Robust mixed H-2/H∞ control for a class of nonlinear stochastic systems

The problem of mixed H2/H∞ control is considered for a class of uncertain discrete-time nonlinear stochastic systems. The nonlinearities are described by statistical means of the stochastic variables and the uncertainties are represented by deterministic norm-bounded parameter perturbations. The mixed H2/H∞ control problem is formulated in terms of the notion of exponentially mean-square quadratic stability and the characterisations of both the H2 control performance and the H∞ robustness performance. A new technique is developed to deal with the matrix trace terms arising from the stochastic nonlinearities and the well-known S-procedure is adopted to handle the deterministic uncertainities. A unified framework is established to solve the addressed mixed H2/H∞ control problem using a linear matrix inequality approach. Within such a framework, two additional optimisation problems are discussed, one is to optimise the H∞ robustness performance, and the other is to optimise the H2 control performance. An illustrative example is provided to demonstrate the effectiveness of the proposed method.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Nuffield Foundation of the UK under Grant NAL/00630/G and the Alexander von Humboldt Foundation of Germany, the National Natural Science Foundation of China under Grant 60474049 and the Fujian provincial Natural Science Foundation of China under Grant A0410012

Robust H2 filtering for a class of systems with stochastic nonlinearities

Copyright [2006] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This paper addresses the robust H2 filtering problem for a class of uncertain discrete-time nonlinear stochastic systems. The nonlinearities described by statistical means in this paper comprise some well-studied classes of nonlinearities in the literature. A technique is developed to tackle the matrix trace terms resulting from the nonlinearities, and the well-known S-procedure technique is adopted to cope with the uncertainties. A unified framework is established to solve the addressed robust H2 filtering problem by using a linear matrix inequality approach. A numerical example is provided to illustrate the usefulness of the proposed method

Robust variance-constrained filtering for a class of nonlinear stochastic systems with missing measurements

The official published version of the article can be found at the link below.This paper is concerned with the robust filtering problem for a class of nonlinear stochastic systems with missing measurements and parameter uncertainties. The missing measurements are described by a binary switching sequence satisfying a conditional probability distribution, and the nonlinearities are expressed by the statistical means. The purpose of the filtering problem is to design a filter such that, for all admissible uncertainties and possible measurements missing, the dynamics of the filtering error is exponentially mean-square stable, and the individual steady-state error variance is not more than prescribed upper bound. A sufficient condition for the exponential mean-square stability of the filtering error system is first derived and an upper bound of the state estimation error variance is then obtained. In terms of certain linear matrix inequalities (LMIs), the solvability of the addressed problem is discussed and the explicit expression of the desired filters is also parameterized. Finally, a simulation example is provided to demonstrate the effectiveness and applicability of the proposed design approach.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Royal Society of the UK and the Alexander von Humboldt Foundation of Germany

Variance-constrained dissipative observer-based control for a class of nonlinear stochastic systems with degraded measurements

The official published version of the article can be obtained from the link below.This paper is concerned with the variance-constrained dissipative control problem for a class of stochastic nonlinear systems with multiple degraded measurements, where the degraded probability for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution over a given interval. The purpose of the problem is to design an observer-based controller such that, for all possible degraded measurements, the closed-loop system is exponentially mean-square stable and strictly dissipative, while the individual steady-state variance is not more than the pre-specified upper bound constraints. A general framework is established so that the required exponential mean-square stability, dissipativity as well as the variance constraints can be easily enforced. A sufficient condition is given for the solvability of the addressed multiobjective control problem, and the desired observer and controller gains are characterized in terms of the solution to a convex optimization problem that can be easily solved by using the semi-definite programming method. Finally, a numerical example is presented to show the effectiveness and applicability of the proposed algorithm.This work was supported in part by the Distinguished Visiting Fellowship of the Royal Academy of Engineering of the UK, the Royal Society of the UK, the GRF HKU 7137/09E, the National Natural Science Foundation of China under Grant 61028008, the International Science and Technology Cooperation Project of China under Grant 2009DFA32050, and the Alexander von Humboldt Foundation of Germany

Variance-constrained H∞ filtering for a class of nonlinear time-varying systems with multiple missing measurements: The finite-horizon case

Copyright [2010] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This paper is concerned with the robust H ∞ finite-horizon filtering problem for a class of uncertain nonlinear discrete time-varying stochastic systems with multiple missing measurements and error variance constraints. All the system parameters are time-varying and the uncertainty enters into the state matrix. The measurement missing phenomenon occurs in a random way, and the missing probability for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution in the interval . The stochastic nonlinearities under consideration here are described by statistical means which can cover several classes of well-studied nonlinearities. Sufficient conditions are derived for a finite-horizon filter to satisfy both the estimation error variance constraints and the prescribed H ∞ performance requirement. These conditions are expressed in terms of the feasibility of a series of recursive linear matrix inequalities (RLMIs). Simulation results demonstrate the effectiveness of the developed filter design scheme.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the U.K. by Grant GR/S27658/01, the Royal Society of the U.K., National Natural Science Foundation of China by Grants 60825303 and 60834003, National 973 Project of China by Grant 2009CB320600, Fok Ying Tung Education Foundation by Grant 111064, the Youth Science Fund of Heilongjiang Province of China by Grant QC2009C63, and by the Alexander von Humboldt Foundation of Germany

Analysis of Implicit Uncertain Systems. Part II: Constant Matrix Problems and Application to Robust H2 Analysis

This paper introduces an implicit framework for the analysis of uncertain systems, of which the general properties were described in Part I. In Part II, the theory is specialized to problems which admit a finite dimensional formulation. A constant matrix version of implicit analysis is presented, leading to a generalization of the structured singular value μ as the stability measure; upper bounds are developed and analyzed in detail. An application of this framework results in a practical method for robust H2 analysis: computing robust performance in the presence of norm-bounded perturbations and white-noise disturbances

Robust H∞ control of time-varying systems with stochastic non-linearities: the finite-horizon case

The official published version can be obtained from the link below.This paper is concerned with the robust H∞ control problem for the class of uncertain non-linear discrete time-varying stochastic systems with a covariance constraint. All the system parameters are time-varying and the uncertainties enter into the state matrix. The non-linearities under consideration are described by statistical means and they cover several classes of well-studied non-linearities. The purpose of the addressed problem is to design a dynamic output-feedback controller such that, the H∞ disturbance rejection attenuation level is achieved in the finite-horizon case while the state covariance is not more than an individual upper bound at each time point. An algorithm is developed to deal with the addressed problem by means of recursive linear matrix inequalities (RLMIs). It is shown that the robust H∞ control problem is solvable if the series of RLMIs is feasible. An illustrative simulation example is given to show the applicability and effectiveness of the proposed algorithm.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under grant GR/S27658/01, the Royal Society of the UK, and the Alexander von Humboldt Foundation of Germany

Tropical Kraus maps for optimal control of switched systems

Kraus maps (completely positive trace preserving maps) arise classically in quantum information, as they describe the evolution of noncommutative probability measures. We introduce tropical analogues of Kraus maps, obtained by replacing the addition of positive semidefinite matrices by a multivalued supremum with respect to the L\"owner order. We show that non-linear eigenvectors of tropical Kraus maps determine piecewise quadratic approximations of the value functions of switched optimal control problems. This leads to a new approximation method, which we illustrate by two applications: 1) approximating the joint spectral radius, 2) computing approximate solutions of Hamilton-Jacobi PDE arising from a class of switched linear quadratic problems studied previously by McEneaney. We report numerical experiments, indicating a major improvement in terms of scalability by comparison with earlier numerical schemes, owing to the "LMI-free" nature of our method.Comment: 15 page