Linear Time-varying Perturbations Research Articles

Stability robustness analysis of state-space models for uncertain Linear time-varying systems

The stability robustness of linear-variant systems in the time domain is considered using the Lyapunov approach and the maximum singular values of a time-variant matrix. Bounds on linear time-varying perturbations that maintain the stability of a uniformly asymptotically stable linear time-variant system are obtained for both unstructured and structured independent perturbations. Bounds are also derived assuming that various elements of the system matrix are perturbed dependently. The result for the structural perturbation case is extended to the stability analysis of time-variant interval matrices.

Time-varying versus time-invariant compensation for rejection of persistent bounded disturbances and robust stabilization

It is shown that time-varying compensation does not improve the optimal rejection of persistent bounded disturbances. This result is obtained by exploiting a key observation that any time-varying compensator which yields a given degree of disturbance rejection must do so uniformly over time, thereby removing any advantage of time-variation. This key observation is exploited to show that time-varying compensation does not improve the optimal rejection of disturbances, regardless of the norm used to measure the disturbances. Thus, absolutely summable, finite-energy, or persistent bounded disturbances may be treated in the same manner. It is shown that time-varying compensation does not help in the bounded-input bounded-output robust stabilization of time-invariant plants with unstructured uncertainty. In doing so, it is also shown that the small-gain theorem is both necessary and sufficient for the bounded-input bounded-output stability of certain linear time-varying plants subject to unstructured linear time-varying perturbations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

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.

The necessity of the small-gain theorem for time-varying and nonlinear systems

It is shown that an appropriately modified version of the small-gain theorem which uses the 'asymptotic gain' of a fading-memory system leads to a necessary and sufficient condition for the stable invertibility of certain feedback operators. This modified small-gain theorem is then used to obtain conditions for robust finite-energy input-output stability of: (1) nonlinear plants subject to nonlinear time-varying perturbations; and (2) linear time-varying plants subject to linear time-varying perturbations. These conditions are shown to be necessary as well as sufficient. >

Improved stability robustness bounds using state transformation for linear discrete systems

Sufficient-condition bounds on the linear time-varying perturbations for robust stability of an asymptotically stable linear time-invariant discrete-time system have been reported recently. The development involves both Schwartz and triangular inequalities, and tends to give conservative results. This paper uses state transformation to obtain improvement in these bounds. Both unstructured and structured perturbation bounds are shown to improve using transformations. The proposed analysis is applied to a fourth-order macroeconomic system model.

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 stability bounds on time-varying perturbations for state-space models of linear discrete-time systems

The stability robustness of linear discrete-time systems in the time domain is addressed using the Lyapunov approach. Bounds on linear time-varying perturbations that maintain the stability of an asymptotically stable linear time-invariant discrete-time nominal system are obtained for both structured and unstructured independent perturbations. Bounds are also derived assuming that various elements of the system matrix are perturbed dependently. The result for the structured perturbation case is extended to the stability analysis of interval matrices.

Aircraft control design using improved time-domain stability robustness bounds

This paper 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 stability condition with respect to the basis of the vector space in which the Lyapunov function is constructed. The proposed analysis is applied to a VTOL aircraft control example to provide a constant linear state feedback control that is stability robust for the full range of parameter variations given.

Decomposition and stability for multiparameter singular perturbation problems

Time-scale separation and stability of linear time-varying and time-invariant multiparameter singular perturbation problems are analyzed. The first problem considered in the paper is that of deriving upper bounds on the small parasitic parameters ensuring the existence of an invertible, bounded transformation exactly separating fast and slow dynamics. This problem is most interesting for the time-varying case. The analysis of this problem in the time-varying case requires the two-time-scale setting introduced by H. K. Khalil and P. V. Kokotovic [16]. This entails that the mutual ratios of the small parameters are bounded by known positive constants. The second problem considered is to derive parameter bounds ensuring that the system in question is uniformly asymptotically stable. The results on decomposition are used to facilitate the derivation of these latter bounds. Fortunately, the analysis of decomposition and stability questions for time-invariant multiparameter singular perturbation problems requires no restriction on the relative magnitudes of the small parameters. A concept of strong D -stability is introduced and shown to greatly simplify the stability analysis of time-invariant multiparameter problems.

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.