Computation of Upper Bounds for the Solution of Continuous Algebraic Riccati Equations
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Computation of Upper Bounds for the Solution of Continuous Algebraic Riccati Equations Wei Zhang · Housheng Su · Jia Wang
Received: 6 April 2012 / Revised: 12 September 2012 / Published online: 28 September 2012 © Springer Science+Business Media New York 2012
Abstract This paper is considered with the computation of upper bounds for the solution of continuous algebraic Riccati equations (CARE). A parameterized upper bound for the solution of CARE is proposed by utilizing some linear algebraic techniques. Based on this bound, more precise estimation can be achieved by means of carefully choosing the bound’s parameters. Iterative algorithm is also developed to obtain more sharper solution bounds. Comparing with some existing results in the literature, the proposed bounds are less restrictive and more effective. The effectiveness and advantages of the proposed approach are illustrated via a numerical example. Keywords Continuous algebraic Riccati equation (CARE) · Lyapunov equation · Matrix bound · Iterative algorithm
1 Introduction In recent years, there has been significant research effort devoted to the analysis and design problems of dynamical systems [1, 7, 18–29]. Generally, the Lyapunov-based W. Zhang · J. Wang Laboratory of Intelligent Control and Robotics, Shanghai University of Engineering Science, Shanghai 201620, China W. Zhang e-mail: [email protected] J. Wang e-mail: [email protected] H. Su () Department of Control Science and Engineering, Key Laboratory of Education Ministry for Image Processing and Intelligent Control, Huazhong University of Science and Technology, Wuhan 430074, China e-mail: [email protected]
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Circuits Syst Signal Process (2013) 32:1477–1488
approach plays a fundamental role in solving these problems [1, 9, 18–21]. Based on the Lyapunov stability theory, many useful results have been addressed in recent literature [18–21, 25–32]. Among these references, two kinds of equation, namely the algebraic Riccati equation and the Lyapunov equation, are frequently encountered [20, 25, 31]. Indeed, it is known that these equations play vital roles in the analysis and synthesis of linear control systems. Therefore, the character of these equations has been received extensive research. In this paper, we focus on the estimation of upper bounds for the solution of these equations. Consider the following continuous algebraic Riccati equation (CARE): AT P + P A − P RP = −Q,
(1)
where A ∈ Rn×n , B ∈ Rn×m , R = BB T , Q ∈ Rn×n is a given positive semi-definite matrix, and P ∈ Rn×n is the unique positive semi-definite solution of the CARE (1). It is assumed that the pair (A, B) is controllable and the pair (A, Q1/2 ) is observable. When B = 0, (1) reduces to the following continuous algebraic Lyapunov equation (CALE) [20]: AT P + P A = −Q.
(2)
As we know, the above equations are frequently involved in optimal control [3], stability analysis [30], observer design [31, 32], and filter design[17], etc. For example, in [31] the observer gain matrix for a class of nonlinear systems was designed
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