A Riccati equation approach to the stabilization of uncertain linear systems
TL;DR: The fundamental idea behind the algorithm presented involves constructing an upper bound for the Lyapunov derivative corresponding to the closed loop system, a quadratic form, which can be found by solving a certain matrix Riccati equation.
About: This article is published in Automatica.The article was published on 1986-07-01. It has received 825 citations till now. The article focuses on the topics: Lyapunov equation & Lyapunov redesign.
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Book•
01 Jan 1994
TL;DR: In this paper, the authors present a brief history of LMIs in control theory and discuss some of the standard problems involved in LMIs, such as linear matrix inequalities, linear differential inequalities, and matrix problems with analytic solutions.
Abstract: Preface 1. Introduction Overview A Brief History of LMIs in Control Theory Notes on the Style of the Book Origin of the Book 2. Some Standard Problems Involving LMIs. Linear Matrix Inequalities Some Standard Problems Ellipsoid Algorithm Interior-Point Methods Strict and Nonstrict LMIs Miscellaneous Results on Matrix Inequalities Some LMI Problems with Analytic Solutions 3. Some Matrix Problems. Minimizing Condition Number by Scaling Minimizing Condition Number of a Positive-Definite Matrix Minimizing Norm by Scaling Rescaling a Matrix Positive-Definite Matrix Completion Problems Quadratic Approximation of a Polytopic Norm Ellipsoidal Approximation 4. Linear Differential Inclusions. Differential Inclusions Some Specific LDIs Nonlinear System Analysis via LDIs 5. Analysis of LDIs: State Properties. Quadratic Stability Invariant Ellipsoids 6. Analysis of LDIs: Input/Output Properties. Input-to-State Properties State-to-Output Properties Input-to-Output Properties 7. State-Feedback Synthesis for LDIs. Static State-Feedback Controllers State Properties Input-to-State Properties State-to-Output Properties Input-to-Output Properties Observer-Based Controllers for Nonlinear Systems 8. Lure and Multiplier Methods. Analysis of Lure Systems Integral Quadratic Constraints Multipliers for Systems with Unknown Parameters 9. Systems with Multiplicative Noise. Analysis of Systems with Multiplicative Noise State-Feedback Synthesis 10. Miscellaneous Problems. Optimization over an Affine Family of Linear Systems Analysis of Systems with LTI Perturbations Positive Orthant Stabilizability Linear Systems with Delays Interpolation Problems The Inverse Problem of Optimal Control System Realization Problems Multi-Criterion LQG Nonconvex Multi-Criterion Quadratic Problems Notation List of Acronyms Bibliography Index.
11,085 citations
Book•
26 Jun 2003
TL;DR: Preface, Notations 1.Introduction to Time-Delay Systems I.Robust Stability Analysis II.Input-output stability A.LMI and Quadratic Integral Inequalities Bibliography Index
Abstract: Preface, Notations 1.Introduction to Time-Delay Systems I.Frequency-Domain Approach 2.Systems with Commensurate Delays 3.Systems withIncommensurate Delays 4.Robust Stability Analysis II.Time Domain Approach 5.Systems with Single Delay 6.Robust Stability Analysis 7.Systems with Multiple and Distributed Delays III.Input-Output Approach 8.Input-output stability A.Matrix Facts B.LMI and Quadratic Integral Inequalities Bibliography Index
4,200 citations
TL;DR: An overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems is provided.
2,186 citations
Cites methods from "A Riccati equation approach to the ..."
...In later works, a Riccati equation approach has been proposed in Petersen and Hollot (1986), Rotea and Khargonekar (1989)....
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Book•
01 Aug 19941,655 citations
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
References
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TL;DR: In this article, a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state, and these feedback controls are continuous functions of the state.
Abstract: We consider a dynamic system containing uncertain elements. Only the set of possible values of these uncertainties is known. Based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state. These feedback controls are continuous functions of the state.
1,546 citations
01 Jan 1978
TL;DR: In this article, a new algorithm for solving algebraic Riccati equations (both continuous-time and discrete-time versions) is presented, which is a variant of the classical eigenvector approach and uses instead an appropriate set of Schur vectors.
Abstract: In this paper a new algorithm for solving algebraic Riccati equations (both continuous-time and discrete-time versions) is presented. The method studied is a variant of the classical eigenvector approach and uses instead an appropriate set of Schur vectors thereby gaining substantial numerical advantages. Complete proofs of the Schur approach are given as well as considerable discussion of numerical issues. The method is apparently quite numerically stable and performs reliably on systems with dense matrices up to order 100 or so, storage being the main limiting factor. The description given below is a considerably abridged version of a complete report given in [0].
1,002 citations
TL;DR: Guaranteed cost control is a method of synthesizing a closed-loop system in which the controlled plant has large parameter uncertainty as mentioned in this paper, and it can be incorporated into an adaptive system by either online measurement and evaluation or prior knowledge on the parametric dependence of a certain easily measured situation parameter.
Abstract: Guaranteed cost control is a method of synthesizing a closed-loop system in which the controlled plant has large parameter uncertainty This paper gives the basic theoretical development of guaranteed cost control, and shows how it can be incorporated into an adaptive system The uncertainty in system parameters is reduced first by either: 1) on-line measurement and evaluation, or 2) prior knowledge on the parametric dependence of a certain easily measured situation parameter Guaranteed cost control is then used to take up the residual uncertainty It is shown that the uncertainty in system parameters can be taken care of by an additional term in the Riccati equation A Fortran program for computing the guaranteed cost matrix and control law is developed and applied to an airframe control problem with large parameter variations
688 citations
TL;DR: In this article, a notion of quadratic stabilizability is defined and the Lyapunov function and the control are constructed using only the bounds ℛ,L.
Abstract: Consider an uncertain system (Σ) described by the equationx(t)=A(r(t))x(t)+B(s(t))u(t), wherex(t) ∈R
n
is the state,u(t) ∈R
m
is the control,r(t) ∈ ℛ ⊂R
p
represents the model parameter uncertainty, ands(t) ∈L ⊂R
l
represents the input connection parameter uncertainty. The matrix functionsA(·),B(·) are assumed to be continuous and the restraint sets ℛ,L are assumed to be compact. Within this framework, a notion of quadratic stabilizability is defined. It is important to note that this type of stabilization is robust in the following sense: The Lyapunov function and the control are constructed using only the bounds ℛ,L. Much of the previous literature has concentrated on a fundamental question: Under what conditions onA(·),B(·), ℛ,L can quadratic stabilizability be assured? In dealing with this question, previous authors have shown that, if (Σ) satisfies certain matching conditions, then quadratic stabilizability is indeed assured (e.g., Refs. 1–2). Given the fact that matching is only a sufficient condition for quadratic stabilizability, the objective here is to characterize the class of systems for which quadratic stabilizability can be guaranteed.
649 citations
TL;DR: In this paper, a stabilizing controller for a class of uncertain dynamical systems is proposed, where the controller is shown to render the closed loop system "practically stable" in a so-called guaranteed sense.
Abstract: This paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems. The vector of uncertain parameters $q( \cdot )$ is time-varying, and its values $q(t)$ lie within a prespecified bounding set Q in $R^p $. Furthermore, no statistical description of $q( \cdot )$ is assumed, and the controller is shown to render the closed loop system “practically stable” in a so-called guaranteed sense; that is, the desired stability properties are assured no matter what admissible uncertainty $q( \cdot )$ is realized. Within the perspective of previous research in this area, this paper contains one salient feature: the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time-invariant. In contrast, in much of the previous literature (see, for example, [1], [2], [7], and [9]), a linear system is stabilized via nonlinear control. Another feature of this paper is the fact that the...
551 citations