Hsi-Han Yeh

Robust Control for Linear Systems with Structured State Space Uncertainty

Abstract A recent sufficient condition for the stability robustness of linear systems with time-varying norm bounded state space uncertainty is extended to include the structure of the uncertainty. Our new result requires that the nominal eigenvalues lie to the left of a vertical line in the complex plane which is determined by a norm involving the structure of the uncertainty and the nominal closed-loop eigenvector matrix. Therefore, this robustness result is especially well suited to the design of control systems using eigenstructure assignment. When the uncertainty is time-invariant, our norm is also an upper bound on the incremental eigenvalue perturbations. We also consider the use of Perron weightings to reduce conservatism and the extension of the results to discrete time systems.

Necessary and sufficient conditions for mixed H/sub 2/ and H/sub infinity / optimal control

Mixed H/sub 2/ and H/sub infinity / optimal control problems are addressed. D.S. Bernstein and W.M. Haddad (1989) considered the case of one exogenous input and two observed outputs. Using a Lagrange multiplier technique, and under the assumption that the order of the controller is specified, they derived a necessary condition for minimizing an upper bound of the H/sub 2/ norm of one transfer matrix, subject to an H/sub infinity / norm constraint on the other. J.C. Doyle et al. (1989) later derived a sufficient condition for minimizing what may be shown to be the dual version of the upper bound defined by Bernstein and Haddad (BH) for the mixed H/sub 2/ and H/sub infinity / optimal control problem. In contrast with the BH system, the system of Doyle et al. (DZB) has two exogenous inputs and one measured output. The conditions are derived under different algebraic frameworks. The results are presented of a study that attempts to unify the two mixed optimality conditions. The sufficient condition for the DZB mixed H/sub 2/ and H/sub infinity / full order optimal control is shown to be the dual of the necessary condition for the BH control. Therefore, both conditions are proved to be necessary and sufficient.<<ETX>>

Robust control systems design using H-infinity optimization theory

In this paper, we show step-by-step procedures for applying the H°° theory to robust control systems design. The objective of the paper is to eliminate the possible difficulties a control engineer may encounter in applying H°° control theory. We will review the physical meanings of the H°° norm, explain how it relates to robustness issues, and show how to formulate H°° optimization problems, including the construction of a state-space realization of the generalized plant. An efficient algorithm is used to compute the optimal H°° norm. The controller formulas of Glover and Doyle are slightly modified and are used to construct an optimal controller without any numerical difficulty.

Robust control design with real-parameter uncertainty and unmodeled dynamics

This paper presents a design methodology for the synthesis of a robust controller that accounts for both unmodeled dynamics and structured real-parameter uncertainty for multiple-input/multiple-output systems. The unmodeled dynamics are assumed to be characterized as a single block dynamic uncertainty at a point in the closed-loop system. In a design aimed at constraining both the H^ norm of a certain disturbance transfer matrix and a quadratic Gaussian performance index under their respective bounds, a surrogate system may be formed by modeling the structured real-parameter uncertainty as additional noise inputs and additional weights at the existing noise inputs and measurement outputs of the system. Application of a Riccati equation approach to this surrogate system then yields a robust controller that, when used in the actual system, will result in a closed-loop system that has the same H^ bound and quadratic Gaussian performance index bound as the surrogate system, even in the presence of given real-parameter variations.

Robust Conrol Design with Real-Parameter Uncertainties

A method is presented for the construction of design equations for low-order controllers to provide robust H¿ norm constraint for systems with real-parameter uncertainties. The multi-goal robust stability and performance problem is broken down into subproblems, each with a simple design goal. Riccati inequalities are easily written for each of the subproblems, then the various terms are collected from each to form Riccati inequalities for the multi-goal problem. This method is based on theorems fundamental to Mu-synthesis so that the results can be used to start the D-K iteration process in Mu-synthesis.

Observer-based compensators for fixed-order H∞ control

This paper present a two-Riccati inequality approach for the design of fixed-order H∞ suboptimal controllers with an observer-based structure. The two Riccati inequalities are only one-way coupled and can be solved consecutively. The fixed-order controller has the structure of a fixed-order estimator of the state feedback control law in the presence of the worst-case disturbances. It retains the observed-based compensator structure of the full-order central controller of the standard H∞ design but adds a fixed-order dynamic and mappings between the full-order and the fixed-order state spaces. In the full-order special case, this controller recovers to the minimal entropy central controller of the standard H∞ design.

Stability robustness measures utilizing structural information

This paper presents techniques of using weighted L1. and L¿ norms to incorporate the structural information of the return difference and the perturbation matrices in the measure of stability robustness of multivariable control systems. The flexibility offered by these norms along with the use of nonnegative matrix theory enables one, in most cases, to reduce the conservatism which has been a typical concern with the use of singular value based stability robustness tests. New stability robustness criteria are derived on the basis of weighted L1 and L¿ norms. Examples are given to demonstrate the merits of these methods over singular value tests.

Robust control design for an aircraft gust attenuation problem

A method that leads to a minimal-order controller that guarantees robust disturbance rejection for uncertain systems via the solution of simple design equations was presented by H.H. Yeh et al. (1992). In the present work, the authors demonstrate the application of this method to an aircraft gust attenuation problem. The design algorithm presented here was successfully applied to synthesize a minimal-order robust controller for attenuating an aircraft lateral gust.<<ETX>>

Computation of the real structured singular value via polytopic polynomials

For a large class of linear time-invariant systems with real parametric perturbations, the coefficient vector of the characteristic polynomial is a multilinear function of the real parameter vector. Based on this multilinear mapping relationship together with the recent developments for polytopic polynomials and parameter domain partition technique, an iterative algorithm for computing the real structured singular value is proposed. The algorithm requires neither frequency search nor Rouths array symbolic manipulations and allows the dependency among the elements of the parameter vector. Moreover, the number of the independent parameters in the parameter vector is not limited to three as is required by many existing structured singular-value computation algorithms.

Robust Design of Multivariable Feedback Systems with Real Parameter Uncertainty and Unmodelled Dynamics

This paper presents a design methodology for the synthesis of a robust controller that accounts for both unmodelled dynamics and structured real parameter uncertainty for multiple-input, multiple-output systems. In a design aimed at constraining both the H¿ norm of a certain disturbance transfer matrix and a quadratic gaussian performance index under their respective bounds, a surrogate system may be formed by modelling the structured real parameter uncertainty as additional noise inputs and additional weights at the existing noise inputs and measurement outputs of the system. Application of a Riccati equation approach to this surrogate system then yields a robust controller which, when used in the actual system, will result in a closed-loop system having the same H¿ bound and quadratic gaussian performance index bound as the surrogate system, even in the presence of given real parameter variations.