Wanquan Liu

Model reduction of singular systems

The model reduction problem for singular systems is investigated. Firstly, the model reduction algorithm reported in Perev and Shafai (1994) is presented and proved to be wrong. Detailed examination of the above algorithm shows that the difficulty of model reduction for singular systems is to retain its impulsive nature. Thus, based on this observation, we investigate the impulsive controllability and impulsive observability of singular systems and propose a new decomposition approach for singular systems. Then a new model reduction algorithm is designed based on the new decomposition via the machinery of Neharis approximation algorithm. This new model reduction algorithm retains the impulsive nature of the original system. Finally, an example is presented to illustrate the effectiveness of the proposed model reduction algorithm.

Iterative solutions to the Kalman–Yakubovich-conjugate matrix equation

Abstract Two operations are introduced for complex matrices. In terms of these two operations an infinite series expression is obtained for the unique solution of the Kalman–Yakubovich-conjugate matrix equation. Based on the obtained explicit solution, some iterative algorithms are given for solving this class of matrix equations. Convergence properties of the proposed algorithms are also analyzed by using some properties of the proposed operations for complex matrices.

The complete solution to the Sylvester-polynomial-conjugate matrix equations

In this paper we propose two new operators for complex polynomial matrices. One is the conjugate product and the other is the Sylvester-conjugate sum. Then some important properties for these operators are proved. Based on these derived results, we propose a unified approach to solving a general class of Sylvester-polynomial-conjugate matrix equations, which include the Yakubovich-conjugate matrix equation as a special case. The complete solution of the Sylvester-polynomial-conjugate matrix equation is obtained in terms of the Sylvester-conjugate sum, and such a proposed solution can provide all the degrees of freedom with an arbitrarily chosen parameter matrix.

An improved suboptimal model reduction for singular systems

An improved suboptimal model reduction for singular systems is proposed. The proposed method is based on Silverman-Ho algorithm as in Wang et al. (2004). A full parametrization of reduced-order models is provided unlike the technique of Wang et al. (2004) where in only partial parametrization is presented. The proposed method is illustrated by a numerical example and is compared with the method of Wang et al. (2004).

Controllability and stability of discrete-time antilinear systems

In this paper, the discrete-time antilinear systems are investigated. Firstly, a closed-form expression for the state response of discrete-time antilinear systems is established. Secondly, the concepts of reachability and controllability are proposed for discrete-time antilinear systems. With the closed-form expression of the state reponse as tools, anti-Gram criteria for reachability are given. In addition, a matrix rank criteria for reachability and controllability is given for time-invariant antilinear systems. In addition, an anti-Lyapnov equation approach is given to check the stability of the time-invariant antilinear systems.

Nonlinear Feedback Control for the Lorenz System

The Lorenz system is well known for its ability to produce chaotic motion and the control problem of this system has attracted much attention in recent years. In this paper, control of the Lorenz chaotic systems based on a nonlinear feedback technique is presented. The objective of control is two-fold: one is to drive the system to one of equilibrium points associated with uncontrolled chaotic motion and the other is to let one of the closed-loop system states track a given signal. The controllers designed here are based on exact linearization theory of nonlinear systems and can regulate the closed-loop system states globally to a given point. Finally, illustrative examples show the effectiveness of the proposed design method.

On the conjugate product of complex polynomial matrices

In this paper, we propose a new operator of conjugate product for complex polynomial matrices. Elementary transformations are first investigated for the conjugate product. It is shown that an arbitrary complex polynomial matrix can be converted into the so-called Smith normal form by elementary transformations in the framework of conjugate product. Then the concepts of greatest common divisors and coprimeness are proposed and investigated, and some necessary and sufficient conditions for the coprimeness are established. Finally, it is revealed that two complex matrices A and B are consimilar if and only if (sI-A) and (sI-B) are conequivalent. Such a fact implies that the Jordan form of a complex matrix A under consimilarity may be obtained by analyzing the Smith normal form of (sI-A).

Parametric solutions to Sylvester-conjugate matrix equations

By two recently proposed operations with respect to complex matrices, a simple explicit solution to the Sylvester-conjugate matrix equation is given in a finite series form. The obtained solution can also be equivalently expressed in terms of the so-called controllability-like matrix and observability-like matrix. The proposed solution can provide all the degrees of freedom which is represented by a free parameter matrix. An illustrative example is employed to show the effectiveness of the proposed method.

Linear quadratic regulation for discrete-time antilinear systems: An anti-Riccati matrix equation approach

Abstract In this paper, the linear quadratic regulation problem is investigated for discrete-time antilinear systems. Two cases are considered: finite time state regulation and infinite time state regulation. First, the discrete minimum principle is generalized to the complex domain. By using the discrete minimum principle and dynamic programming, necessary and sufficient conditions for the existence of the unique optimal control are obtained for the finite time regulation problem in terms of the so-called anti-Riccati matrix equation. Besides, the optimal value of the performance index under the optimal control is provided. Furthermore, the optimal regulation problem on an infinite interval is investigated under the assumption that the considered time-invariant antilinear system is controllable. The resulted closed-loop system under the optimal control turns out to be asymptotically stable.

Implicit Iterative Algorithms for Continuous Markovian Jump Lyapunov Equations

In this technical note, implicit iterative algorithms with some tunable parameters are developed to solve the coupled Lyapunov matrix equations associated with continuous-time Markovian jump linear systems. A significant feature of the proposed algorithms is that the iterative sequences are updated by using not only the information in the last step, but also the information in the current step and the previous steps. Also the convergence rate of the proposed algorithms can be significantly improved by choosing appropriate parameters in the algorithms.