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Journal ArticleDOI

Analysis of stability robustness for generalized state-space systems with structured perturbations

01 Aug 1993-Systems & Control Letters (North-Holland)-Vol. 21, Iss: 2, pp 109-114
TL;DR: In this article, a new approach is proposed to analyze the stability robustness of generalized state-space systems with structured perturbations, which is computationally simple to use and can easily be calculated by computer.
About: This article is published in Systems & Control Letters.The article was published on 1993-08-01. It has received 108 citations till now. The article focuses on the topics: Robustness (computer science).
Citations
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Journal ArticleDOI
TL;DR: A strict linear matrix inequality (LMI) design approach is developed that solves the problems of robust stability and stabilization for uncertain continuous singular systems with state delay via the notions of generalized quadratic stability and generalizedquadratic stabilization.
Abstract: Considers the problems of robust stability and stabilization for uncertain continuous singular systems with state delay. The parametric uncertainty is assumed to be norm bounded. The purpose of the robust stability problem is to give conditions such that the uncertain singular system is regular, impulse free, and stable for all admissible uncertainties, while the purpose of the robust stabilization is to design a state feedback control law such that the resulting closed-loop system is robustly stable. These problems are solved via the notions of generalized quadratic stability and generalized quadratic stabilization, respectively. Necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilization are derived. A strict linear matrix inequality (LMI) design approach is developed. An explicit expression for the desired robust state feedback control law is also given. Finally, a numerical example is provided to demonstrate the application of the proposed method.

759 citations


Cites background from "Analysis of stability robustness fo..."

  • ...It should be pointed out that the robust stability problem for singular systems is much more complicated than that for regular systems because it requires to consider not only stability robustness, but also regularity and absence of impulses (for continuous singular systems) and causality (for discrete-singular systems) at the same time [6], [7], and the latter two need not be considered in regular systems....

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Journal ArticleDOI
TL;DR: The problems of robust stability and robust stabilization are solved with a new necessary and sufficient condition for a discrete-time singular system to be regular, causal and stable in terms of a strict linear matrix inequality (LMI).
Abstract: This note deals with the problems of robust stability and stabilization for uncertain discrete-time singular systems. The parameter uncertainties are assumed to be time-invariant and norm-bounded appearing in both the state and input matrices. A new necessary and sufficient condition for a discrete-time singular system to be regular, causal and stable is proposed in terms of a strict linear matrix inequality (LMI). Based on this, the concepts of generalized quadratic stability and generalized quadratic stabilization for uncertain discrete-time singular systems are introduced. Necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilization are obtained in terms of a strict LMI and a set of matrix inequalities, respectively. With these conditions, the problems of robust stability and robust stabilization are solved. An explicit expression of a desired state feedback controller is also given, which involves no matrix decomposition. Finally, an illustrative example is provided to demonstrate the applicability of the proposed approach.

324 citations

01 Jan 2010
TL;DR: In this paper, a sufficient condition for the existence of a fixed-gain controller is first proposed which guarantees the regularity, impulse-free and stability of the closed-loop system under all possible faults.
Abstract: In this paper, the problem of robust fault tolerant control for a class of singular systems subject to both time-varying state-dependent nonlinear perturbation and actuator saturation is investigated. A sufficient condition for the existence of a fixed-gain controller is first proposed which guarantees the regularity, impulse-free and stability of the closed-loop system under all possible faults. An optimization problem with LMI constraints is formulated to determine the largest contractively invariant ellipsoid. An adaptive fault tolerant controller is then developed to compensate for the failure effects on the system by estimating the fault and updating the design parameter matrices online. Both of these two controllers are in the form of a saturation avoidance feedback with the advantage of relatively small actuator capacities compared with the high gain counterpart. An example is included to illustrate the proposed procedures and their effectiveness.

238 citations

Journal ArticleDOI
TL;DR: The problem of robust fault tolerant control for a class of singular systems subject to both time-varying state-dependent nonlinear perturbation and actuator saturation is investigated and a sufficient condition for the existence of a fixed-gain controller is proposed.

228 citations

Journal ArticleDOI
TL;DR: A sufficient condition for the existence and uniqueness of solution to the singular system is presented and a necessary and sufficient condition is presented in terms of linear matrix inequality, under which the maximal perturbation bound is obtained to guarantee the generalized quadratic stability of the system.
Abstract: This note considers the generalized quadratic stability problem for continuous-time singular system with nonlinear perturbation. The perturbation is a function of time and system state and satisfies a Lipschitz constraint. In this work, a sufficient condition for the existence and uniqueness of solution to the singular system is firstly presented. Then by using S-procedure and matrix inequality approach, a necessary and sufficient condition is presented in terms of linear matrix inequality, under which the maximal perturbation bound is obtained to guarantee the generalized quadratic stability of the system. That is, the system remains exponential stable and the nominal system is regular and impulse free. Furthermore, robust stability for nonsingular systems with perturbation can be obtained as a special case. Finally, the effectiveness of the developed approach for both singular and nonsingular systems is illustrated by numerical examples.

125 citations

References
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Book
01 Jan 1978
TL;DR: This report contains a description of the typical topics covered in a two-semester sequence in Numerical Analysis, and describes the accuracy, efficiency and robustness of these algorithms.
Abstract: Introduction. Mathematical approximations have been used since ancient times to estimate solutions, but with the rise of digital computing the field of numerical analysis has become a discipline in its own right. Numerical analysts develop and study algorithms that provide approximate solutions to various types of numerical problems, and they analyze the accuracy, efficiency and robustness of these algorithms. As technology becomes ever more essential for the study of mathematics, learning algorithms that provide approximate solutions to mathematical problems and understanding the accuracy of such approximations becomes increasingly important. This report contains a description of the typical topics covered in a two-semester sequence in Numerical Analysis.

7,315 citations

Book
01 Apr 1989
TL;DR: This paper presents a meta-analyses of linear singular systems through system analysis via transfer matrix and feedback control of dynamic compensation for singular systems.
Abstract: Solutions of linear singular systems- Time domain analysis- Feedback control- State observation- Dynamic compensation for singular systems- Structurally stable compensation in singular systems- System analysis via transfer matrix- to discrete-time singular systems- Optimal control- Some further topics

3,020 citations


"Analysis of stability robustness fo..." refers background or methods in this paper

  • ...In recent years, there has been a growing interest in the system-theoretic problems of generalized state-space systems (or singular systems, or descriptor systems) due to the extensive applications of generalized state-space systems in large-scale systems, circuits, economics, control theory, and other areas [5,9, 10, 17]....

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  • ...We propose a simple method to evaluate G(t) by the Weierstrass decomposition [5, 17]....

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Journal ArticleDOI
TL;DR: In this paper, a brief historical review of linear singular systems is presented, followed by a survey of results on their solution and properties, and the frequency domain and time domain approaches are discussed together to sketch an overall picture of the current status of the theory.
Abstract: This paper is a brief historical review of linear singular systems, followed by a survey of results on their solution and properties. The frequency domain and time domain approaches are discussed together to sketch an overall picture of the current status of the theory.

1,315 citations

Journal ArticleDOI
TL;DR: In this article, a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed, and concepts of controllability and observability are defined for the impulsive modes.
Abstract: Systems of the form E\dot{x}=Ax + Bu, y=Cx , with E singular, are studied. Of particular interest are the impulsive modes that may appear in the free-response of such systems when arbitrary initial conditions are permitted, modes that are associated with natural system frequencies at infinity. A generalized definition of system order that incorporates these impulsive degrees of freedom is proposed, and concepts of controllability and observability are defined for the impulsive modes. Allowable equivalence transformations of such singular systems are specified. The present framework is shown to overcome several difficulties inherent in other treatments of singular systems, and to extend, in a natural and satisfying way, many results previously known only for regular state-space systems.

1,042 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied a general form of sets of equations that is often the product of problem formulation in large-scale systems, especially when the equations are expressed in terms of the natural describing variables of the system.
Abstract: This paper studies a general form of sets of equations that is often the product of problem formulation in large-scale systems, especially when the equations are expressed in terms of the natural describing variables of the system. Such equations represent a broad class of time-evolutionary phenomena, and include as special cases ordinary static equations of arbitrary dimension, ordinary state-space equations, combinations of static and dynamic equations, and noncausal systems. The main thrust of the paper is to show (for sets of linear equations) that familiar concepts of dynamic system theory can be extended to this more general class, although sometimes with significant modification. Two new (and essentially dual) concepts, that of solvable and conditionable sets of equations, are found to be fundamental to the study of equations of this form. The notion of initial conditions, although not directly related to a state, is used as a general solution method for equations of this type. In addition a set of necessary and sufficient conditions for a set of dynamic equations to contain an embedded state-space representation is derived.

752 citations