Stability Margin of a Feedback System with Structured and Unstructured Uncertainties (original) (raw)

Linear systems with unstructured multiplicative uncertainty: Modeling and robust stability analysis

PLOS ONE, 2017

This article deals with continuous-time Linear Time-Invariant (LTI) Single-Input Single-Output (SISO) systems affected by unstructured multiplicative uncertainty. More specifically, its aim is to present an approach to the construction of uncertain models based on the appropriate selection of a nominal system and a weight function and to apply the fundamentals of robust stability investigation for considered sort of systems. The initial theoretical parts are followed by three extensive illustrative examples in which the first order time-delay, second order and third order plants with parametric uncertainty are modeled as systems with unstructured multiplicative uncertainty and subsequently, the robust stability of selected feedback loops containing constructed models and chosen controllers is analyzed and obtained results are discussed.

On a parameterization of all stabilizing controllers for plants with parametric uncertainty

29th IEEE Conference on Decision and Control, 1990

A parametrization of all (internally) stabilizing controllers for SISO plants characterized by parametric uncertainty is obtained. The cme of a single uncertain parameter is analyzed and necessary and sufficient conditions for robust stability are derived. The results suggest the possibility of using this approach for synthesis of controllers for uncertain plants. The conditions in the multiparameter uncertainty case require more advanced tools for stability verification. A simple, illustrative example is provided.

Systems with structured uncertainty: relations between quadratic and robust stability

IEEE Transactions on Automatic Control, 1993

The purpose of this note is to investigate the relation between the notions of robust stability and quadratic stability for uncertain systems with structured uncertainty due to both real and complex parameter variations. We present examples which demonstrate that for systems containing at least two uncertain blocks, the notions of robust stability for complex parameter variations and quadratic stability for real parameter variations are not equivalent; in fact neither implies the other. A byproduct of these examples is that, for this class of systems, quadratic stability for real perturbations need not imply quadratic stability for complex perturbations. This is in stark contrast with the situation in the case of unstructured uncertainty, for which it is known that quadratic stability for either real or complex perturbations is equivalent to robust stability for complex perturbations, and thus equivalent to a small gain condition on the transfer matrix that the perturbation experiences.

Generalization of the Nyquist robust stability margin and its application to systems with real affine parametric uncertainties

International Journal of Robust and Nonlinear Control, 2001

The critical direction theory for analysing the robust stability of uncertain feedback systems is generalized to include the case of non-convex critical value sets, hence making the approach applicable for a much larger class of relevant systems. A redefinition of the critical perturbation radius is introduced, leading to the formulation of a Nyquist robust stability measure that preserves all the properties of the previous theory. The generalized theory is applied to the case of rational systems with an affine uncertainty structure where the uncertain parameters belong to a real rectangular polytope. Necessary and sufficient conditions for robust stability are developed in terms of the feasibility of a tractable linear-equality problem subject to a set of linear inequalities, leading ultimately to a computable Nyquist robust stability margin. A systematic and numerically tractable algorithm is proposed for computing the critical perturbation radius needed for the calculation of the stability margin, and the approach is illustrated via examples.

Robustness analysis for systems with ellipsoidal uncertainty

International Journal of Robust and Nonlinear Control, 1998

This note derives an explicit expression for computing the robustness margin for affine systems whose real and complex coefficients are related by an ellipsoidal constraint. The expression, which is an application of a result by Chen, Fan, and Nett for rank-one generalized structured singular-value problems, extends and unifies previous results on robustness margin computation for systems with ellipsoidal uncertainty.

WORST-CASE STATE-SPACE ยต-ANALYSIS FOR SYSTEMS SUBJECT TO REAL PARAMETRIC UNCERTAINTY

Abstract This paper considers the application of the structured singular value and the skewed structured singular value to the robust stability of systems subject to strictly real parametric uncertainties. The focus is on the calculation of improved lower bounds for this type of uncertainty using frequency independent techniques necessary to counteract the discontinuous nature of the analysis in this instance.

Stability and Performance Analysis in the Presence of Magnitude Bounded Real Uncertainty

1991

This paper is concerned with stability and performance Riccati equation based analysis methods for perturbed feedback systems. The elements of the state space representation of the systems are assumed to be linearly perturbed with real, magnitude bounded uncertainties. Performance is defined as the value of the H, norm of the transfer function of interest. An analysis method based on simultaneous satisfaction of a set of Riccati inequalities is studied as well as a method based on a single, "overall" Riccati equation. Special attention is paid to the conservatism of these methods. It is shown when the perturbation is treated as real, the result can be at least as conservative as if the perturbation were unstructured, frequency dependent uncertainty. This is further used to show that existing synthesis methods for H. norm minimization may be used for obtaining a compensator that makes the closed loop system satisfy the single Riccati equation criterion.

A measure of robust stability for an identified set of parametrized transfer functions

IEEE Transactions on Automatic Control, 2000

In this paper, we define a measure of robustness for a set of parameterized transfer functions as delivered by classical prediction error identification and that contains the true system at a prescribed probability level. This measure of robustness is the worst case Vinnicombe distance between the model and the plants in the uncertainty region. We show how it can be computed exactly using LMI-based optimization. In addition, we show that this measure is directly connected to the size of the set of controllers that are guaranteed to stabilize all plants in the uncertainty region, i.e., the smaller the worst case Vinnicombe distance for an uncertainty region, the larger the set of model-based controllers that are guaranteed to stabilize all systems in this uncertainty region.

Stability robustness bounds for linear state-space models with structured uncertainty

IEEE Transactions on Automatic Control, 1987

This paper is concerned with the problem of robust stability of linear dynamic systems with structured uncertainty by means of ellipsoidal set-theoretic approach. In this paper, the uncertainty in the physical parameters is expressed in terms of an ellipsoidal set in appropriate vector space. Two ellipsoidal set-theoretic approaches are presented for giving sufficient conditions for robust stability property of the systems with structured uncertainty. The bound produced by the ellipsoidal extension function theorem is shown to be less conservative than the one predicted by the Lagrange multiplier method. In order to introduce the ellipsoidal extension function theorem, in Appendix A of this paper, we try to present the theory of ellipsoidal algebra, following the thought of interval analysis. First of all, we give the concept of ellipsoidal numbers and define their arithmetic operations. Based on them, we finally introduce ellipsoidal vectors and ellipsoidal functions. In terms of the inclusion monotonic property of ellipsoidal functions, we present and prove the ellipsoidal extension function theorem.