Linear Singular Discrete Systems Research Articles

LMI‐based H ∞ control for linear singular discrete systems: a novel method

This study investigates the problem of state feedback H ∞ control for linear singular discrete systems. First, a novel admissibility condition and then a new bounded real lemma are proposed for linear singular discrete systems. Then, the existence condition and explicit expression of a desirable state feedback H ∞ controller are obtained based on the above results. It should be pointed out that all the conditions in this study are necessary and sufficient and in the form of strict linear matrix inequalities that can be solved easily. Finally, some numerical examples are provided to illustrate the effectiveness of the theoretical results in this study.

An LMI approach to -stability and -stabilization of linear discrete singular systems with state delay

Abstract The problems of D -stability and D -stabilization for linear discrete singular systems with state delay are addressed in this paper. By using the state augmentation technique, a necessary and sufficient D -stability condition and a sufficient D -stabilization condition for linear discrete singular time-delay systems are firstly proposed. Then, based on these conditions, quadratic D -stability and D -stabilization conditions for linear discrete singular time-delay systems with norm-bounded uncertainties and interval uncertainties are obtained, respectively. Both the D -stability and D -stabilization conditions are proposed in terms of linear matrix inequalities. Finally, illustrative examples show that the proposed conditions are effective to check the D -stability and to design the D -stabilizing controller for linear discrete singular time-delay systems.

Analysis and design for singular discrete linear systems subject to actuator saturation

A method to estimate the domain of attraction for a singular discrete linear system under a saturated linear feedback is established. Simple conditions are derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractively invariant. These conditions are expressed in terms of linear matrix inequalities. The largest contractively invariant ellipsoid can also be determined by solving an optimization problem with linear matrix inequality constraints. This result is extended to the design of feedback gain that results in the largest contractively invariant ellipsoid, which is also a linear matrix inequality optimization problem. A numerical example demonstrates the applicability and effectiveness of the presented method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Robust strictly dissipative control for discrete singular systems

The problem of strictly dissipative control for discrete singular systems with or without parameter uncertainties is focused. A necessary and sufficient condition in terms of linear matrix inequalities (LMIs) is derived guaranteeing the admissibility and strict dissipativeness of a linear discrete singular system. Then, the existence condition of a state feedback strictly dissipative controller is given by using matrix inequalities (MIs). As for an uncertain discrete singular system, the uncertainties are assumed to be time-invariant and norm-bounded appearing in both the state and input matrices. A sufficient condition is obtained such that the uncertain system is generalised quadratically stable and strictly dissipative. Moreover, a state feedback robust strictly dissipative controller is also constructed by using the solution of MIs.

Robust state feedback H ∞ control for uncertain linear discrete singular systems

A linear matrix inequality (LMI) approach to the robust state feedback H∞ control for linear discrete singular systems with norm-bounded uncertainty is developed. By taking the relationship between slow and fast subsystems of a discrete singular system, a new bounded real lemma is proposed that proves to be sufficient and necessary. On the basis of this, the result on robust H∞ performance is obtained. An explicit expression for the desired controllers is also given that involves no system decomposition. The obtained results are formulated in terms of strict LMIs. Illustrative examples are provided to demonstrate the effectiveness of the proposed approaches.

Reachability analysis of switched linear discrete singular systems

This paper studies the reachability problem of the switched linear discrete singular (SLDS) systems. Under the condition that all subsystems are regular, the reachability of the SLDS systems is characterized based on a peculiar repeatedly introduced switching sequence. The necessary and sufficient conditions are obtained for the reachability of the SLDS systems.

Robust D-stability analysis for linear uncertain discrete singular systems with state delay

In this work, by using the maximum modulus principle and the spectral radii of matrices, a new robust D-stability (i.e., robust eigenvalue clustering in a specified circular region) condition is proposed to ensure that, for all admissible structured parameter uncertainties, the linear discrete singular system with state delay is regular, causal and D-stable. The proposed criterion is mathematically proved to be less conservative than the existing one reported very recently.

D-stability robustness for linear discrete uncertain singular systems with delayed perturbations

In this paper, the robust D-stability problem (i.e. the robust eigenvalue-clustering in a specified circular region problem) of linear discrete singular systems with both structured (elemental) parameter uncertainties and delayed perturbations is investigated. Under the assumptions that the linear discrete nominal singular system is regular and causal, and has all its finite eigenvalues lying inside a specified circular region, by using the maximum modulus principle and the spectral radius of matrices, a new sufficient condition is proposed to preserve the assumed properties when both structured parameter uncertainties and delayed perturbations are added into the linear discrete nominal singular system. When all the finite eigenvalues lie inside the unit circle of the z-plane, the proposed criterion will become the stability robustness criterion. The proposed criterion is mathematically proved and numerically shown, respectively, to be less conservative than the existing ones reported recently.

Basic theory of linear singular discrete systems with delay

This paper discusses existence and uniqueness of solutions of constant coefficient linear singular discrete systems with delay, presents the results that the systems have a unique solution in some cases, and have no solution or have infinite solutions in other cases.

Reachability and controllability of weakly positive singular discrete linear systems

ABSTRACT A notion of weakly positive singular discrete linear time-invariant systems is introduced. Some properties of the fundamental matrix and solutions of the positive singular discrete linear systems are characterised. Necessary and sufficient conditions are given under which the weakly positive system can be reduced to a standard positive system. For regular singular discrete linear system an equivalent (has the same solution) standard system is derived. Necessaiy and sufficient conditions are established for the reachability and controllability of the positive singular systems. Sufficient conditions are given for unreachability of the systems.