Elsevier

Automatica

Volume 37, Issue 2, February 2001, Pages 221-229
Automatica

Brief Paper
LPV Systems with parameter-varying time delays: analysis and control

https://doi.org/10.1016/S0005-1098(00)00156-4Get rights and content

Abstract

In this paper, we address the analysis and state-feedback synthesis problems for linear parameter-varying (LPV) systems with parameter-varying time delays. It is assumed that the state-space data and the time delays are dependent on parameters that are measurable in real-time and vary in a compact set with bounded variation rates. We explore the stability and the induced L2 norm performance of these systems using parameter-dependent Lyapunov functionals. In addition, the design of parameter-dependent state-feedback controllers that guarantee desired L2 gain performance is examined. Both analysis and synthesis conditions are formulated in terms of linear matrix inequalities (LMIs) that can be solved via efficient interior-point algorithms.

Introduction

Time delays are often present in engineering systems due to measurement, transmission and transport lags, computational delays, or unmodeled inertias of system components. The stability analysis and control of these systems has been examined extensively in the controls literature using both state-space and frequency domain methods (e.g., see Malek-Zavarei & Jamshidi, 1987; Watanabe, Nobuyama & Kojima, 1996; Dugard & Verriest, 1998, and the references therein). In many engineering systems, the time delays are known functions of variable operating conditions or system parameters that can be measured in real-time. For example, the transport delay in an internal combustion engine is a known function of the engine speed. Similarly, parameter-dependent time delays often appear in many manufacturing and chemical processes, biomedical systems and robotic systems where changes in the system dynamics result in variable delay times. Motivated by the linear parameter-varying (LPV) control theory, in this work the stabilization and the state-feedback control synthesis of such LPV systems that include parameter-dependent time delays is examined. LPV systems are systems that depend on unknown but measurable time-varying parameters, such that the measurement of these parameters provides real-time information on the variations of the plant's characteristics. Hence, it is desirable to design controllers that are scheduled based on this information. The analysis and control of LPV systems has been investigated recently by Packard (1994), Becker and Packard (1994), Apkarian and Gahinet (1995), Wu, Yang, Packard and Becker (1996), and Gahinet, Apkarian and Chilali (1996). These methods provide a systematic gain-scheduling control approach for nonlinear systems (Rugh, 1991; Shamma and Athans 1990, Shamma and Athans 1992). The LPV analysis and control synthesis problems can be formulated as linear matrix inequality (LMI) constraints that can be solved using recently developed efficient interior-point optimization algorithms (Boyd, El Ghaoui, Feron & Balakrishnan, 1994; Vandenberghe & Boyd, 1994).

Using the LPV framework, in this work we assume that the state-space system matrices and the time delays are functions of time-varying system parameters that are measured in real-time. We seek to synthesize parameter-varying controllers to stabilize the time-delayed LPV system and to provide disturbance attenuation measured in terms of the induced L2 norm of the system. The proposed approach utilizes parameter-dependent Lyapunov functionals to obtain sufficient conditions for stabilization and induced L2 norm performance in terms of LMIs. Although the single delay case is considered, the results can be easily extended to treat systems with multiple delays.

The notation to be used is as follows: R stands for the set of real numbers and R+ for the non-negative real numbers. Rm×n is the set of real m×n matrices. The transpose of a real matrix M is denoted by MT and its orthogonal complements by M. We use Sn×n to denote real, symmetric n×n matrices, and S+n×n for positive-definite n×n matrices. If MSn×n, then M>0(M≥0) indicates that M is a positive-definite (positive-semi-definite) matrix and M<0 (M≤0) denotes a negative-definite (negative-semi-definite) matrix. The matrix norm ||M|| is the maximum singular value of the matrix M, that is ||M||≔σ̄(M)=[λmax(MMT)]1/2. For xRn, its norm is defined as ||x||≔(xTx)1/2. The space of square integrable functions is denoted by L2, that is, for any u∈L2, ||u||2≔[0uT(t)u(t)dt]1/2 is finite. The space of continuous functions will be denoted by C and the corresponding norm is ||φ||=supt||φ(t)||. In a symmetric block matrix, the expression (∗) will be used to denote the submatrices that lie above the diagonal.

Section snippets

Analysis of time-delayed LPV systems

We consider the following state-space model of a time-delayed LPV system:Σd:ẋ(t)=A(ρ(t))x(t)+Ah(ρ(t))x(t−h(ρ(t)))+B(ρ(t))d(t),e(t)=C(ρ(t))x(t)+Ch(ρ(t))x(t−h(ρ(t)))+D(ρ(t))d(t),x(θ)=φ(θ),θ∈[−h(ρ(0)),0],where x(t)∈Rn is the state vector, d(t)∈Rnd is the vector of exogenous inputs, e(t)∈Rne in the output vector and h is a differentiable scalar function representing the parameter-varying delay. We assume that the delay is bounded and that the function th(t) is monotonically increasing, that is h

State-feedback control of time-delayed LPV systems

In this section, the analysis results presented in the previous section are used to design state-feedback controllers for LPV systems with parameter-dependent state and input time delays. We seek to design state-feedback gains that are scheduled based on the real-time measurement of the parameter vector ρ and guarantee prescribed induced L2 norm performance levels for the closed-loop system.

Consider the following time-delayed LPV system:ẋ(t)=A(ρ(t))x(t)+Ah(ρ(t))x(t−h(ρ(t)))+B1(ρ(t))d(t)+B2

Numerical example

Consider the following linear time-varying state-delayed system adopted from Mahmoud and Al-Muthairi (1994):ẋ(t)=01+φsint−2−3+δsintx(t)+φsint0.1−0.2+δsint−0.3x(t−μ|cos(ωt)|)+0.20.2d(t)+φsint0.1+δsintu(t),e(t)=0100x(t)+01u(t),where φ=0.2,δ=0.1,μ=0.09 and ω=5. To validate our proposed time-delayed LPV design methodology, we will assume that the sine and cosine terms in the above model correspond to systems parameters whose functional representation is not known a priori, but they can be measured

Conclusions

In this paper, the analysis and state-feedback control synthesis problems for LPV systems with parameter-dependent state delays are addressed. The corresponding analysis and synthesis conditions for stabilization and induced L2 norm performance are expressed in terms of LMIs that can be solved efficiently using recently developed interior-point algorithms. In addition, we considered the state-feedback control synthesis problem for LPV systems with input delays by augmenting the system dynamics

Karolos Grigoriadis earned his B.S. in Mechanical Engineering from the National Technical University of Athens, Greece (1987), a M.S. in Aerospace Engineering from Virginia Tech (1989), a M.S. in Mathematics from Purdue University (1993) and his Ph.D. in Aerospace Engineering from Purdue (1994). He is currently a Bill D. Cook Associate Professor in the Department of Mechanical Engineering at the University of Houston and the co-director of the Dynamic Systems Control Laboratory. His research

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Karolos Grigoriadis earned his B.S. in Mechanical Engineering from the National Technical University of Athens, Greece (1987), a M.S. in Aerospace Engineering from Virginia Tech (1989), a M.S. in Mathematics from Purdue University (1993) and his Ph.D. in Aerospace Engineering from Purdue (1994). He is currently a Bill D. Cook Associate Professor in the Department of Mechanical Engineering at the University of Houston and the co-director of the Dynamic Systems Control Laboratory. His research interests are in the areas of robust control systems design analysis and design with applications to the control of mechanical and aerospace systems. He is the co-author, with Robert Skelton and Tetsuya Iwasaki of A Unified Algebraic Approach to Linear Control Design (Taylor & Francis, 1998). He is currently serving in the Editorial Board of the IEEE Transactions of Automatic Control, the Systems & Control Letters and the Dynamics and Control. He has received many awards and honors for his research and teaching, including an 1997 NSF CAREER award, a 1997 SAE Ralph Teetor award, a Bill D. Cook Scholar Award, a 1997 Research Excellence and a 1997 Teaching Excellence award from UH.

Fen Wu was born in Chengdu, China, in 1964. He received the BS and the MS degrees in Automatic Control from Beijing University of Aeronautics and Astronautics, Beijing, in 1985 and 1988, respectively, and the Ph.D. degree in Mechanical Engineering from University of California at Berkeley, CA, in 1995. Subsequently, he worked 18 months in the Centre for Process Systems Engineering, Imperial College as a research associate. From 1988 to 1990, Dr. Wu worked at the Chinese Aeronautical Radio Electronics Research Institute, Shanghai, as a control engineer. He also worked as a staff engineer in Dynacs Engineering Company Inc. on the International Space Station plant/controller interaction analysis project. He is currently an Assistant Professor affiliated with the Mechanical and Aerospace Engineering Department, North Carolina State University. His research interests include control of flexible space structures, LMI techniques in systems and control theory, robust H2 and H control, model approximation, gain-scheduling design techniques with its application to aerospace, automotice and industrial problems.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor André Tits under the direction of Editor Tamer Basar

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