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Journal ArticleDOI

Robust control analysis and design for discrete-time singular systems

01 Nov 1994-Automatica (Pergamon Press, Inc.)-Vol. 30, Iss: 11, pp 1741-1750
TL;DR: A simple approach to analyse stability robustness of discrete-time singular systems under structured perturbations is proposed and the developed robustness criteria are then applied to solve robust regional pole-assignment problems of singular systems.
About: This article is published in Automatica.The article was published on 1994-11-01. It has received 103 citations till now. The article focuses on the topics: Robustness (computer science) & Robust control.
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 "Robust control analysis and design ..."

  • ...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

Journal ArticleDOI
TL;DR: In this paper, robust D-stability analysis for uncertain discrete singular systems with state delay and structured uncertainties is investigated and sufficient conditions are developed to ensure that, when the nominal discrete singular delay system is regular, causal and all its finite poles are located within a specified disk, the uncertain system still preserves all these properties when structured uncertainties are added into the nominal system.
Abstract: This work investigates the problem of robust D-stability analysis for uncertain discrete singular systems with state delay and structured uncertainties. Sufficient conditions are developed to ensure that, when the nominal discrete singular delay system is regular, causal and all its finite poles are located within a specified disk, the uncertain system still preserves all these properties when structured uncertainties are added into the nominal system. A computationally simple approach is proposed and a numerical example is given to demonstrate the application of the proposed method.

105 citations


Cites background or methods from "Robust control analysis and design ..."

  • ...is a known positive integer time delay of the system, and are highly structured matrices representing time-invariant parameter uncertainties, and are assumed to have the following properties [ 7 ], [12]:...

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  • ...Both results in [6] and [ 7 ] can be viewed as extensions of those for state‐space systems....

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  • ...When parametric uncertainties appear, the problems of robust D-stability analysis and robust D-pole placement for discrete-time singular systems have been investigated in [ 7 ], and sufficient conditions have been obtained....

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  • ...It is worth pointing out that the robust D-stability problem for singular systems is much more complicated than that for state‐space systems because it requires to consider not only D-stability robustness, but also regularity and impulse immunity (for continuous singular systems) and causality (for discrete singular systems) simultaneously [6], [ 7 ], while for state‐space systems, the latter two issues do not arise....

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Journal ArticleDOI
01 Jul 1999
TL;DR: In this paper, a generalised Lyapunov inequality is described which can be reformulated as nonstrict linear matrix inequalities (LMIs), for checking the regularity, impulse immunity, and stability of discrete-time descriptor systems simultaneously.
Abstract: A generalised Lyapunov inequality is described which can be reformulated as nonstrict linear matrix inequalities (LMIs), for checking the regularity, impulse immunity, and stability of discrete-time descriptor systems simultaneously. Based on this inequality, a bounded real lemma in nonstrict LMIs is obtained which characterises properties of such descriptor systems, including regularity, impulse-free property, stability, and H/sub /spl infin// norm bound condition. The proofs are purely algebraic, therefore they are simple and definite. These results could play a key role in the LMI-based H/sub /spl infin// controller design for discrete-time descriptor systems.

100 citations

Journal ArticleDOI
TL;DR: The problem is the design of state feedback controllers such that the resulting closed-loop system is regular, causal as well as stable for all admissible uncertainties, and the concept of ''generalized quadratic stabilizability'' is defined.

78 citations

References
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Book
01 Jun 1984
TL;DR: In this article, the Routh-Hurwitz problem of singular pencils of matrices has been studied in the context of systems of linear differential equations with variable coefficients, and its applications to the analysis of complex matrices have been discussed.
Abstract: Volume 2: XI. Complex symmetric, skew-symmetric, and orthogonal matrices: 1. Some formulas for complex orthogonal and unitary matrices 2. Polar decomposition of a complex matrix 3. The normal form of a complex symmetric matrix 4. The normal form of a complex skew-symmetric matrix 5. The normal form of a complex orthogonal matrix XII. Singular pencils of matrices: 1. Introduction 2. Regular pencils of matrices 3. Singular pencils. The reduction theorem 4. The canonical form of a singular pencil of matrices 5. The minimal indices of a pencil. Criterion for strong equivalence of pencils 6. Singular pencils of quadratic forms 7. Application to differential equations XIII. Matrices with non-negative elements: 1. General properties 2. Spectral properties of irreducible non-negative matrices 3. Reducible matrices 4. The normal form of a reducible matrix 5. Primitive and imprimitive matrices 6. Stochastic matrices 7. Limiting probabilities for a homogeneous Markov chain with a finite number of states 8. Totally non-negative matrices 9. Oscillatory matrices XIV. Applications of the theory of matrices to the investigation of systems of linear differential equations: 1. Systems of linear differential equations with variable coefficients. General concepts 2. Lyapunov transformations 3. Reducible systems 4. The canonical form of a reducible system. Erugin's theorem 5. The matricant 6. The multiplicative integral. The infinitesimal calculus of Volterra 7. Differential systems in a complex domain. General properties 8. The multiplicative integral in a complex domain 9. Isolated singular points 10. Regular singularities 11. Reducible analytic systems 12. Analytic functions of several matrices and their application to the investigation of differential systems. The papers of Lappo-Danilevskii XV. The problem of Routh-Hurwitz and related questions: 1. Introduction 2. Cauchy indices 3. Routh's algorithm 4. The singular case. Examples 5. Lyapunov's theorem 6. The theorem of Routh-Hurwitz 7. Orlando's formula 8. Singular cases in the Routh-Hurwitz theorem 9. The method of quadratic forms. Determination of the number of distinct real roots of a polynomial 10. Infinite Hankel matrices of finite rank 11. Determination of the index of an arbitrary rational fraction by the coefficients of numerator and denominator 12. Another proof of the Routh-Hurwitz theorem 13. Some supplements to the Routh-Hurwitz theorem. Stability criterion of Lienard and Chipart 14. Some properties of Hurwitz polynomials. Stieltjes' theorem. Representation of Hurwitz polynomials by continued fractions 15. Domain of stability. Markov parameters 16. Connection with the problem of moments 17. Theorems of Markov and Chebyshev 18. The generalized Routh-Hurwitz problem Bibliography Index.

9,334 citations

Book
01 Jan 1969

6,650 citations

Book
07 Jun 1995
TL;DR: Striking a balance between theory and applications, Linear System Theory and Design, 3/e, is ideal for use in advanced undergraduate/first-year graduate courses in linear systems and multivariable system design in electrical, mechanical, chemical, and aeronautical engineering departments.
Abstract: From the Publisher: An extensive revision of the author's highly successful text, this third edition of Linear System Theory and Design has been made more accessible to students from all related backgrounds. After introducing the fundamental properties of linear systems, the text discusses design using state equations and transfer functions. The two main objectives of the text are to: use simple and efficient methods to develop results and design procedures; enable students to employ the results to carry out design. Striking a balance between theory and applications, Linear System Theory and Design, 3/e, is ideal for use in advanced undergraduate/first-year graduate courses in linear systems and multivariable system design in electrical, mechanical, chemical, and aeronautical engineering departments. It assumes a working knowledge of linear algebra and the Laplace transform and an elementary knowledge of differential equations.

4,017 citations


Additional excerpts

  • ...(Chen, 1984, Theorem 3-4)....

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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


"Robust control analysis and design ..." refers background or methods in this paper

  • ...Sometimes the system is called generalized state-space systems, or implicit systems, or descriptor systems or semistate systems (Lewis, 1986)....

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  • ...The system Ex(k + 1) = Ax(k) is said to be asymptotically stable if and only if (Lewis, 1986):...

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  • ...INTRODUCTION In the past ten years, there has been a growing interest in the system-theoretic problems of singular systems due to the extensive applications of singular systems to large-scale systems, circuits, economics, polynomial matrices and other areas (Luenberger, 1977; Verghese et al., 1981; Lewis, 1986; Dai, 1989; Fang and Chang, 1991, 1992)....

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  • ...The system Ex(k + 1) = Ax(k) is said to be asymptotically stable if and only if (Lewis, 1986): all roots of IzE - AI = 0 lie inside the disk D(0, 1)....

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  • ...By the Weierstrass decomposition (Dai, 1989; Lewis, 1986), we propose a simple method to evaluate the matrix T without performing the expansion of ( z E -...

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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


"Robust control analysis and design ..." refers background in this paper

  • ...…growing interest in the system-theoretic problems of singular systems due to the extensive applications of singular systems to large-scale systems, circuits, economics, polynomial matrices and other areas (Luenberger, 1977; Verghese et al., 1981; Lewis, 1986; Dai, 1989; Fang and Chang, 1991, 1992)....

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  • ...We only assume the triple (E, A, B) is strongly controllable (Verghese et al., 1981)....

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  • ...INTRODUCTION In the past ten years, there has been a growing interest in the system-theoretic problems of singular systems due to the extensive applications of singular systems to large-scale systems, circuits, economics, polynomial matrices and other areas (Luenberger, 1977; Verghese et al., 1981; Lewis, 1986; Dai, 1989; Fang and Chang, 1991, 1992)....

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  • ...It has been known that a singular system generally contains three kinds of modes: dynamical finite modes, dynamical infinite modes, and nondynamical infinite modes (Verghese et al., 1981; Bender and Laub, 1987)....

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