H/sub infinity / control and quadratic stabilization of systems with parameter uncertainty via output feedback
TL;DR: In this paper, the robust H/sub infinity / control problem of designing a linear dynamic output feedback controller such that the closed-loop system is quadratically stable and achieves a prescribed level of disturbance attenuation for all admissible parameter uncertainties is considered.
Abstract: The article concerns linear systems which are subject to both time-varying norm-bounded parameter uncertainty and exogenous disturbance It addresses the robust H/sub infinity / control problem of designing a linear dynamic output feedback controller such that the closed-loop system is quadratically stable and achieves a prescribed level of disturbance attenuation for all admissible parameter uncertainties It is shown that such a problem is equivalent to a scaled H/sub infinity / control problem >
Citations
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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
TL;DR: In this paper, the authors dealt with H ∞ control problem for systems with parametric uncertainty in all matrices of the system and output equations and derived necessary and sufficient conditions for quadratic stability with disturbance attenuation.
Abstract: This paper deals with H ∞ control problem for systems with parametric uncertainty in all matrices of the system and output equations. The parametric uncertainty under consideration is of a linear fractional form. Both the continuous and the discrete-time cases are considered. Necessary and sufficient conditions for quadratic stability with H ∞ disturbance attenuation are obtained.
1,557 citations
TL;DR: In this article, the robust control of a class of nonlinear systems with real-time-varying parameter uncertainty is considered and a technique is proposed for designing stabilizing controllers for both problems by converting them into scaled H∞ control problems which do not involve parameter uncertainty.
1,434 citations
20 Dec 2004
TL;DR: In this paper, the output-feedback stabilisation problem is solved for discrete-time systems with time-varying delay in the state, and a stability condition is first proposed, which is dependent on the minimum and maximum delay bounds.
Abstract: The output-feedback stabilisation problem is solved for discrete-time systems with time-varying delay in the state. A stability condition is first proposed, which is dependent on the minimum and maximum delay bounds. Based on this easily verifiable stability condition, the problems of stabilisation by static and dynamic output-feedback controllers are solved within the linear matrix inequality (LMI) framework. Since the obtained conditions for the existence of admissible controllers are not expressed as strict LMI conditions, the cone complementary linearisation procedure is exploited to solve the nonconvex feasibility problem. In addition, the obtained results, including stability analysis, static output-feedback stabilisation and dynamic output-feedback stabilisation are further extended to discrete time-delay systems with norm-bounded uncertain parameters. Numerical examples are also presented to illustrate the applicability of the developed results.
419 citations
TL;DR: In this article, the robust H/sub /spl infin// filtering analysis and synthesis of nonlinear stochastic systems with state and exogenous disturbance-dependent noise is described.
Abstract: This paper describes the robust H/sub /spl infin// filtering analysis and synthesis of nonlinear stochastic systems with state and exogenous disturbance-dependent noise. We assume that the state and measurement are corrupted by stochastic uncertain exogenous disturbance and that the system dynamic is modeled by Ito/spl circ/-type stochastic differential equations. For general nonlinear stochastic systems, the H/sub /spl infin// filter can be obtained by solving second-order nonlinear Hamilton-Jacobi inequalities. When the worst-case disturbance is considered in the design procedure, a mixed H/sub 2//H/sub /spl infin// filtering problem is also solved by minimizing the total estimation error energy. It is found that for a class of special nonlinear stochastic systems, the H/sub /spl infin// filtering design can be given via solving several linear matrix inequalities instead of Hamilton-Jacobi inequalities. A few examples show that the proposed methods are effective.
307 citations
References
More filters
TL;DR: In this article, simple state-space formulas are derived for all controllers solving the following standard H/sub infinity / problem: for a given number gamma > 0, find all controllers such that the H/ sub infinity / norm of the closed-loop transfer function is (strictly) less than gamma.
Abstract: Simple state-space formulas are derived for all controllers solving the following standard H/sub infinity / problem: For a given number gamma >0, find all controllers such that the H/sub infinity / norm of the closed-loop transfer function is (strictly) less than gamma . It is known that a controller exists if and only if the unique stabilizing solutions to two algebraic Riccati equations are positive definite and the spectral radius of their product is less than gamma /sup 2/. Under these conditions, a parameterization of all controllers solving the problem is given as a linear fractional transformation (LFT) on a contractive, stable, free parameter. The state dimension of the coefficient matrix for the LFT, constructed using the two Riccati solutions, equals that of the plant and has a separation structure reminiscent of classical LQG (i.e. H/sub 2/) theory. This paper is intended to be of tutorial value, so a standard H/sub 2/ solution is developed in parallel. >
5,272 citations
TL;DR: In this article, the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problems of stabilization, and the feedback schemes obtainable from a given plant are parameterized.
Abstract: In this paper, the problem of sensitivity, reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization. Stable feedback schemes obtainable from a given plant are parameterized. Salient properties of sensitivity reducing schemes are derived, and it is shown that plant uncertainty reduces the ability, of feedback to reduce sensitivity. The theory is developed for input-output systems in a general setting of Banach algebras, and then specialized to a class of multivariable, time-invariant systems characterized by n \times n matrices of H^{\infty} frequency response functions, either with or without zeros in the right half-plane. The approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity, and on the concept of an approximate inverse. Approximate invertibility, of the plant is shown to be a necessary and sufficient condition for sensitivity reduction. An indicator of approximate invertibility, called a measure of singularity, is introduced. The measure of singularity of a linear time-invariant plant is shown to be determined by the location of its right half-plane zeros. In the absence of plant uncertainty, the sensitivity, to output disturbances can be reduced to an optimal value approaching the singularity, measure. In particular, if there are no right half-plane zeros, sensitivity can be made arbitrarily small. The feedback schemes used in the optimization of sensitivity resemble the lead-lag networks of classical control design. Some of their properties, and methods of constructing them in special cases are presented.
2,203 citations
DOI•
01 Nov 1982TL;DR: In this article, a general approach for analysing linear systems with structured uncertainty based on a new generalised spectral theory for matrices is introduced, which naturally extend techniques based on singular values and eliminate their most serious difficulties.
Abstract: The paper introduces a general approach for analysing linear systems with structured uncertainty based on a new generalised spectral theory for matrices. The results of the paper naturally extend techniques based on singular values and eliminate their most serious difficulties.
1,987 citations
Book•
03 Dec 1986
TL;DR: In this paper, the standard problem and performance bounds of model-matching theory are discussed. But the performance bounds are not defined. And they are not considered in this paper.
Abstract: Background mathematics: Function spaces.- The standard problem.- Stability theory.- Background mathematics: Operators.- Model-matching theory: Part I.- Factorization theory.- Model-matching theory: Part II.- Performance bounds.
1,527 citations
TL;DR: In this paper, the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time-domain results on the quadratic stabilization of uncertain systems and the frequency domain results on H/sup infinity / optimization.
Abstract: The problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time-domain results on the quadratic stabilization of uncertain systems and the frequency-domain results on H/sup infinity / optimization. A complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given. Relations between these robust stabilization problems and H/sup infinity / control theory are explored. It is also shown that in a number of cases, if a robust stabilization problem can be solved via Lyapunov methods, then it can be also be solved via H/sup infinity / control theory-based methods. >
1,464 citations