H∞ control for descriptor systems: A matrix inequalities approach

Masubuchi, Izumi; Kamitane, Yoshiyuki; Ohara, Atsumi; Suda, Nobuhide

doi:10.1016/s0005-1098(96)00193-8

Cited by 622 publications

(261 citation statements)

References 16 publications

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“…The matrix E may be singular and we denote its rank by rank(E) = r ≤ n. It is known that systems having direct transmission paths from w and u to z and y can be transformed by augmenting the descriptor variable as pointed out for example in [5].…”

Section: Preliminariesmentioning

confidence: 99%

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Static Output Feedback Control Design for Descriptor Systems

Yagoubi¹

2010

Journal of Control Science and Engineering

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The static output feedback (SOF) synthesis problem for descriptor systems is considered in this paper. LMI-based algorithms are proposed to find potentially structured SOF gains ensuring admissibility and even H ∞ performance of the closed-loop system. These algorithms are then used to propose a (descriptor) observer-based H ∞ controller design method. An alternative technique for determining such separated estimation/control structure, after the design step, is also proposed. Several numerical examples, throughout the paper, demonstrate the effectiveness of the proposed algorithms.

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Section: Preliminariesmentioning

confidence: 99%

“…Even in the nonsingular case, descriptor systems are very useful to manipulate physical models without losing a physical parameterization or to describe controller/filter implementation. Strict LMI characterizations for admissibility, H ∞ and H 2 norms of descriptor systems are established in [2][3][4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

Static Output Feedback Control Design for Descriptor Systems

Yagoubi¹

2010

Journal of Control Science and Engineering

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“…Although descriptor equations are mathematically less tractable than state equations, many controller design methods for state-space models have been extended to descriptor systems. For linear time-invariant (LTI) descriptor systems, H ∞ controllers [1]- [4], mixed H 2 /H ∞ controllers [5], and general quadratic performance controllers [6], have been proposed. A common feature of these results is that descriptor-type controllers are dealt with, because they are mathematically compatible with descriptor systems.…”

Section: Introductionmentioning

confidence: 99%

State-Space Stabilizing Controllers for Descriptor Systems

Inoue

Wada

Ikeda

et al. 2012

SICE Journal of Control, Measurement, and System Integration

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: This paper considers stabilization of linear time-invariant descriptor systems by dynamic output feedback controllers. We deal with general descriptor systems including those being irregular or impulsive, and derive statespace stabilizing controllers. On the derivation process of the state-space controllers, we first consider descriptor-type controllers. We present a necessary and sufficient condition for the existence of a descriptor-type controller which makes the closed-loop descriptor system regular, impulse-free, and stable. The condition is expressed in terms of linear matrix inequalities (LMIs), and we show that coefficient matrices of any descriptor-type stabilizing controller of the same size as the given descriptor system can be represented by the solution of the LMIs. Then, we present a necessary and sufficient condition for the descriptor-type controller to be transformable to an input-output equivalent state-space controller with the dimension of the dynamic order (the rank of the coefficient matrix for the time-derivative of the descriptor variable) of the given descriptor system, that is, a state-space stabilizing controller. The transformability condition is mild and almost always satisfied by the obtained descriptor-type controller. Furthermore, even if the transformability condition is not satisfied, a slightly modified solution of the LMIs, which always exists, gives a descriptor-type controller being transformable to a state-space controller. The transformation is carried out analytically, thus the coefficient matrices of any such state-space stabilizing controller can be expressed by the solution of the LMIs. We also reveal that if we restrict the classes of descriptor systems or descriptor-type controllers such that their transfer functions are strictly proper, the descriptor-type controllers obtained by the LMI condition are always transformable to state-space controllers.

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“…For these reasons, many studies have investigated the generalized Lyapunov equation and stability of descriptor systems [1][2][3][4]. When the stability and control design problems of a descriptor system are considered, the standard Lyapunov equation is extended to the generalized Lyapunov equation [5][6][7][8][9]. Such descriptor systems naturally occur in many applications, such as multi-body dynamics, electrical circuit simulation, chemical engineering, and semi-discretization of partial differential equations [10][11][12].…”

Section: Introductiuonmentioning

confidence: 99%

Generalized Stability Condition for Descriptor Systems

Oh¹,

Lee²,

Kim³

2012

Journal of Institute of Control, Robotics and Systems

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In this paper, we propose a generalized index independent stability condition for a descriptor systemwithout any transformations of system matrices. First, the generalized Lyapunov equation with a specific right-handed matrix form is considered. Furthermore, the existence theorem and the necessary and sufficient conditions for asymptotically stable descriptor systems are presented. Finally, some suitable examples are used to show the validity of the proposed method.Keywords: generalized Lyapunov equation, descriptor systems, index independent stability condition I. INTRODUCTIUON The singular system model is a natural representation of dynamical systems and can better describe a large class of systems than state-space ones do. Studying the stability problem for singular systems is much more complicated than that for state space systems because it requires consideration not only of stability but also regularity absence of impulses for continuoustime singular systems and causality for discrete-time singular systems which may affect the stability of the system. For these reasons, many studies have investigated the generalized Lyapunov equation and stability of descriptor systems [1][2][3][4]. When the stability and control design problems of a descriptor system are considered, the standard Lyapunov equation is extended to the generalized Lyapunov equation [5][6][7][8][9]. Such descriptor systems naturally occur in many applications, such as multi-body dynamics, electrical circuit simulation, chemical engineering, and semi-discretization of partial differential equations [10][11][12]. The GALE (Generalized Algebraic Lyapunov Equation),has been considered for the stability analysis, wherein E, A, G are given matricesand X is an unknown matrix. Many numerical algorithms have been developed to solve the GALE with a nonsingular matrix E. However, little attention has been paid to the generalized Lyapunov equation for a singular matrix E ([1, [13][14][15][16][17] [14]. Unfortunately, these equations are limited to pencilswith an index of at most one [1,6,14,15]. Recently,a projected Lyapunov equation,the existence of a solution for stability therein, and its applications have been investigated [17,18]. This method is independent of anindex; however, the primary difficulty in applying these approaches [1,6,14,15,17,18] is that the Weierstrass canonical form [19] needs to be found before aa solution of the Lyapunov equation can be obtained. Therefore, the objective of this paper is to propose a generalized index independentstability condition for a descriptor system without transformation of the system matrices into a Weierstrass canonical form.In this paper, we propose the necessary and sufficient conditions for the asymptotic stability of descriptor systemswith any index, without any transformations. The generalized Lyapunov equation with a specific right-handed matrix form is discussed, and the existence of the solution is proved. In section 2, the descriptor system and the mathematical preliminaries are discus...

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H∞ control for descriptor systems: A matrix inequalities approach

Cited by 622 publications

References 16 publications

Static Output Feedback Control Design for Descriptor Systems

Static Output Feedback Control Design for Descriptor Systems

State-Space Stabilizing Controllers for Descriptor Systems

Generalized Stability Condition for Descriptor Systems

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