W.Q. Liu

Model reduction for state-space symmetric systems

In this paper, the model reduction problem for state-space symmetric systems is investigated. First, it is shown that several model reduction methods, such as balanced truncation, balanced truncation which preserves the DC gain, optimal and suboptimal Hankel norm approximations, inherit the state-space symmetric property. Furthermore, for single input and single output (SISO) state-space symmetric systems, we prove that the H∞ norm of its transfer functions can be calculated via two simple formulas. Moreover, the SISO state-space symmetric systems are equivalent to systems with zeros interlacing the poles (ZIP) under mild conditions.

On initial instantaneous jumps of singular systems

This paper is concerned with instantaneous jumps in the state trajectory of a singular system. It is shown that such jumps generically exist at the initial time due to inconsistent initial states. The formula for the jumps is derived, which indicates that the jumps linearly depend on the initial state. Moreover, the set of all initial states which do not cause any state jumps for any output feedback is characterized and is revealed to be a linear subspace. In the meantime, the necessary and sufficient conditions are given for the existence of an output feedback to suppress completely the jumps for a given initial state. Finally, we formulate the problem of designing an output feedback not only to stabilize the plant but also to achieve minimum initial jumps as a minimization problem. >

A new frequency-weighted balanced truncation method and an error bound

In this paper, a new frequency weighted model reduction method with an error bound is proposed. The method is a generalization of Ennss technique and yields stable models even when both input and output weightings are included. The proposed method is compared with other existing methods using numerical examples.

A frequency domain approach to control of singular systems

In this paper, singular control systems are studied from the frequency domain point of view. The proper stable factorization of a singular system is derived. The stability of the closed-loop system consisting of a singular system and controller is addressed. Most significantly, all proper stabilizing controllers are parameterized for a given singular system.

A new approach for frequency weighted L2 model reduction of discrete-time systems

This paper deals with the problem of model reduction based on an optimization technique. The objective function being minimized in the impulse energy of the overall system with unity, single-sided and double-sided weightings. A number of properties of the gradient flows associated with the objective function are obtained. Two examples are presented to illustrate the effectiveness of the proposed method, and results are compared with unweighted and weighted balanced truncation methods.

Identification/reduction to a balanced realization via the extended impulse response gramian

A new identification/reduction algorithm for linear, discrete time-invariant (LDTI) systems is proposed which is based on the extended impulse response gramian first defined here for LDTI systems. The reduction algorithm employs singular value decomposition to retain states corresponding to dominant singular values of these gramians. The proven properties of the reduced order models include convergence to a balanced realization with conditional controllability, observability, and asymptotic stability. A suboptimal property of the model (in minimizing an impulse response error norm) is also demonstrated. The proposed technique can handle impulse response data of deterministic or stochastic nature for system identification application.

Optimal simultaneous stabilization of descriptor systems via output feedback

In this paper, we consider the optimal simultaneous stabilization problem for a collection of linear time-invariant descriptor systems. The objective function for the collection of systems is defined to be the sum of all quadratic objective functions associated with individual systems. The problem is to choose an output feedback gain such that the objective function is minimized. Firstly, the set of all the output feedback gains for which the objective function is finite is characterized and proven to be a compact set. Secondly, an iterative algorithm based on the gradient flow idea is derived for solving this problem and some properties on the gradient flow are obtained to guarantee the success of this method. Thirdly, an initial value for the gradient flow equation is found by solving a standard unconstrained optimization problem. Finally, two numerical examples are solved for illustration.

A new frequency-weighted optimal Hankel norm model reduction and an error bound

A new frequency weighted optimal Hankel norm model reduction method is proposed. The method is an extension of the technique presented in Wang et al. (1997). The method yields stable models even when both input and output weightings are included and also gives an a priori error bound. A numerical example is given to compare the proposed method with other existing methods.

Frequency weighted identification and model reduction via extended impulse response Gramian

A method is presented for identification and model reduction of frequency weighted linear discrete time invariant systems. The method is an extension of the method proposed by Ang et al . (1995), which is based on the extended impulse response Gramian (EIRG). A weighting matrix formed from the impulse response of the frequency weighting is used to weight the EIRG of the original system. The identified/reduced-order models are obtained via singular value decomposition of the frequency weighted EIRG and are shown to converge to a frequency weighted balanced realization under certain conditions. An example is presented to illustrate the proposed method.

Balanced performance preserving controller reduction

Considers balanced performance preserving controller reduction with closed-loop system performance-preserving criteria. The two methods proposed are closed loop block balanced truncation and a closed loop structurally balancing method. Due to the introduction of the performance preserving weightings, new methods have improved traditional balanced controller reduction methods in reducing the closed loop H/sub /spl infin// performance degradation.