Robust PID controller design for non-minimum phase time delay systems
ISA Transactions 40 (2001) 31±39 www.elsevier.com/locate/isatrans Robust PID controller design for non-minimum phase time delay systems Ying J. Huang *, Yuan-Jay Wang Institute of Electrical Engineering, Yuan Ze University, 135 Far-East Road, Chungli, Taiwan Received 10 January 2000; received in revised form 6 July 2000; accepted 6 July 2000Abstract A robust PID controller for a non-minimum phase system subject to uncertain delay time is presented in this paper.Utilizing the gain-phase margin tester method, a speci®cation-oriented parameter region in the parameter plane thatcharacterizes all admissible controller coecient sets can be obtained. The PID controller gains are then directlyselected from the parameter region. Henceforth, the designed controller can guarantee the system at least a pre-speci-®ed safety margin to compensate for the instability induced by the time delay. A compromise between the robustnessand tracking performance of the system in the presence of time delay is achieved. Simulation results indicate that theproposed method performs a good time response, and robustness is obtained eectively. # 2001 Elsevier Science Ltd.All rights reserved.Keywords: PID control; Non-minimum phase; Gain-phase margin tester method1. Introduction processes. Subsequently, parameter plane methods [4, 5] for the evaluation of the PID settings based The PID controllers have been successfully on the given Gm and Pm speci®cations are pre-applied to many industrial control systems. De sented. However, there are few systematic PIDPaor and OMalley  derived PID controllers of tuning formulas for non-minimum phase systems,the Ziegler±Nichols type for unstable processes especially with time delay.with time delay, based on an optimal gain margin Gain margin and phase margin have always(Gm) and an optimal phase margin (Pm). Later, played an important role concerning the robustnessSha®ei and Shenton  presented a graphical tech- of systems. In this paper, the previous achievementnique for calculating PID controller parameters. is extended to the non-minimum phase plant con-According to the gain and phase margin speci®ca- taining an uncertain delay time with speci®cationstions, simple rules were introduced by Ho and Xu in terms of gain and phase. Controllers designed to to tune the PID controller settings for unstable meet the gain phase margin speci®cations have been demonstrated in the literature [4±6]. The gain-phase * Corresponding author. Tel.: +886-3-463-8800, ext. 410; margin tester method  is adopted to test the sta-fax: +886-3-463-3326. bility boundary in the parameter plane [7±9, 11] for E-mail address: firstname.lastname@example.org (Y.J. Huang). any given gain or phase margin speci®cations.0019-0578/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.PII: S0019-0578(00)00036-7
32 Y.J. Huang, Y.-J. Wang / ISA Transactions 40 (2001) 31±39These margins serve as restrictions to scheduling phase systems and endows the system with robustthe controller. Through the de®nition of such safety margins in terms of gain and phase.margins, not only the relative stability margin, but For a high-order non-minimum system whichalso the absolute stability margin, can be guaran- contains a time delay element , its transferteed . Henceforth, the speci®cation of the sys- function is shown as follows,tem in terms of gain margin and phase margin isinterpreted into desired parameter area in a two- 1 À 3:6sexp ÀTsdimensional parameter plane. The method intro- GP s ; 1duced in this paper is based on a search for such 5s 1 s 1 0:2s 1 0:5s 1an aforementioned parameter area to achievecompromise between good tracking performance where T is the delay time of the system. Using aand system robustness with respect to external second-order approximation, the time domain anddisturbance. frequency domain speci®cations are approxi- The advantage of the gain-phase margin tester mately converted into interval gain margins andmethod is that various system performances phase margins . Therefore, the control systemresulting from the tuning of the adjustable para- with a PID controller in a series connection withmeters can be realized completely. A speci®cation- the plant is expected to achieve the speci®cationsoriented parameter area, which characterizes all of 5 dB4Gm410 dB and 30 4Pm460 . Fig. 1admissible stabilizing controller sets, can be shows the block diagram of the considered system.obtained in the parameter plane. PID controller The transfer functions of the process and the con-with coecients selected from the obtained para- troller are denoted as GP s and GC s, respec-meter area stabilizes the non-minimum phase time tively. D s is the external disturbance.delay systems with pre-speci®ed safety margins. An error-actuated PID controller has the gen-Especially when the delay time is uncertain, this eral transfer functionmethod works eectively well. PID controller set-tings could be tuned out o-line in general. It can KIavoid extensive or unnecessary on-line tuning and GC s KP KD s: 2 smakes the implementation of the controller easier.After all, it is noted that this method can be applied The forward open-loop transfer function of theto both stable and unstable systems of high order control system shown in Fig. 1 isand where the controller design has considerable¯exibility. N s G0 s GC sGP s; : 3 D s2. Non-minimum phase time delay control system By letting s j!, and ReG0 j! and ImG0 j! Time delay occurs in the control system when be the real part and imaginary part of the G0 j!,there is a delay between the commanded response respectively, one hasand the start of the output response . Thedelay causes a decreased phase margin whichimplies a lower damping ratio and a more oscilla-tory response for the closed-loop system. Further,it decreases the gain margin, thus moving the sys-tem closer to instability. In this section, a sys-tematic algorithm is introduced for thedetermination of the PID settings. The controlleris designed to compensate for the instabilityinduced by the time delay for the non-minimum Fig. 1. Block diagram of a typical PID control system.
Y.J. Huang, Y.-J. Wang / ISA Transactions 40 (2001) 31±39 33 G0 j! G0 j!ej ; 4 D j! À 1 G0 j!ej N j! 0: 7where Let qG0 j! ReG0 j!2 ImG0 j!2 ; 5 A 1=G0 j!; 8 È É 180: 9 G0 j! tanÀ1 ImG0 j!=ReG0 j! : 6 When 0, A is the gain margin of the system, Substituting (4) into (3), one obtains and when A 1, is the corresponding phase margin. Now we de®ne the gain-phase margin tester function as F j! D j! AeÀj N j!: 10 Eqs. (7)±(10) imply that the function F j!Fig. 2. Block diagram of the control system with a gain-phase should always be equal to zero. This indicates thatmargin tester. the gain margin and the phase margin of the PIDFig. 3. R1 (ABCD) is the user-speci®ed parameter region. P1 (KP 0:4, KI 0:1, and KD 0:10) is the representative point withdelay time T 0:5 s.
34 Y.J. Huang, Y.-J. Wang / ISA Transactions 40 (2001) 31±39control system can be determined from the char- Xa j! 4:1!4 À 6:7!2 j 0:5!5 À 9:3!3 !;acteristic equation. By adding a so-called gain-phase margin tester 13AeÀj into the system as shown in Fig. 2, thecharacteristic equation is Xb j! 3:6!2 j!; 14 KI Xc j! 1 À j3:6!; 151 AeÀj KP KD s s 1 À 3:6sexp ÀTs 0: 11 Xd j! À!2 j3:6!3 : 16 5s 1 s 1 0:2s 1 0:5s 1 From Eq. (12), letting 1 !T, one obtains Noting that AeÀj Acos À jAsin, Eqs. (10) the following two stability equations,and (11) give rise to FR j! KP B1 KI C1 D1 ;F j! Xa j! A cos !T À jsin !T Re Xa Acos1 KP Re Xb Á KP Xb j! KI Xc j! KD Xd j!; 12 KI Re Xc KD Re XD Acos1 KP Im Xb KI Im Xc where KD Im XD ; 0 17 Fig. 4. Bode diagrams with P1 selected (KP 0:40, KI 0:10, and KD 0:10).
Y.J. Huang, Y.-J. Wang / ISA Transactions 40 (2001) 31±39 35FI j! KP B2 KI C2 D2 ; D2 Im Xa Acos1 KD Im Xd Im Xa Acos1 KP Im Xb KI Im Xc KD Im XD À Asin1 KP Re Xb À Asin1 KD Im Xd : 24 KI Re Xc KD Re XD ; 0; 18 Note that Re Xa , Re Xb , Re Xc , and Re Xd where are the real parts of Xa , Xb , Xc , and Xd , respec- tively; and Im Xa , Im Xb , Im Xc , and Im Xd B1 Acos1 Re Xb Asin1 Im Xb ; 19 are the imaginary parts of Xa , Xb , Xc , and Xd , respectively.C1 Acos1 Re Xc Asin1 Im Xc ; 20 Let KD be a constant, and solving Eqs. (17) and (18), one hasD1 Re Xa Acos1 KD Re Xd C1 ÁD2 À C2 ÁD1 KP ; 25 Asin1 KD Im Xd ; 21 B1 ÁC2 À B2 ÁC1B2 Acos1 Im Xb À Asin1 Re Xb ; 22 D1 ÁB2 À D2 ÁB1 KI : 26C2 Acos1 Im Xc À Asin1 Re Xc ; 23 B1 ÁC2 À B2 ÁC1 Fig. 5. Output response and load disturbance response of the controlled system (KP 0:40, KI 0:10, and KD 0:10).
36 Y.J. Huang, Y.-J. Wang / ISA Transactions 40 (2001) 31±393. Parameter plane analysis locus in the plane is a boundary of the constant gain margin. On the other hand, if A 1, and is Let A 1 and 0, and set KD equal to a assumed equal to a constant value, then the locusconstant, then for various values of !, a locus in the plane is a boundary of constant phase mar-representing the stability boundary of the system gin.without the gain-phase margin tester can be plot- By varying one of the parameters, A; and !,ted in the KP ±KI plane. The stability characteristics and ®xing the others, it suces to plot the con-of two sides of the locus are completely dierent. stant gain margin boundary and the constantDe®ne the Jacobian , J, of Eqs. (17) and (18) as phase margin boundary in the parameter plane. Then exploiting the stability equations methodJ B1 ÁC2 À B2 ÁC1 : 27 presented in Ref. , a speci®cation-oriented region enclosed by the constant gain margin boundaries and constant phase margin boundaries could be By resorting to , it is concluded that if J 0, found. The region characterizes all feasible con-then to the left of the stability boundary, facing troller parameter sets which guarantees the con-the direction in which ! increases, is the stable trolled system robust margins, i.e. Gm and Pm ofparameter area. Similarly, to the right of the sta- the system. For every value of KD the parameterbility boundary, facing the direction in which ! area can be found easily in the two-dimensionalincreases, is the stable parameter region while parameter plane. The aforementioned area showsJ 0. Accordingly, the stability boundary isolates a useful relationship between the three parameters,the parameter plane into stable and unstable KP ; KI and KD of the PID controller. The absoluteparameter regions, respectively. Further, if A is and relative stability margins can, in fact, beassumed equal to a constant value and 0, the readily obtained. Trial and error evaluation is Fig. 6. The user-speci®ed parameter regions, R2, R3, and R4 for system with dierent delay time T 0:1, 1, and 2 s, respectively.
Y.J. Huang, Y.-J. Wang / ISA Transactions 40 (2001) 31±39 37avoidable in such a case. Thus a lot of work can be speci®cation-oriented parameter area can also besaved. obtained. A representative point P1 with KP 0:40, KI 0:10, and KD 0:10 is selected. The stability4. Numerical results of the closed-loop system is proved referring to the Bode diagram as shown in Fig. 4. Output response The control result is inspected by the following and disturbance rejection response for step dis-simulation. First, we assume the delay time of the turbance of the controlled system are demon-system is ®xed at T 0:50 s. According to Eqs. strated in Fig. 5. It is seen that tracking error(17)±(26), let KD 0:10, the constant phase mar- approaches zero and disturbance rejection abilitygin boundaries for 30 and 60 can be plotted is obvious.as in Fig. 3. In a same way, for A 5 and 10 dB, Next, a non-minimum phase system subject tothe constant gain margin boundaries can also be uncertain time delay is inspected. By lettingplotted as in Fig. 3. The region ABCD shown in T 0:1, 1 and 2 s, and exploiting Eqs. (17)±(26),Fig. 3 is the parameter area which constitutes of the speci®cation-oriented parameter region can beall the possible parameter sets of the controller found, respectively. Consequently, as seen in Fig.that guarantees the system at-least the pre- 6, one obtains three dierent regions, R2, R3 andspeci®ed safety margins in terms of gain and R4 in the parameter plane. These regions are thephase. For other values of KD , the corresponding speci®cation-oriented areas for dierent delayFig.7. The gain margins and phase margins of the controlled system subject to the variation of the delay time, T, with the designedPID controller (KP 0:41, KI 0:11, and KD 0:10).
38 Y.J. Huang, Y.-J. Wang / ISA Transactions 40 (2001) 31±39Fig. 8. Output responses for three dierent delay time cases: T 0:1, 1, and 2 s. The representative point P2 (KP 0:41, KI 0:11,and KD 0:10) is selected.times. On the intersectional area of those three meter plane for the system with uncertain timeregions, we can freely choose an operation point. delay is introduced in this paper. The advantage ofFor example, P2 (KP 0:41, KI 0:11 and this method is the guaranteed robustness withKD 0:10) is selected. Here one already success- respect to plant variation and external disturbance.fully obtains a robust PID controller for the non- Excessive on-line tuning can be signi®cantly alle-minimum phase plant with uncertain delay time. viated. It promises the control system with goodFig. 7 shows that the designed PID controller tracking and disturbance rejection behavior. Onemaintains the time delay system with known var- can expect that this method of selecting PID con-iation range of delay a robust safety margins. The troller settings can be applied to a wide range oftime responses in Fig. 8 demonstrate the robust- industrial applications.ness of the designed controller in the case ofuncertain delay time. The designed robust PIDcontroller is seen to stabilize the system. References  A.M. De Paor, M. OMally, Controllers of Ziegler± Nichols type for unstable process with time delay, Int. J.5. Conclusions of Control 49 (4) (1989) 1273±1284.  A.T. Shenton, Z. Sha®ei, Relative stability for control systems with adjustable parameters, J. of Guidance, Con- The PID controller for non-minimum phase trol and Dynamics 17 (1994) 304±310.time delay system is less discussed so far. A  W.K. Ho, W. Xu, PID Tuning for unstable processesstraightforward graphical technique for character- based on gain and phase-margin speci®cations, IEE Proc.-izing all admissible PID controllers in the para- Control Theory and Appl 145 (5) (1998) 392±396.
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