Previous Articles     Next Articles

On PID Control Theory for Nonaffine Uncertain Stochastic Systems

ZHANG Jinke1,2, ZHAO Cheng1,2, GUO Lei1,2   

  1. 1. Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
    2. School of Mathematical Science, University of Chinese Academy of Sciences, Beijing 100049, China
  • Received:2021-12-07 Online:2023-01-25 Published:2023-02-09
  • Supported by:
    This research was supported by the National Natural Science Foundation of China under Grant No. 12288201.

ZHANG Jinke, ZHAO Cheng, GUO Lei. On PID Control Theory for Nonaffine Uncertain Stochastic Systems[J]. Journal of Systems Science and Complexity, 2023, 36(1): 165-186.

PID (proportional-integral-derivative) control is recognized to be the most widely and successfully employed control strategy by far. However, there are limited theoretical investigations explaining the rationale why PID can work so well when dealing with nonlinear uncertain systems. This paper continues the previous researches towards establishing a theoretical foundation of PID control, by studying the regulation problem of PID control for nonaffine uncertain nonlinear stochastic systems. To be specific, a three dimensional parameter set will be constructed explicitly based on some prior knowledge on bounds of partial derivatives of both the drift and diffusion terms. It will be shown that the closed-loop control system will achieve exponential stability in the mean square sense under PID control, whenever the controller parameters are chosen from the constructed parameter set. Moreover, similar results can also be obtained for PD (PI) control in some special cases. A numerical example will be provided to illustrate the theoretical results.
[1] Åström K J and Hägglund T, PID Controllers:Theory, Design, and Tuning, Instrument society of America Research Triangle Park, NC, 1995.
[2] Åström K J, Hägglund T, and Astrom K J, Advanced PID Control, ISA-The Instrumentation, Systems, and Automation Society Research Triangle Park, NC, 2006.
[3] Salih A L, Moghavvemi M, Mohamed H A F, et al., Flight PID controller design for a UAV quadrotor, Scientific Research and Essays, 2010, 5(23):3660-3667.
[4] Åström K J and Hägglund T, The future of PID control, Control Engineering Practice, 2001, 9(11):1163-1175.
[5] Samad T, A survey on industry impact and challenges thereof[technical activities], IEEE Control Systems Magazine, 2017, 37(1):17-18.
[6] Guo L, Feedback and uncertainty:Some basic problems and results, Annual Reviews in Control, 2020, 49:27-36.
[7] Ziegler J G and Nichols N B, Optimum settings for automatic controllers, Trans. ASME, 1942, 64(11):759-765.
[8] Chien I L and Fruehauf P S, Consider IMC turning to improve controller performance, Chemical Engineering Progress, 1990, 86(10):33-41.
[9] Bialkowski W L, Dreams versus reality:A view from both sides of the gap:Manufacturing excellence with come only through engineering excellence, Pulp & Paper Canada, 1993, 94(11):19-27.
[10] Ender D B, Process control performance:Not as good as you think, Control Engineering, 1993, 40(10):180-190.
[11] Yu C C, Autotuning of PID Controllers:A Relay Feedback Approach, Springer Science & Business Media, Berlin, 2006.
[12] Killingsworth N J and Krstic M, PID tuning using extremum seeking:Online, model-free performance, optimization, IEEE Control Systems Magazine, 2006, 26(1):70-79.
[13] Romero J G, Donaire A, Ortega R, et al., Global stabilisation of underactuated mechanical systems via PID passivity-based control, Automatica, 2018, 96:178-185.
[14] Fliess M and Join C, Model-free control, International Journal of Control, 2013, 86(12):2228-2252.
[15] O'Dwyer A, PI and PID controller tuning rules:An overview and personal perspective, Proceedings of the IET Irish Signals and Systems Conference, 2006, 161-166.
[16] Zhao C and Guo L, PID controller design for second order nonlinear uncertain systems, Science China, 2017, 60(2):1-13.
[17] Cong X R and Guo L, PID control for a class of nonlinear uncertain stochastic systems, IEEE 56th Annual Conference on Decision and Control (CDC), IEEE, 2017, 612-617.
[18] Zhang J K and Guo L, Theory and design of PID controller for nonlinear uncertain systems, IEEE Control Systems Letters, 2019, 3(3):643-648.
[19] Zhang J K and Guo L, PID control of nonlinear stochastic systems with structural uncertainties, IFAC-PapersOnLine, 2020, 53(2):2189-2194.
[20] Zhao C and Guo L, Towards a theoretical foundation of PID control for uncertain nonlinear systems, Automatica, 2022, 142:110360.
[21] Muniyappa R, Lee S, Chen H, et al., Current approaches for assessing insulin sensitivity and resistance in vivo:Advantages, limitations, and appropriate usage, American Journal of PhysiologyEndocrinology and Metabolism, 2008, 294(1):E15-E26.
[22] Shiriaev A S, Ludvigsen H, Egeland O, et al, Swinging up of non-affine in control pendulum, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251), IEEE, 1999, 6:4039-4044.
[23] Lee J, Seo J, and Choi J, Output feedback control design using extended high-gain observers and dynamic inversion with projection for a small scaled helicopter, Automatica, 2021, 133:109883.
[24] Zhao C and Guo L, On the capability of PID control for nonlinear uncertain systems, IFACPapersOnLine, 2017, 50(1):1521-1526.
[25] Zhao C and Guo L, Control of nonlinear uncertain systems by extended PID, IEEE Transactions on Automatic Control, 2021, 66(8):3840-3847.
[26] Khasminskii R, Stochastic Stability of Differential Equations, Springer Science & Business Media, Berlin, 2011.
[1] WANG Hongxia · ZHANG Huanshui · XIE Lihua. Optimal Control and Stabilization for Itˆo Systems with Input Delay [J]. Journal of Systems Science and Complexity, 2021, 34(5): 1895-1926.
[2] ZHAO Yong,ZHANG Weihai. Observer-Based Controller Design for Singular Stochastic Markov Jump Systems with State Dependent Noise [J]. Journal of Systems Science and Complexity, 2016, 29(4): 946-958.
[3] FANG Bin,LI Xuezhi,MARTCHEVA Maia,CAI Liming. Global Stability for a Heroin Model with Age-Dependent Susceptibility [J]. Journal of Systems Science and Complexity, 2015, 28(6): 1243-1257.
[4] CHEN Fengde,XIE Xiangdong,WANG Haina. Global Stability in a Competition Model of Plankton Allelopathy with Infinite Delay [J]. Journal of Systems Science and Complexity, 2015, 28(5): 1070-1079.
[5] YAN Zhiguo, ZHANG Guoshan, WANG Jiankui,ZHANG Weihai. State and Output Feedback Finite-time Guaranteed Cost Control of Linear Ito Stochastic Systems [J]. Journal of Systems Science and Complexity, 2015, 28(4): 813-829.
[6] LIU Xueliang , XU Bugong , XIE Lihua. DISTRIBUTED TRACKING CONTROL OF SECONDORDER MULTI-AGENT SYSTEMS UNDER MEASU-REMENT NOISES [J]. Journal of Systems Science and Complexity, 2014, 27(5): 853-865.
[7] Yu ZHENG;Lequan MIN;Yu JI;Yongmei SU;Yang KUANG. GLOBAL STABILITY OF ENDEMIC EQUILIBRIUM POINT OF BASICVIRUS INFECTION MODEL WITH APPLICATION TO HBV INFECTION [J]. Journal of Systems Science and Complexity, 2010, 23(6): 1221-1230.
[8] Hongzhi YANG;Huiming WEI;Xuezhi LI. GLOBAL STABILITY OF AN EPIDEMIC MODEL FOR VECTOR-BORNEDISEASE [J]. Journal of Systems Science and Complexity, 2010, 23(2): 279-292.
[9] Li Qiang JIANG;Zhi Een MA;P. Fergola. THE GLOBAL STABILITY OF A COMPETING CHEMOSTAT MODEL WITH DELAYED NUTRIENT RECYCLING [J]. Journal of Systems Science and Complexity, 2005, 18(1): 19-026.
[10] Lei GUO. ADAPTIVE SYSTEMS THEORY: SOME BASIC CONCEPTS, METHODS AND RESULTS [J]. Journal of Systems Science and Complexity, 2003, 16(3): 293-306.
[11] San Ling YUAN;Zhi En MA. STUDY ON AN SIS EPIDEMIC MODEL WITH TIME VARIANT DELAY [J]. Journal of Systems Science and Complexity, 2002, 15(3): 299-306.
[12] Ying Dong LIU;Zheng Yuan LI;Qi Xiao YE. THE GLOBAL STABILITY OF CONSTANT EQUILIBRIUM OF REACTION-DIFFUSION SYSTEMS——DIFFERENTIAL INEQUALITIES AND LIAPUNOV FUNCTIONAL [J]. Journal of Systems Science and Complexity, 2002, 15(1): 89-101.
[13] San Ling YUAN;Zhi En MA. GLOBAL STABILITY AND HOPF BIFURCATION OF AN SIS EPIDEMIC MODEL WITH TIME DELAYS [J]. Journal of Systems Science and Complexity, 2001, 14(3): 327-336.
[14] WU Zhen. MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL PROBLEM OF FULLY COUPLEDFORWARD-BACKWARD STOCHASTIC SYSTEMS [J]. Journal of Systems Science and Complexity, 1998, 11(3): 249-259.
[15] CAO Feng;CHEN Lansun. ASYMPTOTIC BEHAVIOR OF NONAUTONOMOUS DIFFUSIVE LOTKA-VOLTERRA MODEL [J]. Journal of Systems Science and Complexity, 1998, 11(2): 107-111.
Viewed
Full text


Abstract