Robust stability of feedback control systems with uncertain parameters and unmodeled dynamics

Recent results on a generalization of Kharitonov’s interval polynomial problem allow linearly dependent perturbations in the coefficients of a polynomial, and the main objective is to provide computationally tractable criteria for robust stability. In this paper robust-stability criteria are provided in an even more general setting—one which also accommodates unmodeled dynamics. In addition, the test given for robust stability totally avoids so-called “edge combinatorics,” greatly facilitating numerical computation. This research was supported by the National Science Foundation under Grants ECS-8612948 and ECS-8451519 and in part by AFOSR Grant 880020, Honeywell and GE.     

Control effort considerations in the stabilization of uncertain dynamical systems

This paper investigates the problem of designing a state feedback control to stabilize an uncertain nonlinear system. We focus attention on the amplitude (norm) of the controller which is used to achieve this end. The uncertain system is described by a state equation which contains uncertain parameters which are unknown but bounded. A Lyapunov function is used to establish the stability of the closed loop system. The paper gives necessary and sufficient concitions for the uncertain system to be stabilizable with a given Lyapunov function. Furthermore, a procedure is indicated for the construction of the desired feedback control law.     

Robust Schur stability of a polytope of polynomials

The main objective of the authors is to provide a necessary and sufficient condition for a polytope of polynomials to have all its zeros inside the unit circle. The criterion obtained serves as a discrete-time counterpart for results in S. Bialas (1985) and F. Fu and B.R. Barmish (1987) for the continuous case. Also, the results are reduced to operations on (n-1)×(n-1) matrices. It is concluded that, by the edge result of A.C. Bartlett et al. (1987), it suffices to check the exposed edges in order to determine whether a polytope of polynomials has all its zeros in a simply connected region D     

Stability of a polytope of matrices: counterexamples

While there have been significant breakthroughs for the stability of a polytope of polynomials since V.L. Kharitonov's (1978) seminal result on interval polynomials, for a polytope of matrices, the stability problem is considered far from completely resolved. Counterexamples are provided for three conjectures that are directly motivated by the results in the polynomial case. These counterexamples illustrate the fundamental differences between polynomial-stability and matrix-stability problems and indicate that some obvious lines of attack on the matrix polytope stability problem will fail     

A generalization of Kharitonov's four-polynomial concept for robuststability problems with linearly dependent coefficientperturbations

Kharitonov's four-polynomial concept is generalized to the case of linearly dependent coefficient perturbations and more general zero location regions. To this end, a specially constructed scalar function of a scalar variable is instrumental to the robustness analysis. The present work is motivated by two fundamental limitations of Kharitonov's theorem, namely: (1) the theorem only applies to polynomials with independent coefficient perturbations and (2) it only applies to zeros in the left-hand plane     

A Sharp Estimate for the Probability of Stability for Polynomials With Multilinear Uncertainty Structure

This paper addresses the probability of stability for uncertain polynomials that have multilinear functions of real parameters as coefficients. We obtain an estimate for the probability of stability with respect to a class of admissible distributions. This estimate is ldquosharprdquo in the following sense: one obtains a probability of stability of unity when the bounds on the hypercube uncertainty bounding set are below the deterministic robustness radius obtained with the well-known mapping theorem.     

Controlling a constrained linear system to an affine target

We consider the problem of steering the state of a linear system to an affine target when the admissible controls are required to satisfy magnitude constraints. A necessary and sufficient condition for the existence of an admissible control which steers the system to the target from a specified initial condition is presented, as well as a necessary condition and a sufficient condition for global controllability to the target. The conditions are similar to those available in the literature in that they involve a finite dimensional search. Here, however, the affine nature of the target is exploited to reduce the dimensionality of the resulting optimization problem. Hence, the results are more easily applied than those developed in [13].     

On the radius of stabilizability of lti systems: Application to projection implementation in indirect adaptive control

In this paper we solve the problem of finding the ‘largest sphere’ around a linear time-invariant (LTI) stabilizable plant such that all plants of the same order in the sphere are also stabilizable and there exists a plant in its boundary that is not stabilizable. The sphere is described in the plant parameter space and an explicit expression for its radius is given. This result solves the open problem in indirect adaptive control of determining the region to which the parameter search procedure should be constrained so that stabilizability of the estimated plant is preserved.