Dissipative control for nonlinear Markovian jump systems with actuator failures and mixed time-delays

doi:10.1016/j.automatica.2018.09.028

Automatica

Volume 98, December 2018, Pages 358-362

https://doi.org/10.1016/j.automatica.2018.09.028 Get rights and content

Abstract

This paper addresses the dissipative control problem for nonlinear Markovian jump systems subject to actuator failures and mixed time-delays, where the mixed time-delays consist of both discrete and distributed time-delays and are mode-dependent. The purpose of the problem under investigation is to design a state feedback controller such that, in the presence of actuator failures and mixed time-delays, the closed-loop system is asymptotically stable in the mean square sense while achieving the pre-specified dissipativity. By constructing a Lyapunov–Krasovskii functional and using a completing square approach, sufficient conditions are proposed for the existence of the desired controller in terms of the solvability of certain Hamilton–Jacobi inequalities. Finally, an illustrative numerical example is provided to demonstrate the effectiveness of the developed control scheme.

Introduction

Markovian jump systems (MJSs) have been attracting ever-increasing research interest within the systems science and control communities owing primarily to their ability in modeling random variations, see Aberkane and Dragan (2012), Bian, Jiang, and Jiang (2016), Boukas (2006), Terra, Ishihara, Jesus, and Cerri (2013), Vamvoudakis and Safaei (2017) and the references therein. On the other hand, in engineering practice, actuator failure is one of the most frequently encountered phenomena which could give rise to performance degradation or even instability Lunze and Steffen (2006), Ma et al. (2017a), Seo and Kim (1996). Moreover, since time-delays are ubiquitous in a variety of practical systems (e.g. chemical, biological and engineering systems), considerable attention has been paid to the analysis and synthesis issues for systems with time-delays. According to the ways they occur, time-delays can be categorized into discrete and distributed delays Liu et al. (2016), Scarciotti and Astolfi (2016). So far, much effort has been devoted to an investigation on linear systems with discrete and/or distributed delays, see e.g. Basin, Rodriguez-Gonzalez, Fridman, and Acosta (2005), among which the most exploited algorithm is arguably the linear matrix inequality (LMI) framework. Nevertheless, when it comes to the nonlinear delayed systems (especially with Markovian jump parameters), the widely used LMI-based Lyapunov–Krasovskii functional method is no longer applicable. To date, the issues of stability analysis, control and filtering have not been adequately studied for nonlinear MJSs subject to both actuator failures and mixed time-delays, which constitute the first motivation of the current research.

Ever since the seminal work in Willems (1972), the theory of dissipative systems has been playing a paramount role in the study of dynamical systems. In particular, the dissipative control/filtering problems have stirred an increasing research interest leading to a multitude of results reported in the literature. For linear systems, stabilization, control and filtering problems with desired dissipativity have been extensively investigated, see e.g. Feng, Lam, and Shu (2013) and Tan, Soh, and Xie (1999). It should be pointed out that, however, limited work has been done for nonlinear systems with or without Markovian jumping parameters. Those few available results include stability and dissipativity conditions established in (1) Aliyu (1999) for nonlinear Markovian jump systems in virtue of the Hamilton–Jacobi inequality (HJI) approach; and (2) Sheng, Gao, & Zhang (2014) for a class of nonlinear MJSs using an LMI-based method. When the nonlinear MJS with mode-dependent mixed time-delays is concerned, the corresponding dissipative control problem has not been fully examined due mainly to the technical difficulties stemming from the coupling between the nonlinear dynamics and the mode-dependent time-delays. As such, in this paper, we are motivated to design a state feedback controller for a class of nonlinear MJSs to ensure the expected stability and dissipativity in the presence of actuator failures and mode-dependent mixed time-delays.

Section snippets

Problem formulation

Let $r_{t} (t \geq 0)$ be a right-continuous Markovian chain on a probability space $(Ω, F, {F_{t}}_{t \geq 0}, P)$ taking values in a finite state space $M = {1, 2, \dots, q}$ with generator $Π = {π_{i j}}$ given by $P {r_{t + Δ} = j | r_{t} = i} = \{\begin{aligned} π_{i j} Δ + o (Δ), if i \neq j, \\ 1 + π_{i j} Δ + o (Δ), if i = j, \end{aligned}$ where $Δ > 0$ , and $π_{i j} > 0$ is the transition rate from $i$ to $j$ if $j \neq i$ while $π_{i i} = - \sum_{j \neq i} π_{i j}$ .

On a probability space $(Ω, F, {F_{t}}_{t \geq 0}, P)$ , we consider the following class of nonlinear delayed systems: $\{\begin{aligned} \dot{x} (t) = & f (x (t), r_{t}) + m (x (t), r_{t}) φ (u (t)) + D (r_{t}) v (t) \\ + g (x (t - τ_{1, r_{t}})) + \int_{t - τ_{2, r_{t}}}^{t} h (x (s)) d s \\ y (t) = & [\begin{matrix} C (r_{t}) x (t) \\ u (t) \end{matrix}] \\ x \end{aligned}$

Main results

Defining $x_{t} ≜ x (t + θ)$ , we consider a functional $V (x_{t}, t, r_{t}) \in C^{1} (R^{n} \times T \times M)$ associated with system (1) as follows: $V (x_{t}, t, r_{t})$ $≜ V_{1} (x (t), r_{t}) + \int_{t - τ_{1, r_{t}}}^{t} g^{T} (x (s)) P_{1} g (x (s)) d s + \bar{π} \int_{{\underset{̲}{τ}}_{1}}^{{\bar{τ}}_{1}} \int_{t - s}^{t} g^{T} (x (θ)) P_{1} g (x (θ)) d θ d s + \int_{0}^{τ_{2, r_{t}}} \int_{t - s}^{t} h^{T} (x (θ)) P_{2} h (x (θ)) d θ d s + \bar{π} \int_{{\underset{̲}{τ}}_{2}}^{{\bar{τ}}_{2}} \int_{0}^{ϑ} \int_{t - s}^{t} h^{T} (x (θ)) P_{2} h (x (θ)) d θ d s d ϑ$ where $V_{1} (0, r_{t}) = 0$ , $V_{1} (x (t), r_{t}) > 0$ for $x (t) \neq 0$ , and $P_{1}$ and $P_{2}$ are positive definite matrices with compatible dimensions. Moreover, for $V (x_{t}, t, i)$ $(i \in M)$ , we define: $V_{x} (i) ≜ [\begin{matrix} \frac{\partial V (x_{t}, t, i)}{\partial x_{1}} & \frac{\partial V (x_{t}, t, i)}{\partial x_{2}} & \dots & \frac{\partial V (x_{t}, t, i)}{\partial x_{n}} \end{matrix}] .$

Before giving the main

Simulation example

Consider the following mechanical rotational cutting process taken from Wang, Sun, Shi, and Zhao (2013): $\ddot{z} + γ_{1} \dot{z} + γ_{2} (z + z^{3}) = - γ_{3} z (t - 1)$ where $z$ represents the deflection of the machine tool, $γ_{1}$ is the term proportional to the product of natural frequency, the damping ratio $γ_{2}$ represents the tool stiffness and $γ_{3}$ is the delay term proportional to effective cutting stiffness of the workpiece per unit of chip width. It should be pointed out that during the cutting process, the parameter $γ_{3}$ may be

Conclusion

The dissipative control problem has been investigated for the nonlinear time-delay system subject to Markovian jump parameters and actuator failures. The mode-dependent time-delays under consideration include both discrete and distributed time-delays. In terms of Hamilton–Jacobi inequalities, sufficient conditions have been proposed for the existence of the required state feedback controller guaranteeing simultaneously the mean-square asymptotical stability and pre-specified dissipativity of

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grants 61773209, 61773017 and 61873148, the Research Fund for the Taishan Scholar Project of Shandong Province of China, the six talent peaks project in Jiangsu Province under Grant XYDXX-033, the Fundamental Research Funds for the Central Universities under Grant 30916011337, the Postdoctoral Science Foundation of China under Grant 2014M551598, and Alexander von Humboldt Foundation of Germany .

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^☆: The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Hiroshi Ito under the direction of Editor André L. Tits.

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Technical communiqueDissipative control for nonlinear Markovian jump systems with actuator failures and mixed time-delays☆

Abstract

Introduction

Section snippets

Problem formulation

Main results

Simulation example

Conclusion

Acknowledgments

Automatica

Neurocomputing

Automatica

Automatica

Integral sliding mode design for robust filtering and control of linear stochastic time-delay systems

International Journal of Robust and Nonlinear Control

Adaptive dynamic programming for stochastic systems with state and control dependent noise

IEEE Transactions on Automatic Control

Stochastic switching systems: Analysis and design

Dissipative control for linear systems by static output feedback

International Journal of Systems Science

Technical communique
Dissipative control for nonlinear Markovian jump systems with actuator failures and mixed time-delays☆