PID控制器及其设计方法研究
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摘要
过程控制中，大量控制系统是由基本PID控制单元组成。它们的性能直接关系到生产过程的平稳运行和产品的最终质量，与企业的经济效益息息相关，因而其优化设计具有重要的现实意义。
许多传统设计方法往往仅考虑控制问题的一个方面，如经典的Ziegle-Nichols法的设计目标是提高系统抗负载扰动性能，而没有考虑设定值的跟踪问题。新的设计方法需要将控制的各方面问题进行综合考虑。
系统稳定是控制器设计首先需要考虑的问题，只有在系统稳定的前提下，提高其它性能才是有意义的。另外，在控制器实现中，由于各种原因(如环境温度的变化引起电子元件参数变化)，控制器参数往往会偏离理想设计值，这就要求设计的控制器具有非脆弱性，即当控制器的参数发生小的摄动时，仍能保证整个闭环系统稳定。
针对上述问题，通过研究，本文所做的主要工作和研究结果如下：
首先就PID控制器的基本原理及其参数整定方法作简要介绍。接着，对近年来出现的PID控制器优化设计方法作概括性介绍，主要介绍两自由度PID控制器的设计方法和基于线性矩阵不等式(LMI)设计PID控制器的方法。其次，针对一类二阶加纯滞后过程，研究非脆弱PID控制器稳定化设计问题，根据关于准多项式的Hermite-Biehler推广定理，导出使得闭环系统稳定的PID参数区域，并且在这个区域内利用线性规划的方法确定非脆弱的PID控制器参数。最后利用三阶水槽液位控制实验装置，验证了非脆弱PID控制器稳定化设计方法的有效性。
Most of real industrial processes are controlled by PID controllers. The performance of these control systems are directly related to the stable operation of processes, the quality of product and the great benefit of the enterprises. It is very significative for optimization design.
Many traditional design methods consider only one aspect of the control problem. For example, in the classical design rule of Ziegler-Nichols the objective was to reject load disturbances, no aspect was taken to changes in set point. It is necessary for a new design method to consider all aspects of the control problem.
The stability of the closed-loop system is first considered in controller design. If the system is not stable, it is not benefit to improve other performances. When controller implemented, its parameters often have a departure from ideal design value for a variety of reasons. For example, the change of environmental temperature will result in the changes of electronic components parameters. As a result, it is necessary to design a non-fragile PID controller for the stability of the closed-loop system even though there are some disturbances in the parameters of controllers.
Based on the above problems, the main research ideas and results obtained in this paper are as following,
Firstly, the basic principle of PID controller and its parameters tuning method are briefly reviewed. Secondly, some recent optimization design methods are summarily introduced, mainly including the design method of two degrees of freedom PID controller and the approchs of PID controllers design via Linear matrix inequalityies (LMI). Thirdly, based on a suitable extension of the Hermite-Biehler Theorem, the problem of the region of PID parameters for stabilizing a second-order system with time delay is sloved, and the method of Linear Programming is used to determine the non-fragile PID stabilizing
controller. Finally, experiment results for a triple-flume control system prove the effectiveness of the non-fragile PID stabilizing controller.
许多传统设计方法往往仅考虑控制问题的一个方面，如经典的Ziegle-Nichols法的设计目标是提高系统抗负载扰动性能，而没有考虑设定值的跟踪问题。新的设计方法需要将控制的各方面问题进行综合考虑。
系统稳定是控制器设计首先需要考虑的问题，只有在系统稳定的前提下，提高其它性能才是有意义的。另外，在控制器实现中，由于各种原因(如环境温度的变化引起电子元件参数变化)，控制器参数往往会偏离理想设计值，这就要求设计的控制器具有非脆弱性，即当控制器的参数发生小的摄动时，仍能保证整个闭环系统稳定。
针对上述问题，通过研究，本文所做的主要工作和研究结果如下：
首先就PID控制器的基本原理及其参数整定方法作简要介绍。接着，对近年来出现的PID控制器优化设计方法作概括性介绍，主要介绍两自由度PID控制器的设计方法和基于线性矩阵不等式(LMI)设计PID控制器的方法。其次，针对一类二阶加纯滞后过程，研究非脆弱PID控制器稳定化设计问题，根据关于准多项式的Hermite-Biehler推广定理，导出使得闭环系统稳定的PID参数区域，并且在这个区域内利用线性规划的方法确定非脆弱的PID控制器参数。最后利用三阶水槽液位控制实验装置，验证了非脆弱PID控制器稳定化设计方法的有效性。
Most of real industrial processes are controlled by PID controllers. The performance of these control systems are directly related to the stable operation of processes, the quality of product and the great benefit of the enterprises. It is very significative for optimization design.
Many traditional design methods consider only one aspect of the control problem. For example, in the classical design rule of Ziegler-Nichols the objective was to reject load disturbances, no aspect was taken to changes in set point. It is necessary for a new design method to consider all aspects of the control problem.
The stability of the closed-loop system is first considered in controller design. If the system is not stable, it is not benefit to improve other performances. When controller implemented, its parameters often have a departure from ideal design value for a variety of reasons. For example, the change of environmental temperature will result in the changes of electronic components parameters. As a result, it is necessary to design a non-fragile PID controller for the stability of the closed-loop system even though there are some disturbances in the parameters of controllers.
Based on the above problems, the main research ideas and results obtained in this paper are as following,
Firstly, the basic principle of PID controller and its parameters tuning method are briefly reviewed. Secondly, some recent optimization design methods are summarily introduced, mainly including the design method of two degrees of freedom PID controller and the approchs of PID controllers design via Linear matrix inequalityies (LMI). Thirdly, based on a suitable extension of the Hermite-Biehler Theorem, the problem of the region of PID parameters for stabilizing a second-order system with time delay is sloved, and the method of Linear Programming is used to determine the non-fragile PID stabilizing
controller. Finally, experiment results for a triple-flume control system prove the effectiveness of the non-fragile PID stabilizing controller.
引文
[1] Astrom K J and Hagglund T. PID Controllers: Theory, Design and tuning. Research Triangle Park: Instrument Society of America,1995．
[2] Truxal J. Automatic Feedback Control System Synthesis. McGraw-Hill, New York, 1955．
[3] Newton, Gould L A and Kaiser J F. Analytical Design of Linear Feedback Control. John Wiley & Sons, 1957．
[4] Boyd S P and Barratt C H. Linear Controller Design-Limits of Performance. Prentice Hall Inc.,Englewood Cliffs, New Jersey, 1991．
[5] Mayne D Q and Polak E . Optimization based design and control. IFAC 12th World Congress, 1993,3:129-138． Sydney, Australia.
[6] Smith C L, Corripio A B and Martin J J. Controller tuning from simple process models. Instrumentation Technology, 1975,12: 39-44．
[7] Ziegler J G and Nichols N B. Optimum Settings for automatic controllers. Trans.ASME, 1942, 64: 759-768．
[8] Cohen G H and Coon G A. Theoretical consideration of retarded control. Trans.ASME, 1953,75:827-834．
[9] Wallen A. A Tool for Rapid system Identification. Conference on Control Applications, 1999:1555-1560, Kohala Coast Island of Hawaii, Hawaii.
[10] Van Overschee P and Moor B D. Optimal PID Control of Chemical Batch Reactor. Proceedings of the 1999 European Control Conference. Karlsruhe, Germany.
[11] Astrom K J and Hagglund T. The future of PID control. Control Engineering Practice,2001,9:1163-1175．
[12] 田保峡，工业PID控制器及其参数整定方法研究，浙江大学博士学位论文，2001．
[13] Keel L H, Bhattacharyya S P. Robust, fragile or Optimal ? IEEE Trans, on Auto. Control, 1997,42(8) : 1098-1105．
[14] Hang C C, Astrom K J and Ho W K. Refmemente of the Ziegler-Nichols tuning formula.IEE.Proceedings,Part D,1991,138(2) : 111-118．
[15] 陶永华，尹怡欣，葛芦生编著，新型PID控制及其应用．机械工业出版社，1999．
[16] Astrom K J. Toward intelligent control. IEEE control Systems Magazine, 1989(April): 60-64．
[17] 王伟，张晶涛，柴天佑．PID参数先进整定方法综述．自动化学报，2000，26(3) ：347-355．
[18] Astrom K J and Hagglund T. Automatic tuning of simple regulators with specifications on phase and amplitude margins. Automatica, 1984,20: 645-651．
[19] Kraus T W, Myron T J. Self-tuning PID controller uses pattern recognition approach. Control Engineering, 1984(6) : 106-111．
[20] Bristol E H. Pattern recognition: An alternative to parameter identification in adaptive control. Automatica, 1997,13: 197-202．
[21] Yu C C. Autotuning of PID controllers : relay feedback approach. Springer-Verlag London Limited 1999．
[22] Chang R C, Shen S H, Yu C C. Derivation of Transfer Funtion from Relay Feedback System. Ind. Eng. Chem. Res. 1992,31,855-859．
[23] Astrom K J, Panagopoulos H and Hagglund T. Design of PI controllers based on non-convex optimization. Automatica, 1998,34(4) : 585-601．
[24] Panagopoulos H , Astrom K J and Hagglund T. Design of PID controllers based on constrained optimization. Proceedings of the 1999 American Control Conference, 3858-3862． San Diego, California.
[25] Panagoploulos H, K. J. Astrom. PID Control Desgin and H∞Loop Shaping. International Journal of Robust and Nonlinear Control. 2000,10: 1249-1261．
[26] Panagopoulos H. PID control design, extension, application. Ph.D.thesis, Department of Automatic Control, Lund Institute of Technology, Lund, Sweden.2000．
[27] Hwang C, Hsiao C Y. A new approach to mixed H2/H∞ optimal PI / PID contoller design. Department of Chemical Engineering National Chung Cheng University, TaiWan.2001．
[28] Boyd S, Ghaoui L E, Feron E, Balkrishnan V. Linear matrix inequalityies in systems and control theory. Philadelphia; SIAM.1994．
[29] Gahinet P, Nemirovski A, Laub J, Chilali M. LMI Toolbox for Use With Matlab, The Mathworks Inc., 1995．
[30] Chen C L, Wang T C, Lee W C, Tsai H W. H∞PI controller design: an LMI approach. Department of Chemical Engineering. National Taiwan University.
[31 ] Ge M, Chiu M S, Wang Q G. Robust PID controller design via LMI approach. Journal of processs control, 2002,12: 3-13．
[32] Zheng F , Wang Q G, Lee T H. (2002) . On the design of multivariable PID controllers via
LMI approach. Automatica ,2002,38: 517-526．
[33] Mattei M. Robust multivariable PID controller for Linear parameter varying systems. Automatica, 2001,37: 1997-2003．
[34] Scherer C. The Riccati Inequality and State-Space H∞ optimal control, Ph.D thesis, Wurzburg University, Germany. 1990．
[35] Packard A, Zhou K, Pandey P, Becker G. A collection of robust control problems leading to LMI's. Proceeding CDC,1991,1245-1250．
[36] Gahinet P, Apkarian P. A linear matrix inequality approach to H∞control. Int. J.Robust and Nonlinear control, 1994,4: 421-448．
[37] Chilali M , Gahinet P. H∞design with pole placement constraints: an LMI approach, IEEE Trans. On Auto. Control 1996,41: 358-367．
[38] Datta A, Ho Ming-Tzu and Bhattacharyya S P. Structure and Synthesis of PID Controller. Springer-Verlag London Limited.2000．
[39] Silva G J, Datta A, Bhattacharyya S P. PI stabilization of first-order systems with time delay. Automatica, 2001, 37(12) : 2025-2031．
[40] Pontryagin L S. On the zeros of some elementary transcenden-tal function. American Mathematical Society Translation, 1995,2: 95-110．
[41] Bellman R and Cooke K L. Differential-difference equations. London: Academic Press Inc.1963．
[42] Bhattacharyya S P, Chapellat H, Keel L H. Robust Control: The Parametric Approach, Prentice Hall, 1995．
[43] Wang Q G, Lee T H, Fung H W, et al . PID Tuning for Improved Performance. IEEE Trans. Contr. Syst. Technol., 1999, 7(4) : 457-465．
[44] 组态王Version 5． 1用户手册北京亚控自动化软件科技有限公司．
[45] 冯培悌编著，系统辨识．浙江大学出版社，1999．
[2] Truxal J. Automatic Feedback Control System Synthesis. McGraw-Hill, New York, 1955．
[3] Newton, Gould L A and Kaiser J F. Analytical Design of Linear Feedback Control. John Wiley & Sons, 1957．
[4] Boyd S P and Barratt C H. Linear Controller Design-Limits of Performance. Prentice Hall Inc.,Englewood Cliffs, New Jersey, 1991．
[5] Mayne D Q and Polak E . Optimization based design and control. IFAC 12th World Congress, 1993,3:129-138． Sydney, Australia.
[6] Smith C L, Corripio A B and Martin J J. Controller tuning from simple process models. Instrumentation Technology, 1975,12: 39-44．
[7] Ziegler J G and Nichols N B. Optimum Settings for automatic controllers. Trans.ASME, 1942, 64: 759-768．
[8] Cohen G H and Coon G A. Theoretical consideration of retarded control. Trans.ASME, 1953,75:827-834．
[9] Wallen A. A Tool for Rapid system Identification. Conference on Control Applications, 1999:1555-1560, Kohala Coast Island of Hawaii, Hawaii.
[10] Van Overschee P and Moor B D. Optimal PID Control of Chemical Batch Reactor. Proceedings of the 1999 European Control Conference. Karlsruhe, Germany.
[11] Astrom K J and Hagglund T. The future of PID control. Control Engineering Practice,2001,9:1163-1175．
[12] 田保峡，工业PID控制器及其参数整定方法研究，浙江大学博士学位论文，2001．
[13] Keel L H, Bhattacharyya S P. Robust, fragile or Optimal ? IEEE Trans, on Auto. Control, 1997,42(8) : 1098-1105．
[14] Hang C C, Astrom K J and Ho W K. Refmemente of the Ziegler-Nichols tuning formula.IEE.Proceedings,Part D,1991,138(2) : 111-118．
[15] 陶永华，尹怡欣，葛芦生编著，新型PID控制及其应用．机械工业出版社，1999．
[16] Astrom K J. Toward intelligent control. IEEE control Systems Magazine, 1989(April): 60-64．
[17] 王伟，张晶涛，柴天佑．PID参数先进整定方法综述．自动化学报，2000，26(3) ：347-355．
[18] Astrom K J and Hagglund T. Automatic tuning of simple regulators with specifications on phase and amplitude margins. Automatica, 1984,20: 645-651．
[19] Kraus T W, Myron T J. Self-tuning PID controller uses pattern recognition approach. Control Engineering, 1984(6) : 106-111．
[20] Bristol E H. Pattern recognition: An alternative to parameter identification in adaptive control. Automatica, 1997,13: 197-202．
[21] Yu C C. Autotuning of PID controllers : relay feedback approach. Springer-Verlag London Limited 1999．
[22] Chang R C, Shen S H, Yu C C. Derivation of Transfer Funtion from Relay Feedback System. Ind. Eng. Chem. Res. 1992,31,855-859．
[23] Astrom K J, Panagopoulos H and Hagglund T. Design of PI controllers based on non-convex optimization. Automatica, 1998,34(4) : 585-601．
[24] Panagopoulos H , Astrom K J and Hagglund T. Design of PID controllers based on constrained optimization. Proceedings of the 1999 American Control Conference, 3858-3862． San Diego, California.
[25] Panagoploulos H, K. J. Astrom. PID Control Desgin and H∞Loop Shaping. International Journal of Robust and Nonlinear Control. 2000,10: 1249-1261．
[26] Panagopoulos H. PID control design, extension, application. Ph.D.thesis, Department of Automatic Control, Lund Institute of Technology, Lund, Sweden.2000．
[27] Hwang C, Hsiao C Y. A new approach to mixed H2/H∞ optimal PI / PID contoller design. Department of Chemical Engineering National Chung Cheng University, TaiWan.2001．
[28] Boyd S, Ghaoui L E, Feron E, Balkrishnan V. Linear matrix inequalityies in systems and control theory. Philadelphia; SIAM.1994．
[29] Gahinet P, Nemirovski A, Laub J, Chilali M. LMI Toolbox for Use With Matlab, The Mathworks Inc., 1995．
[30] Chen C L, Wang T C, Lee W C, Tsai H W. H∞PI controller design: an LMI approach. Department of Chemical Engineering. National Taiwan University.
[31 ] Ge M, Chiu M S, Wang Q G. Robust PID controller design via LMI approach. Journal of processs control, 2002,12: 3-13．
[32] Zheng F , Wang Q G, Lee T H. (2002) . On the design of multivariable PID controllers via
LMI approach. Automatica ,2002,38: 517-526．
[33] Mattei M. Robust multivariable PID controller for Linear parameter varying systems. Automatica, 2001,37: 1997-2003．
[34] Scherer C. The Riccati Inequality and State-Space H∞ optimal control, Ph.D thesis, Wurzburg University, Germany. 1990．
[35] Packard A, Zhou K, Pandey P, Becker G. A collection of robust control problems leading to LMI's. Proceeding CDC,1991,1245-1250．
[36] Gahinet P, Apkarian P. A linear matrix inequality approach to H∞control. Int. J.Robust and Nonlinear control, 1994,4: 421-448．
[37] Chilali M , Gahinet P. H∞design with pole placement constraints: an LMI approach, IEEE Trans. On Auto. Control 1996,41: 358-367．
[38] Datta A, Ho Ming-Tzu and Bhattacharyya S P. Structure and Synthesis of PID Controller. Springer-Verlag London Limited.2000．
[39] Silva G J, Datta A, Bhattacharyya S P. PI stabilization of first-order systems with time delay. Automatica, 2001, 37(12) : 2025-2031．
[40] Pontryagin L S. On the zeros of some elementary transcenden-tal function. American Mathematical Society Translation, 1995,2: 95-110．
[41] Bellman R and Cooke K L. Differential-difference equations. London: Academic Press Inc.1963．
[42] Bhattacharyya S P, Chapellat H, Keel L H. Robust Control: The Parametric Approach, Prentice Hall, 1995．
[43] Wang Q G, Lee T H, Fung H W, et al . PID Tuning for Improved Performance. IEEE Trans. Contr. Syst. Technol., 1999, 7(4) : 457-465．
[44] 组态王Version 5． 1用户手册北京亚控自动化软件科技有限公司．
[45] 冯培悌编著，系统辨识．浙江大学出版社，1999．