Convex Duality Approach to Robust Stabilization of Uncertain Plants

In this thesis we are study the problem of designing the controllers that are robust with respect to the parametric uncertainty. In Part I "The Rank-One Problem" we consider the class of systems with restriction that the structure of uncertainty is limited to a vector. In Chapter " Canonical Parametrization of the Dual Problem in Robust Optimization: Non-Rational Case" we extend the class of allowed systems. The main result is the canonical parametrization of all destabilizing uncertainties in the dual problem. The corresponding result in the rational case was previously stated in terms of unstable zero-pole cancellations. For non-rational systems the situation with common zeros is more complicated. The nominal factors can contain a singular component and cannot be treated by unstable cancellations. We have shown that in the general case the common zeros of the plant factors are naturally replaced by a scalar function with the positive winding number. To illustrate the duality principle, the result is applied to a system with delay. By dual parametrization we can easily calculate the optimal uncertainty bound and the optimal controller. Since the optimal controller is not robustly stabilizing in the strong sense,as it is only a limit of suboptimal robustly stabilizing controllers,we have to regularize the limiting controller. In Chapter "Regularization of the Limiting Optimal Controller in Robust Stabilization" we present a method of obtaining the suboptimal controller of lower order that provides the stability margin as close to the optimal one as we wish. The method is illustrated with some scalar examples. In Chapter "Robust Control via Linear Programming" we propose the numerical algorithm for the optimal robust control synthesis. The algorithm proposed is a sequence of the standard linear programming problems of growing dimensions which approximate the initial problem. In the special case, when the uncertainty parameter is real-valued, it is shown that the initial problem can be considered as finite-dimensional in the space of variables. In Part II "Convex Duality: Matrix Case" we generalize the results to the system with matrix uncertainties. We obtain a canonical factorization of a plant with unstructured uncertainty in terms of an unitary matrix function with finite winding number and an outer matrix function. We introduce a metric in the space of factorization and discuss connection with nu-gap metric

Sliding mode adaptive state observation for time-delay uncertain nonlinear systems

In this paper a method to design robust adaptive sliding mode observers (ASMO) for a class of nonlinear time- delay systems with uncertainties, is proposed. The objective is to achieve insensitivity and robustness of the proposed sliding mode observer to matched disturbances. A novel systematic design method is synthesized to solve matching conditions and compute observer stabilizing gains. The Lyapunov-Krasovskii theorem is employed to prove the ultimate stability with arbitrary boundedness radius of the estimation error of the proposed filter. Finally, the ability of ASMO for fault reconstruction is studied

SMC design for robust H∞ control of uncertain stochastic delay systems

Recently, sliding mode control method has been extended to accommodate stochastic systems. However, the existing results employ an assumption that may be too restrictive for many stochastic systems. This paper aims to remove this assumption and present in terms of LMIs a sliding mode control design method for stochastic systems with state delay. In some cases, the proposed method provides a control scheme for finite-time stabilization of stochastic delay systems

Stability analysis and observer design for neutral delay systems

Copyright [2002] 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 deals with the observer design problem for a class of linear delay systems of the neutral-type. The problem addressed is that of designing a full-order observer that guarantees the exponential stability of the error dynamic system. An effective algebraic matrix equation approach is developed to solve this problem. In particular, both the observer analysis and design problems are investigated. By using the singular value decomposition technique and the generalized inverse theory, sufficient conditions for a neutral-type delay system to be exponentially stable are first established. Then, an explicit expression of the desired observers is derived in terms of some free parameters. Furthermore, an illustrative example is used to demonstrate the validity of the proposed design procedur

Robust filtering for a class of stochastic uncertain nonlinear time-delay systems via exponential state estimation

Copyright [2001] 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.We investigate the robust filter design problem for a class of nonlinear time-delay stochastic systems. The system under study involves stochastics, unknown state time-delay, parameter uncertainties, and unknown nonlinear disturbances, which are all often encountered in practice and the sources of instability. The aim of this problem is to design a linear, delayless, uncertainty-independent state estimator such that for all admissible uncertainties as well as nonlinear disturbances, the dynamics of the estimation error is stochastically exponentially stable in the mean square, independent of the time delay. Sufficient conditions are proposed to guarantee the existence of desired robust exponential filters, which are derived in terms of the solutions to algebraic Riccati inequalities. The developed theory is illustrated by numerical simulatio