DETERMINISTIC LEAD/LAG COMPENSATOR AND ITERATIVE LEARNING CONTROLLER DESIGN FOR HIGH PRECISION SERVO MECHANISMS NALIN DARSHANA KARUNASINGHE B.Sc(Hons.) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2003 Acknowledgement First I would like to express my sincere gratitude to my supervisor, Prof Xu Jianxin for his guidance, patience and support during my M.Eng research Prof Xu’s successive and endless enthusiasms in research helped me to arouse my interest in various aspects of control engineering; his constant encouragement and stimulating discussions throughout the graduate program helped me complete this thesis I deeply appreciate the research position awarded to me by the Center for Intelligent Control, National University of Singapore, without which I could not have completed my M.Eng research I would like to thank Prof Ai Poh Loh for her encouragement and valuable advices throughout my stay at National University of Singapore I would also like to thank Ms Sara and Mr Zhang Hengwei at Control and Simulation Lab for helping me in logistics I also like to thank my wife Geethanjali, my parents and my family for their love, support and encouragement during the long period of study from my childhood and for taking other burdens on behalf of me I wish to dedicate this thesis to my beloved father, Mr Muniprema Karunasinghe i Table of Contents Acknowledgement i Table of Contents ii Summary iv List of Figures v Introduction 1.1 Overview and Past Work 1.2 Motivation 1.3 Organization of the Thesis A Deterministic Technique for ‘Most Favorable Lead Compensator Design’ 2.1 Introduction 2.2 Conventional Compensator Design 2.2.1 Case study 10 2.3 Deterministic Maximum Phase Compensator Design 13 2.4 New Compensator Design with Maximum Bandwidth 16 2.5 Proposed Graphical Method 18 2.6 Case Study for the Graphical Method 24 2.7 Simulation Results 27 2.8 Experimental Results 31 A Deterministic Technique for ‘Most Favourable Lag Compensator Design’ 38 3.1 Introduction 38 3.2 Conventional Lag Compensator Design 38 3.2.1 Case Study 40 ii 3.3 Deterministic Lag Compensator Design 43 3.3.1 Case Study 45 Design and Realization of Iterative Learning Control for High Precision Servomechanism 50 4.1 Introduction 50 4.2 Modeling and Most Favorable Indices 55 4.2.1 Experimental Setup and Modeling 55 4.2.2 Objective Functions for Sampled-Date ILC Servomechanism 56 4.3 PCL Design 60 4.4 CCL Design 65 4.5 PCCL Design 70 4.6 Most Favorable Robust PCCL Design 75 4.7 Conclusion 79 Conclusion and Recommendations 81 5.1 Contributions from This Thesis 81 5.2 Future Studies 82 References 83 Appendix A (Maximum Phase Lead Compensator) 87 Appendix B 89 iii Summary This research focused on the controller design for a high precision servo mechanism An XY table powered by two DC servo motors was used as a test bed for the control algorithms Firstly, a new simplified lead/lag compensator design was used The advantage of this design over existing ones was the ability to develop an explicit expression of the controller for a given phase margin This design could be performed with only knowledge of the frequency response of the plant The exact model and the model parameters are not required However a major disadvantage of this type of controller is its inability to perform satisfactorily in high precision control applications, mainly due to inherent non-linearities of the plant To improve performance, ILCs (Iterative Learning Controllers) ware used to compensate for the non-linearities Previous cycle learning (PCL), current cycle learning (CCL) and highbred version, previous-current cycle learning (PCCL) mechanisms were used to update the learning process Robustness of these controllers against parameter variations, uncertainties and non-linearities was studied, with tracking trajectory used as the controlling task With CCL algorithm, it was possible to reduce the tracking error by 98% with 10 iterations Furthermore, with the PCCL algorithm, an increase of convergence speed was observed; it was able to reduce the tracking error by 98% with only iterations iv List of Figures Figure 2.1 Bode plot for open loop system 11 Figure 2.2 Bode plot of the system with conventional compensator 13 Figure 2.3 Plot of φc (ω0 )Vs GP ( jω0 ) 20 Figure 2.4 Modified phase plot 21 Figure 2.5 Bode plot of the system with deterministic max phase compensator 24 Figure 2.6 Bode plot of the system with deterministic max bandwidth compensator 27 Figure 2.7 Normalized open loop step response of lead compensators .28 Figure 2.8 Close loop step response 29 Figure 2.9 Bode diagram of the plant with compensators 30 Figure 2.10 Bode diagram comparison of different compensators 31 Figure 2.11 Bode diagram for X-axis 33 Figure 2.12 Simulation results for step response of the system with different compensators 35 Figure 2.13 Simulation results of the tracking error X-axis 36 Figure 2.14 Experimental results of tracking error x-axis 36 Figure 3.1 Bode plot for open loop plant 40 Figure 3.2 Bode plot of conventionally designed lag compensator 42 Figure 3.3 Bode plot comparison of the plant 43 Figure 3.4 Plot of β Vs frequency ( ω ) .45 Figure 3.5 Plot of τ Vs frequency ( ω ) 46 Figure 3.6 Bode plot of deterministic lag compensator 47 Figure 3.7 Phase and gain margin with deterministic lag compensator 47 Figure 3.8 Open loop step response of the plant with deferent compensators 48 v Figure 3.9 Close loop step response of the plant 49 Figure 4.1 Block diagram of the system 55 Figure 4.2 The target trajectory 57 Figure 4.3 Tracking error x axis 58 Figure 4.4 The block-diagram of the sampled-data PCL 60 Figure 4.5 The convergence speeds of PCL algorithm in frequency domain 63 Figure 4.6 The maximum tracking errors with sampled-data PCL algorithm 65 Figure 4.7 The block-diagram of CCL learning algorithm .66 Figure 4.8 The convergence speeds of CCL algorithm in frequency domain 68 Figure 4.9 The maximum tracking errors with sampled-data CCL algorithm 69 Figure 4.10 The diagram of PCCL algorithm 71 Figure 4.11 The convergence speeds of sampled-data PCCL algorithm in frequency domain 73 Figure 4.12 The maximum tracking errors with sampled-data PCCL algorithm .73 Figure 4.13 The control profile in X-axis and Y-axis of PCCL algorithm at the seventh iteration 74 Table 4.1 Experimental Results of Model Variations .76 Figure 4.14 Comparison of tracking errors for case 78 Figure 4.15 Comparison of tracking errors for case 79 X and Y position (m) Vs time(s) 91 X’ and Y’ velocity (ms-1) Vs time(s) 91 X’’ and Y’’ acceleration (ms-2) Vs time(s) 91 X-Y Trajectory 92 vi 1.1 Introduction Overview and Past Work Precision servo systems have been widely used in the manufacturing industry, such as fiber optics, IC welding processes, polishing and grinding of fine surfaces, and production of miscellaneous precision machine tools Thus there is a need to develop an effective control strategy to control these plants Control system design using the frequency domain approach in general and Bode diagram in particular, has been used from the early stage of control engineering (James 1947, Toro 1960, Ogata 1990) Among compensators for linear time invariant systems, PID and lead/lag compensators are the most widely used compensator schemes in the industry Although these techniques were widely used to solve control system problems, most of the design methods involved trial and error techniques (James 1947, Toro 1960, Ogata 1990) Recently some attempts were made to eliminate the trial and error nature of the design process Wakeland (1976) was a pioneer in proposing the one-step design for a phase lead compensator Mitchell (1977) developed a similar technique to solve phase lag compensator design problem Yeung et al (1995, 2000) has developed a few chart based techniques to design compensators in frequency domain On the other hand, Iterative Learning Control (ILC) has evolved from the idea of using time-history of previous motion by Uchyama (Uchiyama, 1978) However, the first steps to rigorous treatments of learning control were made simultaneously and independently by Aritomo et al (1984), Casalino and Bartolini (1984), and Craig (1984) After almost two decades since Uchyama’s idea (Uchiyama, 1978), ILC has made a progressive advance Most of the efforts in the literature focused on the P and D type of learning update law Nowadays ILC have become one of the most active research areas in control theory and applications Differing from many existing intelligent control methods such as fuzzy logic control or neural control, the effectiveness of ILC schemes are guaranteed with convergence analysis Most of the ILC algorithms currently available adopt all or some of the following axioms: 1.) Each trial ends in a fixed time of duration (T>0), 2.) A desired output, yd (t ) is given a priori over that time with duration t ∈ [0, T ] 3.) Repetition of initial setting is satisfied, that is, the initial state xk (0) of the objective system can be set the same at the beginning of the each trial: xk (0) = x0 for k=1,2,… 4.) Invariance of system dynamics is ensured throughout these repeated trials 5.) Each output error, ∆ek = yk (t ) − yd (t ) ,can be utilized in the construction of the next input uk +1 (t ) 6.) The system dynamics are invertible, that is, for a given desired output yd (t ) with a piecewise continuous derivative, there is a unique input ud (t ) that exists for the system and yield the output yd (t ) However, most of the plants in the real world not behave as expected in the above axioms Thus there is a need to develop a robust iterative learning controller for practical plants 1.2 Motivation None of the above methods give a solution that satisfy a given design criteria such as phase margin, which is a measure of the stability In this work we have developed a non-trial and error technique to develop a lead/lag compensator to give the ‘most favorable’ performance in the time domain with respect to the given parameters including; rise time, overshoot and settling time In the design of the lead compensator we selected one with maximum phase added system and within that one with the maximum bandwidth In the lag compensator design, the design goal is not only to maximize the bandwidth, but also to find a solution to the additional delay contributed by the lag compensator This additional delay degrades the performance of the plant Thus, the selection of a time constant for the compensator has to be compatible with the time constants of the open loop plant Hence these two problems have to be viewed separately In this work, the above two problems are solved separately using two different techniques to achieve ‘most favorable’ results In a real industrial problem, what is required in design is a simple and realizable solution In a plant one of the advantage is that it is possible without difficulty to obtain the frequency response data We have developed a methodology that employs frequency response data to design a ‘most favorable’ lag/lead compensator for a plant in order to achieve a given phase margin show that the robust most favourable PCCL can work equally well and achieve almost the same performance The convergence of the tracking error in X−axis −2 10 non−robust optimal design robust optimal design −3 ess 10 −4 10 −5 10 Iteration Number The convergence of the tracking error in Y−axis −2 10 non−robust optimal design robust optimal design −3 ess 10 −4 10 −5 10 Iteration Number Figure 4.14 Comparison of tracking errors for case In the second case, the amplitude of the target trajectory in Equation(4.4) is reduced from 0.05 to 0.03 meters The amplitude of servo control signal is also scaled down from 0.2 to about 0.15, and the servo parameters a and b likely correspond to the second last row of Table 4.1 Again, the experimental results in Figure 4.15 show that the robust most favorable PCCL works equally well or better than the most favorable PCCL 78 The convergence of the tracking error in X−axis −2 10 non−robust optimal design robust optimal design −3 ess 10 −4 10 −5 10 Iteration Number The convergence of the tracking error in Y−axis −2 10 non−robust optimal design robust optimal design −3 ess 10 −4 10 −5 10 Iteration Number Figure 4.15 Comparison of tracking errors for case 4.7 Conclusion In this work we addressed one important and practical issue: Whether a high precision servomechanism can be realized without using an equally high precision model Through both theoretical analysis and intensive experimental investigation, we demonstrate the possibility of realizing such a servo control system by means of ILC techniques, in particular the PCCL algorithm The most favorable design as well as the robust most favorable design, on the other hand, warrants the achievement of either the best tracking performance or the most robust design Finally, it is worth 79 pointing out that the ILC based servomechanism possesses very useful characteristics It uses only a low gain feedback but achieves a high precision tracking performance This is because the learning nature converts the control system from one which is initially feedback dominated to one which is feed forward dominated 80 5.1 Conclusion and Recommendations Contributions from This Thesis This work proposes a simple but an accurate technique to design a unique ‘most favorable’ lead compensator for a given phase margin It also discusses methods of overcoming implementation limitations of lead compensator in some modes It was also possible to design a ‘most favorable’ lead compensator while maximizing the gain crossover frequency, which also producing the desired phase margin Finally we have presented a simple graphics based solution for use for the control system plants when only the Bode plot of the process is available This is a very useful method specially for designing industrial control plants in real life With the application of the available computation power this task has become very simple This algorithm was tested successfully on a high precision DC servomechanism to verify its effectiveness It is obvious in the real world that most of the plants show non-linearity in their behaviors and that most of the controlling algorithms such as simple lead/lag compensator are unable to achieve high accuracy due to this problem Mainly because of the limitations imposed by the experimental test bed leading to high non linearity behavior One reason could be the motor could be not the most appropriate model for this test bed To overcome the limitations of simple lag and lead compensator designs for the control of high precision servomechanisms, an iterative learning control principle was used In this section, it was observed that PCL (Previous Cycle Learning) is not very stable due to uncertainties of the plant and further it was observed that the convergence speed is not satisfactory In order to improve the robustness of the controller, the Current Cycle Learning (CCL) algorithm 81 was used The closed loop nature the system was able to track the path in the presence of the uncertainties in the system Experimental results verified that the CCL algorithm was able to reduce the tracking error to 98% in 10 iterations In order to improve the convergence speed as well as to keep the final tracking error at the same level the combined PCL and CCL algorithm was used The PCCL used current and previous cycle error coefficients, thus making it more robust as well as giving a higher convergence speed compared to CCL alone With our experimental results, it was shown that with only iterations the tracking error was able to reduce to 98% level The other advantage of using the iterative learning control is its ability to achieve high precision control without the trade off with a higher control loop gain 5.2 Future Studies In our experimental work it was seen that by using ILC algorithms most of the nonlinearities of the plants could be overcome by repetitive learning In the actual world it would be more beneficial if an algorithm could be found, which could directly learn in the presence of non-linearity and uncertainties Furthermore, such a system will be more advantages if it could adjust to varying loads and new tasks using the constantly updated database 82 References (Aritomo 1984) Aritomo,S., Kawamura,S., and Miyazaki, F., Bettering operation of Robots by learning, , Journal of Robotic Systems, Vol pp 123-14., 1984 (Bien 1998) Bien,Z., Xu,J.X., Iterative Learning Control, Analysis, Design, Integration and Applications, Kluwer academic publishers, 1998 (Burdet 1998) Burdet,E., A.Codourey, L.Rey, Experimental evaluation of nonlinear adaptive controllers, IEEE control systems magazine, Vol.18, pp.39–47, 1998 (Ccsalino 1984) Casalino, G and Bartolini, G., A learning procedure for the control of movements of robotic manipulators, IASTED Symp On Robotics and Automation, pp 108-111, 1984 (Chen 1998) Chen, Y.O, Z.M.Gong, C.Y.Wen, Analysis of a high-order iterative learning control algorithm for uncertain non-linear systems with state delays, Automatic, Vol.34, pp.345–353, 1998 (Chien 2000) Chien, C.J., A sampled-data iterative learning control using fuzzy network design, International Journal of Control, Vol.73, pp.902– 913, 2000 (Craig 1984) Craig, J.J., Adaptive controllers of manipulators through repeated trails, Proc ACC, Sandiago, CA, June 1984 (Ferreira 1993) Ferreira, P.M.G.,The tracking problem revisited: the internal model principle, Systems, Man and cybernetics, 1993 Vol pp 624-629, 1993 83 (Ham 2001) Ham, C., Qu, Z., Kaloust, J., Non-linear learning control for a class of non-linear systems, Automatica, Vol.37, pp.419–428, 2001 (James 1947) James, H.M., Nichols, N B., and Phillips, R S., Theory of Servomechanisms New York: McGraw-Hill Book Co Inc 1947 (Kawamura 1987)Kawamura, S., Miyazaki, F., Arimoto, S., Intelligent control of robot motion based on learning control, Proc of the 1987 IEEE Int Symp on Intelligent Control Philadelphia, Pennsylvania, USA, pp.365–370, 1987 (Kue 1992) Kuc, Y.Y., Lee, S.J., Nam, K.H., An iterative learning control theory for a class of non-linear dynamic systems, Automatica, Vol.28, pp.1215–1221, 1992 (Longman 1998)Longman,R.W, Designing Iterative Learning and Repetitive Controllers, In Iterative Learning Control - Analysis, Design, Integration and Application, Kluwer Academic Publishers, pp.107145, 1998 (Marro 1998) Marro G and Zanasi R New Formulae and Graphics for Compensator Design, In Proc 1998 IEEE International Conference on Control Applications, September 1998, Trieste, Italy, pp.129-133 (Mitchell 1977) Mitchell J.R Comments on ‘Bode Compensator Design’, IEEE Transactions on Automatic Control, Vol AC-22, pp.869-870, October 1977 (Moore 1998) Moore, K.L., Iterative Learning Control - An Expository Overview, Applied & Computational Controls, Signal Processing and Circuits, Vol.1, pp.1-42, 1998 84 (Norrlof 2001) Norrlof, M, Gunnarsson, M.S., Disturbance aspects of iterative learning control, Engineering Applications of Artificial Intelligence, Vol.14, pp.87–94, 2001 (Ogata 1990) Ogata, K Modern Control Engineering (Second Ediion) pp.540-592, New Jersey: Prentice-Hall Inc 1990 (Ortega 1970) Ortega, J.M., Rheinboldt, W.C., Iterative solution of non-linear equations in several variables, Academic Press, New York, U.S.A, 1970 (Phan 1996) Phan, M.Q., Juang, J.N., Designs of learning controllers based on an auto-regressive representation of a linear system, AIAA Journal of Guidance, Control, and Dynamics Vol.19, pp.355–362, 1996 (Roover 2000) de Roover, D., Bosgra, O.H., Steinbuch, M., Internal-model-based design of repetitive and iterative learning controllers for linear multivariable systems, International Journal of Control, Vol 73, pp 914–929, 2000 (Saab 1995) Saab, S.S., A discrete-time learning control algorithm for a class of linear time-invariant, IEEE Trans Automat Control, Vol.46, pp.1138–1142, 1995 (Sinha 1986) Sinha, N.K Control Systems pp.233-239, New York: CBS College Publishing 1986 (Uchiyama 1978) Uchiyama, M Formation of high-speed motion pattern of a mechanical arm by trial, SICE (Society for Implimentatiopn and Control Engineers) Transactions Vol 14, pp 706-712 1978 (Wakeland 1976) Wakeland W.R Bode Compensator Design, IEEE Transactions on Automatic Control, Vol AC-21, pp.771-773, October 1976 85 (Wang 2000) Wang, D.W., On D-type and P-type ILC designs and anticipatory approach, Int J Control, Vol.73, pp.890-901, 2000 (Xu 2000a) Xu,J.X and Song,Y., Yeung, Multi-scale direct learning control of linear time-varying high-order systems, Automatica Vol 36, pp 6168, 2000 (Xu 2000b) Xu, J.X., Lee, T.H., Chen, Y.Q., Knowledge learning in discrete-time control systems, Knowledge-Base Systems-Techniques and Applications Academic Press, pp.943-976, 2000 (Xu 2003) J.-X Xu and Y Tan, Linear and Nonlinear Iterative Learning Control, in series of Lecture Notes in Control and Information Sciences 291, Springer Verlag, Germany, 2003 (Yeung 1995) Yeung, S.K., Chaid, K.Q and Dinh, T.X Bode Design Charts for Continuous-Time and Discrete-Time Compensators, IEEE Transactions on Education, Vol 38, No.3, pp.252-257, August 1995 (Yeung 1998) Yeung, S.K., Wong K.W and Chen K.L A Non-Trial-and-Error Method for Lag-Lead Compensator Design, IEEE Transactions on Education, Vol 41, No.1, pp.76-80, February 1998 (Yeung 2000) Yeung, S.K and Lee, K.H A Universal Design Chart for Linear Time-Invariant Continuous-Time and Discrete-Time Compensators, IEEE Transactions on Education, Vol 43, No.3, pp.309-315 August 2000 86 Appendix A (Maximum Phase Lead Compensator) In general, assume the compensator to be in the form GC = + ατ s 1+τ s (A.1) Then, phase angle of the compensator at the angular frequency of (ω ) is as follows (φ ) = tan −1 (ατω ) − tan −1 (τω ) (A.2) Replacing τω =x (>0 ∵ ω > 0;τ > ), (A.3) Then, applying tangent difference equation on equation(A.2), it can be rewritten as tan(φ ) = αx − x And we know that < φ < π / + αx (A.4) ∴ By partial differentiating equation (A.4) with respect to x we get, ∂ (tan φ ) (α − 1)(1 − αx ) = ∂x (α + x ) (A.5) ∂ (tan φ ) (α − 1)(1 − αx ) =0 = ∂x (α + x ) (A.6) then, we can find tangent by solving (A.6) for x we will get x = ± α but, x>0 ∴ The frequency which will maximize the phase will be at x = (A.7) + α or ω = τ α To confirm this is a maxima, we can find the 2nd derivative of (A.4) with respect to x αx(α − 1)(αx − 3) ∂ (tan φ ) = ∂x (1 + αx ) (A.8) 87 After applying x = α , Equation (A.8) becomes, ∂ (tan φ ) (1 − α ) α = ∂x (A.9) When α > , Equation (A.9)