Unstructured Perturbations Research Articles

State space measures of robustness of pole locations for structured and unstructured perturbations

Measures of pole location robustness for linear feedback systems are derived from a state space model of the system. The robustness tests ensure that the eigenvalues of the perturbed systems matrix A + E remain in a desired region D of the complex plane containing the eigenvalues of the nominal system matrix A. The region D may be any open set of the complex plane whatsoever. The results are expressed in terms of induced matrix norms and apply to structured perturbations of the form E = BΔC, where B and C define the structure of E, and may be nonsquare matrices. Rank one perturbations E of minimal norm and with the given structure that will cause A + E to have an eigenvalue outside of D are constructed for the cases when the matrix norm is induced by the vector 1-norm or the vector ∞-norm. The advantages of having robustness measures for several matrix norms that can be computed are illustrated with a simple example that demonstrates how the conservatism of single tests can be reduced using several tests (i.e. several matrix norms). A method for computing numerically the robustness measures for particular norms is presented. It can be used to compute, with a guaranteed degree of accuracy, the maximum of the norm of the frequency response of a system.

Relationship of singular value stability robustness bounds to spectral radius for discrete systems with application to digital filters

This paper presents additional results and an application for recently obtained stability robustness bounds on linear time-varying perturbations of an asymptotically stable linear timeinvariant discrete-time system. The bounds were developed using Lyapunov theory and singular value decomposition, and provide sufficient conditions for maintaining asymptotic stability for both unstructured and structured perturbations in the state-space discrete system model. A derivation is presented for the relationship between the unstructured perturbation bound and the spectral radius of the system model matrix, showing that the normal form realisation of the system gives the largest unstructured perturbation bound. Both the unstructured and structured perturbation bounds are applied to a recursive digital filter and to coefficient truncation as time-invariant examples to illustrate the use of these relations.

Reduced-order dynamic compensator design for stability robustness of linear discrete-time systems

A reduced-order dynamic compensator design is presented with stability robustness for linear discrete systems, by including a stability robustness component in addition to the standard quadratic state and control terms in the performance criterion. The robustness component is based on an unstructured perturbation stability bound for time varying perturbations. The controller design is developed by the parameter optimization technique and involves the solution of five algebraic matrix equations, four of which are discrete-time Lyapunov matrix equations. >

Robust Stability Against Structured and Unstructured Perturbations: New Results

The classical Lure problem considers the stability of a fixed linear time invariant dynamic system perturbed by sector bounded nonlinear feedback. The small gain theorem on the other hand, considers a fixed linear time invariant dynamic system perturbed by stable functions with bounded Hoo norm in the feedback loop. These results provide a way to account for unstructured perturbations of the fixed linear system. In this paper we are concerned with the same problems when the linear system is also subject to parameter perturbations or uncertainties which are not necessarily small. This motivates us to consider 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 we obtain precise and non-conservative bounds on the level of unstructured perturbations that can be tolerated. We also develop auxiliary results on passivity and strict positive realness which are of interest in their own right.

Improved robust stability bounds for discrete-time linear regulators with computational delays

Recently obtained stability robustness bounds on linear time-varying perturbations of an asymptotically stable linear time-invariant discrete-time system are applied to linear regulators with computational delays. The bounds were developed using Lyapunov theory and singular value decomposition for unstructured and structured time-varying perturbations. The new structured perturbation bound may provide improved results for the elemental perturbations over a bound based on the unstructured perturbation bound of Ishihara (1988, Automatica, 24, 696–700), as indicated both by an example and by sufficient conditions.

Robust disturbance rejection in l1 optimal control systems

Given a class of plants formed by perturbing a nominal discrete-time linear shift-invariant plant with norm bounded unstructured perturbation, the problem of finding a single compensator that will stabilize all plants in this class and at the same time minimize the worst case norm of a certain performance measure is addressed. Assuming the performance measure of interest is the sensitivity function, first an expression for the supremum, over all plants in this class, of such a measure is derived for a system with a robustly stabilizing compensator. This is achieved by first showing that the proposed expression is an upper bound for the worst case sensitivity norm, and then by constructing a plant in the allowable class that will achieve this upper bound. Combined with already known necessary and sufficient conditions for robust stability, the derived expression provides an effective way of combining both robust stability and performance in one, easy to compute, measure. Secondly, a scheme for finding a controller that minimizes the worst case norm of the sensitivity function while achieving robust stability, is provided. This makes possible the incorporation of stability and performance robustness requirements in one design procedure.

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.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Quadratic stability with real and complex perturbations

It is shown that the equivalence between real and complex perturbations in the context of quadratic stability to linear, fractional, unstructured perturbations does not hold when the perturbations are block structured. For a limited class of problems, quadratic stability in the face of structured complex perturbations is equivalent to a particular class of scaled norms, and hence appropriate synthesis techniques, coupled with diagonal constant scalings, can be used to design quadratically stable systems. >

Robustness measures for integrated structural/control systems

Several stability robustness methods are compared in an effort to employ an appropriate approach in the integrated design of space structures and control systems. The motivation of this work is to identify a method which gives a realistic stability bound on the parameter uncertainties of the physical model in designing a control system for aerospace structures. Four different methods are compared for structured and unstructured perturbation matrices. A suitable method is suggested based on the results obtained for a two-bar truss, a five-bar truss, and ACOSS-four structures.

Robust stability bounds for a class of discrete-time regulators with computation delays

Robust stability of discrete-time regulators which utilize state predictors to compensate computation delays is considered. Novel expressions for the return difference matrices and the complementary sensitivity matrices at the input and the output of the regulator are found to obtain simple bounds for unstructured perturbations. Robust stability for perturbations of the system matrix and/or the gain matrix is also considered. Under certain restrictions on the nominal system, simple bounds for the perturbations are obtained directly from the characteristic equation. It is shown that, as far as the effect of the computation delays is concerned, these bounds are explicitly related to those for the unstructured perturbations.

Comments on "Reduced conservatism in stability robustness bounds by state transformation" [with reply

Reference is made to the paper by R.K. Yedavalli and Z. Liang (see ibid., vol.AC-31, p.863-6, 1986) in which it is shown that state transformation can reduce the conservatism in stability robustness bounds. An attempt is made by the commenters to prove that for unstructured perturbations there always exists a better bound than that reported in the above paper, which can be obtained without using state transformations. In replying, the original authors point out that the result given by the commenters is useful in the sense that it provides the 'best' answer as far as unstructured perturbations are concerned. However, there exist some limitations. They claim that, as far as Lyapunov-based time domain methods are concerned, the question of finding the 'best' transformation, either for unstructured or structured, time-varying perturbations is still an open problem. >

Robust stability for linear time-delay systems with linear parameter perturbations

Abstract A new robust state feedback controller design in the time domain is developed for parametrically perturbed linear systems with time delay. The properties of the matrix measure and comparison theorem are employed to investigate the robust stability conditions of systems with linear structured or unstructured perturbations. A method is proposed to design the robust state feedback controller to satisfy the robustness requirement. Examples are given to illustrate the proposed method.

Reduced conservatism in stability robustness bounds by state transformation

This note addresses the issue of in the time domain stability robustness bounds obtained by the Lyapunov approach. A state transformation is employed to improve the upper bounds on the linear time-varying perturbation of an asymptotically stable linear time-invariant system for robust stability. This improvement is due to the variance of the conservatism of the Lyapunov approach with respect to the basis of the vector space in which the Lyapunov function is constructed. Improved bounds are obtained, using a transformation, on elemental and vector norms of perturbations (i.e., structured perturbations) as well as on a matrix norm of perturbations (i.e., unstructured perturbations). For the case of a diagonal transformation, an algorithm is proposed to find the optimal transformation. Several examples are presented to illustrate the proposed analysis.

Perturbation bounds for robust stability in linear state space models

Abstract This paper addresses the aspect of stability robustness of linear systems in the time domain. Upper bounds on the linear perturbation of an asymptotically stable linear system are obtained to maintain stability, both for structured as well as unstructured perturbations using the Lyapunov approach. For structured perturbation the resulting bound is such that it garners the structural information about the nominal (as well as the perturbation) matrix into a single unified expression. In the case of unstructured perturbations, special features of the nominally stable matrix are exploited resulting in simpler expressions for the bound (without the need to solve the Lyapunov equation). Improvement of the proposed measures is illustrated with the help of examples.

A new robust stability criterion for linear and non-linear multivariable feedback systems

Earlier eigenvalue inclusion regions are replaced by new regions which are tight and thus lead to sufficient and necessary stability conditions for uncertain linear multivariate systems in the presence of unstructured perturbations. The approach provides an exact and immediate assessment of gain and phase margins. When applied to systems with sector bounded but unstructured non-linearities, the inclusion regions produce a necessary and sufficient test for the circle criterion conditions to hold in the multivariable case; as such the resulting stability criterion gives the best possible Nyquist-type extension of the circle criterion to the multivariable case for unstructured non-linearities.

Eigenvalue Inclusion Regions and their Use in the Analysis of Linear and Nonlinear Multivariable Feedback System

The derivation of tight eigenvalue inclusion regions help extend the Generalised Nyquist criterion to the case of uncertain systems and lead to a necessary and sufficient Nyquist type of criterion for the case of unstructured perturbations. The approach provides an exact and immediate assessment of gain/phase margins. When applied to systems with nonlinearities which are sector bounded but otherwise unstructured, it produces a necessary and sufficient test for the circle criterion conditions to hold in the multivariable case; as such the resulting criterion is an improvement on the currently available extensions of the circle criterion.

On Logarithmic Norms

This paper discusses some normlike properties of the logarithmic norm and related efficiency questions. Some computational questions are raised and some examples of use are indicated.