Reachable Set Estimation and Decentralized Controller Design for Large-Scale Nonlinear Systems With Time-Varying Delay and Input Constraint

This paper investigates the problem of reachable set estimation and decentralized controller design for large-scale nonlinear systems, where time-varying delays and nonlinearities appear in interconnections connected to other subsystems. The Takagi–Sugeno model is used to describe each nonlinear subsystem. The aim of this paper is to design a decentralized state-feedback fuzzy controller such that the reachable set of the resulting closed-loop system with input constraint is bounded by an intersection of ellipsoids. First, a model transformation is proposed to reformulate the closed-loop system as several feedback interconnections with extra inputs and outputs. Then, based on a combined application of the Lyapunov–Krasovskii functional and the scaled small-gain theorem, the input–output approach is developed for the reachable set estimation and synthesis. Several conditions for the existence of a decentralized state-feedback fuzzy controller that ensures an ellipsoidal bound of reachable sets for the closed-loop system with input constraint are derived in terms of linear matrix inequalities. Finally, two numerical examples are given to validate the effectiveness of the proposed strategy.

[1]  Hieu Minh Trinh,et al.  A new approach to state bounding for linear time-varying systems with delay and bounded disturbances , 2014, Autom..

[2]  Bor-Sen Chen,et al.  H∞ decentralized fuzzy model reference tracking control design for nonlinear interconnected systems , 2001, IEEE Trans. Fuzzy Syst..

[3]  Yongsheng Ding,et al.  Comparison of necessary conditions for typical Takagi-Sugeno and Mamdani fuzzy systems as universal approximators , 1999, IEEE Trans. Syst. Man Cybern. Part A.

[4]  Wei Chang,et al.  $H_{\infty} $ Fuzzy Control Synthesis for a Large-Scale System With a Reduced Number of LMIs , 2015, IEEE Transactions on Fuzzy Systems.

[5]  K. Gu,et al.  Small gain problem in coupled differential‐difference equations, time‐varying delays, and direct Lyapunov method , 2011 .

[6]  Tong Heng Lee,et al.  LMI Approach to Analysis and Control of Takagi-Sugeno Fuzzy Systems with Time Delay (Lecture Notes in Control and Information Sciences) , 2007 .

[7]  Chung-Shi Tseng,et al.  A Novel Approach to $H_{\bm \infty}$ Decentralized Fuzzy-Observer-Based Fuzzy Control Design for Nonlinear Interconnected Systems , 2008, IEEE Transactions on Fuzzy Systems.

[8]  Fei Liu,et al.  H∞ Filtering for Discrete-Time Systems With Stochastic Incomplete Measurement and Mixed Delays , 2012, IEEE Trans. Ind. Electron..

[9]  James Lam,et al.  On reachable set estimation of singular systems , 2015, Autom..

[10]  Aryan Saadat Mehr,et al.  On ${\cal H}_{\infty }$ Filtering for Discrete-Time Takagi–Sugeno Fuzzy Systems , 2012, IEEE Transactions on Fuzzy Systems.

[11]  Cheng-Wu Chen,et al.  Robust stabilization of nonlinear multiple time-delay large-scale systems via decentralized fuzzy control , 2005, IEEE Trans. Fuzzy Syst..

[12]  Emilia Fridman,et al.  On reachable sets for linear systems with delay and bounded peak inputs , 2003, Autom..

[13]  Emilia Fridman,et al.  Input-output approach to stability and L2-gain analysis of systems with time-varying delays , 2006, Syst. Control. Lett..

[14]  Hongbin Zhang,et al.  Delay-Dependent Decentralized $H_\infty$ Filtering for Discrete-Time Nonlinear Interconnected Systems With Time-Varying Delay Based on the T–S Fuzzy Model , 2012, IEEE Transactions on Fuzzy Systems.

[15]  Shaocheng Tong,et al.  Observer-Based Adaptive Fuzzy Backstepping Output Feedback Control of Uncertain MIMO Pure-Feedback Nonlinear Systems , 2012, IEEE Transactions on Fuzzy Systems.

[16]  Ligang Wu,et al.  A New Approach to Stability Analysis and Stabilization of Discrete-Time T-S Fuzzy Time-Varying Delay Systems , 2011, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[17]  James Lam,et al.  Estimation and synthesis of reachable set for switched linear systems , 2016, Autom..

[18]  Yijing Wang,et al.  A non-ellipsoidal reachable set estimation for uncertain neural networks with time-varying delay , 2014, Commun. Nonlinear Sci. Numer. Simul..

[19]  Huijun Gao,et al.  A New Model Transformation of Discrete-Time Systems With Time-Varying Delay and Its Application to Stability Analysis , 2011, IEEE Transactions on Automatic Control.

[20]  Hieu Minh Trinh,et al.  Reachable set bounding for nonlinear perturbed time-delay systems: The smallest bound , 2015, Appl. Math. Lett..

[21]  Vladimir L. Kharitonov,et al.  Stability of Time-Delay Systems , 2003, Control Engineering.

[22]  Hieu Minh Trinh,et al.  On backwards and forwards reachable sets bounding for perturbed time-delay systems , 2015, Appl. Math. Comput..

[23]  Shengyuan Xu,et al.  Reachable set estimation and controller design for distributed delay systems with bounded disturbances , 2014, J. Frankl. Inst..

[24]  K. Gu.  An integral inequality in the stability problem of time-delay systems , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[25]  J. Lam,et al.  New insight into delay‐dependent stability of time‐delay systems , 2015 .

[26]  J. Lam,et al.  Estimation and synthesis of reachable set for discrete‐time periodic systems , 2016 .

[27]  Panagiotis Tsiotras,et al.  Stability of time-delay systems: equivalence between Lyapunov and scaled small-gain conditions , 2001, IEEE Trans. Autom. Control..

[28]  Ju H. Park,et al.  Stability and stabilization for discrete-time systems with time-varying delays via augmented Lyapunov-Krasovskii functional , 2013, J. Frankl. Inst..

[29]  James Lam,et al.  An improved result on reachable set estimation and synthesis of time-delay systems , 2014, Appl. Math. Comput..

[30]  Dragoslav D. Šiljak,et al.  Large-Scale Dynamic Systems: Stability and Structure , 1978 .

[31]  Jianbin Qiu,et al.  Nonsynchronized-State Estimation of Multichannel Networked Nonlinear Systems With Multiple Packet Dropouts Via T–S Fuzzy-Affine Dynamic Models , 2011, IEEE Transactions on Fuzzy Systems.

[32]  Guang-Hong Yang,et al.  Control Synthesis of Singularly Perturbed Fuzzy Systems , 2008, IEEE Transactions on Fuzzy Systems.

[33]  G. Feng,et al.  A Survey on Analysis and Design of Model-Based Fuzzy Control Systems , 2006, IEEE Transactions on Fuzzy Systems.

[34]  Hongbin Zhang,et al.  Delay-dependent decentralised H α filtering for fuzzy interconnected systems with time-varying delay based on Takagi¿Sugeno fuzzy model , 2013 .

[35]  Magdi S. Mahmoud,et al.  Decentralized sliding-mode output-feedback control of interconnected discrete-delay systems , 2012, Autom..

[36]  Chung-Yao Kao,et al.  Simple stability criteria for systems with time-varying delays , 2004, Autom..

[37]  Qiong Huang,et al.  New delay-dependent stability analysis for fuzzy time-delay interconnected systems , 2013, Int. J. Gen. Syst..

[38]  Lubomír Bakule,et al.  Decentralized control: Status and outlook , 2014, Annu. Rev. Control..

[39]  Shengyuan Xu,et al.  Reachable set estimation for discrete‐time linear systems with time delays , 2015 .

[40]  Bing Chen,et al.  New Decentralized $H_{\infty }$ Filter Design for Nonlinear Interconnected Systems Based on Takagi–Sugeno Fuzzy Models , 2015, IEEE Transactions on Cybernetics.

[41]  Wei Chang,et al.  Hα fuzzy controller synthesis for large-scale systems with non-linear interconnections and disturbance , 2014 .

[42]  Bo Yu,et al.  Robust mixed H2/H∞ control of networked control systems with random time delays in both forward and backward communication links , 2011, Autom..

[43]  James Lam,et al.  $H_{\bm \infty}$ Fuzzy Filtering of Nonlinear Systems With Intermittent Measurements , 2009, IEEE Transactions on Fuzzy Systems.

[44]  Guoping Liu,et al.  Improved delay-range-dependent stability criteria for linear systems with time-varying delays , 2010, Autom..

[45]  Kazuo Tanaka,et al.  Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach , 2008 .