Brief paperAnalysis of SDC matrices for successfully implementing the SDRE scheme☆
Introduction
Recently, the state-dependent Riccati equation (SDRE) approach for nonlinear system stabilization has attracted considerable attention (Bogdanov and Wan, 2007, Bracci et al., 2006, Çimen, 2010, Cloutier et al., 1996, Erdem and Alleyne, 2004, Hammett et al., 1998, Lam et al., 2012, Liang and Lin, 2011, Shamma and Cloutier, 2003, Sznaier et al., 2000). The SDRE scheme is known to include the following benefits (Çimen, 2010): (i) the concept is intuitive and simple, and directly adopts the LQR design at every nonzero state; (ii) the design can directly affect system performance with predictable results by adjusting the state and the control weightings to specify the performance index (for instance, the engineer may modulate the weighting of the system state to speed up the response, although at the expense of increased control effort); (iii) the scheme possesses an extra design degree of freedom arising from the non-unique state-dependent coefficient (SDC) matrix representation of the nonlinear drift term, which can be utilized to enhance controller performance; and (iv) the approach preserves the essential system nonlinearities because it does not truncate any nonlinear terms. Many practical and meaningful applications successfully performed by the SDRE design have been reported (see Çimen, 2010 and the references therein). The first solid theoretical contributions on SDRE control have been provided by Cloutier et al. (1996) and Mracek and Cloutier (1998). The current study attempts to provide further theoretical support of the SDRE control strategy, as discussed in the recent survey by Çimen (2012), with rigorous mathematical proofs.
The SDRE design for nonlinear systems can be described as follows. Consider a class of nonlinear control systems and a quadratic-like performance index as (1)–(2) below: where and denote the system states and control inputs, respectively, , , , , and denotes the transpose of a vector or a matrix. Note that the weighting matrices and are in general state-dependent. The procedure of the SDRE scheme is summarized as the following three steps (Çimen, 2010):
- (i)
Factorize into the SDC matrix representation as , where .
- (ii)
Symbolically check the stabilizability of and the observability (resp., detectability) of to ensure the existence of a unique positive definite (resp., semi-definite) solution of the following SDRE: where has full rank and satisfies .
- (iii)
Solve for from (3) to produce the SDRE controller
It is known that a unique positive definite (resp., semi-definite) solution in (3) exists, rendering pointwise Hurwitz, if (resp., if and only if) both the conditions “ is stabilizable” and “ is observable (resp., has no unobservable mode on the -axis)” are satisfied (Zhou & Doyle, 1998). To avoid the difficulty of symbolic checking conditions, stated above, of the SDRE approach, in this article we will study the following three problems: Problem 1 Let be given. Denote and . Explore the existence condition and, if the existence condition is satisfied, present all that satisfy the conditions that is stabilizable and is observable.
Problem 2 Same as Problem 1, except that the condition “ is observable” is replaced with “ is detectable”.
Problem 3 Same as Problem 1, except that the condition “ is observable” is replaced with “ has no unobservable mode on the -axis”.
From the discussions above, this study may also provide an auxiliary means to successfully continue the SDRE scheme at states in which a specific SDC matrix representation fails to operate, but where Problems 1, 2 or 3 is solvable.
To explore the existence condition of Problem 1, Problem 2, Problem 3 and characterize their solution matrices, we introduce the notations and as follows. Let be given with and . We define , null space of , and as a selected constant matrix having orthonormal columns and satisfying . Clearly, is a vector space of dimension , and the column vectors of form an orthonormal basis of . Similarly, if and , we define and as a selected constant matrix having orthonormal rows and satisfying . Additionally, we denote , known as the dual space of , and as the set of negative real numbers.
The rest of this article is organized as follows: Section 2 presents the necessary and sufficient existence conditions for Problem 1, Problem 2, Problem 3; Section 3 includes a description of the parameterization of the solution matrices for the planar case when the existence conditions are satisfied; Section 4 presents an illustrative example; and Section 5 provides the conclusions.
Section snippets
Necessary and sufficient existence conditions
Necessary and sufficient existence conditions for Problem 1, Problem 2, Problem 3 are stated as Theorem 1 below: Theorem 1 Problem 1 is unsolvable if and only if are linearly dependent (LD) and . Problem 2 is unsolvable if and only if for some and . Problem 3 is unsolvable if and only if and .
Proof The proofs of (i) and (ii) can be found from Liang and Lin (2011), while (iii) is easily derived from the proof of (ii). Details are omitted. □
Parameterization of all solution matrices
Given that the existence condition of Problems 1, 2 or 3 is satisfied, this section explores their solution matrices. To this end, we denote , and as the sets of such that is controllable, is stabilizable, is observable, is detectable and has no unobservable mode on the -axis, respectively. Additionally, we assume hereafter that, without loss of any generality, both and have full rank.
An illustrative example
Consider the following system Clearly, this system is in the form of (1) with and . System (7) is stabilizable and two global stabilizers, one using the Sontag formula with the control Lyapunov function (Sontag, 1989) and the other adopting the backstepping scheme (Khalil, 1996), have the following forms: and To demonstrate the SDRE
Conclusions
This article has presented necessary and sufficient conditions for the existence of SDC matrices in a nonlinear system such that the SDRE scheme can be successfully implemented. These existence conditions are easy to verify, and when they are satisfied, all of the feasible SDC matrices are explicitly parameterized for the planar case. An example is also given to demonstrate the use of the main results. Nevertheless, the application of this study in SDRE design for better system performance,
Yew-Wen Liang (M’02) was born in Taiwan in 1960. He received the B.S. degree in Mathematics from the Tung Hai University, Taichung, Taiwan, Republic of China, in 1982, the M.S. degree in Applied Mathematics in 1984 and the Ph.D. degree in Electrical and Computer Engineering in 1998 from the National Chiao Tung University, Hshinchu, Taiwan, Republic of China. Since August 1987, he has been with the National Chiao Tung University, where he is currently an Associate Professor of Electrical and
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Yew-Wen Liang (M’02) was born in Taiwan in 1960. He received the B.S. degree in Mathematics from the Tung Hai University, Taichung, Taiwan, Republic of China, in 1982, the M.S. degree in Applied Mathematics in 1984 and the Ph.D. degree in Electrical and Computer Engineering in 1998 from the National Chiao Tung University, Hshinchu, Taiwan, Republic of China. Since August 1987, he has been with the National Chiao Tung University, where he is currently an Associate Professor of Electrical and Control Engineering. His research interests include nonlinear control systems, reliable control, and fault detection and diagnosis issues.
Li-Gang Lin received the B.S. degree and M.S. degree in Electrical Control Engineering (ECE) from National Chiao Tung University (NCTU), Hsinchu, Taiwan, in 2008 and in 2010, respectively. He is currently working toward his joint/double Ph.D. degrees from ESAT of Katholieke Universiteit Leuven, Belgium, and institute of ECE in NCTU. His research interests include nonlinear control systems, state-dependent (differential/difference) Riccati equation, reliable and robust control.
- ☆
This work was supported by the National Science Council, Taiwan, under Grants 99-2218-E-009-004, 100-2221-E-009-026-MY2, and 101-2623-E-009-005-D. The material in this paper was partially presented at the 18th IFAC World Congress, August 28–September 2, 2011, Milano, Italy. This paper was recommended for publication in revised form by Associate Editor Constantino M. Lagoa under the direction of Editor Roberto Tempo.
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