Decentralised adaptive output feedback stabilisation for stochastic time-delay systems via LaSalle-Yoshizawa-type theorem

ABSTRACT This paper deals with the decentralised output feedback stabilisation problem for a class of large-scale stochastic time-delay nonlinear systems. A general theorem is firstly given to guarantee the global existence and uniqueness of the solution for stochastic time-delay systems. In addition, a stochastic version of the well-known LaSalle-Yoshizawa theorem with time-varying delay is initially proposed for the controller design and stability analysis. Then, for a class of large-scale stochastic systems with time-varying delays, totally decentralised adaptive delay-dependent controllers are designed by using K-filter and backstepping approach. Via LaSalle-Yoshizawa-type theorem and constructing a general Lyapunov function, it is shown that all signals in the closed-loop system are bounded almost surely and the solution is almost surely asymptotically stable. Finally, a simulation example is given to illustrate the effectiveness of the results of this paper.

[1]  Li Xu,et al.  Decentralized adaptive output-feedback controller design for stochastic nonlinear interconnected systems , 2012, Autom..

[2]  Liang Liu,et al.  Output-feedback stabilization for stochastic high-order nonlinear systems with time-varying delay , 2011, Autom..

[3]  Ju H. Park,et al.  Decentralized guaranteed cost dynamic control for synchronization of a complex dynamical network , 2012, 2012 12th International Conference on Control, Automation and Systems.

[4]  P. Krishnamurthy,et al.  A generalized scaling based control design for nonlinear nontriangular systems with input and state time delays , 2011, Proceedings of the 2011 American Control Conference.

[5]  Ju H. Park,et al.  Robust Nonfragile Decentralized Controller Design for Uncertain Large-Scale Interconnected Systems With Time-Delays , 2002 .

[6]  Xuerong Mao,et al.  The Improved LaSalle-Type Theorems for Stochastic Differential Delay Equations , 2012 .

[7]  Miroslav Krstic,et al.  Stabilization of stochastic nonlinear systems driven by noise of unknown covariance , 2001, IEEE Trans. Autom. Control..

[8]  H. Kushner.  Stochastic Stability and Control , 2012 .

[9]  Xuerong Mao,et al.  Stochastic differential equations and their applications , 1997 .

[10]  Hamid Reza Karimi,et al.  Stability of Stochastic Nonlinear Systems With State-Dependent Switching , 2013, IEEE Transactions on Automatic Control.

[11]  P. Kokotovic,et al.  Adaptive nonlinear design with controller-identifier separation and swapping , 1995, IEEE Trans. Autom. Control..

[12]  Xue‐Jun Xie,et al.  Backstepping controller design for a class of stochastic nonlinear systems with Markovian switching , 2008, CCC 2008.

[13]  Jack K. Hale,et al.  Sufficient conditions for stability and instability of autonomous functional- Differential equations. , 1965 .

[14]  Zhihong Man,et al.  Finite-time stability and instability of stochastic nonlinear systems , 2011, Autom..

[15]  Songjiao Shi,et al.  State feedback stabilization for a class of stochastic time-delay nonlinear systems , 2003, IEEE Trans. Autom. Control..

[16]  Zhong-Ping Jiang,et al.  A small-gain condition for iISS of interconnected retarded systems based on Lyapunov-Krasovskii functionals , 2010, Autom..

[17]  W. Zheng,et al.  Finite‐time H ∞  control for discrete‐time switched nonlinear systems with time delay , 2015 .

[18]  N. Bershad,et al.  Random differential equations in science and engineering , 1975, Proceedings of the IEEE.

[19]  Fuad E. Alsaadi,et al.  Adaptive partial-state feedback control for stochastic high-order nonlinear systems with stochastic input-to-state stable inverse dynamics , 2015, Autom..

[20]  R. Khasminskii.  Stochastic Stability of Differential Equations , 1980 .

[21]  P. Pepe,et al.  A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems , 2006, Syst. Control. Lett..

[22]  Zheng-Guang Wu,et al.  Robust $\mathcal{H}_{\infty}$ decentralized dynamic control for synchronization of a complex dynamical network with randomly occurring uncertainties , 2012 .

[23]  Liang Liu,et al.  State feedback stabilization for stochastic feedforward nonlinear systems with time-varying delay , 2013, Autom..

[24]  J. Tsinias.  Stochastic input-to-state stability and applications to global feedback stabilization , 1998 .

[25]  Peter Kuster,et al.  Nonlinear And Adaptive Control Design , 2016 .

[26]  Xudong Ye,et al.  Decentralized Adaptive Stabilization of Large-Scale Nonlinear Time-Delay Systems With Unknown High-Frequency-Gain Signs , 2011, IEEE Transactions on Automatic Control.

[27]  Shengyuan Xu,et al.  Delay-Dependent Stability Criteria for Reaction–Diffusion Neural Networks With Time-Varying Delays , 2013, IEEE Transactions on Cybernetics.

[28]  X. Mao,et al.  A note on the LaSalle-type theorems for stochastic differential delay equations , 2002 .

[29]  Jing Zhou,et al.  Decentralized Backstepping Adaptive Output Tracking of Interconnected Nonlinear Systems , 2008, IEEE Transactions on Automatic Control.

[30]  Xin Yu,et al.  Output Feedback Regulation of Stochastic Nonlinear Systems With Stochastic iISS Inverse Dynamics , 2010, IEEE Transactions on Automatic Control.

[31]  Shuzhi Sam Ge,et al.  Adaptive output-feedback control for a class of uncertain stochastic non-linear systems with time delays , 2008, Int. J. Control.

[32]  Yuanqing Xia,et al.  Backstepping controller design for a class of stochastic nonlinear systems with Markovian switching , 2008, 2008 27th Chinese Control Conference.

[33]  X. Mao.  LaSalle-Type Theorems for Stochastic Differential Delay Equations , 1999 .

[34]  D. Feng,et al.  Stochastic functional differential equations with infinite delay , 2009 .

[35]  Junwei Lu,et al.  Tuning functions‐based robust adaptive tracking control of a class of nonlinear systems with time delays , 2012 .

[36]  Xuerong Mao,et al.  Robust stability and controllability of stochastic differential delay equations with Markovian switching , 2004, Autom..