H. Chapellat

A generalization of Kharitonov's theorem; Robust stability of interval plants

The robust stability problem is considered for interval plants, in the case of single input (multioutput) or single output (multi-input) systems. A necessary and sufficient condition for the robust stabilization of such plants is developed, using a generalization of V. L. Kharitonovs theorem (1978). The generalization given provides necessary and sufficient conditions for the stability of a family of polynomials delta (s)=Q/sub 1/(s)P/sub 1/(s)+ . . . +Q/sub m/(s)P/sub m/(s), where the Q/sub i/ are fixed and the P/sub i/ are interval polynomials, the coefficients of which are regarded as a point in parameter space which varies within a prescribed box. This generalization, called the box theorem, reduces the question of the stability of the box, in parameter space to the equivalent problem of the stability of a prescribed set of line segments. It is shown that for special classes of polynomials Q/sub i/(s) the set of line segments collapses to a set of points, and this version of the box theorem in turn reduces to Kharitonovs original theorem. >

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/sub infinity /) 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/sub infinity /-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. >

Analysis of time-varying control strategies for optimal disturbance rejection 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/sub infinity / 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. >

An alternative proof of Kharitonov's theorem

An alternative proof is presented of Kharitonovs 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 of a family of disc polynomials

In his well-known theorem, V. L. Kharitonov established that Hurwitz stability of a set f1 of interval polynomials with complex coefficients (polynomials where each coefficient varies in an arbitrary but prescribed rectangle of the complex plane) is equivalent to the Hurwitz stability of only eight polynomials in this set. In this paper we consider an alternative but equally meaningful model of uncertainty by introducing a set fD of disc polynomials, characterized by the fact that each coefficient of a typical element P(s) in fD can be any complex number in an arbitrary but fixed disc of the complex plane. Our result shows that the entire set is Hurwitz stable if and only if the ‘center’ polynomial is stable, and the H ∞-norms of two specific stable rational functions are less than one. Our result can be readily extended to deal with the Schur stability problem and the resulting condition is equally simple.

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. >

Extremal robustness properties of multilinear interval systems

This paper deals with robustness properties of a control system subject to parameter uncertainty when the parameters appear affine multilinearly in the coefficients of the characteristic polynomial. The robust Hurwitz stability of such a multilinear interval polynomial family reduces to that of the stability of a set of manifolds derived from the Kharitonov polynomials. Several extremal properties of these stability manifolds are derived. These properties are useful in determining the worst case parametric, H∞ and nonlinear sector bounded stability margins for control systems containing interval parametric uncertainty. They generalize previous results obtained by the authors for the linear case.

Stability margins for multilinear interval control systems

A multilinear extension of the result of H. Chapellat and S.P. Bhattacharyya (1989) is given. This result, called the multilinear CB theorem, is useful in determining robust parametric stability and stability margins. The result constitutes a generalization of Kharitonovs theorem to the case of multilinear interval uncertainty. It provides certain manifolds which capture all the extremal information regarding the uncertainty structure. These extremal manifolds play the same role in analysis and design as the CB segments of S.P. Bhattacharyya (1991) do in the linear case. The usefulness of these CB manifolds is illustrated by an example where it is shown how parametric stability margins can be efficiently calculated.<<ETX>>

Stability margins for Hurwitz polynomials

The authors treat the robust stability issue using the characteristic polynomial, for two different cases: first in coefficient space with respect to perturbations in the coefficient of the characteristic polynomial; and then for a control system containing perturbed parameters in the transfer function description of the plant. In coefficient space, a simple expression is first given for the l/sup 2/-stability margin for both the monic and nonmonic cases. Following this, a method is given to find the l/sup infinity /-margin, and the method is extended to reveal much larger stability regions. In parameter space the authors consider all single-input (multi-output) or single-output (multi-input) systems with a fixed controller and a plant described by a set of transfer functions which are ratios of polynomials with variable coefficients. A procedure is presented to calculate the radius of the largest stability ball in the space of these variable parameters. The calculation serves as a stability margin for the control system. The formulas that result are quasi-closed-form expressions for the stability margin and are computationally efficient.<<ETX>>

Robust stability against structured and unstructured perturbations

The problem of unstructured perturbations is considered when the linear system is also subject to parameter perturbations or uncertainties which are not necessarily small. This motivates the consideration of a family of interval plants, i.e. plants with transfer function coefficients varying in prescribed ranges. For such families of systems subject to prescribed ranges of parameter perturbations precise and nonconservative bounds on the level of unstructured perturbations can be tolerated. Auxiliary results on passivity and strict positive realness that are of interest in their own right are also developed.<<ETX>>