Robust stability and stabilization for singular systems with state delay and parameter uncertainty
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.
Citations
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TL;DR: A new necessary and sufficient condition is proposed in terms of strict linear matrix inequality (LMI), which guarantees the stochastic admissibility of the unforced Markovian jump singular system.
Abstract: This paper is concerned with the state estimation and sliding-mode control problems for continuous-time Markovian jump singular systems with unmeasured states. Firstly, a new necessary and sufficient condition is proposed in terms of strict linear matrix inequality (LMI), which guarantees the stochastic admissibility of the unforced Markovian jump singular system. Then, the sliding-mode control problem is considered by designing an integral sliding surface function. An observer is designed to estimate the system states, and a sliding-mode control scheme is synthesized for the reaching motion based on the state estimates. It is shown that the sliding mode in the estimation space can be attained in a finite time. Some conditions for the stochastic admissibility of the overall closed-loop system are derived. Finally, a numerical example is provided to illustrate the effectiveness of the proposed theory.
596 citations
Additional excerpts
..., [1], [2], [5], [8], [11], [26], [27], and [28] and references therein....
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TL;DR: This note provides an improved asymptotic stability condition for time-delay systems in terms of a strict linear matrix inequality that avoids bounding certain cross terms which often leads to conservatism.
Abstract: This note provides an improved asymptotic stability condition for time-delay systems in terms of a strict linear matrix inequality. Unlike previous methods, the mathematical development avoids bounding certain cross terms which often leads to conservatism. When time-varying norm-bounded uncertainties appear in a delay system, an improved robust delay-dependent stability condition is also given. Examples are provided to demonstrate the reduced conservatism of the proposed conditions.
536 citations
TL;DR: A new model based on the updating instants of the holder is formulated, and a linear matrix inequality (LMI)-based procedure is proposed for designing state-feedback controllers, which guarantee that the output of the closed-loop networked control system tracks theoutput of a given reference model well in the Hinfin sense.
Abstract: This paper is concerned with the problem of Hinfin output tracking for network-based control systems. The physical plant and the controller are, respectively, in continuous time and discrete time. By using a sampled-data approach, a new model based on the updating instants of the holder is formulated, and a linear matrix inequality (LMI)-based procedure is proposed for designing state-feedback controllers, which guarantee that the output of the closed-loop networked control system tracks the output of a given reference model well in the Hinfin sense. Both network-induced delays and data packet dropouts have been taken into consideration in the controller design. The network-induced delays are assumed to have both an upper bound and a lower bound, which is more general than those used in the literature. The introduction of the lower bound is shown to be advantageous for reducing conservatism. Moreover, the controller design method is further extended to more general cases, where the system matrices of the physical plant contain parameter uncertainties, represented in either polytopic or norm-bounded frameworks. Finally, an illustrative example is presented to show the usefulness and effectiveness of the proposed Hinfin output tracking design.
389 citations
TL;DR: The purpose of this article is to survey the recent results developed to analyse the asymptotic stability of time-delay systems and give special emphases to the issues of conservatism of the results and computational complexity.
Abstract: Recent years have witnessed a resurgence of research interests in analysing the stability of time-delay systems. Many results have been reported using a variety of approaches and techniques. However, much of the focus has been laid on the use of the Lyapunov-Krasovskii theory to derive sufficient stability conditions in the form of linear matrix inequalities. The purpose of this article is to survey the recent results developed to analyse the asymptotic stability of time-delay systems. Both delay-independent and delay-dependent results are reported in the article. Special emphases are given to the issues of conservatism of the results and computational complexity. Connections of certain delay-dependent stability results are also discussed.
385 citations
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
References
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01 Jan 1994
11,176 citations
"Robust stability and stabilization ..." refers methods in this paper
...The desired robustly stabilizing state feedback for uncertain singular system ( ) can be obtained by solving the strict LMI (38), which can be solved numerically very efficiently by using interior-point algorithm, and no tuning of parameters is involved [2]....
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Book•
01 Jan 1995
TL;DR: In this paper, the focus is on adaptive nonlinear control results introduced with the new recursive design methodology -adaptive backstepping, and basic tools for nonadaptive BackStepping design with state and output feedbacks.
Abstract: From the Publisher:
Using a pedagogical style along with detailed proofs and illustrative examples, this book opens a view to the largely unexplored area of nonlinear systems with uncertainties. The focus is on adaptive nonlinear control results introduced with the new recursive design methodology--adaptive backstepping. Describes basic tools for nonadaptive backstepping design with state and output feedbacks.
6,923 citations
Book•
05 Apr 1977
TL;DR: In this paper, Liapunov functional for autonomous systems is used to define the saddle point property near equilibrium and periodic orbits for linear systems, which is a generalization of the notion of stable D operators.
Abstract: 1 Linear differential difference equations.- 1.1 Differential and difference equations.- 1.2 Retarded differential difference equations.- 1.3 Exponential estimates of x(?, f).- 1.4 The characteristic equation.- 1.5 The fundamental solution.- 1.6 The variation-of-constants formula.- 1.7 Neutral differential difference equations.- 1.8 Supplementary remarks.- 2 Retarded functional differential equations : basic theory.- 2.1 Definition.- 2.2 Existence, uniqueness, and continuous dependence.- 2.3 Continuation of solutions.- 2.4 Differentiability of solutions.- 2.5 Backward continuation.- 2.6 Caratheodory conditions.- 2.7 Supplementary remarks.- 3 Properties of the solution map.- 3.1 Finite- or infinite-dimensional problem?.- 3.2 Equivalence classes of solutions.- 3.3 Exponential decrease for linear systems.- 3.4 Unique backward extensions.- 3.5 Range in ?n.- 3.6 Compactness and representation.- 3.7 Supplementary remarks.- 4 Autonomous and periodic processes.- 4.1 Processes.- 4.2 Invariance.- 4.3 Discrete systems-maximal compact invariant sets.- 4.4 Fixed points of discrete dissipative processes.- 4.5 Stability and maximal invariant sets in processes.- 4.6 Periodic trajectories of ?-periodic processes.- 4.7 Convergent systems.- 4.8 Supplementary remarks.- 5 Stability theory.- 5.1 Definitions.- 5.2 The method of Liapunov functional.- 5.3 Liapunov functional for autonomous systems.- 5.4 Razumikhin-type theorems.- 5.5 Supplementary remarks.- 6 General linear systems.- 6.1 Global existence and exponential estimates.- 6.2 Variation-of-constants formula.- 6.3 The formal adjoint equation.- 6.4 The true adjoint.- 6.5 Boundary-value problems.- 6.6 Stability and boundedness.- 6.7 Supplementary remarks.- 7 Linear autonomous equations.- 7.1 The semigroup and infinitesimal generator.- 7.2 Spectrum of the generator-decomposition of C.- 7.3 Decomposing C with the formal adjoint equation.- 7.4 Estimates on the complementary subspace.- 7.5 An example.- 7.6 The decomposition in the variation-of-constants formula.- 7.7 Supplementary remarks.- 8 Linear periodic systems.- 8.1 General theory.- 8.2 Decomposition.- 8.3 Supplementary remarks.- 9 Perturbed linear systems.- 9.1 Forced linear systems.- 9.2 Bounded, almost-periodic, and periodic solutions stable and unstable manifolds.- 9.3 Periodic solutions-critical cases.- 9.4 Averaging.- 9.5 Asymptotic behavior.- 9.6 Boundary-value problems.- 9.7 Supplementary remarks.- 10 Behavior near equilibrium and periodic orbits for autonomous equations.- 10.1 The saddle-point property near equilibrium.- 10.2 Nondegenerate periodic orbits.- 10.3 Hyperbolic periodic orbits.- 10.4 Supplementary remarks.- 11 Periodic solutions of autonomous equations.- 11.1 Hopf bifurcation.- 11.2 A periodicity theorem.- 11.3 Range of the period.- 11.4 The equation $$\dot x(t) = - \alpha x(t - 1)[1 + x(t)]$$.- 11.5 The equation $$\dot x(t) = - \alpha x(t - 1)[1 - {x^2}(t)]$$.- 11.6 The equation $$\ddot x(t) + f(x(t))\dot x(t) + g(x(t - r)) = 0$$.- 11.7 Supplementary remarks.- 12 Equations of neutral type.- 12.1 Definition of a neutral equation.- 12.2 Fundamental properties.- 12.3 Linear autonomous D operators.- 12.4 Stable D operators.- 12.5 Strongly stable D operators.- 12.6 Properties of equations with stable D operators.- 12.7 Stability theory.- 12.8 General linear equations.- 12.9 Stability of autonomous perturbed linear systems.- 12.10 Linear autonomous and periodic equations.- 12.11 Nonhomogeneous linear equations.- 12.12 Supplementary remarks.- 13 Global theory.- 13.1 Generic properties of retarded equations.- 13.2 The set of global solutions.- 13.3 Equations on manifolds : definitions.- 13.4 Retraded equations on compact manifolds.- 13.5 Further properties of the attractor.- 13.6 Supplementary remarks.- Appendix Stability of characteristic equations.
5,799 citations
"Robust stability and stabilization ..." refers background in this paper
...INTRODUCTION Control of delay systems has been a topic of recurring interest over the past decades since time delays are often the main causes for instability and poor performance of systems and encountered in various engineering systems such as chemical processes, long transmission lines in pneumatic systems, and so on [8]....
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...[8] X....
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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
"Robust stability and stabilization ..." refers background or methods in this paper
...To this end, we note that the regularity and the absence of impulses of the pair (E;A) implies that there exist two invertible matricesG andH 2 n n such that [4]...
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...Proof: Noting the regularity and the absence of impulses of the pair (E;A) and using the decomposition as in [4], the desired result follows immediately....
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...Singular systems are also referred to as descriptor systems, implicit systems, generalized statespace systems, differential-algebraic systems, or semistate systems [4], [11]....
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...E _ x(t) = Ax(t) +Adx(t ): (5) Definition 1: [4], [11]: 1) The pair(E;A) is said to be regular if det(sE A) is not identically zero....
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...A great number of results based on the theory of regular systems (or state-space systems) have been extended to the area of singular systems [4], [11]....
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TL;DR: In this paper, the authors present an algorithm for the stabilization of a class of uncertain linear systems, which is described by state equations which depend on time-varying unknown-but-bounded uncertain parameters.
1,483 citations
"Robust stability and stabilization ..." refers background in this paper
...Lemma 6 [14]: Given matrices , and of appropriate dimensions and with symmetrical, then + F ( ) + ( F ( ) )T < 0 for all F ( ) satisfying F ( )F ( ) I , if and only if there exists a scalar > 0 such that + T + 1 T < 0: For simplicity we introduce the matrix 2 n (n r) satisfying E = 0 and rank = n r....
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