u β − u tan φ m − φ p ωc
(
(
)
, 10 . +
ωu ω tan φ m − φ p ≤ β ≤ 1.2 + u tan φ m − φ p ωc ωc
(
)
(
)
)
βω u − ω c tan φm − φ p 66 Td ( ) = , ω c = frequency when the open loop gain equals unity. 2 2 ωu − ωc Crossing point of the Nyquist curve and relay with hysteresis is within the unit circle: 2
Ti ( 66) =
ωu −1 ωc ω ω u β + u tan φ p − φ m ωc
(
(
)
)
,
ωu − ωr ω − ωr − 0.4 ≤ β ≤ u + 0.4 ωr ωr
βω u − ω c tan φm − φ p 66 Td ( ) = , ω r = oscillation frequency when a pure relay is switched into closed loop. 2 2 ωu − ωc
Kc
Rule Garcia and Castelo [127a]
3
Kc
Td
( 66a )
( 66a )
=
=
Kc
( 66a ) 3
(
cos 180 0 + φm − ∠G p ( jω1 )
(
G p ( jω1 )
Td
( 66a )
( 66a ) d
Ti
), T
( 66a )
i
)
sin 180 0 + φm − ∠G p ( jω1 ) + 1
(
Ti
2ω1 cos 180 0 + φm − ∠G p ( jω1 )
=
(
T
)
2[sin 180 0 + φ m − ∠G p ( jω1 ) + 1]
(
Comment
ω1 cos 180 + φ m − ∠G p ( jω1 ) 0
)
,
) , ω = oscillation frequency when a sine function is placed in series
with the process in closed loop; ω1 < ωu .
1
Table 42: PID tuning rules - non-model specific – controller with filtered derivative 1 Td s . 8 tuning rules. Gc ( s) = Kc 1 + + Ti s 1 + Td s N
Kc
Ti
Ti 2( A 1 − Ti )
A3
Rule
Td
Comment
Td ( 67) 4
N < 10
A3A 4 − A2 A 5 2 A 3 − A1A 5
N ≥ 10
Direct synthesis
A 2 − Td
( 67)
Td
A1 −
N
Vrancic [72] A3
(
2 A 1A 2 − A 3K m − Td A
2 1
A3 A 2 − Td A1
)
05 . Ti A1 − Km Ti
( 67 ) 2
A2 − A 2 2 − 4χ A1A 3 2χ A1
χTi
N = 10 χ = [ 02 . , 0.25]
Td (67 a)
8 ≤ N ≤ 20
A3
Vrancic [73]
Kc
− A3 − A 5A1 + 2
Td ( 67) =
4
(A
2 3
− A 5A1
A 2 − Td (67 a) A 1 −
)
2
−
Td (67 a) Km N
4 ( A 3A 2 − A5 )( A5 A2 − A4 A 3 ) N
2 ( A 3A2 − A5 ) N
t
t
y( τ ) y1( t) = Km − dτ , y2 ( t) = ∆u
∫ 0 t
y5 ( t) =
2
( 67a )
∫ [A
4
∫ (A 1 − y1 (τ))dτ ,
t
y3 ( t) =
0
∫ ( A 2 − y2 (τ))dτ ,
t
y 4 (t ) =
0
∫ [A
3
]
− y 3 ( τ ) dτ ,
0
]
− y 4 ( τ) dτ , A1 = y1( ∞) , A 2 = y 2 ( ∞) , A 3 = y3 ( ∞) , A4 = y 4 ( ∞) , A 5 = y5 ( ∞)
0
Alternatively, if the process model is G m ( s) = K m
(
1 + b1s + b2 s 2 + b3 s3 + b4 s 4 + b5 s5 − sτ e , then 1+ a 1s + a 2 s2 + a 3s 3 + a 4 s4 + a 5s 5 m
)
A1 = Km ( a1 − b1 + τ m ) , A2 = K m b 2 − a2 + A1a1 − b1 τm + 05 . τ m2 ,
( (b − a (a − b
)
A3 = Km a 3 − b3 + A 2 a1 − A1a 2 + b2 τ m − 05 . b1τ m 2 + 0167 . τ m3 ,
)
A4 = Km
4
4
+ A 3a 1 − A 2a 2 + A 1a 3 − b3 τ m + 05 . b2 τ m 2 + 0167 . b1τ m 3 + 0.042 τ m 4 ,
A5 = Km
5
5
+ A 4 a1 − A 3a 2 + A2 a 3 − A1a 4 + b 4 τm − 05 . b3 τ m2 + 0167 . b2 τ m 3 − 0.042b1τ m 4 + 0.008τm 5
K c ( 67a ) =
A3 [ T (67 a ) ]2 ( 67 a ) 2 2 A 1A 2 − A 0A 3 − Td A1 − d A 0A 1 N
Td (67a ) is obtained from a solution of the following equation:
[
A0 A3 ( 67a ) Td N3
]
4
+
[
A1A 3 ( 67 a ) Td N2
]
3
−
[
A 0A 5 − A 2A 3 ( 67 a ) Td N
] + (A 2
2 3
)[
− A 1A5 Td
( 67a )
] + (A A 2
5
− A 3A4 ) = 0
)
Rule
Kc
Ti
Td
Comment
Direct synthesis Lennartson and Kristiansson [157] Model: Method 1
Kc(111) 5
Ti (111)
0.4Ti (111)
Ku Km ≤ 0.6
Kc(111a )
Ti (111a)
0.4Ti (111a)
Ku Km > 10 ω u K u Km ≤ 0.4
Kc(111b )
Ti (111b )
0.4Ti (111b)
Ku K m > 10 ω u Ku K m ≥ 0.4
Kc(111c) 6
Ti (111c)
0.4Ti (111c)
Ku Km > 10 ω u K u Km ≤ 0.4
Kc(111d )
Ti (111d)
0.4Ti (111d )
Ku Km > 10 ω u Ku K m ≥ 0.4
( 111e)
( 111e)
( 111e)
Kristiansson and Lennartson [157] Model: Method 1
Kc
5
12Ku K m − 35K u Km + 30 2
Kc (111) = Ku K m
N=
0.4Ti
2
[
Ku K m + 25 . 12K u Km − 35Ku K m + 30 2
2
]
, Ti
( 111)
K c (111)
=
−0.053ω u + 0.47ω u − 014 . ω u + 0.11 3
2
,
2.5 35 30 + 12 − Km K u K m K u K m 2 Ku 2
Kc (111a ) = Ku Km N=
Ti
Ku K m > 167 . ω u Ku K m > 0.45 N = 2.5
2 2 K c (111a ) 12 Ku Km − 35Ku Km + 30 (111a ) , T = , i −0.525ω u3 + 0.473ω u2 − 0.143ω u + 0.113 Ku Km + 3 12Ku 2K m2 − 35KuK m + 30
[
]
3 35 30 + 12 − 2 2 Km Ku K mK u Km Ku 2 2 Kc 12Ku Km − 35Ku Km + 30 ( 111b) , Ti = , 3 2 −0185 . ω u + 1052 . ω u − 0854 . ω u + 0.309 K uKm + 3 12Ku2 Km 2 − 35Ku Km + 30
( 111b)
Kc (111b ) = Ku Km N=
6
3 Km Ku
[
35 30 + 12 − 2 2 K mK u Km Ku
Kc (111c ) = Ku K m
N=
]
( 111c )
12Ku K m − 35K u K m + 30 2
2
[
Ku K m + 25 . 12K u Km − 35Ku Km + 30 2
2
]
, Ti
( 111c )
=
−0525 . ωu
3
Kc , 2 + 0.473ω u − 0143 . ω u + 0113 .
2.5 35 30 + 12 − Km K u K m K u K m 2 Ku 2
Kc (111d ) = Ku K m
12Ku K m − 35K u Km + 30 2
[
( 111d)
2
Ku K m + 25 . 12K u Km − 35Ku K m + 30 2
2
]
, Ti
( 111d )
=
−0185 . ωu
3
Kc , 2 + 1052 . ω u − 0.854ω u + 0309 .
2.5 35 30 + 12 − Km K u K m K u K m 2 Ku 2 7.71 914 . K c (111e ) ( 111e ) (111e ) Kc = + 314 . , Ti = 2 2 − 3 K m Ku −0.63ω u + 0.39ω u 2 + 0.15ω u + 0.0082 Km K u N=
Rule
Kc
Ti
Td
Kristiansson and Lennartson [158] Model: Method 1
Kc(111f) 7
Ti (111f )
0.4Ti (111f )
Comment
(111g)
Kristiansson and Lennartson [158a]
Kc Kc
Model: Method not specified
7
Kc (111f ) =
N=
2.5 Km K u
Ti
(111g )
(111g )
Td
(111g )
Tf
(111g)
3
2
Kc
Ti
Td
(111h )
Tf
(111h )
Td
(111g )
Td
(111h )
( 111f )
]
Tf given below; 0.1 ≤ K u K m ≤ 0.5 Tf
[
3
2
]
1. 6( −20 + 13 K m K u )(1+ 0.37 K m K u ) − 1 2 2 2 K m K u ( −20 + 13K m K u )(1 + 0.37 K m K u ) 2 1.1K m K u − 2.3K m K u + 1. 6 2 2 K K (1. 1K u K m − 2. 3K u K m + 1.6) 1.6( −20 + 13K m K u )(1 + 0. 37K m K u ) = m u − 1 ω u ( −20 + 13K m K u )(1 + 0. 37K m K u ) 2 1. 1K m 2 K u 2 − 2. 3K m K u + 1.6 2
2
( −20 + 13K m K u ) 2 (1 + 0.37 K m K u ) 2 2 2 2 2 2 K K (1. 1K u K m − 2.3K u K m + 1. 6) (1.1K m K u − 2.3K m K u + 1.6) = m u − 1 ω u ( −20 + 13K m K u )(1 + 0.37 K m K u ) 2 1.6( −20 + 13K m K u )(1 + 0. 37K m K u ) − 1 2 2 1.1K m K u − 2. 3K m K u + 1.6
=
K m K u (1.1K u K m − 2.3K u K m + 1.6)
=
2
ω u ( − 20 + 13 K m K u )(1 + 0.37 K m K u ) 2
=
2 2 2 (1.1K u K m − 2.3K u K m + 1. 6) 4. 8K m K u (1 + 0.37 K m K u ) − 1 2 2 2 2 2 3ω u K m K u (1+ 0.37 K m K u ) 1. 1K m K u − 2. 3K m K u + 1.6
2 2 (1.1K u K m − 2.3K u K m + 1.6) 4.8K m K u (1+ 0.37 K m K u ) − 1 2 2 2 3ω u (1 + 0.37 K m K u ) 1.1K m K u − 2.3K m K u + 1.6
3K m 2 K u 2 (1+ 0.37 K m K u ) 2 2 2 2 2 2 (1.1K u K m − 2.3K u K m + 1.6) (1.1K m K u − 2.3K m K u + 1. 6) = − 1 2 4 . 8 K K ( 1 + 0 . 37 K K ) 3ω u (1 + 0. 37K m K u ) m u m u −1 1. 1K m 2 K u 2 − 2.3K m K u + 1.6
1.1K u K m − 2. 3K u K m + 1.6 2
=
2
3ω u (1 + 0.37 K m K u ) 2
(111h )
given below;
K u K m > 0.5
35 30 + 12 − K m K u K m 2 Ku 2
(1. 1K u K m − 2. 3K u K m + 1.6)
=
(111h )
(111h )
Ti
(111h )
2
[
2
8
(111h ) 8
Ti
(111g )
12 Ku Km − 35Ku Km + 30Ku Kc Km Ku , Ti (111f ) = , 3 3 2 2 2 2 Km Ku + 25 . 12K u Km − 35Ku Km + 30 ω u 095 . Km Ku − 2Km Ku + 14 .
2
Kc
(111g )
Kc
Rule Kristiansson and Lennartson [158a] (continued) Model: Method not specified Other
Kc
Ti
(111i) 9
Ti
Leva [67]
(1 − ω
2
G p ( jω )
(1 − ω
)
2
+ ω Ti 2 2
Ti Td
)
2
+ ω Ti 2
ω 1+
2π β
)
(
tan φ m − φ ω − 0.5π
)
αTd (68) 10
Comment
Tf
(111i )
[
2 tan φ m + 1 + tan 2 φ m ωu
]
(111i )
given below;
K u K m < 0.1
2π βω
β = 10 φω > φm − π
Td ( 68)
6 ≤ α ≤ 10 φω < φm − π
Ti 4
Parameters determined at φ m = 300 ,450 ,600
2
Ku cosφ m
Astrom [68]
Td
tan φm − φ ω − 0.5π
Ti Td
ωTi 2
(111i )
(
ωTi G p ( jω )
Td
20 .8(1.8 + 0.3K m K135 0 ) − 1 13 K m (1.8 + 0.3K m K 1350 ) ( −6 + 3. 7K m K 1350 ) ( −6 + 3.7K 1350 K m ) K 1350 K m 20. 8(1.8 + 0. 3K m K 1350 ) (111i ) Ti = − 1 13ω1350 (1. 8 + 0.3K m K135 0 ) 2 (− 6 + 3.7K m K 1350 ) 2 169 (1. 8 + 0.3K m K135 ) ( −6 + 3. 7K m K135 )K m K135 ( −6 + 3.7K m K 135 ) K m K 135 ( −6 + 3. 7K m K 135 ) 2 (111i ) ( 111i) Td = − 1 , Tf = 2 20 .8(1. 8 + 0 .3K K 13ω 135 (1.8 + 0.3K m K 135 ) 2 13ω135 (1. 8 + 0.3K m K135 ) m 135 ) −1 ( −6 + 3.7K m K135 )
9
Kc
(111i )
=
( −6 + 3.7K 1350 K m ) 2
2
0
0
0
0
0
0
0
10
Td ( 68) =
− αω + α 2ω 2 + 4αω 2 tan 2 ( φ m − φ ω − 05 . π) 2αω 2 tan(φ m − φω − 0.5π )
0
0
0
0
0
Table 43: PID tuning rules - non-model specific – ideal controller with set-point weighting 1 U( s) = Kc Fp R(s) − Y(s) + ( Fi R(s) − Y(s)) + Td s( Fd R(s) − Y(s)) . 1 tuning rule. Ts i
(
)
Rule
Kc
Ti
Td
Comment
Ultimate cycle Mantz and Tacconi [118]
0.6K u
0.5Tu Fi = 1
0125 . Tu Fd = 0.654
Quarter decay ratio
Fp = 017 .
Table 44: PID tuning rules - non-model specific – ideal controller with proportional weighting 1 G c ( s) = Kc b + + Td s . 1 tuning rule. Ti s
Kc
Rule Direct synthesis
Astrom and Hagglund [3] – page 217
(0.33e
− 0.31κ − κ
Ti 2
)K
u
κ = 1 Km Ku
,
Td
Comment
)T
b = 058 . e −1.3κ + 3.5κ 0 < Km Ku < ∞ maximum Ms = 1.4
)T
b = 025 . e 0.56 κ − 0.12 κ 0 < Km Ku < ∞ maximum Ms = 2.0
2
(0.76e
−1.6κ − 0.36κ
(0.59e
−1.3κ + 0.38 κ
2
)T
. e (017
−0.46κ − 2.1κ
)T
. e (015
−1.4κ + 0.56κ
u
2
u
2
(0.72e
− 1.6κ + 1.2 κ
2
)K
u
2
u
2
u
Rule Ultimate cycle Fu et al. [128]
Table 45: PID tuning rules - non-model specific – non-interacting controller 1 K Ts U( s) = Kc 1 + E( s) − c d Y( s) . 1 tuning rule. sT Ti s 1+ d N Kc Ti Td
0.5K u
0.34Tu
0.08Tu
Comment
1 Table 46: PID tuning rules - non-model specific – series controller U( s) = Kc 1 + (1 + sTd ) . 3 tuning rules. Ti s Rule
Kc
Ti
Td
Comment
025 . Ku
033 . Tu
05 . Tu
‘optimum’ servo response
0.2K u
0. 25Tu
05 . Tu
0.2K u
0.5Tu
0. 33Tu
0. 33Ku
0. 33Tu
0.5Tu
‘optimum’ regulator response - step changes No overshoot; close to optimum regulator ‘Some’ overshoot
025 . Ku
033 . Tu
0.5Tu
Ultimate cycle Pessen [131]
Pessen [129] Grabbe et al. [130]
Table 47: PID tuning rules - non-model specific – series controller with filtered derivative 1 sTd . 1 tuning rule. U( s) = Kc 1 + 1 + sT Ti s 1 + d N Rule
Kc
Ti
Td
Ultimate cycle Hang et al. [36] - page 58
0.35Ku
113 . Tu
0. 20Tu
Comment
1 1 + sTd Table 48: PID tuning rules - non-model specific – classical controller U( s) = Kc 1 + . 1 tuning rule. T Ti s 1+ s d N Rule
Kc
Ti
Td
Comment
Ultimate cycle Corripio [117] – page 27
0.6K u
05 . Tu
0125 . Tu
10 ≤ N ≤ 20
Table 49: PID tuning rules - non-model specific – non-interacting controller 1 U( s) = Kc 1 + E ( s) − Kc Td sY (s) . 1 tuning rule. Ti s Rule
Kc
Ti
Td
0.75K u
0.625Tu
01 . Tu
Ultimate cycle VanDoren [114]
Comment
Table 50: PID tuning rules - IPD model
K m e− sτ m s
Kc
Rule
- ideal controller Gc ( s) = K c 1 + 1 + Td s . 5 tuning rules.
Ti s
Ti
Td
βτ m
χτ m
Comment
Direct synthesis
α Wang and Cluett [76] K mτ m – deduced from Closed loop Damping graph time Factor, ξ constant Model: Method 2 τm 0.707
Gain margin Phase margin Am φ m [degrees]
α
β
χ
2.0
32
0.9056
2.6096
0.3209
2τm
0.707
3.1
40
0.5501
4.0116
0.2205
3τ m
0.707
4.4
46
0.3950
5.4136
0.1681
4τm
0.707
5.5
49
0.3081
6.8156
0.1357
5τ m
0.707
6.7
52
0.2526
8.2176
0.1139
6τ m
0.707
7.8
54
0.2140
9.6196
0.0980
7τm
0.707
8.9
55
0.1856
11.0216
0.0861
8τ m
0.707
10.0
56
0.1639
12.4236
0.0767
9τm
0.707
11.2
57
0.1467
13.8256
0.0692
10τ m
0.707
12.2
58
0.1328
15.2276
0.0630
11τm
0.707
13.4
59
0.1213
16.6296
0.0579
12τ m
0.707
14.5
59
0.1117
18.0316
0.0535
13τ m
0.707
15.6
59
0.1034
19.4336
0.0497
14τ m
0.707
16.7
60
0.0963
20.8356
0.0464
15τ m
0.707
17.8
60
0.0901
22.2376
0.0436
16τ m
0.707
19.0
60
0.0847
23.6396
0.0410
τm
1.0
2.0
37
0.8859
3.2120
0.3541
2τm
1.0
2.9
46
0.6109
5.2005
0.2612
3τ m
1.0
3.8
52
0.4662
7.1890
0.2069
4τm
1.0
4.6
56
0.3770
9.1775
0.1713
5τ m
1.0
5.5
58
0.3164
11.1660
0.1462
6τ m
1.0
6.4
61
0.2726
13.1545
0.1275
7τm
1.0
7.1
62
0.2394
15.1430
0.1311
8τ m
1.0
8.0
64
0.2135
17.1315
0.1015
9τm
1.0
8.7
65
0.1926
19.1200
0.0921
10τ m
1.0
9.5
66
0.1754
21.1085
0.0843
11τm
1.0
10.4
67
0.1611
23.0970
0.0777
12τ m
1.0
11.1
67
0.1489
25.0855
0.0721
13τ m
1.0
12.0
68
0.1384
27.0740
0.0672
14τ m
1.0
12.8
68
0.1293
29.0625
0.0630
15τ m
1.0
13.4
69
0.1213
31.0510
0.0592
16τ m
1.0
14.4
69
0.1143
33.0395
0.0559
Rule
Kc
Ti
Td
Comment
Cluett and Wang [44]
0.9588 Km τ m
30425 . τm
0.3912 τ m
Closed loop time constant = τ m
Model: Method 2
0.6232 Km τ m
52586 . τm
0.2632 τ m
Closed loop time constant = 2τm
0.4668 Km τ m
7.2291τ m
0.2058τ m
Closed loop time constant = 3τ m
0.3752 Km τ m
91925 . τm
01702 . τm
Closed loop time constant = 4τ m
0.3144 Km τ m
111637 . τm
01453 . τm
Closed loop time constant = 5τ m
0.2709 Km τ m
131416 . τm
01269 . τm
Closed loop time constant = 6τ m
160 . τm
0.48τ m
1.48 K mτm
2τ m
0.37τ m
Decay ratio: 2.7:1
094 . Km τ m
2τ m
0.5τ m
Ultimate cycle ZieglerNichols equivalent
Rotach [77] Model: Method 4 Process reaction Ford [132] Model: Method 2 Astrom and Hagglund [3] – page 139 Model: Method not relevant
121 . K mτ m
Damping factor for oscillations to a disturbance input = 0.75.
Table 51: PID tuning rules - IPD model
K m e− sτ m s
- Ideal controller with first order filter, set-point weighting and
1 1 1 + 0.4Trs output feedback U(s ) = K c 1 + + Tds − Y(s ) − K 0 Y(s) . 1 tuning rule. R( s) Ti s 1 + sTr Tf s + 1
Rule Direct synthesis Normey-Rico et al. [106a] Model: Method not specified
Kc
Ti
Td
Comment
Tf = 0. 13τm 0.563 K m τm
1.5τm
0.667 τm
1 2Km τ m Tr = 0.75τ m
K0 =
Table 52: PID tuning rules - IPD model
K m e− sτ m s
- ideal controller with filtered derivative
1 Td s G c ( s) = Kc 1 + + sT Ts i 1+ d N
Rule Robust Chien [50] Model: Method 2
. 1 tuning rule.
Kc
Ti
Td
2 K m ( λ + 0.5τm )
2λ + τm
τ m ( λ + 025 . τm ) 2 λ + τm
Comment
λ=
1 ; N=10 Km
K m e− sτ m - series controller with filtered derivative s 1 Td s . 1 tuning rule. G c ( s) = Kc 1 + 1 + Ts Ts i 1 + d N
Table 53: PID tuning rules - IPD model
Rule
Kc
Ti
Td
Comment
1 2λ + 0.5τ m 2 K m [ λ + 05 . τ m ]
2λ + 05 . τm
05 . τm
1 λ= , τ m ; N=10 K m
1 05 . τm 2 K m [ λ + 05 . τ m ]
05τ m
2λ + 05 . τm
1 λ= , τ m ; N=10 K m
Robust Chien [50] Model: Method 2
1 1 + Td s . - classical controller G c ( s) = Kc 1 + Ti s 1 + Td s N 5 tuning rules.
K e− sτ m Table 54: PID tuning rules - IPD model m s
Rule
Kc
Ti
Td
Comment
Ultimate cycle Luyben [133] Model: Method 2
0.46Ku
2.2Tu
016 . Tu
maximum closed loop log modulus of +2dB ; N=10
311 . Ku
2.2Tu
364T . u
N=10
0.5τm
2λ + 05 . τm
1 λ= , τ m ; N=10 K m
2λ + 0.5τ m
0.5τm
1 λ= , τ m ; N=10 K m
Belanger and Luyben [134] Model: Method 2 Robust Chien [50] Model: Method 2
Regulator tuning Minimum IAE Shinskey [17] – page 121. Model: Method not specified Minimum IAE Shinskey [17] – page 117. Model: Method 1 Minimum IAE – Shinskey [59] – page 74 Model: Method not specified
1 Km
05 . τ m λ + 0.5τ 2 [ m]
1 2λ + 0.5τ m 2 K m [ λ + 05 . τ m ]
Minimum performance index
056 . Ku
0.39Tu
015 . Tu
0.93 Km τ m
157 . τm
056 . τm
09259 . Km τ m
160 . τm
0.58τ m
N=10
09259 . Km τ m
1.48τ m
0.63τm
N=20
Table 55: PID tuning rules - IPD model
K m e− sτ m s
- Alternative non-interacting controller 1 -
1 U( s) = Kc 1 + E( s) − Kc Td sY ( s) . 2 tuning rules. Ti s Rule Regulator tuning Minimum IAE Shinskey [59] – page 74. Model: Method not specified Minimum IAE – Shinskey [17] – page 121. Model: Method not specified Minimum IAE – Shinskey [17] – page 117. Model: Method 1
Kc
Ti
Td
Minimum performance index
12821 . K m τm
19 . τm
046 . τm
0. 77K u
0. 48Tu
015 . τm
128 . K mτ m
1.90τ m
0.48τ m
Comment
Table 56: PID tuning rules - IPD model
Kc
Rule Direct synthesis Chien et al. [74] Model: Method 1
1
Kc
( 68a )
(
Kc
( 68a ) 1
1.414 TCL + τ m 2 CL
Km T
+ 0.707TCL τ m + 0.25τ m
2
)
K m e− sτ m - I-PD controller U(s) = Kc E (s) − K c(1 + Td s)Y(s) . s Ti s 1 tuning rule.
Ti
Td
Comment
1414 . TCL + τ m
0.25τ m2 + 0.707TC Lτ m 1.414TCL + τ m
Underdamped system response - ξ = 0.707 . τ m ≤ 0.2Tm
Table 57: PID tuning rules - IPD model G m ( s) =
U(s) = Kc (1 +
Rule Direct synthesis Hansen [91a] Model: Method not specified
K me− sτ m s
- controller
1 ) E( s) + Kc ( b − 1) R (s) − Kc TdsY( s) . 1 tuning rule. Tis
Kc
Ti
Td
Comment
0.938 K m τm
2. 7τm
0.313 τm
b = 0.167
Table 58: PID tuning rules - IPD model
K m e− sτ m s
- Two degree of freedom controller:
1 Td s U (s) = K c 1 + + Tis T 1+ d N Rule
Kc
Servo/regulator tuning
1 1.253 K m τ m
Model: ideal process
α = 0. 6642
φ c = phase corresponding to the crossover frequency; Km =1
Td
Comment
Minimum performance index
Taguchi and Araki [61a]
Minimum ITAE Pecharroman and Pagola [134a]
Ti
β T s d E( s) − K c α + R (s) . 2 tuning rules. Td s 1+ s N
2.388 τ m
0.4137 τm
β = 0.6797
1.672 K u
0.366 Tu
0.136 Tu
1.236 K u
0.427 Tu
0.149 Tu
0.994 Ku
0.486 Tu
0.155 Tu
0.842 Ku
0.538 Tu
0.154 Tu
0.752 Ku
0.567 Tu
0.157 Tu
0.679 Ku
0.610 Tu
0.149 Tu
0.635 K u
0.637 Tu
0.142 Tu
0.590 Ku
0.669 Tu
0.133 Tu
0.551K u
0.690 Tu
0.114 Tu
0.520 K u
0.776 Tu
0.087 Tu
0.509 Ku
0.810 Tu
0.068 Tu
τm ≤ 1.0 ; Tm Overshoot (servo step) ≤ 20% ; settling time ≤ settling time of tuning rules of Chien et al. [10] α = 0. 601 , β = 1 , N = 10, φ c = −1640
α = 0.607 , β = 1 , N = 10, φ c = −1600
α = 0.610 , β = 1 , N = 10, φ c = −1550
α = 0.616 , β = 1 ,
Model: Method 6
N = 10, φ c = −1500
α = 0.605 , β = 1 , N = 10, φ c = −1450
α = 0.610 , β = 1 , N = 10, φ c = −1400
α = 0.612 , β = 1 , N = 10, φ c = −1350
α = 0.610 , β = 1 , N = 10, φ c = −1300
α = 0.616 , β = 1 , N = 10, φ c = −1250
α = 0.609 , β = 1 , N = 10, φ c = −1200
α = 0. 611 , β = 1 , N = 10, φ c = −1180
Km e − sτ - ideal controller Gc ( s) = K c 1 + 1 + Td s . s(1 + sTm ) Ti s m
Table 59: PID tuning rules - FOLIPD model
1 tuning rule. Rule
Kc
Ti
Td
Comment
Ultimate cycle McMillan [58] Model: Method not relevant
Kc( 69) 1
Ti ( 69 )
Td ( 69)
Tuning rules developed from Ku , Tu
2
1
Kc ( 69)
T 0.65 T 0.65 1111 . Tm 1 ( 69 ) ( 69 ) m = . τ m 1 + m , Ti = 2τ m 1 + , Td = 05 K m τ m 2 T 0.65 τm τ m m 1 + τ m
Km e − sτ - ideal controller with filter s(1 + sTm ) m
Table 60: PID tuning rules - FOLIPD model
1 1 G c ( s) = Kc 1 + + Td s . 3 tuning rules. Ti s 1 + Tf s Kc
Ti
Td
0.463λ + 0277 . [ ] 2 Km τ m
τm Tm + 0.238λ + 0123 .
Tm τ m 0238 . λ + 0123 . ) Tm + τ m (
Rule
Comment
Robust
Tan et al. [81] Model: Method 2 Zhang et al. [135] Model: Method 2
Tan et al. [136] Model: Method 2
([ 0238 . λ + 0123 . ]Tm + τ m )
(
3λ + Tm + τ m
K m 3λ
2
+ 3λτ m + τ m2
)
3λ + Tm + τm
( 3λ + τ m ) Tm 3λ + τ m + Tm
λ = 15 . τ m ….Overshoot = 58%, Settling time = 6τ m λ = 25 . τ m ….Overshoot = 35%, Settling time = 11τ m λ = 35 . τ m ….Overshoot = 26%, Settling time = 16τm λ = 45 . τm ….Overshoot = 22%, Settling time = 20τm
τm 5.750λ + 0.590 λ = 0.5 - performance λ = 0.1 - robustness λ = 0.25 - acceptable Tf =
Tf =
λ3 3λ 2 + 3λτ m + τ m 2
15 . τ m ≤ λ ≤ 4.5τ m Obtained from graph
Tmτ m
0. 0337Tm τm 1 + 2 01225 . T Km τ m m
Tm + 81633 . τm
01225 . Tm + τ m
0. 0754Tm τm 1+ 2 0 . 1863 T Km τ m m
Tm + 53677 . τm
01863 . Tm + τ m
01344 . Tm τm 1+ 0.2523Tm K mτ m2
Tm + 3.9635τ m
0.2523Tm + τ m
Tf = 05549 . τm
Tm τ m
Tf = 0. 4482τ m
Tm τ m
Tf = 0. 2863τm
Km e − sτ - ideal controller with set-point weighting s(1 + sTm ) m
Table 61: PID tuning rules - FOLIPD model
1 G c ( s) = Kc b + + Td s . 1 tuning rule. Ti s Rule
Kc
Ti
Td
Comment
Direct synthesis Astrom and Hagglund [3] - pages 212-213
56 . e −8 .8 τ + 6.8τ , K m ( Tm + τ m )
11 . τm e 6.7 τ − 4.4 τ
τ = τ m ( τ m + Tm )
Model: Method 1 or Method 2.
86 . e −7.1τ + 5.4 τ K m ( Tm + τ m )
10 . τ m e3.3τ − 2 .33τ
2
2
17 . τm e −6.4 τ + 2.0 τ
2
Maximum Ms = 1.4
2
b = 012 . e 6.9 τ − 6.6 τ 2
0.38τ me 0.056τ − 0.60 τ
2
2
Maximum Ms = 2.0
b = 056 . e −2 .2τ + 1.2 τ
2
Km e − sτ 1 1 + Td s . Table 62: PID tuning rules - FOLIPD model - classical controller G c ( s) = Kc 1 + s(1 + sTm ) Ti s 1 + Td s N 5 tuning rules. m
Kc
Rule
Ti
Td
Comment
Tm
2λ + τm
λ = [ τ m , Tm ] ; N=10
2λ + τm
Tm
λ = [ τ m , Tm ] ; N=10
Robust Chien [50] Model: Method 2
Regulator tuning Minimum IAE – Shinskey [59] – page 75. Model: Method not specified
1 Km
T m λ+τ 2 [ ] m
1 2λ + τ m K m [ λ + τ m ] 2
Minimum performance index
1.38( τ m + Tm )
0. 78 K m ( τ m + Tm ) 100
(
Minimum IAE 108K m τ m 122 . − 0.03 τ m Shinskey [59] – pages 158-159 100 Model: Open loop T 108K m τ m 1 + 0. 4 τ m m method not specified Tm
(
Minimum ITAE Poulin and Pomerleau [82], [92] – deduced from graph Model: Method 2
Output step load disturbance
Input step load disturbance
)
b
K m (τ m + Tm )
)
Tm 2
a (τ m + T m )
2
0.66(τ m + Tm )
τ m = Tm ; N=10
T − m 157 . τ m 1 + 12 . 1 − e τ m
0.56τ m + 0.75Tm
Tm > 05 . τm
T − m 157 . τ m 1 + 12 . 1 − e τ m
0.56τ m + 075 . Tm
Tm ≤ 05 . τm
a(τ m + Tm )
+1
0≤
Tm
τm ≤2; ( Td N)
01 . Tm ≤
Td ≤ 033 . Tm N
τm ( Td N)
a
b
τm ( Td N)
a
b
0.2 0.4 0.6 0.8 1.0
5.0728 4.9688 4.8983 4.8218 4.7839
0.5231 0.5237 0.5241 0.5245 0.5249
1.2 1.4 1.6 1.8 2.0
4.7565 4.7293 4.7107 4.6837 4.6669
0.5250 0.5252 0.5254 0.5256 0.5257
0.2 0.4 0.6 0.8 1.0
3.9465 3.9981 4.0397 4.0397 4.0397
0.5320 0.5315 0.5311 0.5311 0.5311
1.2 1.4 1.6 1.8 2.0
4.0397 4.0278 4.0278 4.0218 4.0099
0.5312 0.5312 0.5312 0.5313 0.5314
Km e − sτ - series controller with derivative filtering s(1 + sTm ) m
Table 63: PID tuning rules - FOLIPD model
1 Td s . 1 tuning rule. G c ( s) = Kc 1 + 1 + Ts Ts i 1 + d N Kc
Rule
Ti
Td
2λ + τm
Tm
Tm
2λ + τm
Robust Chien [50] Model: Method 2
1 Km
2λ + τ m λ + τ 2 [ m]
1 Tm 2 K m [ λ + τ m ]
Comment
λ = [ τ m , Tm ] ; N=10 (Chien and Fruehauf [137])
λ = [ τ m , Tm ] ; N=10
Km e − sτ - alternative non-interacting controller 1 s(1 + sTm ) m
Table 64: PID tuning rules - FOLIPD model
1 U(s) = Kc 1 + E (s) − K c Td sY ( s) . 2 tuning rules. Ti s Rule Regulator tuning Minimum IAE Shinskey [59] – page 75. Model: Method not specified Minimum IAE Shinskey [59] – page 159. Model: Open loop method not defined
Kc
Ti
Td
Minimum performance index
118 . K m (τ m + Tm )
1.28 K m τ m (1 + 0.24
Tm τm
2
T − 014 . m ) τ m
1.38( τ m + Tm )
0.55( τ m + Tm )
Tm − 19 . τ m 1 + 0.75[1 − e τ m ]
0.48τ m + 0.7Tm
Comment
Km e − sτ - ideal controller with filtered derivative s(1 + sTm ) m
Table 65: PID tuning rules - FOLIPD model
1 Td s . 1 tuning rule. G c ( s) = Kc 1 + + Ti s 1 + s Td N Rule
Kc
Robust Chien [50]
2λ + τ m + Tm
Model: Method 2
K m( λ + τ m)
Ti
2
2λ + Tm + τ m
Td
Tm ( 2λ + τ m ) 2λ + Tm + τ m
Comment
λ = Tm ; N = 10
Km e − sτ - ideal controller with set-point weighting s(1 + sTm ) m
Table 66: PID tuning rules - FOLIPD model
(
)
U(s) = Kc Fp R ( s) − Y( s) +
Kc [ Fi R(s) − Y(s)] + Kc Td s[ Fd R (s) − Y(s)] . 1 tuning rule. Ti s
Rule
Kc
Ti
Td
Ultimate cycle Oubrahim and Leonard [138] Model: Method not relevant
06 . Ku Fp = 01 .
05 . Tu Fi = 1
0125 . Tu Fd = 0.01
Comment
τm < 0.8 ; Tm 20% overshoot
0.05 <
Km e − sτ - Alternative classical controller s(1 + sTm ) m
Table 67: PID tuning rules - FOLIPD model
1+ T s 1 + T s i d G c ( s) = Kc . 1 tuning rule. 1 + Td s 1 + Td s N N Rule Direct synthesis Tsang and Rad [109] Model: Method 3
Kc
Ti
Td
Comment
0.809 K mτ m
Tm
0.5τ m
Maximum overshoot = 16%; N = 8.33
Km e − sτ - Two degree of freedom controller: s(1 + sTm ) m
Table 68: PID tuning rules – FOLIPD model
1 Td s U (s) = K c 1 + + Tis T 1+ d N Rule
Kc
Servo/regulator tuning Taguchi and Araki [61a]
Ti
β T s d E( s) − K c α + R (s) . 2 tuning rules. Td s 1+ s N
Td
Minimum performance index
Kc
( 69a ) 2
Ti
( 69a )
Td
( 69a )
Model: ideal process
Minimum ITAE Pecharroman and Pagola [134a]
φ c = phase corresponding to the crossover frequency; Km = 1 ; Tm = 1 ; 0.05 < τm < 0. 8 .
Comment
1.672 K u
0.366 Tu
0.136 Tu
1.236 K u
0.427 Tu
0.149 Tu
0.994 Ku
0.486 Tu
0.155 Tu
0.842 Ku
0.538 Tu
0.154 Tu
0.752 Ku
0.567 Tu
0.157 Tu
0.679 Ku
0.610 Tu
0.149 Tu
0.635 K u
0.637 Tu
0.142 Tu
0.590 Ku
0.669 Tu
0.133 Tu
0.551K u
0.690 Tu
0.114 Tu
τm ≤ 1.0 ; Tm Overshoot (servo step) ≤ 20% ; settling time ≤ settling time of tuning rules of Chien et al. [10] α = 0. 601 , β = 1 , N = 10, φ c = −1640
α = 0.607 , β = 1 , N = 10, φ c = −1600
α = 0.610 , β = 1 , N = 10, φ c = −1550
α = 0.616 , β = 1 , N = 10, φ c = −1500
α = 0.605 , β = 1 ,
Model: Method 4
N = 10, φ c = −1450
α = 0.610 , β = 1 , N = 10, φ c = −1400
α = 0.612 , β = 1 , N = 10, φ c = −1350
α = 0.610 , β = 1 , N = 10, φ c = −1300
α = 0.616 , β = 1 ,
2
Kc
Td
( 69a )
( 69 a )
1 0.5184 = 0.7608 + τm Kc [ − 0.01308 ] 2 Tm
2 ( 69 a ) 0.03330 + 3.997 τ m − 0.5517 τm , T = T m i Tm Tm
2 3 τm τm τm = Tm 0.03432 + 2. 058 − 1. 774 + 0.6878 , Tm Tm Tm
α = 0. 6647 , β = 0. 8653 − 0. 1277
N = 10, φ c = −1250
τm τ + 0.03330 m Tm Tm
2
Rule Minimum ITAE Pecharroman and Pagola [134a] continued
Kc
Ti
Td
0.520 K u
0.776 Tu
0.087 Tu
0.509 Ku
0.810 Tu
0.068 Tu
Comment
α = 0.609 , β = 1 , N = 10, φ c = −1200
α = 0. 611 , β = 1 , N = 10, φ c = −1180
Km e− sτ K m e− sτ or - ideal controller (1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1 m
Table 69: PID tuning rules - SOSPD model
m
1 Gc ( s) = K c 1 + + Td s . 27 tuning rules. Ti s Kc
Rule Servo tuning Minimum ITAE – Sung et al. [139] Model: Method 2 Regulator tuning Minimum ITAE – Sung et al. [139] Model: Method 2
Ti
Td
Comment
Minimum performance index
Kc( 70)
Ti (70 )
Td
( 70)
0.05 <
τm ≤2 Tm1
0.05 <
τm ≤2 Tm1
Minimum performance index
Kc( 71)
Ti (71)
Td
( 71)
1
1
Kc
Kc
( 70)
( 70)
=
− 0.983 τ 1 −0.04 + 0333 ξ m , ξ m ≤ 0.9 or . + 0.949 m Km Tm1
−0.832 τm 1 τm = −0.544 + 0308 . + 1408 . ξ m , ξ m > 0.9 . Km Tm1 Tm1
τ τ τ τ Ti ( 70) = Tm1 2.055 + 0.072 m ξ m , m ≤ 1 or Ti ( 70) = Tm1 1768 . + 0.329 m ξ m , m > 1 T T T T m1 m1 m1 m1 Td ( 70) =
τ ξ T − 0.870 1 − e 1.060
m
m1
Kc
( 71)
m
Tm1 1.090 T 0.55 + 1.683 m1 τm
−2.001 − 0.766 τ τm τm 1 = −067 . + 0297 . + 2189 . ξ m , m < 0.9 or Km Tm1 Tm1 Tm1
Kc ( 71) =
2 −0.766 τ τ τ 1 − 0365 . + 0.260 m − 14 . + 2189 . m ξ m , m ≥ 0.9 Km Tm1 Tm1 Tm1
τ Ti ( 71) = Tm1 2.212 m Tm1
Ti
( 71)
0.520
τ − 03 . , m < 0.4 or Tm1
ξm − 2 τ 0.15+0.33 m τm Tm1 = Tm1{−0.975 + 0.910 − 1.845 + 1 − e Tm1
Td ( 71) =
2 τm τ } , m ≥ 0.4 5 . 25 − 0 . 88 − 2 . 8 Tm1 Tm1
Tm1 ξm − −1.121 τ 1.171 −0.530 − 0.15+ 0.939 m T τ Tm 1 1 − e 145 . + 0.969 m1 − 19 . + 1576 . m τm Tm1
Kc
Rule
Minimum ITAE – Lopez et al. [84] - taken from plots
Ti
Td
Comment
ξm
τm Tm1
Kc
Ti
Td
0.5
0.1
25 Km
05 . Tm1
0.25Tm1
0.5
1.0
07 . Km
1.3Tm1
12 . Tm1
0.5
10.0
0.35 Km
5Tm1
1.0Tm1
1.0
0.1
25 Km
0.5Tm1
0.2Tm1
1.0
1.0
18 . Km
1.7Tm1
0.7Tm1
4.0
1.0
9.0 Km
2Tm1
0.45Tm1
Model: Method 12
Representative results
Ultimate cycle Regulator – nearly minimum IAE, ISE, ITAE – Hwang [60]
Kc
( 72) 2
Ti
( 72 )
1.45(1 + Ku K m ) 116 . 1 − 2 ε0 Km ω u
Decay ratio = 0.15 τ 0.2 ≤ m ≤ 2.0 and Tm1
. ω u2τ m2 ) ( 1 − 0.612ω u τ m + 0103 0.6 ≤ ξ m ≤ 4.2 ;
Model: Method 3
Kc
( 73)
Ti
(73)
1.45(1 + Ku K m ) 116 . 1 − 2 ε0 Km ω u . ω u2τ m2 ) ( 1 − 0.612ω u τ m + 0103
ε < 2.4
Decay ratio = 0.15 τ 0.2 ≤ m ≤ 2.0 and Tm1
0.6 ≤ ξ m ≤ 4.2 ; 2.4 ≤ ε < 3
2
KH =
τm 2 ξ τ T 4 K mTd Kc τ m 9 2 − Tm1 − m m m + + 2 9 9ω u 2 K m τm 18
4 2 2 2 2 2 3 3 3 2 2 2 2 9 T 4 + τm + 49 τm ξ m Tm − τm Tm1 + 7Tmξ m τ m + 10 τm1 Tm1 ξ m − K T K 10 τm + 4 Tm1 τm (ξ m τ m + Tm1 ) − 8Kc Km Td τ m m1 c d m 2 2 324 81 9 81 9 81 9 2 Km τm 81ω u
ωH = Tm12
ε=
Kc
1 + K HK m 6T 2 + 4Tm1ξ mτ m + K uK mτ m 2 , ε 0 = m1 2 2τ m Tm12ω u 2ξ τ T K K τ 2 Kc KmTd τ m + m m m+ H m m − 3 6 3ω u
6Tm12 + 4 Tm1ξm τ m + KH Km τ m 2 − 4 τ mK mK cTd ( Km Kc Td τ m 2 + 2 τ mTm12 )ω H
( 72)
[
] , T
[
] , T
0674 . 1 − 0.447ω H τ m + 0.0607ω H2 τ m 2 = KH − K m (1 + KH K m )
0.778 1 − 0.467ω H τ m + 0.0609ω H 2 τ m 2 Kc ( 73) = KH − K m (1 + K H K m )
i
i
( 72)
( 73)
=
=
Kc
( 72)
(
(1 + K H K m )
ω H K m 00607 . 1 + 105 . ω H τ m − 0.233ω H 2 τ m 2 Kc
(
( 73)
)
(1 + K H K m )
ω H K m 0.0309 1 + 2.84ω H τ m − 0.532ω H 2 τ m 2
)
Rule
Kc
Ti
Td
Comment
Regulator – nearly minimum IAE, ISE, ITAE – Hwang [60] – continued
Kc( 74) 3
Ti (74 )
1.45(1 + Ku K m ) 116 . 1 − 2 ε0 Km ω u
Decay ratio = 0.15 τ 0.2 ≤ m ≤ 2.0 and Tm1
. ω u2τ m2 ) ( 1 − 0.612ω u τ m + 0103
0.6 ≤ ξ m ≤ 4.2 ; 3 ≤ ε < 20
Model: Method 3
Kc( 75)
Decay ratio = 0.15 τ 0.2 ≤ m ≤ 2.0 and Tm1
1.45(1 + Ku K m ) 116 . 1 − 2 ε0 Km ω u
Ti (75)
. ω u2τ m2 ) ( 1 − 0.612ω u τ m + 0103 0.6 ≤ ξ m ≤ 4.2 ; ε ≥ 20
Kc( 76)
Decay ratio = 0.2 τ 0.2 ≤ m ≤ 2.0 and Tm1
1.45(1 + Ku K m ) 116 . 1 − 2 ε0 Km ω u
Ti (76 )
. ω u2τ m2 ) ( 1 − 0.612ω u τ m + 0103 0.6 ≤ ξ m ≤ 4.2 ; ε < 2.4
Kc
( 77)
Ti
1.45(1 + Ku K m ) 116 . 1 − 2 ε0 Km ω u
( 77 )
. ω u2τ m2 ) ( 1 − 0.612ω u τ m + 0103
Decay ratio = 0.2 τ 0.2 ≤ m ≤ 2.0 and Tm1
0.6 ≤ ξ m ≤ 4.2 ; 2.4 ≤ ε < 3
Kc( 78)
1.45(1 + Ku K m ) 116 . 1 − 2 ε0 Km ω u
Ti (78)
. ω u2τ m2 ) ( 1 − 0.612ω u τ m + 0103
Decay ratio = 0.2 τ 0.2 ≤ m ≤ 2.0 and Tm1
0.6 ≤ ξ m ≤ 4.2 ; 3 ≤ ε < 20
Kc( 79)
Decay ratio = 0.2 τ 0.2 ≤ m ≤ 2.0 and Tm1
1.45(1 + Ku K m ) 116 . 1 − 2 ε0 Km ω u
Ti (79 )
. ω u2τ m2 ) ( 1 − 0.612ω u τ m + 0103 0.6 ≤ ξ m ≤ 4.2 ; ε ≥ 20
3
[
]
( 74) 131 . ( 0519 . . ε + 0514 . ε 2 ) ω H τm 1 − 103 K c (1 + K H K m ) Kc ( 74) = K H − , Ti (74) = K m (1 + KH K m ) ω H K m 0.0603 1 + 0.929 ln[ ω H τ m ] 1 + 2.01 ε − 12 . ε2
[
114 . 1 − 0.482ω H τ m + 0.068ω H 2 τ m 2 Kc ( 75) = KH − K m (1 + K H K m ) Kc
( 76)
Kc
( 77)
Kc
( 78)
[
] , T
0.622 1 − 0.435ω H τ m + 0.052ω H τ m = KH − K m (1 + K H K m ) 2
2
=
( 75)
i
] , T
[
0.724 1 − 0.469ω H τ m + 0.0609ω H 2 τ m 2 = KH − K m (1 + K H K m )
[
)(
(
( 76)
i
] , T
Kc
( 75)
(
(1 + K H K m )
ω H K m 0.0694 − 1 + 21 . ω H τ m − 0.367ω H2 τ m 2
=
( 77)
i
]
)
K c (76) (1 + K H Km )
(
ω H Km 0.0697 1 + 0.752ω H τ m − 0145 . ω H 2 τm2 =
Kc
( 77 )
(
(1 + KH K m )
ω H K m 00405 . 1 + 193 . ω H τ m − 0363 . ωH 2 τm2
)
)
ω τ ( 78) 126 . ( 0.506) H m 1 − 1.07 ε + 0.616 ε 2 K c (1 + K H K m ) = KH − , Ti (78) = K m (1 + KH K m ) ω H K m 0.0661 1 + 0.824 ln[ ω H τ m ] 1 + 171 . ε − 117 . ε2
[
109 . 1 − 0497 . ω H τ m + 00724 . ω H 2 τm2 Kc ( 79) = K H − Km (1 + KH K m )
)(
(
] , T
i
( 79)
=
(
Kc
( 79 )
(1 + KH K m )
ω H K m 0.054 − 1 + 2.54ω H τ m − 0457 . ω H 2 τm2
)
)
)
Rule
Kc
Ti
Td
Regulator – nearly minimum IAE, ISE, ITAE – Hwang [60] – continued
Kc(8 0) 4
Ti (80)
Comment
1.45(1 + Ku K m ) 116 . 1 − 2 ε0 Km ω u . ω u2τ m2 ) ( 1 − 0.612ω u τ m + 0103
Decay ratio = 0.25 τ 0.2 ≤ m ≤ 2.0 and Tm1
0.6 ≤ ξ m ≤ 4.2 ; ε < 2.4
Kc
Model: Method 3
( 81)
Ti
1.45(1 + Ku K m ) 116 . 1 − 2 ε0 Km ω u
( 81)
. ω u2τ m2 ) ( 1 − 0.612ω u τ m + 0103
Decay ratio = 0.25 τ 0.2 ≤ m ≤ 2.0 and Tm1
0.6 ≤ ξ m ≤ 4.2 ; 2.4 ≤ ε < 3
Kc(8 2)
1.45(1 + Ku K m ) 116 . 1 − 2 ε0 Km ω u
Ti (82)
. ω u2τ m2 ) ( 1 − 0.612ω u τ m + 0103
Decay ratio = 0.25 τ 0.2 ≤ m ≤ 2.0 and Tm1
0.6 ≤ ξ m ≤ 4.2 ; 3 ≤ ε < 20
Kc(83)
1.45(1 + Ku K m ) 116 . 1 − 2 ε0 Km ω u
Ti (83)
Decay ratio = 0.25 τ 0.2 ≤ m ≤ 2.0 and Tm1
. ω u2τ m2 ) ( 1 − 0.612ω u τ m + 0103 0.6 ≤ ξ m ≤ 4.2 ; ε ≥ 20
Servo– nearly minimum IAE, ISE, ITAE – Hwang [60]. In all, decay ratio = 0.1 with τ 0.2 ≤ m ≤ 2.0 and Tm1
Kc(8 4)
Ti (84 )
0471 . Ku K mω u
Kc(85)
Ti (85)
0471 . Ku K mω u
τ τm + 0117 . m Tm1 Tm1
2
ξ ≤ 0613 . + 0613 .
τ τm + 0117 . m Tm1 Tm1
2
ξ ≤ 0613 . + 0613 .
0.6 ≤ ξ m ≤ 4.2 Model: Method 3
4
Kc
( 80)
[
0.584 1 − 0.439ω H τ m + 0.0514ω H 2 τ m 2 = KH − K m (1 + K H K m )
[
( 80)
i
] , T
0675 . 1 − 0.472ω H τ m + 0061 . ω H 2 τm2 Kc (81) = K H − K m (1 + KH K m )
[
] , T ( 81)
i
=
Kc
(
(1 + KH K m )
ω H K m 0.0714 1 + 0.685ω H τ m − 0.131ω H 2 τ m 2 Kc
=
( 80)
( 81)
(
(1 + K H K m )
ω H K m 00484 . 1 + 143 . ω H τ m − 0.273ω H 2 τ m 2
]
)
ω τ ( 82) 12 . ( 0.495) H m 1 − 11 . ε + 0.698 ε 2 K c (1 + K H K m ) Kc (82 ) = K H − , Ti (82) = K m (1 + K H K m ) ω H K m 0.0702 1 + 0.734 ln[ ω H τ m ] 1 + 148 . ε − 11 . ε2
[
103 . 1 − 0.51ω H τ m + 0.0759ω H 2 τ m 2 Kc (83) = KH − K m (1 + K H K m )
[
] , T
0.822 1 − 0.549ω H τ m + 0112 . ωH τm Kc (84 ) = K H − K m (1 + K H K m )
[
2
)(
(
2
( 83)
i
] , T
0.786 1 − 0.441ω H τ m + 0.0569ω H 2 τ m 2 Kc (85) = KH − K m (1 + K H K m )
( 84)
i
] , T
i
Kc
=
( 83)
(
(1 + KH K m )
ω H K m 0.0386 −1 + 3.26ω H τ m − 0.6ω H 2 τ m 2 =
( 85)
Kc
( 84 )
(
)
(1 + K H K m )
ω H K m 0.0142 1 + 6.96ω H τ m − 177 . ω H 2 τm2
=
Kc
(
( 85)
)
)
(1 + K H Km )
ω H K m 0.0172 1 + 4.62ω H τ m − 0.823ω H 2 τ m 2
)
)
Kc
Rule Servo– nearly minimum IAE, ISE, ITAE – Hwang [60] (continued)
Kc
Servo – nearly minimum IAE, ISE, ITAE – Hwang [60]
Ti
( 8 6) 5
Ti
Td
Ti (87 )
( 8 8)
( 88)
Kc
Ti
0471 . Ku K mω u
( 86)
Kc(8 7)
0471 . Ku K mω u 0471 . Ku K mω u
Ti (89)
0471 . Ku K mω u
Kc( 90)
Ti (90)
0471 . Ku K mω u
Kc
( 91)
Ti
( 91)
0471 . Ku K mω u
Kc
( 92)
Ti
( 92 )
0471 . Ku K mω u
Kc(8 9)
Comment τ τm + 0117 . m Tm1 Tm1
2
ξ ≤ 0613 . + 0613 .
τ τm + 0117 . m Tm1 Tm1
2
ξ ≤ 0613 . + 0613 .
ξ > 0649 . + 058 .
τm Tm1
ξ > 0649 . + 058 .
6
Model: Method 3
Kc( 93)
5
ξ ≤ 0649 . + 058 .
]
τm Tm1 τm Tm1
τm Tm1
2
τm − 0005 . Tm1
2
τm − 0005 . Tm1
2
τ 2 τm − 0005 . m Tm1 Tm1
ξ > 0613 . + 0613 .
ξ ≤ 0649 . + 058 .
τ 2 τm − 0005 . m Tm1 Tm1
ξ > 0613 . + 0613 .
[
Kc
( 88)
] , T
[
0.794 1 − 0.541ω H τ m + 0126 . ω H 2 τ m2 = KH − K m (1 + K H K m )
[
0.738 1 − 0415 . ω H τ m + 0.0575ω H 2 τ m 2 Kc (89 ) = K H − K m (1 + K H K m )
[
( 87)
i
] , T
=
(
=
( 89)
i
( 87)
(1 + K H K m )
ω H K m 0.0609 − 1 + 197 . ω H τ m − 0.323ω H 2 τ m 2 =
( 88)
i
] , T
Kc
Kc
( 88)
(
(1 + KH K m )
ω H K m 0.0078 1 + 8.38ω H τ m − 197 . ω H 2 τm2 Kc
( 89 )
(
(1 + K H K m )
ω H K m 0.0124 1 + 4.05ω H τ m − 0.63ω H 2 τ m 2
]
τ 2 τm + 0117 . m Tm1 Tm1
)
) )
ω τ ( 90) 115 . ( 0.564) H m 1 − 0.959 ε + 0.773 ε 2 K c (1 + K H K m ) Kc ( 90) = K H − , Ti (90) = K m (1 + K H K m ) ω H K m 0.0335 1 + 0947 . ln[ ω H τ m ] 1 + 19 . ε − 1.07 ε 2
[
107 . 1 − 0.466ω H τ m + 0.0667ω H 2 τ m 2 Kc ( 91) = KH − K m (1 + K H K m )
[
0.789 1 − 0.527ω H τ m + 011 . ω H τm Kc ( 92) = K H − K m (1 + KH K m )
[
2
2
] , T
0.76 1 − 0.426ω H τ m + 0.0551ω H 2 τ m 2 Kc ( 93) = KH − K m (1 + K H K m )
( 92)
i
] , T
( 91)
i
] , T
)(
(
i
=
=
( 93)
Kc
( 91)
(
(1 + KH K m )
ω H K m 0.0328 −1 + 2.21ω H τ m − 0.338ω H 2 τ m 2 Kc
( 92)
(
(1 + K H K m )
ω H K m 0009 . 1 + 9.7ω H τ m − 2.4ω H2 τ m 2
=
Kc
(
( 93)
)
(1 + KH K m )
ω H K m 0.0153 1 + 4.37ω H τ m − 0.743ω H 2 τ m 2
)
)
,
τ 2 τm + 0117 . m Tm1 Tm1
)(
(
2
τm − 0005 . Tm1
ω τ ( 86) 128 . ( 0.542) H m 1 − 0986 . ε + 0558 . ε 2 K c (1 + K H K m ) Kc (86) = KH − , Ti (86) = K m (1 + K H Km ) ω H K m 0.0476 1 + 0.996 ln[ ω H τ m ] 1 + 2.13 ε − 113 . ε2
114 . 1 − 0466 . ω H τ m + 0.0647ω H 2 τ m 2 Kc (87 ) = K H − K m (1 + K H K m ) 6
ξ > 0649 . + 058 .
0471 . Ku K mω u
Ti (93)
[
ξ > 0649 . + 058 .
τm − 0005 . Tm1
)
)
,
Kc
Rule Servo – nearly minimum IAE, ISE, ITAE – Hwang [60] (continued)
Kc
( 94) 7
Kc Minimum IAE regulator – ultimate cycle - Shinskey [16] – page 151. Model: Method 10. Direct synthesis Gain and phase margin – Hang et al. [35] Model: Method 4 Gain and phase margin – Ho et al. [140]
Ti
Ti
( 95)
Ti
Gain and phase margin – Ho et al. [142] Model: Method 1
7
Comment
0471 . Ku K mω u
( 94 )
ξ ≤ 0649 . + 058 .
ξ > 0613 . + 0613 .
0471 . Ku K mω u
(95)
τ 2 τm − 0005 . m Tm1 Tm1
ξ ≤ 0649 . + 058 .
τ τm + 0117 . m Tm1 Tm1
τ 2 τm − 0005 . m Tm1 Tm1
ξ > 0613 . + 0613 .
0.38Tu
015 . Tu
Tm 2 Tm2 + τ m = 0.25
0.6766Ku
0.33Tu
019T . u
Tm 2 Tm2 + τ m = 0.5
0.7874K u
0. 26Tu
0.21Tu
Tm 2 Tm2 + τ m = 0.75
πTm1 , Am Km τ m
2Tm1 ,
0.5Tm1 Sample A m ,φ m
A m = 1.5, φm = 30 o
A m = 3.0 , φm = 60 o
A m = 2 .0, φm = 45o
A m = 4.0 , φ m = 67 .5o
ω p Tm1
1
A m Km
4ω p τ m 2
2ω p −
A m = 2 .0, φm = 45
π
+
A m = 5.0, φ m = 72 o
provided. Model has a repeated pole ( Tm1 )
Tm 2
Tm1 > Tm 2 . Sample A m ,φ m provided A m φ m + 05 . πA m (A m − 1)
1 Tm1
A m = 3.0 , φm = 60 o
A m = 4.0, φ m = 70 o
Kc( 96)
Ti (96 )
Td ( 96)
A m = 2 .0, φm = 45o
A m = 3.0 , φm = 60 o
A m = 4.0, φ m = 70 o
Kc (97)
Ti (97 )
Td ( 97)
A m = 2 .0, φm = 45o
A m = 3.0 , φm = 60 o
A m = 4.0, φ m = 60 o
[
]
[
111 . 1 − 0467 . ω H τ m + 0.0657ω H 2 τ m 2 Kc ( 95) = K H − K m (1 + KH K m )
ωp =
(A
2 m
)
−1 τm
τm τm , ≤1 Tm Tm Sample A m ,φ m provided * ω nτ m < 2ξ m ; Sample A m ,φ m provided 2ξ m ≥
] , T
( 97)
Td ( 97)
)(
(
( 95)
i
=
Kc
(
( 95)
(1 + KH K m )
ω H K m 0.0477 − 1 + 2.07ω H τ m − 0.333ω H 2 τ m 2
πξ m ( 97) 2 2 πξ m + π − 2ω p τm , Ti = Tm1 + π − 2ω p τ m , π ω p Tm1 Tm1ω p πTm1 = πξ 2 m + π Tm1 − 2Tm1ω p τm ωp =
)
2 2 τ π Tm1 ( 96) ( 96) πξ m + π − 2 m , Ti = Tm1( πξ m + π − 2τ m ) , Td = π πA m K m Tm1 2( πξ mTm1 + π Tm1 − 2τ m )
2
2ω pTm1 π Am
2
Kc
,
τ 2 τm + 0117 . m Tm1 Tm1
ω τ ( 94) 122 . ( 0.55) H m 1 − 0.978 ε + 0.659 ε 2 K c (1 + K H K m ) Kc ( 94) = K H − , Ti (94) = K m (1 + K H K m ) ω H K m 0.0421 1 + 0969 . ln[ ω H τ m ] 1 + 2.02 ε − 111 . ε2
Kc ( 96) =
, 2
0.6173Ku
o
Model: Method 1 Gain and phase margin – Ho et al. [141] Model: Method 1
Td
8πτm ξ m π + π2 + A m 2 − 1 − 2πA m ( A m − 1) T m1 * φm < 4A m
(
)
)
Kc
Rule
Ti
Td
Comment
ξ m > 0.7071 or ξ m Tm1 Km τ m
Wang et al. [143]
0.05 <
05 . Tm1 ξm
2ξ m Tm1
07071 . τm Tm1 2ξ m 2 − 1
0.7071τm Tm1 2 ξ m 2 − 1
Model: Method 8
or 0.05 < ξmτ m Tm1
Minimum of 2
2 Tm1 e Km
−
Gain and phase margin – Leva et al. [34]
Tm τ m ξm
Kc
,
07358 . ξ m Tm1 Km τ m
( 98) 8
Ti
( 98 )
0.25Ti
> 1, ξ m ≥ 1
ξ mτ m Tm1
< 015 . ,
> 1, ξ m < 1
ξ m ≤ 0.7071 and
05 . Tm1 ξm
2ξ m Tm1
< 015 .
0.15 ≤
ξmτ m Tm1
≤1
ξ m ≤ 1 or ξ m > 1 with 05 . π − φm > 3τ m 2 ξ m + ξ m − 1 Tm1
( 98)
Model: Method 5
ξ m > 1 with
Kc( 99)
Ti (99 )
3τ m 2 ξ m − ξ m −1 Tm1 < 0.5π − φm ≤ 3τ m 2 ξ m + ξ m − 1 , Tm1
Td ( 99)
φ m = 70 0 at least
8
Kc
( 98)
Ti( 98)
z=
=
ω cn Ti Km
( 1 + ω T ) + 4ξ ω (1 − T T ω ) + T ω 2
cn
2
2
2
m1
m
2
i d
cn
2
cn
2
2
i
Tm1 2
2
, ω cn =
cn
2ξ τ T ( 05 1 m m m1 . π − φm ) 4.07 − φ m + tan −1 2 2 2 τm ( 0.5π − φ m ) Tm1 − τ m
2 2ξ mω cnTm1 2 1 , Kc ( 99) = ω cnTm(99 = tan 05 . ω cn τ m + φ m − 05 . π − tan− 1 2 2 ω cn ω cn Tm1 − 1 K mTd ) ω cn
ω cn Tm1 −1 tan φ m − 05 . π + ω cnτ m + tan ξ m + ξm 2 − 1
, Ti (99 ) =
Tm1z + ξ m + ξ m 2 − 1 z ξ m + ξ m 2 − 1
ω cn 2 +
1 2ξ m ξ m 2 − 1 − 1 Tm12 , 2 2 ω cn + z
, Td (99) =
Tm1 Tm1z + ξ m + ξ m2 − 1
Rule Gain and phase margin – Wang and Shao [144]
Kc
other tuning rules obtained using authors method
Td
ξ m Tm1 Km τ m
2ξ m Tm1
05 .
Tm1 ξm
Am = 3 , φ m = 60 0
2.094
ξ mTm1 K mτ m
2ξ m Tm1
05 .
Tm1 ξm
A m = 15 . , φ m = 30 0
1571 .
ξ mTm1 K mτ m
2ξ m Tm1
05 .
Tm1 ξm
A m = 2 , φ m = 450
0.785
ξ m Tm1 K mτ m
2ξ m Tm1
05 .
Tm1 ξm
A m = 4 , φ m = 67.50
0.628
ξ mTm1 K mτ m
2ξ m Tm1
05 .
Tm1 ξm
A m = 5 , φ m = 72 0
Pemberton [145] Model: Method 1
2( Tm1 + Tm2 ) 3Km τ m
Tm1 + Tm2
Tm1Tm2 Tm1 + Tm 2
Pemberton [24]
(Tm1 + Tm2 ) K mτ m
Tm1 + Tm2
Tm 1Tm2 Tm1 + Tm 2
2( Tm1 + Tm 2 ) 3Km τ m
Tm1 + Tm2
Model: Method 1
Pemberton [145] Model: Method 1
Comment
1047 .
Model: Method 8 Authors quote tuning rule for Am = 3 , φ m = 60 0 ;
Ti
Tm1 + Tm 2 4
01 . ≤
τm ≤ 10 . ; Tm1
0.2 ≤
τm ≤ 10 . Tm 2
01 . ≤
τm ≤ 10 . Tm1
0.2 ≤
τm ≤ 10 . Tm 2
Suyama [100] Model: Method 1
Tm1 + Tm 2 2K m τ m
Tm1 + Tm2
Tm 1Tm2 Tm1 + Tm 2
Smith et al. [146] Model: Method not stated Chiu et al. [29]
Tm1 + Tm 2 K m ( λ + τm )
Tm1 + Tm2
Tm 1Tm2 Tm1 + Tm 2
λTm1 + Tm2 K m (1 + λτm )
Tm1 + Tm2
Tm 1Tm2 Tm1 + Tm 2
λTm 1 K m (1 + λτ m )
λ variable; suggested values are 0.2, 0.4, 0.6 and 1.0. Tm1 > Tm2 . λ = pole of
Tm1
Tm2
2λξ m Tm1 K m (1 + λτ m )
2ξ m Tm1
Tm1 2ξ m
specified closed loop overdamped response. Underdamped response
Gorez and Klan [147a] Model: Ideal Miluse et al. [27a]
2ξ m Tm1 K m (2ξ m Tm1 + τ m )
2ξ m Tm1
Tm1 2ξ m
Non-dominant time delay
Tm1 + Tm2
Tm 1Tm2 Tm1 + Tm 2
Closed loop overshoot = 0%
Model: Method not specified
0. 368 (Tm1 + Tm 2 ) K mτ m 0. 514 (Tm1 + Tm 2 ) K mτ m
Tm1 + Tm2
Tm 1Tm2 Tm1 + Tm 2
Closed loop overshoot = 5%
0. 581( Tm1 + Tm 2 ) Km τ m
Tm1 + Tm2
Tm 1Tm2 Tm1 + Tm 2
Closed loop overshoot = 10%
0. 641( Tm1 + Tm 2 ) Km τ m
Tm1 + Tm2
Tm 1Tm2 Tm1 + Tm 2
Closed loop overshoot = 15%
Model: Method 6 Wang and Clemens [147] Model: Method 9
Overdamped process; Tm1 > Tm 2
Rule
Kc
Ti
Td
Comment
Miluse et al. [27a] (continued)
0. 696 (Tm1 + Tm 2 ) K mτ m
Tm1 + Tm2
Tm 1Tm2 Tm1 + Tm 2
Closed loop overshoot = 20%
0. 748 (Tm1 + Tm 2 ) K m τm
Tm1 + Tm2
Tm 1Tm2 Tm1 + Tm 2
Closed loop overshoot = 25%
0. 801( Tm1 + Tm 2 ) Km τ m
Tm1 + Tm2
Tm 1Tm2 Tm1 + Tm 2
Closed loop overshoot = 30%
0. 853( Tm1 + Tm2 ) K m τm
Tm1 + Tm2
Tm 1Tm2 Tm1 + Tm 2
Closed loop overshoot = 35%
0. 906 (Tm1 + Tm 2 ) K mτ m
Tm1 + Tm2
Tm 1Tm2 Tm1 + Tm 2
Closed loop overshoot = 40%
0. 957 (Tm1 + Tm 2 ) K mτ m
Tm1 + Tm2
Tm 1Tm2 Tm1 + Tm 2
Closed loop overshoot = 45%
1.008 ( Tm1 + Tm 2 ) K m τm
Tm1 + Tm2
Tm 1Tm2 Tm1 + Tm 2
Closed loop overshoot = 50%
2ξ m Tm1
Tm1 2ξ m
Closed loop overshoot = 0%
Model: Method not specified
0. 736ξ mTm1 Km τ m 1.028 ξ m Tm1 K m τm
2ξ m Tm1
Tm1 2ξ m
Closed loop overshoot = 5%
Underdamped process; 0.5 < ξ m ≤ 1
1.162 ξ m Tm1 K m τm
2ξ m Tm1
Tm1 2ξ m
Closed loop overshoot = 10%
1.282 ξ m Tm1 K m τm
2ξ m Tm1
Tm1 2ξ m
Closed loop overshoot = 15%
1.392 ξ m Tm1 K m τm
2ξ m Tm1
Tm1 2ξ m
Closed loop overshoot = 20%
1.496 ξ m Tm1 K m τm
2ξ m Tm1
Tm1 2ξ m
Closed loop overshoot = 25%
1.602 ξ m Tm1 K m τm
2ξ m Tm1
Tm1 2ξ m
Closed loop overshoot = 30%
1.706 ξ m Tm1 K m τm
2ξ m Tm1
Tm1 2ξ m
Closed loop overshoot = 35%
1.812 ξ m Tm1 K m τm
2ξ m Tm1
Tm1 2ξ m
Closed loop overshoot = 40%
1.914 ξ m Tm1 K m τm
2ξ m Tm1
Tm1 2ξ m
Closed loop overshoot = 45%
2.016 ξ m Tm1 K mτ m
2ξ m Tm1
Tm1 2ξ m
Closed loop overshoot = 50%
0. 736 Tm1 K m τm
2Tm1
0.5Tm1
Tm1 = Tm 2
Miluse et al. [27a]
Miluse et al. [27b] Model: Method 14
‘Re-tuning’ rule. ω 0 = 2 Seki et al. [147b] ωc tan θ c + 2 + tan 2 θ c cos θ Model: Method not G c ( jω 0 ) c 2 ω0 ωc specified
0.5Ti
freq. when controlled system goes unstable (crossover freq.); ω c = new crossover freq. ; θ c = phase lag
Rule Autotuning – Landau and Voda [148] Model: Method not relevant
Kc
Ti
Td
Comment
4+β β 4 2 2 G p ( jω ) 135
4+β 1 β ω135
4 1 4 + β ω135
1≤ β ≤ 2 ; τm < 0.25Tm1
3
3.2 ωu
08 . ωu
ω uτ m ≤ 016 .
4 ωu
1 ωu
016 . < ω u τ m ≤ 0.2
Tm 1 + Tm2 + 05 . τm
Tm1 Tm2 + 0.5( Tm1 + Tm2 ) τ m
λ varies graphically with τm (Tm1 + Tm 2 )
0
G p ( jω u ) 19 . G p ( jω u )
0
0
Robust Brambilla et al. [48] – values deduced from graph Model: Method 1
Chen et al. [53]
Tm1 + Tm 2 + 0.5τ m K m τ m ( 2λ + 1) τ m (Tm1 + Tm 2 )
λ
0.1 0.50 0.2 0.47 0.5 0.39 1.00 ξ m Tm τ mK m
Tm1 + Tm 2 + 05 . τm
0.1 ≤ τ m ( Tm1 + Tm 2 ) ≤ 10
τ m (Tm1 + Tm 2 )
λ
τ m (Tm1 + Tm 2 )
λ
1.0 2.0 5.0
0.29 0.22 0.16
10.0
0.14
A m = 3.14 ,
2ξ m τm
τm 2ξ m
1.22 ξ m Tm τ mK m
2ξ m τm
τm 2ξ m
A m = 2.58 ,
1.34 ξ m Tm τ mK m
2ξ m τm
τm 2ξ m
A m = 2. 34 ,
1.40 ξ m Tm τ mK m
2ξ m τm
τm 2ξ m
A m = 2. 24 ,
1.44 ξ m Tm τ mK m
2ξ m τm
τm 2ξ m
A m = 2.18 ,
1.52 ξ m Tm τ mK m
2ξ m τm
τm 2ξ m
1.60 ξ m Tm τ mK m
2ξ m τm
τm 2ξ m
φ m = 61. 40 , Ms = 1. 00
Model: Method 1
φ m = 55. 00 , Ms = 1. 10 φ m = 51.6 0 , Ms = 1. 20 φ m = 50.0 0 , M s = 1. 26 φ m = 48. 70 , Ms = 1. 30 A m = 2. 07 ,
φ m = 46. 50 , Ms = 1. 40 A m = 1.96 ,
φ m = 44.10 , Ms = 1. 50 2
Lee et al. [55] Model: Method 1
Ti K m ( 2λ + τ m )
2ξ mTm1 −
2 λ − τm 2( 2λ + τ m ) 2
2
Ti − 2ξ mTm1 +
Tm1 − Ti
τm 6Ti ( 2λ + τm ) 3
Desired closed loop e− τ m s response = ( λs + 1) 2
Rule
Kc
Lee et al. [55] (continued)
Ti K m ( 2λ + τ m )
Ti K m( λ + τ m ) Ti K m( λ + τ m )
Ti 2 λ2 − τ m 2 − 2( 2λ + τ m )
Tm1 + Tm2
2ξ mTm1
Td
τm + 2( λ + τ m )
Tm1 + Tm2
2
τm + 2 ( λ + τm ) 2
Comment
Ti − Tm1 − Tm2 +
Tm1Tm2 Ti
Desired closed loop e− τ m s response = ( λs + 1) 2
τm 6Ti ( 2λ + τ m ) 3
−
τ m3 Tm1 2 − 6(λ + τ m ) Ti − 2ξ m Tm1 Ti
Desired closed loop e− τ m s response = ( λ s + 1)
τ m3 Tm1Tm 2 − 6 λ + τm Ti − ( Tm1 + Tm2 ) Ti
Desired closed loop e− τ m s response = ( λ s + 1)
(
)
Km e− sτ K m e− sτ or - filtered controller (1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1 m
Table 70: PID tuning rules - SOSPD model
m
1 1 G c ( s) = Kc 1 + + Td s . 1 tuning rule. Ti s Tf s + 1 Rule
Kc
Ti
2Tm1 + τ m 2( λ + τ m ) K m
Tm1 + 05 . τm
Td
Comment
Robust Hang et al. [35] Model: Method 4
Tm1τm 2Tm1 + τm
Model has a repeated pole (Tm1 ) .
λ > 025 . τm . Tf =
λτ m 2( λ + τ m )
Km e− sτ K m e− sτ or - filtered controller (1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1 m
Table 71: PID tuning rules - SOSPD model
m
b s+1 1 G c ( s) = Kc 1 + + Td s 1 . 1 tuning rule. Ti s a 1s + 1 Rule Robust Jahanmiri and Fallahi [149] Model: Method 6
Kc
Ti
Td
2ξ m Tm1 K m (τ m + λ) λ = 0.25τ m + 01 . ξ m Tm1
2ξ m Tm1
Tm1 2ξ m
Comment
b1 = 0.5τ m a1 =
λτ m 2( τ m + λ )
Km e− sτ K m e− sτ or - classical controller (1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1 m
Table 72: PID tuning rules - SOSPD model
m
1 1 + Td s . 7 tuning rules. G c ( s) = Kc 1 + Ti s 1 + Td s N Kc
Rule Regulator tuning
Ti
Td
T − 1.5τ τm 15 . −e
m1
Minimum IAE Shinskey [59] – page 159
1
Kc
( 100 )
0800 . Tm1 Km τ m 0770 . Tm1 K mτ m 0833 . Tm1 Km τ m
** Minimum IAE Shinskey [17]
059 . Ku 085 . Tm1 K mτ m 087 . Tm1 Km τm 100 . Tm1 K mτ m 125 . Tm1 K mτ m
1
Kc (100) =
48 + 57 1 − e
m2
25 . Tm1 K mτ m
** Minimum IAE Shinskey [17]
m
T − 1 + 0.9 1 − e τ
Model: Open loop method not specified
** Minimum IAE Shinskey [59]
Comment
Minimum performance index
1.2 Tm1 − τm
m
1.2 T − 0.56τ m 1 − e τ
m1
m
Tm2 ≤3 τm
0.6Tm2
τm + 0.2Tm2
τm + 0.2Tm2
15 . ( Tm2 + τ m )
060 . ( Tm2 + τ m )
12 . ( Tm2 + τ m )
0.70( Tm 2 + τ m )
075 . ( Tm2 + τ m )
060 . ( Tm2 + τ m )
0. 36Tu
0. 26Tu
198 . τm
086 . τm
2.30τ m
165 . τm
2.50τm
200 . τm
2.75τm
2.75τ m
100 2 Tm 2 Km τ m Tm2 1 + 0.34 − 0.2 Tm1 τm τ m
+
Tm2 >3 τm Tm 2 = 025 . Tm 2 + τm Tm 2 = 0.5 Tm2 + τ m Tm 2 = 0.75 Tm2 + τ m τm T = 0.2 , m 2 = 0.2 Tm1 Tm1 τm T = 0.2 , m 2 = 01 . Tm1 Tm1 Tm 2 = 0.2 Tm1 Tm 2 = 0.5 Tm1 Tm 2 = 10 . Tm1
Rule
Kc
Ti
Td
Minimum ISE – McAvoy and Johnson [83] – deduced from graph
α Km
βτ m
χτ m
ξm
Model: Method 1
α
τm Tm1
β
χ
ξm
τm Tm1
α
β
χ
Comment
ξm
τm Tm1
α
β
χ
N = 20
1 1 1
0.5 0.7 0.97 0.75 4.0 7.6 3.33 2.03 10.0 34.3 5.00 2.7
4 4
0.5 4.0
3.0 1.16 0.85 22.7 1.89 1.28
7 7
0.5 4.0
5.4 1.19 0.85 40.0 1.64 1.14
N = 10
1 1 1
0.5 0.9 1.10 0.64 4.0 8.0 4.00 1.83 10.0 33.5 6.25 2.43
4 4
0.5 4.0
3.2 1.33 0.78 23.9 2.17 1.17
7 7
0.5 4.0
5.9 1.39 1.04 42.9 1.89 1.37
Direct synthesis
3 Astrom et al. [30] – Method 13
Smith et al. [26] – deduced from graph [ ρ = τ m ( Tm1 + Tm2 ) ] Model: Method not stated
3τ m K m 1 + Tm1 + Tm 2
Tm1 + Tm2
Tm 1Tm2 Tm1 + Tm 2
Tm1
Tm2
N = 8 - Honeywell UDC6000 controller
ξm
αTm1 K mτ m ρ
α
ξm
ρ
α
ξm
ρ
α
6 1.75 1.75 1.5 1.0
≥ 0.02 0.27 0.07 0.11 0.25
0.51 0.46 0.36 0.33 0.46
3 1.75 1.5 1.5 1.0
≥ 004 . 0.13 0.33 0.08 0.17
0.50 0.42 0.44 0.28 0.48
2 1.75 1.5 1.0 1.0
≥ 006 . 0.07 0.17 0.50 0.13
0.45 0.39 0.38 0.40 0.49
N = 10
Km e− sτ K m e− sτ or - alternative classical (1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1 m
Table 73: PID tuning rules - SOSPD model
m
1 1 + NTd s controller G c ( s) = Kc 1 + . 1 tuning rule. Ti s 1 + Td s Rule
Kc
Ti
Td
Comment
****** 0.7
Hougen [85] Model: Method 1
08 . Tm1 Tm2 τm 0.8
0. 3
084 . Tm1 Tm 2 τm
05 . Tm1 + Tm 2
3
τm Tm1 Tm 2
N=10
0.2
0.53Tm1 + 1.3Tm2
0.08( τ m Tm1Tm2 )
0.28
N=30
Km e− sτ K m e− sτ or - alternative non(1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1 m
Table 74: PID tuning rules - SOSPD model
m
1 interacting controller 1: U( s) = K c 1 + E ( s) − Kc Td sY ( s) . 3 tuning rules. Ti s Kc
Rule Regulator tuning Minimum IAE Shinskey [59] – page 159 Model: open loop method not specified
Minimum IAE Shinskey [17] – page 119. Model: method not specified.
Kc (101) =
38 + 401 − e
Td
Comment
Minimum performance index 1
Kc
( 101)
T − m1 τ m 0.5 + 1. 4[ 1 − e 1.5 τ m ]
T − m2 1 + 0.48 1 − e τ m
1.2 Tm1 − 0.42τ m 1 − e τm +
0.6Tm2
Tm2 ≤3 τm
τ m + 0.2Tm2
τ m + 0.2Tm2
118 . Tm1 Km τ m
2.20τ m
0.72τ m
τm T = 0.2 , m 2 = 01 . Tm1 Tm1
125 . Tm1 K m τm
2.20τm
110 . τm
Tm 2 = 0.2 Tm1
167 . Tm1 K mτ m
2.40τ m
165 . τm
Tm 2 = 0.5 Tm1
25 . Tm1 K mτ m
2.15τ m
2.15τ m
Tm 2 = 10 . Tm1
085 . Ku
035 . Tu
017 . Tu
τm T = 0.2 , m 2 = 0.2 Tm1 Tm1
333 . Tm 1 Km τm
Minimum IAE Shinskey [17] – page 121. Model: method not specified.
1
Ti
1.5Tm1 − τm
100 2 Tm 2 Km τ m Tm2 1 + 0.34 − 0.2 Tm1 τm τ m
Tm 2 >3 τm
Km e− sτ K m e− sτ or - series controller (1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1 m
Table 75: PID tuning rules - SOSPD model
m
1 G c ( s) = K c 1 + (1 + Td s) . 1 tuning rule. Ti s Rule
Kc
Regulator tuning Least mean square error - Haalman [23] Model: Method 1
Ti
Td
Comment
Minimum performance index
2Tm1 3τ m K m
Tm1
Tm2
Tm1 > Tm2 . Maximum sensitivity = 1.9, Gain margin = 2.36, Phase margin = 50 0
Km e− sτ K m e− sτ or - non-interacting (1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1 m
Table 76: PID tuning rules - SOSPD model
1 Td s . 2 tuning rules. E( s) − controller U (s) = K c 1 + Y ( s ) T s T s d i 1+ N Kc Ti Td
Rule Servo – Min. IAE Huang et al. [18] Model: Method 1
2
Kc (102 ) =
m
Kc
( 102 ) 2
Ti
( 102)
Td
Comment 0<
( 102 )
0.1 ≤
Tm 2 ≤1; Tm1
τm Tm1
≤ 1 ; N =10
0.0865 τm 1 τ T τ T 7.0636 + 66.6512 m − 137 .8937 m2 − 122 .7832 m m2 2 + 261928 . Km Tm1 Tm1 Tm1 Tm1
2.6405 1.0309 2.345 1.0570 τm Tm 2 T T T 1 336578 . + 30098 . − 10.9347 m 2 + 141511 . m2 + 29.4068 m2 Km Tm1 Tm1 Tm1 Tm1 τm
+
− 0.9450 −0.9282 0.8866 Tm2 τ m Tm 2 τ m τ m Tm2 1 34.3156 + − 701035 . + 152.6392 Km Tm 1 Tm1 Tm1 Tm1 Tm1 Tm1
1 + Km Ti
( 102)
0 .8148 τ T τ T T − 47.9791 m m 2 − 57.9370e + 10.4002e T Tm1 Tm1 m
m2
τ m Tm 2
m1
m1
Tm1 2
+ 6.7646e
τ + 7.3453 m Tm
− 0.4062
2 2 3 τm Tm2 τm τm Tm2 τ m Tm2 = Tm1 0.9923 + 0.2819 − 0.2679 − 1.4510 + 0.6424 − 0.6712 + 2 .504 2 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1
2 2 3 4 4 T τ τ T T τm τ m Tm2 + Tm1 2.5324 m 2 m + 2.3641 m m 2 + 2.0500 m 2 − 18759 . + 08519 . Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 3 2 2 3 4 τ m Tm2 Tm 2 Tm2 τm τ T + Tm 1 − 13496 . . . − 34972 − 2.4216 m m 2 − 31142 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 5 4 2 3 2 3 5 τm Tm2 τ m τm Tm 2 Tm 2 Tm2 τ m + Tm 1 05862 . . . + 0.0797 + 0.985 + 12892 + 12108 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1
Td
( 102)
2 2 3 τm Tm2 τm τm Tm 2 τm Tm 2 = Tm1 0.0075 + 0.3449 + 0.3924 − 0.0793 + 0.6485 + 2.7495 + 0.8089 2 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1
2 T τ τ + Tm1 − 9.7483 m 2 m + 3.4679 m Tm1 Tm1 Tm1
2
3
4
Tm2 Tm2 τm . . − 58194 − 10884 Tm1 Tm1 Tm1
3 2 2 3 4 τm Tm 2 Tm 2 T τ τ T + Tm 1 12.0049 m 2 m − 14056 . . − 3.7055 m m 2 + 100045 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 5 4 2 3 2 3 τm Tm2 τ m τ m Tm 2 Tm 2 τ m + Tm 1 0.3520 . − 6.3603 − 32980 + 7.0404 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 4 5 6 τ m Tm 2 Tm 2 τm Tm 2 + Tm1 14294 . − 69064 . + 0 . 0471 . + 11839 T T T T Tm1 m1 m1 m1 m1
5
τm Tm 2 . + 17087 T m1 Tm1
6
Kc
Rule Servo-Min. IAE Huang et al. [18]
Kc
( 103) 3
Ti
(N=10)
Ti
Td
( 103)
( 103)
Td
Comment 0 .4 ≤ ξ m ≤ 1 ; 0 .05 ≤
4 2 3 3 2 4 5 τ m Tm 2 τ m Tm2 τ m Tm 2 τ m Tm 2 + Tm1 1. 7444 . . . − 12817 − 21281 + 15121 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1
3
Kc (103) =
−1.9009 τ τ 1 τ −81727 . − 32.9042 m + 319179 . ξ m + 38.3405ξ m m + 0.2079 m Km Tm1 Tm1 Tm1
+
0.1571 1.2234 τ τm 1 29.3215 m + 359456 . − 214045 . ξ m0 .1311 + 51159 . ξ m1.9814 − 219381ξ m1.737 Km Tm1 Tm 1
−0.1303 1.2008 τm τm 1 τ − 17.7448ξ m + + 268655 . ξ m − 52.9156 m ξ m1.1207 Km Tm1 Tm1 Tm1
1 + Km Ti
( 103)
τ τ T − 22.4297 m ξ m 0.3626 − 33331 . e Tm1
m m1
+ 85175 . e
ξm
− 15312 . e
τ mξm Tm1
+ 08906 .
ξ m Tm1 τm
2 2 τ τ τ τ 2 = Tm1 11731 . + 6.3082 m − 0.6937ξ m + 8.5271 m − 24 .7291ξ m m − 6.7123ξ m m + 7 .9559ξ m Tm1 Tm1 Tm1 Tm1
3 4 3 τm τm τm τ + Tm1 − 32.3937 . . ξ m2 m − 271372 + 166.9272ξ m + 363954 Tm1 Tm1 Tm1 Tm1 2 τm 2 τm 3 + Tm1 − 94.8879ξ m . ξ m 3 + 29.9159ξ m 4 − 22.6065 ξ m − 16084 Tm1 Tm1 5 4 3 2 τ τ τ τm 3 + Tm 1 49.6314 m − 84.3776ξ m m − 938912 . ξ m 2 m + 1101706 . ξm Tm1 Tm1 Tm1 Tm1 6 5 τm 4 τ τ 5 + Tm 1 − 251896 . . ξ m m + 55268 . ξ m6 ξ m − 19.7569 ξ m − 12.4348 m − 117589 Tm1 Tm1 Tm1 4 3 2 τm τm τm τm 2 3 4 + Tm 1 68.3097 . . ξm 5 ξ m − 17.8663 ξ m − 225926 ξ m + 95061 Tm1 Tm1 Tm1 Tm1
Td
( 104)
2 3 τm τm τm τm 2 = Tm1 0.0904 + 0.8637 − 0.1301ξ m + 4 .9601 + 0.7170ξ m − 12.5311 + 14 .3899ξ m Tm1 Tm1 Tm1 Tm1
2 4 τ τ τ + Tm1 − 42.5012ξ m m − 214907 . ξ m2 m − 69555 . ξ m 3 − 12.3016 m Tm1 Tm1 Tm1 3 2 τ τ τ + Tm1 102.9447ξ m m + 7.5855ξ m 2 m + 19.1257 m ξ m 3 + 17.0952 ξ m 4 Tm1 Tm1 Tm1 5 4 3 2 τm τm τm 2 τm 3 + Tm1 108688 . . ξm − 17.2130ξ m − 1100342 + 50.6455 ξm Tm1 Tm1 Tm1 Tm1 6 5 τ τ τ + Tm 1 − 16.7073 m ξ m 4 − 16.2013ξ m5 − 0. 0979 m − 109260 . ξ m m + 54409 . ξ m6 Tm1 Tm1 Tm1
τm ≤1 Tm1
Rule
Kc
Ti
Td
Regulator – Min. IAE - Huang et al. [18] Model: Method 1
Kc(104 ) 4
Ti (104)
Td (104 )
Comment Tm 2 ≤1; Tm1
0< τm
0.1 ≤
Tm1
≤ 1 ; N =10
4 3 2 τm τm τm τm 5 2 3 4 + Tm 1 29.4445 . . . ξm ξ m + 216061 ξ m − 241917 ξ m + 62798 Tm1 Tm1 Tm1 Tm1
4
Kc (104) =
− 0.8058 τm 1 τm Tm2 τ m Tm2 01098 . − 86290 . + 766760 . − 33397 . + 11863 . 2 Km Tm1 Tm1 Tm1 Tm1
+
0.6642 2 .1482 0.8405 2.1123 τm τm T Tm2 1 231098 . + 203519 . − 52.0778 m 2 − 121033 . Km Tm1 Tm1 Tm1 Tm1
+
0.5306 −1.0781 −0.4500 1 τ T T τ T τ 9.4709 m m2 + 13.6581 m 2 m − 19.4944 m2 m Km Tm1 Tm1 Tm1 Tm1 Tm1 Tm1
1.1427 τ T 1 τ T − 28.2766 m m2 + − 191463 . e T + 8.8420e T Km Tm1 Tm1
Ti
( 104)
m
m2
m1
m1
τ m Tm 2 Tm1 2
+ 7.4298e
− 114753 .
Tm 2 τm
2 2 3 τ T τ τ T τ T = Tm1 − 0.0145 + 2 .0555 m + 0.7435 m2 − 4.4805 m + 1.2069 m m2 2 + 0.2584 m 2 + 7.7916 m Tm1 Tm1 Tm1 Tm1 Tm1 Tm1
2 2 3 4 3 Tm 2 τm Tm 2 τ m τ m Tm2 Tm 2 τ m + Tm1 − 6.0330 . + 3.9585 − 30626 − 7.0263 + 7.0004 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 2 2 3 4 5 T τ τm Tm 2 Tm 2 τm + Tm1 − 2.7755 m 2 m − 15769 . + 31663 . + 2 . 4311 T m1 Tm1 Tm1 Tm1 Tm1 Tm1
T T τ τ m Tm2 T τ τ + Tm 1 − 0.9439 m 2 − 2.4506 m 2 m − 0.2227 m2 m + 19228 . − 0.5494 m Tm1 Tm 1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 5
Td
( 104)
4
2
3
2
3
2 2 3 τm Tm2 τm τm Tm 2 τ mTm 2 = Tm1 −0.0206 + 0.9385 + 0.7759 − 2.3820 − 3.2336 + 2 .9230 + 7.2774 2 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1
T + Tm 1 − 9.9017 m 2 Tm1
2
τm τ + 2.7095 m T T m1 m1
2
3
Tm 2 Tm2 τm . . + 61539 − 111018 T T m1 m1 Tm1
3 2 2 τ m Tm2 T τ τ + Tm 1 10.6303 m2 m + 57105 . − 7.9490 m Tm1 Tm1 Tm1 Tm1 Tm1
3
4
Tm 2 T − 6.6597 m 2 Tm1 Tm1
4
5 4 2 3 2 3 τm T τ τ m Tm2 T τ + Tm 1 80849 . . − 4.4897 m 2 m − 7.6469 m2 m + 21155 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm 1 4 5 6 5 6 τ m Tm 2 Tm 2 τm Tm 2 Tm 2 τ m + Tm 1 50694 . . . + 4.1225 − 2.274 + 0519 − 11295 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 4 2 3 3 2 4 τ T τ T τm Tm 2 τm + Tm 1 2.2875 m m2 + 0.9524 m m 2 − 16307 . . − 09321 T T T T T T T m1 m1 m1 m1 m1 m1 m1
Tm2 Tm1
5
Tm2 Tm1
5
Rule Regulator - Min. IAE Huang et al. [18] Model: Method 1
5
Kc
( 105)
Kc
Kc
( 105) 5
Ti
Ti
Td
( 105)
Td
( 105)
Comment
[N=10]
0 .4 ≤ ξ m ≤ 1 ; 0 .05 ≤
τm ≤1 Tm1
3 0.086 τm τm τm 1 τm = −357307 . − 1419 . + 14023 . ξ m + 6.8618ξ m . ξ m − 0.9773 + 555898 Km Tm1 Tm1 Tm1 Tm
+
2 −1. 6624 −0.6951 τ τm τ 1 τm − 33093 . ξ m m + 538651 . ξ m 2 + 114911 . ξ m3 + 08778 . − 29.8822 m Km Tm 1 Tm1 Tm1 Tm1
+
− 0.4762 −2.1208 τ τ 1 53535 . m − 16.9807 ξ m1.1197 − 254293 . ξ m1.4622 − 01671 . ξ m 58981 + 0.0034ξ m m Km Tm1 Tm
τ τ ξ 1 τm τ m 1.2103 ξ m Tm1 3.0836 T ξ − 250355 + . ξm − 54.9617 ξm − 01398 . e − 82721 . e + 63542 . e T + 10479 . Km Tm1 Tm1 τm m
m1
Ti
( 105)
m
m
m
m1
2 3 τm τm τm τm 2 = Tm1 0.2563 + 11.8737 − 1.6547ξ m − 16.1913 + 3.5927 ξ m + 19.5201 − 9 .7061ξ m Tm1 Tm1 Tm1 Tm1
2 4 τ τ τ + Tm 1 − 14.5581ξ m m + 2.939ξ m 2 m − 0.4592ξ m 3 − 34.6273 m Tm1 Tm1 Tm1 3 2 τm τ 2 τm 3 4 + Tm 1 50.5163ξ m + 8.9259ξ m + 8.6966 m ξ m − 6.9436ξ m Tm1 Tm1 Tm 1 5 4 3 2 τm τm τm 2 τm 3 + Tm 1 27.2386 . − 20.0697ξ m − 42.2833ξ m + 85019 ξm Tm1 Tm1 Tm1 Tm1 6 5 τ τ τ + Tm 1 − 12.2957 m ξ m 4 + 8.0694 ξ m 5 − 7.7887 m + 2.3012 ξ m m − 2.7691ξ m 6 Tm1 Tm1 Tm1 4 3 2 τm τm τm τ 2 3 4 5 + Tm 1 88984 . . . ξ m + 102494 ξ m − 54906 ξ m + 4.6594 m ξ m Tm1 Tm1 Tm1 Tm1
Td
( 105)
2 3 τm τm τm τm 2 = Tm1 −0.021 + 3.3385 + 0.185ξ m − 0.5164 − 0.8815ξ m + 0.584 − 0.9643ξ m Tm1 Tm1 Tm1 Tm1
2 4 τ τm 2 τ 3 + Tm1 − 12513 . ξ m m + 13468 . ξ m m + 2.3181ξ m − 52368 . Tm1 Tm1 Tm1 3 2 τ τ 2 τ 3 4 + Tm1 153014 . ξ m m + 119607 . ξ m m − 2.0411 m ξ m − 31988 . ξm Tm1 Tm1 Tm1 5 4 3 2 τm τm τm 2 τm 3 + Tm1 34675 . − 0.8219ξ m − 15.0718ξm − 18859 . ξm Tm1 Tm1 Tm1 Tm1 6 5 τ 4 τ τ 5 6 + Tm1 0.4841 m ξ m + 2.2821ξ m − 0.9315 m + 0529 . ξ m m − 06772 . ξm Tm1 Tm1 Tm1
4 3 2 τm τm τ τ 2 3 4 5 + Tm1 −14212 . ξ m + 71176 . ξ m − 2.3636 m ξ m + 0.5497 m ξ m Tm1 Tm1 Tm1 Tm1
Km e− sτ K m e− sτ or - ideal controller with (1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1 m
Table 77: PID tuning rules - SOSPD model
(
)
set-point weighting U(s) = Kc Fp R ( s) − Y( s) +
Kc
Rule
m
Kc [ Fi R(s) − Y(s)] + Kc Td s[ Fd R (s) − Y(s)] . 1 tuning rule. Ti s Ti
Td
Comment
05 . Tu Fi = 1
0125 . Tu
Repeated pole τ 01 . < m < 3; Tm
Ultimate cycle
06 . Ku Oubrahim and Leonard [138]
Fp = 13 .
Model: Method not specified
16 − Km Ku 17 + Km Ku
Fd = Fp
2
06 . Ku Fp =
38 29 + 35Km K u
05 . Tu Fi = 1
0125 . Tu Fd = Fp
2
10% overshoot Repeated pole τ 01 . < m < 3; Tm 20% overshoot
Km e− sτ K m e− sτ or - non-interacting (1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1 m
Table 78: PID tuning rules - SOSPD model
m
1 controller U( s) = Kc b + [ R (s) − Y(s) ] − ( c + Tds)Y( s) . 1 tuning rule. Ts i Rule
Kc
Ti
Td
Comment
Kc(105a) 6
154 . Ti ( 105a )
Td (105a)
c = K c ( 105a ) − 1 K m ;
Direct synthesis b = 0.198 Hansen [150]
zero overshoot b = 0.289
Model: Method 1 127 . Ti (105a )
Kc(105a)
Td (105a)
c = K c ( 105a ) − 1 K m ;
minimum IAE b = 0.143 175 . Ti (105a )
Kc(105a)
Td (105a)
c = K c ( 105a ) − 1 K m ;
conservative tuning
6
Kc
Td
(105a )
( 105a )
=
=
[
2 Tm1Tm 2 + (Tm1 + Tm2 )τ m + 0.5τ m2
[ 2[ T
]
3
9 Km Tm1Tm2 τ m + 0.5(Tm1 + Tm2 )τ m + 0167 . τm
[
m1Tm2
2
+ (Tm1 + Tm 2 )τ m + 05 . τ m2
]
[
3 Tm1Tm2 τ m + 05 . ( Tm1 + Tm 2 )τ m 2 + 0167 . τm3
2
3K m Tm1Tm 2 τ m + 0.5(Tm1 + Tm2 )τ m + 0167 . τm 2
]
3 2
, Ti (105a ) =
3
]
−
Tm1 + Tm2 − τ m Km
[T
m1Tm 2
+ (Tm1 + Tm2 )τ m + 05 . τm
2
]
],
Km e− sτ K m e− sτ or - ideal controller with (1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1 m
Table 79: PID tuning rules - SOSPD model
m
1 Td s filtered derivative G c ( s) = Kc 1 + + sT Ts i 1+ d N Rule
Kc
Ti
ω p Tm 1
1
A m Km
4ω p 2 τ m
. 2 tuning rules. Td
Comment
Tm2
provided. N = 20. Tm1 > Tm2
Direct synthesis Hang et al. [151]
2ω p −
Model: Method 1
A m = 2 .0 , φ m = 45o A m = 3.0 ,φ m = 60 Robust Hang et al. [151] Model: Method 1
Tm1 K m( λ + τ m)
π
Sample A m , φ m 1 + Tm1
A m = 4.0 , φ m = 67.5 o
A m = 5.0, φ m = 72 o
Tm1
Tm2
o
ωp =
A m φ m + 0.5πA m (A m − 1)
(A
2 m
)
−1 τm
N = 20. Tm1 > Tm2
Table 80: PID tuning rules – SOSPD model
K m e− sτ
m
Tm1 s2 + 2ξ m Tm1s + 1 2
1 Td s U (s) = K c 1 + + Tis T 1+ d N
Kc
Rule
- Two degree of freedom controller: β T s d E( s) − K c α + R (s) . 3 tuning rules. Td s 1+ s N
Ti
Servo/regulator tuning
Td
Minimum performance index
Taguchi and Araki [61a]
Kc
(105 b) 7
Ti
(105 b )
Td
(105b )
(105 c )
Ti
(105 c )
Td
(105 c )
Model: ideal process
Kc
Minimum ITAE Pecharroman and Pagola [134a]
0.7236 K u
0.5247 Tu
0.1650 Tu
Model: Method 15
7
Kc
Td
(105 b)
(105 b)
1 0. 6978 = 1.389 + τm Kc [ − 0.02295 ] 2 Tm
2
Td
(105c )
(105 c )
τm ≤ 1.0 ; ξ m = 1. 0 Tm Overshoot (servo step) ≤ 20% ; settling time ≤ settling time of tuning rules of Chien et al. [10] τm ≤ 1.0 ; ξ m = 0.5 Tm Overshoot (servo step) ≤ 20% ; settling time ≤ settling time of tuning rules of Chien et al. [10] α = 0. 5840 , β = 1 , N = 10, φ c = − 139.65 0 K m = 1 ; Tm = 1 ; ξm =1
2 3 (105 b) 0.02453 + 4.104 τ m − 3. 434 τm + 1.231 τ m , T = T i m Tm Tm Tm
2 3 4 τm τm τm τ m = Tm 0.03459 + 1. 852 − 2.741 + 2.359 − 0. 7962 , Tm Tm Tm T m
α = 0. 6726 − 0.1285
Kc
Comment
3
τm τ τ τ τ − 0.1371 m + 0.07345 m , β = 0. 8665 − 0.2679 m + 0.02724 m Tm T T T m m m Tm
1 0.5013 = 0.3363 + τm Kc [ − 0.01147 ] 2 Tm
2 3 (105 c ) − 0.02337 + 4.858 τm − 5.522 τm + 2.054 τ m , T = T m i Tm Tm Tm
2 3 τm τm τ m = Tm 0.03392 + 2. 023 − 1.161 + 0.2826 , Tm Tm Tm
2
α = 0. 6678 − 0. 05413
2
3
τm τ τ τ τ − 0. 5680 m + 0. 1699 m , β = 0. 8646 − 0.1205 m − 0.1212 m Tm Tm Tm Tm Tm
2
Rule
Kc
Ti
Td
Minimum ITAE Pecharroman and Pagola [134b]
0.803 K u
0.509 Tu
0.167 Tu
φ c = phase corresponding to the crossover frequency; K m = 1 ; Tm = 1 ; 0.1 < τm < 10
0.727 K u
0.524 Tu
0.165 Tu
0.672 Ku
0.532 Tu
0.161Tu
0.669 Ku
0.486 Tu
0.170 Tu
Model: Method 15
0.600 Ku
0.498 Tu
0.157 Tu
0.578 K u
0.481Tu
0.154 Tu
0.557 K u
0.467 Tu
0.149 Tu
0.544 Ku
0.466 Tu
0.141Tu
0.537 K u
0.444 Tu
0.144 Tu
0.527 K u
0.450 Tu
0.131Tu
0.521K u
0.440 Tu
0.129 Tu
0.515 K u
0.429 Tu
0.126 Tu
0.509 Ku
0.399 Tu
0.132 Tu
0.496 Ku
0.374 Tu
0.123 Tu
0.480 Ku
0.315 Tu
0.112 Tu
0.430 Ku
0.242 Tu
0.084 Tu
Comment
α = 0.585 , β = 1 , N = 10, φ c = −1460
α = 0.584 , β = 1 , N = 10, φ c = −1400
α = 0.577 , β = 1 , N = 10, φ c = −1340
α = 0.550 , β = 1 , N = 10, φ c = −1250
α = 0.543 , β = 1 , N = 10, φ c = −1150
α = 0.528 , β = 1 , N = 10, φ c = −1050
α = 0.504 , β = 1 , N = 10, φ c = −930
α = 0.495 , β = 1 , N = 10, φ c = −840
α = 0.484 , β = 1 , N = 10, φ c = −730
α = 0.477 , β = 1 , N = 10, φ c = −630
α = 0.454 , β = 1 , N = 10, φ c = −520
α = 0.445 , β = 1 , N = 10, φ c = −410
α = 0.433 , β = 1 , N = 10, φ c = −300
α = 0.385 , β = 1 , N = 10, φ c = −190
α = 0.286 , β = 1 , N = 10, φ c = −100
α = 0.158 , β = 1 , N = 10, φ c = −6 0
Table 81: PID tuning rules - I 2PD model G m ( s) =
U(s) = Kc (1 +
K m e− sτ m - controller s2
1 ) E( s) + Kc ( b − 1) R (s) − Kc TdsY( s) . 1 tuning rule. Tis
Rule
Kc
Direct synthesis Hansen [91a] Model: Method 1
3.75K m τ m
2
Ti
Td
Comment
5.4τ m
2. 5τ m
b = 0.167
Table 82: PID tuning rules – SOSIPD model (repeated pole)
K m e −s τ
s (1 + Tm1s) 2
1 Td s U (s) = K c 1 + + Tis T 1+ d N Rule
Kc
- Two degree of freedom controller:
β T s d E( s) − K c α + R (s) . 2 tuning rules. Td s 1+ s N
Ti
Servo/regulator tuning Taguchi and Araki [61a]
m
Td
Comment
Minimum performance index
Kc
(105 d) 8
Ti
(105 d )
Td
τm ≤ 1.0 ; Tm Overshoot (servo step) ≤ 20% ; settling time ≤ settling time of tuning rules of Chien et al. [10] α = 0. 601 , β = 1 ,
(105d )
Model: Method 2
Minimum ITAE Pecharroman and Pagola [134a]
1.672 K u
0.366 Tu
0.136 Tu
φ c = phase corresponding to the crossover frequency; K m = 1 ; Tm = 1 ; 0.1 < τm < 10
1.236 K u
0.427 Tu
0.149 Tu
0.994 Ku
0.486 Tu
0.155 Tu
0.842 Ku
0.538 Tu
0.154 Tu
Model: Method 1
0.752 Ku
0.567 Tu
0.157 Tu
0.679 Ku
0.610 Tu
0.149 Tu
0.635 K u
0.637 Tu
0.142 Tu
0.590 Ku
0.669 Tu
0.133 Tu
0.551K u
0.690 Tu
0.114 Tu
N = 10, φ c = −1640
α = 0.607 , β = 1 , N = 10, φ c = −1600
α = 0.610 , β = 1 , N = 10, φ c = −1550
α = 0.616 , β = 1 , N = 10, φ c = −1500
α = 0.605 , β = 1 , N = 10, φ c = −1450
α = 0.610 , β = 1 , N = 10, φ c = −1400
α = 0.612 , β = 1 , N = 10, φ c = −1350
α = 0.610 , β = 1 , N = 10, φ c = −1300
α = 0.616 , β = 1 ,
8
Kc
Ti
(105d )
(105 d)
Td
(105 d)
1 0. 5667 = 0.1778 + τm Kc + 0. 002325 Tm
N = 10, φ c = −1250
,
2 3 4 τm τm τm τm = Tm 0.2011 + 11.16 − 14. 98 + 13. 70 − 4.835 Tm Tm Tm Tm
τ τ τ = Tm 1. 262 + 0.3620 m , α = 0. 6666 , β = 0. 8206 − 0.09750 m + 0.03845 m Tm Tm Tm
2
Rule Minimum ITAE Pecharroman and Pagola [134a] – continued
Kc
Ti
Td
0.520 K u
0.776 Tu
0.087 Tu
0.509 Ku
0.810 Tu
0.068 Tu
Comment
α = 0.609 , β = 1 , N = 10, φ c = −1200
α = 0. 611 , β = 1 , N = 10, φ c = −1180
Table 83: PID tuning rules - SOSPD model with a negative zero
K m (1 − sTm3 ) e− s τ
m
(1 + sTm1 )(1 + sTm2 )
- controller with filtered
1 Td s . 1 tuning rule. derivative G c ( s) = Kc 1 + + Ti s 1 + Td s N Rule
Kc
Ti
Td
Tm1 + Tm2 − Tm 3 K m( λ + τ m )
Tm1 + Tm2 − Tm 3
Tm1 Tm2 − (Tm1 + T m2 − Tm3 )Tm 3 Tm1 + T m2 − Tm3
N=10; λ = [ Tm1 , τm ]
2ξTm 2 − Tm3
2ξTm1 − Tm 3
Tm2 1 − (2 ξTm1 − Tm3 )Tm3
N=10; λ = [ Tm1 , τm ]
Comment
Robust Chien [50] Model: Method 1
Km (λ + τ m )
2ξ Tm1 − Tm3
Tm1 > Tm 2 > Tm3
Table 84: PID tuning rules - SOSPD model with a negative zero
K m (1 − sTm3 ) e− s τ
m
(1 + sTm1 )(1 + sTm2 )
- classical controller
1 1 + Td s . 1 tuning rule. G c ( s) = Kc 1 + Ti s 1 + Td s N Rule
Kc
Ti
Td
Comment
Tm 2 K m( λ + τ m )
Tm2
Tm1
Tm1 > Tm 2
Tm1 K m( λ + τ m )
Tm1
Tm2
Robust Chien [50] Model: Method 1
N=10; λ = [ Tm1 , τm ] Tm1 > Tm 2
N=10; λ = [ Tm1 , τm ]
Table 85: PID tuning rules - SOSPD model with a negative zero
K m (1 − sTm3 ) e− s τ
m
(1 + sTm1 )(1 + sTm2 )
- series controller with
1 Td s . 1 tuning rule. filtered derivative G c ( s) = Kc 1 + 1 + Ts Ts i 1 + d N Rule
Kc
Ti
Td
Comment
Tm 2 K m( λ + τ m )
Tm2
Tm1 − Tm3
Tm1 > Tm 2 > Tm3
Tm1 K m( λ + τ m )
Tm1
Tm 2 − Tm3
Robust Chien [50] Model: Method 1
N=10; λ = [ Tm1 , τm ]
Tm1 > Tm 2 > Tm3
N=10; λ = [ Tm1 , τm ]
Table 86: PID tuning rules - SOSPD model with a positive zero
K m (1 + sTm 3 ) e− sτ
m
(1 + sTm1 )(1 + sTm 2 )
- ideal controller
1 Gc ( s) = K c 1 + + Td s . 1 tuning rule. T s i Kc
Rule
ρ
Ti
( p0 q1 − p1 )
,
Minimum IAE, ISE 2 p0 and ITAE – Wang et p 0 = Tm1 + Tm2 + Tm3 − τ m al. [97] q 1 = Tm1 + Tm 2 + 05 . τm
Model: Method 1
Td
Comment
Minimum performance index p p0 q 2 − p 2 p1 q1 − 1 , − , p0 p 0 q1 − p1 p 0
p2 = 05 . Tm1Tm 2 τ m 1
01 . ≤
τm ≤1 Tm1
p1 = Tm1Tm2 +
01 . ≤
Tm 2 ≤1 Tm 1
q 2 = Tm1Tm2
01 . ≤
Tm 3 ≤1 Tm1
0.5τ m (Tm1 + Tm2 ) − 0.5Tm3 τ m
+0.5τ m ( Tm1 + Tm 2 )
1
ρ = 35550 . − 3.6167
+
Tm 2 Tm 1
τm T T − 0.4918 m 2 − 0.3318 m 3 − 0.4685 Tm 1 Tm1 Tm 1
ρ = 39395 . − 3.2164
+
+
Tm3 Tm1
τm T T . − 0.3318 m 2 + 2.5356 m 3 , minimum IAE 14746 T T Tm1 m1 m1
τm Tm 2 Tm 3 τ m τm Tm2 Tm3 + 16185 . − 58240 . + . − 02383 . + 13508 . 10933 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1
Tm 2 τm T T − 0.6679 m 2 − 0.0564 m3 − 0.2383 Tm 1 Tm1 Tm1 Tm1
ρ = 32950 . − 34779 .
+
τm Tm 2 Tm 3 τ τm Tm2 Tm3 + 21781 . − 55203 . + m 14704 . − 04685 . + 14746 . Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1
+
Tm3 τm T T . − 0.0564 m2 + 2.5648 m3 , minimum ISE 13508 Tm1 Tm1 Tm1 Tm1
τm Tm2 Tm3 τ τm T Tm3 + 25336 . − 55929 . + m 14407 . − 0.5712 m2 + 15340 . Tm1 Tm 1 Tm1 Tm 1 Tm1 Tm1 Tm1
Tm 2 τm T T − 0.3268 m2 − 0.5790 m 3 − 0.5712 Tm 1 Tm1 Tm1 Tm 1
+
Tm3 Tm1
τm Tm2 T . − 05790 . + 2.7129 m 3 , minimum ITAE 15340 Tm1 Tm1 Tm1
Table 87: PID tuning rules - SOSPD model with a positive zero
K m (1 + sTm 3 ) e− sτ
m
(1 + sTm1 )(1 + sTm 2 )
- ideal controller with filtered
1 Td s . 1 tuning rule. derivative G c ( s) = Kc 1 + + Ti s 1 + Td s N Kc
Rule
Ti
Td
Comment
Robust Tm1 + Tm2 +
Chien [50]
T m3 τ m λ + T m3 + τ m
K m ( λ + Tm 3 + τ m )
Tm3 τ m Tm1 + Tm2 +
Tm3 τ m λ + Tm 3 + τ m
λ + Tm 3 + τ m + Tm1
Model: Method 1 2ξTm1 +
Tm3 τ m λ + Tm3 + τ m
K m ( λ + Tm 3 + τ m )
2 ξTm1 +
Tm1 Tm2 Tm3 τ m + Tm2 + λ + Tm 3 + τ m
Tm3 τ m
Tm3 τ m
λ + Tm3 + τ m
λ + Tm 3 + τ m +
Tm2 3 Tm 3 τ m 2ξ Tm1 + λ + Tm3 + τ m
Tm1 > Tm 2 > Tm3
N=10; λ = [ Tm1 , τm ]
Tm1 > Tm 2 > Tm3
N=10; λ = [ Tm1 , τm ]
Table 88: PID tuning rules - SOSPD model with a positive zero
K m (1 + sTm 3 ) e− sτ
m
(1 + sTm1 )(1 + sTm 2 )
- classical controller
1 1 + Td s . 1 tuning rule. G c ( s) = Kc 1 + Ti s 1 + Td s N Rule
Kc
Ti
Td
Comment
Tm 2 K m ( λ + Tm3 + τm )
Tm2
Tm1
Tm1 > Tm 2 > Tm3
Tm1 K m ( λ + Tm3 + τm )
Tm1
Tm2
Robust Chien [50] Model: Method 1
N=10; λ = [ Tm1 , τm ]
Tm1 > Tm 2 > Tm3
N=10; λ = [ Tm1 , τm ]
Table 89: PID tuning rules - SOSPD model with a positive zero
K m (1 + sTm 3 ) e− sτ
m
(1 + sTm1 )(1 + sTm 2 )
- series controller with
1 Td s . 1 tuning rule. filtered derivative G c ( s) = Kc 1 + 1 + Ts Ts i 1 + d N Rule
Kc
Ti
Tm 2 K m ( λ + Tm3 + τm )
Tm2
Tm1 K m ( λ + Tm3 + τm )
Tm1
Td
Robust Chien [50] Model: Method 1
Tm1 +
Tm 3τ m λ + Tm 3 + τ m
Tm 2 +
Tm 3τ m λ + Tm3 + τ m
Comment N=10; λ = [ T, τ m ] , T = dominant time constant N=10; λ = [ T, τ m ] , T = dominant time constant
Km e− sτ - ideal controller (1 + sTm1 )(1 + sTm 2 )(1 + sTm 3 ) m
Table 90: PID tuning rules - TOLPD model
1 Gc ( s) = K c 1 + + Td s . 1 tuning rule. Ti s Rule
Kc
Ti
Td
Comment
Minimum performance index Standard form optimisation binomial - Polonyi [153] Model: Method 1 Standard form optimisation – minimum ITAE Polonyi [153] Model: Method 1
1 − 4 Tm 2 + Tm2 Tm1 6 τ T m m3 τ m
4 6Tm2 τ m + τ m
τm
Tm1 > 10( Tm2 + τm )
Tm2 T T 1 − 21 . + m2 m1 3 . 4 τ Tm3 τ m m
2.7 3.4Tm2 τ m + τ m
τm
Tm1 > 10( Tm2 + τm )
Table 91: PID tuning rules - TOLPD model - G m (s ) =
K m e− sτ
m
(1 + sTm ) 3
– Two degree of freedom controller:
1 Td s U (s) = K c 1 + + Tis T 1+ d N
Kc
Rule
Ti
Servo/regulator tuning
Kc
(105 e ) 2
Ti
Model: ideal process
Kc
(105e )
Td
Comment
Minimum performance index
Taguchi and Araki [61a]
2
β T s d E( s) − K c α + R (s) . 1 tuning rule. Td s 1+ s N
1 1.275 = 0.4020 + τm Kc − 0.003273 Tm
(105 e )
Td
(105 e )
τm ≤ 1.0 ; Tm Overshoot (servo step) ≤ 20% ; settling time ≤ settling time of tuning rules of Chien et al. [10]
,
2 3 4 τm τm τm τm Ti = Tm 0. 3572 + 7. 647 − 12. 86 + 11 .77 − 4. 146 Tm Tm Tm Tm 2 2 τm τm τm τm (105 e ) Td = Tm 0.8335 + 0.2910 − 0. 04000 , α = 0. 6661 − 0.2509 + 0. 04773 , Tm Tm Tm Tm (105 e )
β = 0. 8131 − 0.2303
τm τ + 0.03621 m Tm Tm
2
K m e− sτ 1 - ideal controller Gc ( s) = K c 1 + + Td s . 1 − sTm Ti s m
Table 92: PID tuning rules - unstable FOLPD model
3 tuning rules.
Kc
Rule
Ti
Td
Comment
Direct synthesis De Paor and O’Malley [86]
Tm cos K m
(1 − T τ )T
Tm 1 − Tm τ m sin Km Tm τ m
m
m
m τm
(1 − T τ )T m
m
m
τm
Model: Method 1 Chidambaram [88] Model: Method 1 Valentine and Chidambaram [154] dominant pole placement Model: Method 1
1 T . + 0.3 m 13 Km τm
Tm 1165 . Km τ m
0.245
1
+
1− T τ m m Tm Tm τ m φ m = tan − 1
0.245
Tm 1165 . Km τ m
0.245
(
,
)
1 Tm τ m 2 Tm 1 − Tm τ m
1 − Tm τ m − Tm τm 1 − Tm τ m Tm τm
(
τ Tm 25 − 27 m Tm τm 0176 T . + 0.36 Tm m 0.179 − 0.324
Tm 1165 . Km τ m
tan 0.75φ m
τm Tm
τm + 0.161 Tm
2
1 Ti
)
046 . τm
τm < 0.6 Tm
τ . + 0.36 m Tm 0176 Tm
τm < 06 . Tm
τ τ . + 0.36 m 25Tm 0176 . + 0.36 m Tm 0176 Tm Tm 0.176Tm + 0.36τ m τ 0.12 − 0.1 m Tm
τm <1 Tm
τ . + 0.36 m Tm 0176 Tm
0.6 ≤
08 . <
τm Tm τm Tm
≤ 08 .
≤1
K m e− sτ - non-interacting controller 1 − sTm m
Table 93: PID tuning rules - unstable FOLPD model
1 KTs U( s) = Kc 1 + E ( s) − c d Y(s) . 2 tuning rules. sT Ti s 1+ d N Kc
Rule Servo tuning Huang and Lin [155] - minimum IAE Model: Method 2 Regulator tuning Huang and Lin [155] - minimum IAE Model: Method 2
Ti
Td
Comment
Minimum performance index −
−1 .0251 τm τ m 1 − 0 .433 + 0 .2056 + 0 .3135 Km Tm T m
6.6423 7.6983τ m τ τm T − 0. 0312+ 1. 6333 τm + 00399 Tm − 00018 . + 08193 . + 7. 7853 m . e T m m Tm Tm Tm
0.01 ≤
τm ≤ 0.8 ; N=10 Tm
0.01 ≤
τm ≤ 0.8 ; N=10 Tm
Minimum performance index −
−1.004 1 τm τ 0.2675 + 01226 . + 0.8781 m Km Tm Tm
2. 9123 τ m τ Tm 00005 . + 2.4631 m + 95795 . T T m m
τ Tm 0.0011 + 0.4759 m Tm
K m e− sτ - classical controller 1 − sTm m
Table 94: PID tuning rules - unstable FOLPD model
1 1 + Td s . 1 tuning rule. G c ( s) = Kc 1 + Ti s 1 + Td s N Rule
Kc
Regulator tuning Shinskey [16] minimum IAE – page 381. Method: Model 1
Ti
Td
Comment
Minimum performance index 170 . τm 0.60τm
τ m Tm = 01 .
Tm K mτ m
1.90τ m
0.60τm
τ m Tm = 0.2
08929 . Tm Km τ m
2.00τ m
0.80τ m
τ m Tm = 0.5
08621 . Tm K m τm
2.25τm
0.90τm
τ m Tm = 0.67
08333 . Tm K m τm
2.40τ m
100 . τm
τ m Tm = 0.8
0.9091Tm Km τ m
Table 95: PID tuning rules - unstable SOSPD model G m ( s) =
Km e − sτ - ideal controller (1 − sTm1 )(1 + sTm 2 ) m
1 Gc ( s) = K c 1 + + Td s . 2 tuning rules. Ti s Kc
Ti
Td
Comment
Kc(106 ) 3
Ti (106)
Td (106 )
Tuning rules developed from Ku , Tu
λ Tm1 λ + 2 + Tm 2 Tm1 2 λ
λ λ + 2 + Tm 2 Tm1
Km uncertainty = 50%
τm Tm = 02 .
λ = [0.5Tm ,19 . Tm ]
τm Tm = 0.4
λ = [1.3Tm ,19 . Tm ]
τm Tm = 02 .
λ = [0.4Tm ,4. 3Tm ]
τm Tm = 0.4
λ = [11 . Tm ,4.3Tm ]
τm Tm = 0.6
λ = [ 2.2Tm , 4.3Tm ]
Rule Ultimate cycle McMillan [58] Model: Method not relevant Robust Rotstein and Lewin [89] Model: Method 1
Km uncertainty = 30%
λ λ + 2 Tm 2 Tm1 λ λ + 2 + Tm 2 Tm1
λ determined graphically – sample values provided
2
3
Kc
Ti
( 106)
( 106)
=
1111 . Tm1Tm2 Km τ m 2
1 0.65 , ( Tm1 + Tm 2 )Tm1Tm2 1 + T − T T − τ τ m2 )( m1 m) m ( m1
0.65 0.65 (T + T ) T T (T + T )T T m1 m2 m1 m2 ( 106) m1 m2 m1 m 2 = 2τ m 1 + = 05 . τ m 1 + , Td ( Tm1 − Tm2 )( Tm1 − τ m ) τ m (Tm1 − Tm 2 )( Tm1 − τ m ) τ m
Table 96: PID tuning rules - unstable SOSPD model G m ( s) =
Km e − sτ - classical controller (1 − sTm1 )(1 + sTm 2 ) m
1 1 + Tds . 2 tuning rules. G c ( s) = Kc 1 + Ti s 1 + Td s N Kc
Rule
Ti
Regulator tuning bTm1 1 +
aTm2 2
Minimum ITAE 2 4( τm + Tm 2 ) Poulin and Pomerleau [82], [92] – K m ( aTm1 − 4[ τ m + Tm2 ]) deduced from graph Model: Method 1 (τ m + Td N) Tm1
Output step load disturbance
Input step load disturbance
Td
Comment
Minimum performance index
4Tm1 ( τ m + Tm2 )
aTm1 − 4( τ m + Tm2 )
0≤
Tm2
τm ≤2; T ( d N)
01 . Tm ≤
Td ≤ 033 . Tm N
a 0.9479 1.0799 1.2013 1.3485 1.4905
b 2.3546 2.4111 2.4646 2.5318 2.5992
(τ m + Td N) Tm1
0.05 0.10 0.15 0.20 0.25
0.30 0.35 0.40 0.45 0.50
a 1.6163 1.7650 1.9139 2.0658 2.2080
b 2.6612 2.7368 2.8161 2.9004 2.9826
0.05 0.10 0.15 0.20 0.25
1.1075 1.2013 1.3132 1.4384 1.5698
2.4230 2.4646 2.5154 2.5742 2.6381
0.30 0.35 0.40 0.45 0.50
1.6943 1.8161 1.9658 2.1022 2.2379
2.7007 2.7637 2.8445 2.9210 3.0003
Table 97: PID tuning rules - unstable SOSPD model G m ( s) =
Km e − sτ - series controller (1 − sTm1 )(1 + sTm 2 ) m
1 Gc ( s) = Kc 1 + (1 + Td s) . 1 tuning rule. Ti s Rule
Kc
Direct synthesis Ho and Xu [90]
ω p Tm1
Model: Method 1
A m Km
Ti
Td
157 . ω p − ω p τm − 2
1 Tm1
Tm2
Comment
ωp =
A m φ m + 157 . A m (A m − 1)
(A
2 m
)
− 1 τm
Table 98: PID tuning rules - unstable SOSPD model G m ( s) =
Km e − sτ - non-interacting controller (1 − sTm1 )(1 + sTm 2 ) m
1 KTs U( s) = Kc 1 + E ( s) − c d Y(s) . 2 tuning rules. sT Ti s 1+ d N Rule Servo tuning Huang and Lin [155] - minimum IAE
Kc
Ti
Td
Comment
Minimum performance index
Tm2 ≤ Tm1 ;
Kc(107 ) 4
Ti (107)
Td (107 )
0.05 ≤
Model: Method 2
4
Kc
( 107 )
τm ≤ 0.4 Tm1
−1.344 0.995 τm Tm 2 1 τm Tm2 T τ =− 10.741 − 13363 . + 0099 . + 727.914 − 708.481 + 9.915 m 2 2m Km Tm1 Tm1 Tm1 Tm1 Tm1
−
1.031 0.997 τ T T T Tm 2 Tm 2 1 T T T 84.273 m1 − 90 . 959 + 9 . 034 e − 2 . 386 e − 16 . 304 e Km τm Tm1 τm m
m2
m1
m1
m 2 τm 2 m1
2.12 0.985 τ T τm T T τ Ti (107 ) = Tm1 −149 .685 − 141418 . − 88.717 m − 17.29 m 2 + 20518 . m2 − 12.82 m 2 2m Tm1 Tm1 Tm1 Tm1 Tm1 0.286 1.988 τ T T τ Tm 2 T + Tm1 3.611 m + 0.000805 m2 + 141.702e T − 2.032e T + 10.006e T Tm1 Tm1 τm
Td
( 107 )
m
m2
m1
m1
m 2τ m 2 m1
2 2 τm Tm2 τm Tm2 T τ = Tm1 −0.4144 + 15805 . − 142.327 + 01123 . + 0.7287 − 18.317 m 2 2m Tm1 Tm1 Tm1 Tm1 Tm1 3 3 τ 2T τ T 2 τ T + Tm1 48695 . m − 10542 . m2 + 204.009 m 3m 2 + 47.26 m m32 Tm1 Tm1 Tm1 Tm1
τ + 396349 . m Tm1
4
τ 3T τ T 3 τ 2T 2 T τ + Tm1 − 138.038 m 4m 2 + 52155 . m m42 − 646.848 m m4 2 + 19.302 m 2 − 4731.72 m Tm1 Tm1 Tm1 Tm1 Tm1 4
τ 4T τ T 4 τ 3T 2 + Tm1 425789 . m 5m 2 − 289 .746 m m52 − 841807 . m m5 2 Tm1 Tm1 Tm1
5
τ 2T 3 Tm 2 + 1313.72 m m5 2 − 37688 . Tm1 Tm1
6 τ 5T τ T 5 τ 4T 2 τm + Tm1 6264 .79 . m 6m2 + 204.689 m m62 + 25706 . m m6 2 − 161469 Tm1 Tm1 Tm1 Tm1 3 6 τ 2T 4 τ T T + Tm1 − 791857 . m m6 2 + 648.217 m m2 2 − 5.71 m2 Tm1 Tm1 Tm1
5
Rule
Kc
Ti
Td
Huang and Lin [155] - minimum IAE continued Model: Method 2
Kc(108 ) 5
Ti (108)
Td (108 )
5
Kc (108) = −
Ti
Tm 1 < Tm 2 ≤ 10Tm1 ; 0.05 ≤
τm ≤ 025 . Tm1
− 0.3055 0.5174 τ T 1 τm T T τ −1302 . + 85914 . + 34.82 m + 10.442 m 2 − 22.547 m2 − 14.698 m 2 2m Km Tm1 Tm1 Tm1 Tm1 Tm2
1 − Km ( 108)
Comment
1.0077 0.9879 τ T T Tm 2 Tm 2 T 52.408 m1 − 5147 . + 53.378e − 0.000001e T τ T τ m m1 m m
m2
m1
m1
Tm2 τ m
+ 0.286e
Tm1 2
2 2 Tm 2 τm Tm2 τm Tm2 τ m = Tm1 72.806 − 268.746 − 4.9221 . . + 246819 + 0.6724 + 151351 Tm1 Tm1 Tm1 Tm1 Tm12 3 3 4 τ 2T τ T 2 τ Tm 2 τ + Tm1 − 6914.46 m − 00092 . . m 3m 2 − 14.27 m m32 + 558017 . m − 795465 Tm1 Tm1 Tm1 Tm1 Tm1
τ 3T τ T 3 τ 2T 2 Tm2 + Tm 1 1417 .65 m 4m 2 + 0.4057 m m42 + 55536 . m m4 2 − 0001119 . Tm1 Tm1 Tm1 Tm1 T τ T T T + Tm 1 − 44.903e + 0.000034e − 15694 . eT m
m2
m1
m1
m2 τ m 2 m1
τ + 678778 m Tm1
19.056
Tm2 Tm1
4
7.3464
1.1798 0.1064 τ T τ T T τ Td (108) = Tm1 175515 . − 86.2 m + 348.727 m − 0.008207 m 2 − 55619 . m2 + 0.0418 m 2 2m Tm1 Tm1 Tm1 Tm1 Tm1
0.0355 0.0827 τ τ Tm2 T + Tm 1 78959 . m + 0.005048 m 2 − 187 .01e T Tm 1 Tm1 τm
Tm 2 τ m
Tm 2
m
m1
+ 0000001 . e
Tm1
− 00149 . e
Tm1 2
Kc
Rule Regulator tuning Huang and Lin [155] – minimum IAE
Ti
Td
Comment
Minimum performance index
Tm2 ≤ Tm1 ;
Kc
( 109 ) 6
Ti
( 109)
Td
( 109 )
0.05 ≤
Model: Method 2
6
Kc
( 109 )
−1.164 2.54 τm Tm2 1 τm Tm 2 T τ =− − 174167 . − 31364 . + 0.4642 − 103069 . − 83916 . − 66.962 m2 2m Km Tm1 Tm1 Tm1 Tm1 Tm1
1 − Km Ti
( 109)
τm ≤ 0.4 Tm1
1.065 1.014 τ T T Tm 2 T 59.496 m1 m 2 − 7079 . + 23121 . e τm τm Tm1
Tm 2 τ m
Tm 2
m
m1
+ 126.924e
Tm 1
+ 26.944e
Tm 1 2
2 2 3 τm Tm2 τm τm Tm 2 Tm2 τ m = Tm1 0.008 + 2.0718 + 6.431 + 0.7503 − 18686 . + 0.4556 + 2.4484 Tm1 Tm1 Tm1 Tm1 Tm12 Tm1 3 4 T τ 2 τ T 2 T τ 3 T τ + Tm 1 − 2.9978 m 2 − 21135 . m2 m3 + 12.822 m m32 + 39.001 m + 22848 . m 2 4m Tm1 Tm1 Tm1 Tm1 Tm1
T 3τ T 2τ 2 T + Tm1 − 4.754 m 2 4 m − 0.527 m2 4m + 164 . m2 Tm1 Tm1 Tm1
4
2 2 3 τ Tm2 τ τm Tm 2 T τ Td (109) = Tm1 −0.0301 + 11766 . − 4.4623 m + 05284 . − 14281 . . m + 4.6 m 2 2m + 11176 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 3 4 τ m2 Tm 2 τ m Tm2 2 Tm 2 τm 3 Tm 2 τm + Tm 1 10886 . − 05039 . − 98564 . − 7528 . − 5.0229 3 3 4 Tm1 Tm1 Tm1 Tm1 Tm1 4 τ T 3 τ 2T 2 Tm 2 + Tm1 − 2.3542 m m42 + 9.3804 m m4 2 − 01457 . Tm1 Tm1 Tm1
Rule
Kc
Ti
Td
Huang and Lin [155] - minimum IAE – continued Model: Method 2
Kc(110 ) 7
Ti (110)
Td (110 )
7
Kc
( 110)
Comment
Tm 1 < Tm 2 ≤ 10Tm1 ; 0.05 ≤
τm ≤ 025 . Tm1
2.1984 0.791 τm Tm 2 1 τm Tm 2 T τ =− 1750.08 + 1637.76 + 1533.91 − 7.917 + 6187 . − 6.451 m2 2m Km Tm1 Tm1 Tm1 Tm1 Tm1
3.2927 1.0757 τ T Tm1 Tm 2 1 Tm 2 T 0002452 − . + 13729 . − 1739.77e − 0.000296e T Km τm τm Tm1 m
m2
m1
m1
Tm 2 τ m
+ 2.311e
Tm1 2
2 2 3 τ Tm 2 τ τ Tm 2 T τ Ti (110) = Tm1 51678 . − 57.043 m + 1337.29 m + 01742 . − 01524 . + 7.7266 m 2 2m − 6011.57 m Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 3 4 Tm2 τ m 2 τ m Tm2 2 Tm2 τ m 3 Tm2 τm + Tm 1 0.0213 + 0 . 0645 + 913552 . + 274 . 851 − 65.283 3 3 4 Tm1 Tm1 Tm1 Tm1 Tm1 4 τ T T 3τ T 2τ 2 Tm 2 T T + Tm 1 0.003926 m 2 4 m − 2.0997 m 2 4m − 0001077 . − 49 . 007 e + 0 . 000026 e T T T m1 m1 m1 T τ + Tm1 0.2977e T m2
m1
m
m
m2
m1
m1
2 2 3 τ T τ τ T T τ Td (110) = Tm1 −0.0605 + 4.6998 m − 29.478 m + 0.0117 m 2 − 0.0129 m 2 + 0.6874 m2 2m + 140.135 m Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 3 4 τ m 2 Tm 2 τ mTm 2 2 Tm 2 τm 3 Tm 2 τm + Tm 1 0.002455 . − 01289 . − 238 . 864 + 06884 . − 14712 3 3 4 Tm1 Tm1 Tm1 Tm1 Tm1 4 τ T 3 τ m 2Tm 2 2 Tm 2 + Tm 1 0.007725 m m42 − 01222 . − 0 . 000135 4 Tm1 Tm1 Tm1
Table 99: PID tuning rules – general model with a repeated pole G m ( s) =
K m e − sτ - ideal controller (1 + sTm ) n m
1 Gc ( s) = K c 1 + + Td s . 1 tuning rule. Ti s Kc
Rule Direct synthesis Skoczowski and Tarasiejski [156] Model: Method 1
1
≤
ω g Tm Km
(1 + ω
2 g
Tm 2
)
Ti
Td
Tm
n−1 Tm , n ≥ 2 n+2
Comment
n− 1
1 + ω g 2 Td 2 1
2n + 4 τ m 4n + 2 − Tm n 2 − 2n − 2 + + φm ± b π Tm π ωg = 2 4 n + 2 τ m 2n − 2 2 2Tm n − 4n + 3 + + φm π Tm π
with 2
b = Tm
2 2n + 4 τ m 4 n + 2 2 2 4 n + 2 τm 2 n − 2 + φ m + 4( n + 2) 1 − φ m n2 − 4n + 3 + + φm n − 2n − 2 + π Tm π π π Tm π
Table 100: PID tuning rules – general stable non-oscillating model with a time delay - ideal controller 1 Gc ( s) = K c 1 + + Td s . 1 tuning rule. Ti s
Kc
Rule
Ti
Td
Comment
Direct synthesis
2
Kc
(110 a )
=
Ti Ti
(110 a ) 2
Ti
(110 a )
Kc
(110 a )
Ti
(110 a )
Kc
(110 a )
Ti
(110 a )
Kc
Gorez and Klan [147a] Model: not specified
( 110a )
( 110a )
+ τm
, Ti
(110 a )
τ 1+ 1 + 2 m Tar = Tar 2
Td Ti
(110 a )
(110a )
τm Tar
0.25Ti
τm 1 − T ar
(110 a )
2
τ −2 m Tar
,
2 T τ 2τ m Ti (110a ) + Tcr cr + Ti (110a ) − Taa − m 1 − K c (110a ) 1 + (110 a ) Tar 2Tar 3Ti (110 a ) Ti Tar = average residence time of the process (which equals Tm + τ m for a FOLPD process, for example); T aa =
Td
(110 a )
=
Tar
(
)
(110 a ) 1 + τ m additional apparent time constant; Tcr = 1 − K c 2T (110a ) i
τ m
Table 101: PID tuning rules – fifth order model with delay K m (1 + b1s + b2 s2 + b3s3 + b4 s4 + b ss5 ) e− s τ 1 G m ( s) = – ideal controller Gc ( s) = K c 1 + + Td s . 2 3 4 5 Ti s 1 + a 1s + a 2 s + a 3s + a 4 s + a 5s m
(
)
1 tuning rule.
Kc
Rule Direct synthesis Magnitude optimum - Vrancic et al. [159] Model: Method 1
Ti
Td
Ti (112)
Td (112 )
Comment
3
Kc
( 112 )
3
Kc ( 112 ) =
Td
( 112)
)
[
2 Km − a1 b 1 + a1a2 + a1b1 − a3 − b1b 2 + b 3 + ( a1 − b 1 ) τ m + ( a1 − b 1 ) τ m2 + 0.333τ m 3 − ( a1 − b 1 + τ m ) Td 2
2
2
(
2
)
a1 − a1 b 1 + a 1b 2 − 2 a1a 2 + a2 b 1 + a 3 − b 3 + τ m a 1 − a1b 1 − a 2 + b 2 + 0.5( a1 − b1 ) τm + 0.167τ m 3
Ti (112 ) =
(
a13 − a1 2b 1 + a1 b 2 − 2 a1a2 + a2 b 1 + a 3 − b 3 + τ m a1 2 − a1b1 − a 2 + b 2 + 0.5( a1 − b1 ) τ m 2 + 0167 . τm 3
2
[a
2 1
2
− a 1b 1 − a 2 + b 2 + ( a1 − b 1 ) τ m + 0.5τ m − ( a1 − b1 + τ m ) Td
… see attached sheet with δ = 0.
2
2
]
3
]
Table 102: PID tuning rules – fifth order model with delay K m (1 + b1s + b2 s2 + b3s3 + b4 s4 + b ss5 ) e− s τ G m ( s) = – controller with filtered derivative 1 + a1s + a2 s2 + a3s 3 + a 4 s4 + a 5s5 m
(
)
1 Tds . 1 tuning rule. G c ( s) = Kc 1 + + Ts Ts i 1 + d N Rule Direct synthesis Magnitude optimum - Vrancic et al. [73] Model: Method 1
Kc
Ti
Td
Comment
Kc(113) 4
Ti (113)
Td (113)
8 ≤ N ≤ 20
4
Kc ( 113) =
Ti( 113 ) =
(
)
a13 − a12 b1 + a1b2 − 2a1a2 + a 2b 1 + a 3 − b 3 + τ m a12 − a1b 1 − a2 + b 2 + 0.5( a1 − b1 ) τ m2 + 0.167τ m3 2 T2 2 2 2 2 3 2 K m −a1 b 1 + a1a2 + a1b 1 − a3 − b1b 2 + b 3 + ( a1 − b1) τ m + ( a1 − b1 ) τ m + 0.333τ m − ( a1 − b1 + τ m ) Td − d ( a1 − b1 + τ m ) N
(
)
a13 − a12 b1 + a1b 2 − 2a1a2 + a2 b1 + a 3 − b 3 + τ m a12 − a1b 1 − a 2 + b 2 + 0.5( a1 − b1 ) τ m2 + 0.167τ m3 2 Td 2 2 a1 − a1b 1 − a 2 + b 2 + ( a1 − b 1) τ m + 0.5τ m − ( a1 − b1 + τ m ) Td − N
Td (113) = see attached sheet
4. Conclusions
The report has presented a comprehensive summary of the tuning rules for PI and PID controllers that have been developed to compensate SISO processes with time delay. Further work will concentrate on evaluating the applicability of these tuning rules to the compensation of processes with time delays, as the value of the time delay varies compared with the other dynamic variables.
Year
1942 1950 1951 1952 1953 1954 1961 1964 1965 1967 1968 1969 1972 1973 1975 1979 1980 1982 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 1941-1950
Table 103: Tuning rules published by year and medium Journal articles Conference Books/ Ph.D. papers/ thesis correspondence 1 1 1 1 3 1 1 1 1 1 1 1 2 2 1 1 2 1 2 1 2 1 1 1 1 1 3 4 2 4 3 4 1 5 1 2 5 4 1 5 2 2 14 2 1 8 6 2 11 4 1 6 7 12 1 1 1 16 2
0
0
Total
1 1 1 1 3 1 1 1 1 2 1 2 2 2 2 1 2 1 3 2 1 1 9 7 5 5 3 10 9 17 16 16 13 14 17 2
1951-1960 1961-1970 1971-1980 1981-1990 1991-2000 TOTAL
6 5 7 15 68 103
0 0 0 7 44 51
0 3 2 7 8 20
6 8 9 29 120 174
List of journals in which tuning rules were published and number of tuning rules published Advances in Modelling and Analysis C, ASME Press 1 AIChE Journal 4 Automatica 12 British Chemical Engineering 1 Chemical Engineering Communications 5 Chemical Engineering Progress 1 Chemical Engineering Science 1 Control 1 Control and Computers 1 Control Engineering 7 Control Engineering Practice 4 European Journal of Control 1 Hydrocarbon Processing 2 Hungarian Journal of Industrial Chemistry 1 IEE Proceedings, Part D 9 (including IEE Proceedings - Control Theory and Applications, Proceedings of the IEE, Part 2) IEEE Control Systems Magazine 1 IEEE Transactions on Control Systems Technology 4 IEEE Transactions on Industrial Electronics and Control Instrumentation 1 IEEE Transactions on Industrial Electronics 1 IEEE Transactions on Industry Applications 1 Industrial and Engineering Chemistry Process Design and Development 2 Industrial Engineering Chemistry Research 12 International Journal of Adaptive Control and Signal Processing 1 International Journal of Control 3 International Journal of Electrical Engineering Education 1 International Journal of Systems Science 1 Instrumentation 1 Instrumentation Technology 2 Instruments and Control Systems 2 ISA Transactions 2 Journal of Chemical Engineering of Japan 2 Journal of the Chinese Institute of Chemical Engineers 3 Pulp and Paper Canada 1 Process Control and Quality 1 Thermal Engineering (Russia) 2 Transactions of the ASME 7 (including Transactions of the ASME. Journal of Dynamic Systems, Measurement and Control) Transactions of the Institute of Chemical Engineers 1
Classification of journals in which tuning rules were published and number of tuning rules published Chemical Engineering Journals 32 (AIChE Journal, British Chemical Engineering, Chemical Engineering Communications, Chemical Engineering Progress, Chemical Engineering Science, Transactions of the Institute of Chemical Engineers, Hungarian Journal of Industrial Chemistry, Industrial and Engineering Chemistry Process Design and Development, Industrial
Engineering Chemistry Research, Journal of Chemical Engineering of Japan, Journal of the Chinese Institute of Chemical Engineers) Control Engineering Journals 53 (Automatica, Control, Control and Computers, Control Engineering, Control Engineering Practice, European Journal of Control, IEE Proceedings, Part D, IEE Proceedings - Control Theory and Applications, Proceedings of the IEE, Part 2, IEEE Control Systems Magazine, IEEE Transactions on Control Systems Technology, International Journal of Adaptive Control and Signal Processing, International Journal of Control, International Journal of Systems Science, Instrumentation, Instrumentation Technology, Instruments and Control Systems, ISA Transactions, Process Control and Quality) Mechanical Engineering Journals 8 (Advances in Modelling and Analysis C, ASME Press, Transactions of the ASME, Transactions of the ASME. Journal of Dynamic Systems, Measurement and Control) Electrical/Electronic Engineering Journals 4 (EE Transactions on Industrial Electronics and Control Instrumentation, IEEE Transactions on Industrial Electronics, IEEE Transactions on Industry Applications, International Journal of Electrical Engineering Education) Trade Journals 5 (Hydrocarbon Processing, Pulp and Paper Canada, Thermal Engineering (Russia))
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2.
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3.
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4.
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7.
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8.
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9.
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Appendix 1: List of symbols used (more than once) in the paper. a 1, a2 = PID filter parameters A m = gain margin b = setpoint weighting factor
b1 = PID filter parameter c = derivative term weighting factor E(s) = Desired variable, R(s), minus controlled variable, Y(s) FOLPD model = First Order Lag Plus time Delay model FOLIPD model = First Order Lag plus Integral Plus time Delay model
G c ( s) = PID controller transfer function G p ( jω ) = process transfer function at frequency
G p ( jω ) = magnitude of G p ( jω ) ,
ω
∠Gp ( jω) = phase of G p ( jω )
IAE = integral of absolute error IMC = internal model controller IPD model = Integral Plus time Delay model ISE = integral of squared error ISTES = integral of squared time multiplied by error, all to be squared ISTSE = integral of squared time multiplied by squared error ITAE = integral of time multiplied by absolute error
ki = integral gain of the parallel PID controller k p = proportional gain of the parallel PID controller
kd = derivative gain of the parallel PID controller Kc = Proportional gain of the controller Km = Gain of the process model Ku = Ultimate gain K25% = proportional gain required to achieve a quarter decay ratio m = multiplication parameter in the lead controller
Ms = closed loop sensitivity N = determination of the amount of filtering on the derivative term PP = perturbance peak = peak of system output when unit step disturbance is added R(s) = Desired variable RT= recovery time = time for perturbed system output (when a unit step disturbance is added) to come to its final value SOSPD model = Second Order System Plus time Delay model
Td = Derivative time of the controller TCL = desired closed loop system time constant Tf = Time constant of the filter in series with the PID controller Ti = Integral time of the controller Tm = Time constant of the FOLPD process model Tm1, Tm2 ,Tm3 = Time constants of the higher order process models Tu = Ultimate time constant T25% = period of the quarter decay ratio waveform TOLPD model = Third Order Lag Plus time Delay model TS = settling time U(s) = manipulated variable Y(s) = controlled variable
λ = Parameter that determines robustness of compensated system.
ξ m = damping factor of an underdamped process model, ξ = damping factor of the compensated system κ = 1 K m Ku
φ = phase lag,
φ m = phase margin,
φ ω = phase lag at an angular frequency of
ω
φ c = phase corresponding to the crossover frequency τ m = time delay of the process model, ω = angular frequency,
τ = τ m ( τ m + Tm )
ω u = ultimate frequency,
ω φ = angular frequency at a phase lag of
φ
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