Analysis of time-varying control strategies for optimal disturbancerejection and robustness

Analysis of linear time-invariant systems that are controlled by time-varying controllers is given. The problems of disturbance rejection and robustness are formulated and analyzed using functional analytic methods, which reveal the properties that are common to these problems. The theoretical development presented uses the theory and techniques of nuclear operators and their relation with the duality theory of tensor products of Banach spaces. The authors show how time-varying compensation offers no advantage over time-invariant compensation for the problem of disturbance rejection over general signal spaces, in both continuous and discrete time, and for the problem of L_∞ robust stabilization of time-invariant plants. In addition, another application of the theory for the problem of norm minimization subject to a norm constraint is presented     

Robust stability under structured and unstructured perturbations

The problem of robust stability for linear time-invariant single-output control systems subject to both structured (parametric) and unstructured (H_∞) perturbations is studied. A generalization of the small gain theorem which yields necessary and sufficient conditions for robust stability of a linear time-invariant dynamic system under perturbations of mixed type is presented. The solution involves calculating the H_∞ -norm of a finite number of extremal plants. The problem of calculating the exact structured and unstructured stability margins is then constructively solved. A feedback control system containing a linear time-invariant plant which is subject to both structured and unstructured perturbations is considered. The case where the system to be controlled is interval is treated, and a nonconservative, easily verifiable necessary and sufficient condition for robust stability is given. The solution is based on the extremal of a finite number of line segments in the plant parameter property of a finite number of line segments in the plant parameter space along which the points closest to instability are encountered     

On robust stability of interval control systems

The authors develop results on the robust stability of a nonlinear control system containing both parametric as well as unstructured uncertainty. The basic system considered is that of the classical Lur'e problem of nonlinear control theory. A robust version of the Lur'e problem consisting of a family of linear time-invariant systems subjected simultaneously to bounded parameter variations and feedback perturbations from a family of sector-bounded nonlinear gains is presently treated. By using the Kharitonov theorem to develop some extremal results on positive realness of interval transfer functions (i.e. a family of rational transfer functions with bounded independent coefficient perturbations), the authors determine the size of a sector of nonlinear feedback gains for which absolute stability can be guaranteed. These calculations amount to the determination of the stability margin of the system under joint parametric and nonlinear feedback perturbations     

Some recent results in structured robust stability: An overview

Some recent results on robust stability with structured perturbations, using the polynomial framework, are presented in this paper without proof. As background we first describe Kharitonov's theorem and give an interpretation of it as a generalization of the Hermite—Bieler interlacing theorem. The need for a generalization of this result for tackling the control problem is explained, and our new results are then presented. An important generalization of Kharitonov's theorem that solves the box problem in parameter space is described. Some efficient formulae for the l²-stability margin in parameter space are also given. The results are illustrated by examples.     

An alternative proof of Kharitonov's theorem

An alternative proof is presented of Kharitonov's theorem for real polynomials. The proof shows that if an unstable root exists in the interval family, then another unstable root must also show up in what is called the Kharitonov plane, which is delimited by the four Kharitonov polynomials. This fact is proved by using a simple lemma dealing with convex combinations of polynomials. Then a well-known result is utilized to prove that when the four Kharitonov polynomials are stable, the Kharitonov plane must also be stable, and this contradiction proves the theorem     

Robust stability manifolds for multilinear interval systems

The stability of a class of multilinearly perturbed families of systems is considered. It is shown how the problem of checking the stability of the entire family can be reduced to that of checking certain subsets that are independent of the degrees of the polynomials involved. The extremal property of these subsets is established. The results point to the need for a complete study of the stability of manifolds of polynomials composed of products of simple surfaces