Liqian Zhang

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

This work investigates the problem of robust D-stability analysis for uncertain discrete singular systems with state delay and structured uncertainties. Sufficient conditions are developed to ensure that, when the nominal discrete singular delay system is regular, causal and all its finite poles are located within a specified disk, the uncertain system still preserves all these properties when structured uncertainties are added into the nominal system. A computationally simple approach is proposed and a numerical example is given to demonstrate the application of the proposed method.

Lyapunov and Riccati equations of discrete-time descriptor systems

In this paper, we further develop the generalized Lyapunov equations for discrete-time descriptor systems given by Bender. We associate a stable discrete-time descriptor system with a Lyapunov equation which has a unique solution. Furthermore, under the assumptions of reachability and observability, the solutions are guaranteed to be positive definite. All results are valid for causal and noncausal descriptor systems. This provides a unification of Lyapunov equations and theories established for both normal and descriptor systems. Based on the developed Lyapunov equation, a Riccati equation is also obtained for solving the state-feedback stabilization problem.

Optimal model reduction of discrete-time descriptor systems

An optimal model reduction method is presented to obtain stable reduced-order models for discrete-time descriptor systems. A parametrization based on the bilinear Routh approximation of linear normal discrete-time systems is used to parametrize the causal subsystems of the reduced-order models. The expressions for the error and its gradient are explicitly given. They are then employed to solve an unconstrained optimization problem for the model reduction problem. The descriptor system structure is preserved in the reduced-order models.

New lyapunov and riccati equations for discrete-time descriptor systems

Abstract In this paper, we further develop the generalized Lyapunov equations for discrete-time descriptor systems given by Bender. We associate a stable discrete-time descriptor system with a Lyapunov equation which has unique solution. Furthermore, under the assumptions of reachability and observability, the solutions are guaranteed to be positive definite. All results are valid for causal and noncausal descriptor systems. This provides a unification of Lyapunov equations and theories established for both normal and descriptor systems. Based on the developed Lyapunov equation, a Riccati equation is also obtained for solving the state-feedback stabilization problem.

Generalized lyapunov equations for analyzing the stability of descriptor systems

Abstract In this paper, a necessary and sufficient condition for asymptotic stability is derived for continuous-time linear time-invariant descriptor systems using the Lyapunov approach. The result is presented in terms of the solutions of a generalized Lyapunov equation which is valid for both impulsive and non-impulsive descriptor systems. Conditions for asymptotic stability based on R-controllability or R-observability are also provided. Numerical examples are employed to illustrate the results.

Delay-Dependent γ-Suboptimal H∞ Model Reduction for Neutral Systems With Time-Varying Delays

In this paper, the model reduction problem of neutral systems with time-varying delays is studied with y suboptimality under the H ∞ measure. A delay-dependent bounded realness condition of the H ∞ norm is given via linear matrix inequalities (LMIs). Based on such a condition, a sufficient condition to characterize the existence of the reduced-order models is given in terms of LMIs with inverse constraints. By employing a sequential convex optimization approach, a reduced-order model can be computed with Has error less than some prescribed scalar y.

A parametric optimization approach to discrete-time model reduction

Abstract A new optimization method is presented to obtain stable reduced order models of a linear stable discrete-time system under l 2 optimality. The number of delay operators may be imposed in advance in the reduced order models. The state-space bilinear Routh canonical realization is used to parametrize the reduced model and the optimal parameters are obtained by solving a gradient-based unconstrained optimization problem. Explicit gradient formulas are available for numerical implementation. The stability of the reduced models is ensured in the iteration process. The formulation can be easily modified to achieve zero steady-state response error for a unit step without affecting the unconstrained nature of the optimization. Numerical examples are used to illustrate the effectiveness of the proposed method.

Optimal model reduction via delay approximation

Abstract A model reduction method for SISO and MIMO time delay systems is studied in this paper. The proposed method is based on L2 minimality on the reduced order models via the approximation of the delay element by a finite dimensional model in the original system. The technique is developed in terms of state-space matrices and is formulated as an unconstrained optimization problem. To faciliate numerical implementation, analytical gradient formulas involved in the method are derived. A numerical example is presented to demonstrate the effectiveness of the proposed method.

On Analyzing the Stability of Discrete Descriptor Systems via Generalized Lyapunov Equations

In this paper, the asymptotic stability of linear discrete time-invariant descriptor systems is studied via a generalized Lyapunov equation. The analysis covers both the causal and noncausal cases. In particular, the asymptotic stability of a discrete descriptor system (DDS) is related to the existence of a positive semidefinite solution of the generalized Lyapunov equation. The results strengthened those of earlier works for causal descriptor systems.

Optimal model reduction of stable delay systems

A model reduction method for stable delay systems under L/sub 2/ optimality is introduced in this paper. The reduced models may take the form of either a stable finite dimensional system or a delay system with reduced order finite dimensional part. Based on the Routh parametrization of stable systems, the two cases are studied under a unified framework of unconstrained optimization. Numerical examples are used to illustrate the effectiveness of the proposed method.