New results in relay feedback analysis and multivariable stability margins

NEW RESULTS IN RELAY FEEDBACK ANALYSIS AND MULTIVARIABLE STABILITY MARGINS YE ZHEN NATIONAL UNIVERSITY OF SINGAPORE 2007 NEW RESULTS IN RELAY FEEDBACK ANALYSIS AND MULTIVARIABLE STABILITY MARGINS YE ZHEN (B.Eng., M.Eng., WHU) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2007 Acknowledgments I would like to express my deepest gratitude to my main supervisor, Prof. Wang Qing-Guo, who has not only taught me a lot on my research, but also cared about my life through my Ph.D. study. Without his gracious encouragement and generous guidance, I would not be able to finish the work so smoothly. His unwavering confidence and patience have aided me tremendously. His wealth of knowledge and accurate foresight have greatly impressed and benefited me. I am also grateful to my co-supervisor Prof. Hang Chang Chieh, for his excellent guidance. In spite of busy work, he always takes his golden time to give me valuable advices and helps. I am indebted to both of them for their care and advises in my academic research and other personal aspects. I would like to extend special thanks to Dr. Lin Chong, Dr. He Yong, Dr. Wen Guilin and Prof. Andrey E. Barabanov of St. Petersburg State University, for their comments, advice and the inspiration given, which have played a very important role in this piece of work. Special gratitude goes to Prof. Shuzhi Sam Ge, Prof. Ben M Chen, Prof. Xu Jian-Xin and Dr. Xiang Cheng who have taught me in class and given me their kind help in one way or another. Not forgetting my friends and colleagues, I would ii Acknowledgments iii like to express my thanks to Dr. Lu Xiang, Mr. Liu Min, Mr. Zhang Zhiping, Miss Gao Hanqiao, Mr. Lee See Chek, Mr. Wu Dongrui, Mr. Wu Xiaodong, Ms. Hu Ni and many others in the Advanced Control Technology Lab (Center for Intelligent Control) for making the everyday work so enjoyable. I enjoyed very much the time spent with them. I am also grateful to the National University of Singapore for the research scholarship. Finally, this thesis would not have been possible without the love, patience and support from my family and girl friend, Miss Lim Lihong Idris. The encouragement from them has been invaluable. I would like to dedicate this thesis to them and hope that they will find joy in this humble achievement. Ye Zhen August, 2007 Contents Acknowledgments ii Summary vii List of Tables x List of Figures xi Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 13 Relay Analysis for A Class of Servo Plants 14 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.1 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . 24 2.3.2 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . 40 iv Contents v 2.4 52 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tuning of Lead/Lag Compensators 3.1 3.2 53 Relay Auto-tuning for A Class of Servo Plants . . . . . . . . . . . . 53 3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.1.2 Relay Feedback System . . . . . . . . . . . . . . . . . . . . . 54 3.1.3 Parameter Identification from Limit Cycles . . . . . . . . . . 56 3.1.4 Controller Tuning . . . . . . . . . . . . . . . . . . . . . . . . 61 3.1.5 Real Time Implementation . . . . . . . . . . . . . . . . . . . 65 3.1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Tuning of Phase Lead Compensators . . . . . . . . . . . . . . . . . 71 3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.2.2 Tuning Method . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.2.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Multiloop Gain Margins and PID Stabilization 80 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3 The Proposed Approach . . . . . . . . . . . . . . . . . . . . . . . . 87 4.4 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.4.1 Proportional Control . . . . . . . . . . . . . . . . . . . . . . 94 4.4.2 PD Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.4.3 PI Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Contents vi 4.5 Computational Algorithm . . . . . . . . . . . . . . . . . . . . . . . 98 4.6 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Multiloop Phase Margins 118 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.3 Time Domain Method . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.4 5.5 5.3.1 Finding Allowable Diagonal Delays . . . . . . . . . . . . . . 128 5.3.2 Evaluating Phase Margins . . . . . . . . . . . . . . . . . . . 135 Frequency Domain Method . . . . . . . . . . . . . . . . . . . . . . . 144 5.4.1 The Proposed Approach . . . . . . . . . . . . . . . . . . . . 144 5.4.2 Illustration Examples . . . . . . . . . . . . . . . . . . . . . . 151 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Conclusions 159 6.1 Main Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.2 Suggestions for Further Work . . . . . . . . . . . . . . . . . . . . . 161 Author’s Publications 163 Bibliography 165 Summary The present thesis contributes to the control literature in the following four topics: (1) Relay analysis for a class of servo systems, (2) Tuning of lead/lag compensators, (3) Multiloop gain margins and PID stabilization, and (4) Multiloop phase margins. Relay analysis is to determine whether or not the limit cycle exists for a given plant, and if so, to reveal the relationship between the amplitude/period of the limit cycle and the plant parameters. In this thesis, complete results on the uniqueness of solutions, existence and stability of the limit cycles are established for a class of servo plants described by G(s) = Ke−Ls /(s(s+a)) under a relay feedback using the point transformation method and the Poincare map. Newton-Raphson’s method is used for determining the amplitude and period of a stable limit cycle from the plant parameters. Identifying its transfer function from the limit cycle observed for the above servo plant and designing a proper lead/lag/PD controller are the other side of the problem, and called relay auto-tuning. Closed-form formulas are obtained for directly computing the plant parameters from a limit cycle in time domain, vii Summary viii and an analytical technique is developed for tuning lead/lag/PD compensators for minimization of the integral squared error (ISE) instead of normally used gain and phase margins. A real time implementation of the proposed method on a DC motor is made to show its effectiveness. In a more general case for the tuning of phase lead compensators with specifications of gain and phase margins, a simple graphical method is proposed, which can achieve the given margins exactly regardless of the plant order, time delay or damping nature. The method transforms the problem of solving a set of nonlinear coupled equations into finding the intersection points of two graphs plotted using the frequency response information of the plant only. The solvability of the problem can be easily observed from the plot because it is related to the existence and number of intersection points of two graphs. A criterion is also established to decide the right one from possible multiple intersection solutions. The effectiveness of the method is then demonstrated with an example. Loop gain margins of a multivariable system are defined as the allowed perturbation ranges of gains for each loop such that the closed-loop system remains stable. A more general case is the problem of determining the parameter ranges of multi-loop proportional-integral-derivative (PID) controllers which stabilize a given process. An effective computational scheme is established by converting the considered problem to a quasi-LMI problem connected with robust stability test. The descriptor model approach is employed together with linearly parameterdependent Lyapunov function method. Numerical examples are given for illustration. The results are believed to facilitate real time tuning of multi-loop PID controllers for practical applications. Summary ix Loop phase margins of a multivariable system are defined as the allowable individual loop phase perturbations within which stability of the closed-loop system is guaranteed. Two approaches using time and frequency domain information are proposed for computing the loop phase margins. The time domain algorithm is composed of two steps: Firstly, find the stabilizing ranges of loop time delays using delay-dependent stability criteria; Secondly, convert these stabilizing ranges of loop delays into respective loop phase margins by multiplying a fixed frequency. The frequency domain algorithm makes use of unitary mapping between frequency responses of the system output and input, which is then converted, using the Nyquist stability analysis, to a simple constrained optimization problem solved numerically with the Lagrange multiplier and Newton-Raphson method. This frequency domain approach provides exact loop phase margins and thus improves the LMI results obtained by time domain algorithm, which could be conservative. Chapter 6. Conclusions 161 ranges of multiloop time delays, where plenty of results on delay-dependent/independent stability criteria are used. Frequency domain method is to convert the same problem to a constrained optimization, which improves the method of Bar-on and Jonckheere (1998) to guarantee the diagonal structure of loop phase perturbations. Comparing with time domain method, frequency domain method can provide exact margins and thus improves the LMI results by reducing the possible conservativeness. 6.2 Suggestions for Further Work The thesis has taken the full route from initial ideas, via theoretical developments, to methodologies that can be applied to relevant engineering problems. Several new results have been obtained but some topics remain open and are recommended for future work. A. Relay Analysis for Higher Order Plants It is natural and interesting at this point to see if the proposed relay analysis can be extended to other classes of plants. Consider a class of 2nd-order plants without integrator, i.e. G(s) = K/[(s + a)(s + b)], (a > 0, b > 0) . Let the initial condition, x(t0 ), be such that e(t0 ) = −Kx(t0 ) < ε+ with u(t˜) = u− . Then Ku− < will cause e(t) to monotonically increase for t > t0 with the potential maximum value of K/(ab). If ε+ ≥ K/(ab), no switching on S+ can occur and no limit cycle exists. This shows that the conditions for the existence of limit cycles will involve relay hysteresis and cannot be as neat as the original class of Chapter 6. Conclusions 162 plants. Now if G(s) has a pair of complex poles with oscillatory step response, the resultant e(t) is no longer piecewise monotonic. This will cause an essential difficulty: e(t) may reach a switching plane but does not pass through it (Lin et al., 2004b). These cases need further research. B. Multiloop Gain and Phase Margins for Time Delay Systems If the multivariable plant has time delay, one may use Pade approximation for time delay so as to apply the proposed procedure. For instance, consider the time delay system ⎡ s−1 ⎢ (s + 1)(s + 3) G(s) = ⎢ ⎣ s+2 ⎤ −0.4s e ⎥ s+3 ⎥. −0.4s ⎦ e s+2 Using the Pade approximation e−Ls ≈ (1−Ls/2)/(1+Ls/2), G(s) is approximated ⎡ as ⎢ ⎢ ˆ G(s) = ⎢ ⎣ s−1 (s + 1)(s + 3) s+2 4(1 − 0.2s) (s + 3)(1 + 0.2s) 3(1 − 0.2s) (s + 2)(1 + 0.2s) ⎤ ⎥ ⎥ ⎥. ⎦ For a proportional control K(s) = diag {k1 , k2 }, the stabilizing region for k1 and k2 are calculated as: k1 ∈ [−1.7174, −0.1079] and k2 ∈ [0.5580, 0.8910]. The timedelay case for our problem will lead to a different system description. The feedback of delay output gives rise to a more complicated state equation, for which the stabilizing ranges of PID parameters may not be transformed into a polytopic problem, whereas the technique used in this thesis is suitable for a polytopic problem. One needs to find a totally different technique to solve the delay problem, which could be another future study. Author’s Publications [1] Qing-Guo Wang, Zhen Ye, Chang Chieh Hang (2006). Tuning of phase-lead compensators for exact gain and phase margins. Automatica 42(2), 349–352. [2] Qing-Guo Wang, Xue-Ping Yang, Min Liu, Zhen Ye, Xiang Lu (2007). Stable Model Reduction for Time Delay Systems. Journal Of Chemical Engineering Of Japan 40(2), 139–144. [3] Qing-Guo Wang, Chong Lin, Zhen Ye, Guilin Wen, Yong He, Chang Chieh Hang (2007). A Quasi-LMI Approach to Computing Stabilizing Parameter Ranges of Multi-loop PID Controllers. Journal of Process Control 17(1), 59–72. [4] Zhen Ye, Qing-Guo Wang, Chong Lin, Chang Chieh Hang, Andrey E. Barabanov (2007). Relay feedback analysis for a class of servo plants. Journal of Mathematical Analysis and Applications 334(1), 28–42. [5] Guilin Wen, Qing-Guo Wang, Yong He, Zhen Ye (2007). Multivariable PD controller design for chaos synchronization of Lur’e systems. Physics Letters A 363(3), 192–196. 163 Author’s Publications 164 [6] Tang, Wei, Qing-Guo Wang, Zhen Ye and Zhiping Zhang (2007). PID Tuning for Dominant Poles and Phase Margin. Asian Journal of Control 9(4). [7] Qing-Guo Wang, Yong He, Zhen Ye, Chong Lin, Chang Chieh Hang (2007). On loop phase margins of multivariable control systems. accepted by Journal of Process Control. [8] Chang Chieh Hang, Qing-Guo Wang and Zhen Ye (2005). Tuning of lead compensators with gain and phase margin specifications. 16th IFAC World Congress, July 4–8, Prague, Czech. [9] Qing-Guo Wang, Chang Chieh Hang and Zhen Ye (2004). 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[...]... of both gain and phase margins to get a good performance and robustness In this context, several tuning methods have been reported (Yeung et al., 1998; Yeung and Lee, 2000; Ogata, 2002), and Ogata’s method remains the best known and is most broadly employed among all of them However, owing to the trial -and- error nature in Ogata’s method, gain and phase margins cannot be satisfied exactly, and errors... 1 Introduction 1.1 Motivation Since emerged in 1940s, control theory has been well developed and broadly applied in engineering practice Fruitful results and great achievements have made it a reality that one can deal with not only the single-input-single-output (SISO)/linear/determined system but also the multi-input-multi-output (MIMO)/nonlinear/uncertain system Nevertheless, some problems remain... of the loop gain margins based on Gershgorin bands or other frequency domain techniques are inevitably conservative, which brings some limitations to their applications Doyle (1982) developed the μ -analysis, which is utilized as an effective tool for robust stabilizing analysis in multivariable feedback control (Skogestad and Postlethwaite, 2005) As a method in frequency domain, the μ -analysis treats... novel auto-tuning approach and a real-time implementation For a general system, tuning of phase lead compensators is addressed with specifications of exact gain and phase margins A new definition of multiloop gain margins is proposed and the algorithm of computing stabilizing ranges of multiloop PID controller parameters is developed as well Likewise, a new definition of multiloop phase margins is given... plants under relay feedback In the subsequent Chapter 3, the idea is extended to relay auto-tuning of lead/lag/PD controllers for the same servo plants For general plants, the tuning of phase lead compensators with exact gain and phase margins is also addressed Chapter 4 is concerned with computing stabilizing parameter ranges within multi-loop PID controllers, from which loop gain margins of multivariable. .. 2005a) Since the major tuning difficulty lies in the nonlinearity and coupling of lead/lag parameters, graphical method may be applicable to find solutions apparently C Multiloop Gain Margins and PID Stabilization PID controllers have dominated industrial applications for more than fifty years because of their simplicity in controller structure, robustness to modeling errors and disturbances, and the availability... gain and phase margins is proposed It will achieve the given margins exactly regardless of the plant order, time delay or damping nature The solutions are found from the intersections of the curves of two real functions plotted using the frequency response of the plant only An example is provided for illustration and comparison C Multiloop Gain Margins and PID Stabilization The problem of determining... (Tsypkin, 1958) Although the solution procedures of Hamel and Tsypkin are almost identical, the frequency domain approach is more 1 Chapter 1 Introduction 2 popular in engineering practice because of its ease of manipulation With Afunction and incremental gain (Atherton, 1975), a limit cycle can be determined as well as its stability However, as a general method for relay analysis, the frequency domain... The results are believed to facilitate real time tuning of multi-loop PID controllers for practical applications D Multiloop Phase Margins New definition of loop phase margins is given, which extends the concept of phase margin from SISO systems to MIMO systems Two algorithms from time and frequency domains for computing the loop phase margins are developed For the time domain algorithm, the stabilizing... together with algorithms in time and frequency domain In Chapter 1 Introduction 11 particular, the thesis has investigated and contributed to the following areas: A Relay Analysis for A Class of Servo Systems A class of second-order servo plants, described by G(s) = Ke−Ls /[s(s + a)], a > 0, under relay feedback is studied Complete results on the uniqueness of solutions, existence and stability of the limit . NEW RESULTS IN RELAY FEEDBACK ANALYSIS AND MULTIVARIABLE STABILITY MARGINS YE ZHEN NATIONAL UNIVERSITY OF SINGAPORE 2007 NEW RESULTS IN RELAY FEEDBACK ANALYSIS AND MULTIVARIABLE STABILITY MARGINS YE. topics: (1) Relay analysis for a class of servo systems, (2) Tuning of lead/lag compen- sators, (3) Multiloop gain margins and PID stabilization, and (4) Multiloop phase margins. Relay analysis. system and further results can be obtained. B. Tuning of Lead/Lag Compensators In control engineering, a notable modern and wide application of relay analysis is in control system auto-tuning.