Elsevier

Automatica

Volume 48, Issue 3, March 2012, Pages 569-576
Automatica

Brief paper
Stochastically exponential stability and stabilization of uncertain linear hyperbolic PDE systems with Markov jumping parameters

https://doi.org/10.1016/j.automatica.2012.01.006Get rights and content

Abstract

This paper is concerned with the problem of robustly stochastically exponential stability and stabilization for a class of distributed parameter systems described by uncertain linear first-order hyperbolic partial differential equations (FOHPDEs) with Markov jumping parameters, for which the manipulated input is distributed in space. Based on an integral-type stochastic Lyapunov functional (ISLF), the sufficient condition of robustly stochastically exponential stability with a given decay rate is first derived in terms of spatial differential linear matrix inequalities (SDLMIs). Then, an SDLMI approach to the design of robust stabilizing controllers via state feedback is developed from the resulting stability condition. Furthermore, using the finite difference method and the standard linear matrix inequality (LMI) optimization techniques, recursive LMI algorithms for solving the SDLMIs in the analysis and synthesis are provided. Finally, a simulation example is given to demonstrate the effectiveness of the developed design method.

Introduction

Markov jump systems are a special class of hybrid systems (Krasovskii & Lidskii, 1961) with two complements in the state vector: the mode and the state, where the system mode is governed by a continuous Markov chain taking values in a finite set, and the state in each mode is represented by a system of differential equations. This family of systems may be emerging as an effective framework for various control problems in different fields such as target tracking, manufacturing processes, fault-tolerant control systems, communication systems, and economic systems (Mariton, 1990). Over the past two decades, a great number of important issues have been extensively studied for linear lumped parameter systems with Markov jumping parameters (MJPs), such as controllability and observability (Ji & Chezick, 1990), stability and stabilization (Bolzern et al., 2006, Fang and Loparo, 2002, Feng et al., 1992, Tanelli et al., 2010), H2 and H control (Costa et al., 1999, Li and Ugrinovskii, 2007, Shi and Boukas, 1997, Souza and Fragoso, 1993), robustness (Boukas et al., 1999, Ghaoui and Aitrami, 1996, Souza, 2006), guaranteed cost control (Boukas, Liu, & Al-Sunni, 2003), slide-mode control (Shi, Xia, Liu, & Rees, 2006), etc. Despite these efforts, it remains an open issue how to solve the problem of stochastic stability analysis and controller synthesis for uncertain distributed parameter systems described by partial differential equations (PDEs) with MJPs.

The well-known classification of PDE systems could be hyperbolic, parabolic, and elliptic, etc. (Ray, 1981), according to the properties of the spatial differential operator (SDO). The type of PDEs essentially determines the approach to solve the control problem for PDE systems. For a parabolic PDE system, motivated by the fact that its dominant dynamic behavior can be approximately described by a low-dimensional ordinary differential equation (ODE) system, control methods predominantly apply Galerkin’s procedure to derive a low-dimensional ODE approximation of the original PDE system (Balas, 1979, Curtain, 1982), which is then used for controller design purposes. In contrast to parabolic PDE systems, hyperbolic PDE systems involve SDOs, whose eigenvalues cluster along vertical or nearly vertical asymptotes in the complex plane and thus cannot be accurately represented by a finite number of modes. These systems can represent the dynamics of industrial processes involved in convection with negligible diffusion effects, e.g., fluid heat exchanger, plug-flow reactor (PFR) and fiber spinlines (Christofides, 2001). Having recognized that the infinite-dimensional nature of hyperbolic PDE systems has to be involved in the control development, some control approaches have been proposed for hyperbolic PDE systems in the past few decades. For example, Ray (1981) introduced the optimal control methods on the basis of the original PDE model, and the control methods on the basis of equivalent ODE realizations obtained by the method of characteristics for linear first-order hyperbolic PDE (FOHPDE) systems; Curtain and Zwart (1995) developed linear-quadratic (LQ) optimal control methods for linear infinite-dimensional systems, where the control gain operator was obtained by the solution of an algebraic operator Riccati equation for state-space model. More recently, Aksikas et al. presented an LQ optimal control design via spectral factorization for nonlinear FOHPDE model of a nonisothermal PFR (Aksikas, Winkin, & Dochain, 2007), and a general LQ control design for a class of linear FOHPDE systems (Aksikas, Fusman, Forbes, & Winkin, 2009), where the corresponding control gain operators were all obtained by the solutions of a matrix Riccati differential equation in the space variables. However, these control methods on linear FOHPDE systems are developed under the assumption that the system parameters are deterministic, which are not suitable for the control design of the PDE systems with MJPs. In practice, many industrial processes considered may experience random changes in their structures and parameters caused by abrupt phenomena such as failures and repairs of the components, sudden environment changes, changes in the interconnection of subsystems, etc. In such situations, it is more realistic to model the dynamic behaviors of these processes by PDEs with MJPs. However, to the best of the authors’ knowledge, few results are available for the analysis and synthesis of uncertain linear FOHPDE systems with MJPs.

Motivated by the aforementioned considerations, this paper aims to develop the sufficient conditions of robustly stochastically exponential stability and stabilization for a class of uncertain linear FOHPDE systems with MJPs, for which the manipulated input is distributed in space. The main contribution of this paper is that the stochastic stability analysis and distributed state feedback control design problem for a class of uncertain linear FOHPDE systems with MJPs is handled by employing the PDE theory and a novel integral-type stochastic Lyapunov functional (ISLF). Different from the previous results of ODE systems with MJPs (Boukas et al., 1999, Fang and Loparo, 2002, Ghaoui and Aitrami, 1996) and linear FOHPDE systems (Aksikas et al., 2009), the methods of stochastic stability analysis and the state feedback control design are developed in terms of spatial differential linear matrix inequalities (SDLMIs) by utilizing the ISLF without resorting to semigroup theory or functional analysis. Furthermore, recursive algorithms for solving the SDLMIs in the analysis and synthesis are presented on the basis of the existing linear matrix inequality (LMI) optimization techniques.

Notations. , n and m×n denote the set of all real numbers, n-dimensional Euclidean space and the set of all m×n matrices, respectively. Identity matrix, of appropriate dimensions, will be denoted by I. If not explicitly stated, all matrix functions of x defined on interval [z1,z2] are C1 continuous. The superscript T is used for the transpose of a matrix or a vector. For a symmetric matrix M, M>(,<,)0 means that it is positive definite (semi-positive definite, negative definite, semi-negative definite, respectively). The space-varying symmetric matrix function M(x)>(,<,)0, x[z1,z2] means that it is positive definite (semi-positive definite, negative definite, semi-negative definite, respectively) for each x[z1,z2]. The symbol is used as an ellipsis for terms in matrix expressions that are induced by symmetry, e.g., [S+[M+]XY]=Δ[S+[M+MT]XXTY].

Section snippets

Problem formulation and preliminaries

Given a probability space (Ω,F,P), where Ω is the sample space, F is the algebra of events and P is the probability measure defined on F, ηt is the system mode and let {ηt,t0} be an homogeneous, finite-state Markov process with right continuous trajectories and taking values in a finite set S=Δ{1,2,,s} with generator Π=(πij)s×s. The transition probability from mode i at time t to mode j at time t+Δt, i,jS is pij(Δt)=ΔPr(ηt+Δt=j|ηt=i)={πijΔt+o(Δt)ij1+πiiΔt+o(Δt)i=j where Δt>0, limΔt0(o(Δt)/Δ

Robust stability analysis

In this section, instead of directly dealing with the robust stability analysis of the free system of (2) when ηt=iS, i.e., yt(x,t)=Aiy(x,t)+[Ai(x)+ΔAi(x,t)]y(x,t) we first consider the nominal system of (17) and present a stochastically exponential stability condition. Using the functional (11) and the operator [] defined in (12), for the nominal system of (17), we find that [V(ηt,t)]=z1z2yT(x,t)Pi(x)yt(x,t)dx+z1z2ytT(x,t)Pi(x)y(x,t)dx+Γ1,i=Γ1,i+Γ2,i+Γ3,i+Γ4,i where Γ1,i=Δj=1sπijz1z2yT(

Robust state feedback stabilization

In this section, the results in the previous section will be extended to give an SDLMI-based robust stabilization condition for the uncertain hyperbolic PDE system (2) via the controller (8). Likewise, we first give the result for the nominal case, and then generalize it to the uncertain case.

Theorem 3

For a given scalar χ>0, the nominal system of (2) is stochastically exponentially stabilizable with χ-DR via the controller (8) if there exist n×n diagonal matrices Qi(x)=Δdiag{q1,i(x),,qn,i(x)}>0 and m×n

Recursive LMI algorithms

In this section, we will provide recursive LMI algorithms for solving the SDLMIs in (28), (34) by employing the finite difference method (with a backward difference for the spatial derivative), respectively. More specially, these SDLMIs can be solved by discretizing the interval [z1,z2] into space instances {xk,kN,x0=z1,xnz=z2} of the same distance, where N=Δ{0,1,2,,nz}, xkxk1=Δε1=(z2z1)/nz, and using the backward difference for the spatial derivative, i.e., dLi(x)/dx(Li(xk)Li(xk1))/ε1,i

Numerical example

In this section, an example is presented to illustrate the effectiveness of the proposed stabilization method. We consider a steam-jacketed tubular heat exchanger (Ray, 1981), whose dynamic model is of the form: Tt(x,t)=υ(x)Tx(x,t)a(x,t)(T(x,t)+δ(ηt)Tu(x,t)) subject to the boundary and initial conditions T(0,t)=Tin(t),T(x,0)=T0(x),η0=r0 where T(x,t) denotes the temperature of the reactor, x[0,1], Tu(x,t) denotes the jacket temperature, υ(x) denotes the fluid velocity in the exchanger, a(x,t)

Conclusion

In this paper, we have investigated the problem of robustly stochastically exponential stability and stabilization for a class of uncertain linear FOHPDE systems with MJPs, for which the manipulated input is distributed in space. By using the ISLF, the robust stability criterion is first derived in terms of SDLMIs. Then, the SDLMI-based robust stabilization condition via state feedback is developed from the resulting stability criterion. Furthermore, the recursive LMI algorithms are proposed to

Acknowledgments

The authors also gratefully acknowledge the helpful comments and suggestions of the Associate Editor and anonymous reviewers, which have improved the presentation.

Jun-Wei Wang received his B.Sc. degree in Mathematics and Applied Mathematics and his M.Sc. degree in System Theory from Harbin Engineering University, Harbin, China, in 2007 and 2009, respectively. He is now studying for his Ph. D. degree in Control Science and Engineering in Beihang University (Beijing University of Aeronautics and Astronautics), Beijing, China. His research interests include robust control and filtering, stochastic systems, distributed parameter systems, and fuzzy modeling

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    Jun-Wei Wang received his B.Sc. degree in Mathematics and Applied Mathematics and his M.Sc. degree in System Theory from Harbin Engineering University, Harbin, China, in 2007 and 2009, respectively. He is now studying for his Ph. D. degree in Control Science and Engineering in Beihang University (Beijing University of Aeronautics and Astronautics), Beijing, China. His research interests include robust control and filtering, stochastic systems, distributed parameter systems, and fuzzy modeling and control.

    Huai-Ning Wu was born in Anhui, China, on November 15, 1972. He received his B.E. degree in automation from Shandong Institute of Building Materials Industry, Jinan, China and his Ph.D. degree in control theory and control engineering from Xi’an Jiaotong University, Xi’an, China, in 1992 and 1997, respectively.

    From August 1997 to July 1999, he was a Postdoctoral Researcher in the Department of Electronic Engineering at Beijing Institute of Technology, Beijing, China. In August 1999, he joined the School of Automation Science and Electrical Engineering, Beihang University (formerly Beijing University of Aeronautics and Astronautics). From December 2005 to May 2006, he was a Senior Research Associate in the Department of Manufacturing Engineering and Engineering Management (MEEM), City University of Hong Kong, Hong Kong. From October to December between 2006–2008 and from July to August in 2010 and 2011, he was a Research Fellow in the Department of MEEM, City University of Hong Kong. He is currently a Professor at Beihang University. His current research interests include robust control and filtering, fault-tolerant control, distributed parameter systems, and fuzzy/neural modeling and control. He is a member of the Committee of Technical Process Failure Diagnosis and Safety, Chinese Association of Automation.

    Han-Xiong Li received his B.E. degree in aerospace engineering from the National University of Defence Technology, Changsha, China, in 1982, his M.E. degree in electrical engineering from Delft University of Technology, Delft, The Netherlands, in 1991, and his Ph.D. degree in electrical engineering from the University of Auckland, Auckland, New Zealand, in 1997.

    Currently, he is a full professor in the Department of Systems Engineering and Engineering Management, the City University of Hong Kong. Over the last twenty years, he has had opportunities to work in different fields, including military service, industry, and academia. His research experience and accomplishment include, fuzzy-PID for process control, a pioneering 3-domain fuzzy logic system for modeling and control, intelligent modeling and control of spatio-temporal dynamic system, with application to thermal cure process and fluid dispensing for IC packaging. He has published over 100 SCI journal papers (nearly half of them in IEEE Transactions and ASME Transactions) with h-index 22. His current research interests are in intelligent modeling and control, integrated process design and control, and distributed parameter systems.

    Dr. Li serves as Associate Editor of IEEE Transactions on Systems, Man & Cybernetics, part-B, and IEEE Transactions on Industrial Electronics. He was awarded the Distinguished Young Scholar (overseas) by the China National Science Foundation in 2004, a Chang Jiang chair professor by the Ministry of Education, China in 2006. He is a fellow of the IEEE.

    This work was supported by the National Basic Research Program of China (973 Program) under Grant 2012CB720003, the National Natural Science Foundation of China under Grants 61074057, 61121003, and 51175519, the General Research Fund project from Research Grants Council of Hong Kong, SAR, under CityU: 117310, the Fundamental Research Funds for the Central Universities and the Innovation Foundation of BUAA for PhD Graduates, China. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Huijun Gao under the direction of Editor Ian R. Petersen.

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