Observer design and stabilization of the dominant state of discrete-time linear systems

K e y w o r d s M o d e l reduction, reduced-order observers, Stabilization, Dominant state, Discretetime linear systems, Separation principle. 1. I N T R O D U C T I O N During the past four decades, significant at tention has been paid to the construction of reducedorder observers and stabilization using reduced-order controllers for linear control systems [1-7]. The s tandard observer design for linear systems either est imates the full s ta te vector or a linear functional of the state vector as originally proposed by Luenberger [1]. As far as the stabilization of linear control systems is concerned, the s tate vector may not be available for measurement and so when the linear control system is both stabilizable and observable, we use the separation principle for linear control systems and use an est imate of the s tate in lieu of the state vector. This approach works well with small-scale linear systems. However, for large-scale systems, the observer design problem for a full-state vector or a linear functional of the state vector leads to a design problem with high dimensionality. For such large-scale systems, the order of the observer is comparable with the order of the observed state dynamics. As a consequence, the observer design problem for large-scaie systems involves potential numerical and practical difficulties, and so the s tate feedback laws using an est imate of the s tate in lieu of the s tate vector may not yield the desired stabilization results. 0895-7177/05/$ see front matter O 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2003.06.011 Typeset by A~v~-TEX 582 V. SUNDARAPANDIAN In this paper, we first derive the reduced-order model for a given discrete-time linear system keeping only the dominant state of the given linear plant. The dominant state of a linear dynamical system corresponds to the slow modes of the system, while the nondominant state of a linear dynamical system corresponds to the fast modes of the system [4,5,7-9]. As an application of our recent work [9], we first derive the reduced-order model of the given linear discrete-time plant. Using the reduced-order model thus obtained, we characterize the existence of a reduced-order exponential observer that tracks the state of the reduced-order model, i.e., the dominant state of the original linear plant. We note that the model reduction and the reduced-order observer design detailed in this paper are discrete-time analogs of the results of Aldeen and Trinh [7]. Using the reduced-order model of the original plant, we also characterize the existence of a stabilizing state feedback control law that uses only the dominant state of the original plant. Also, when the plant is stabilizable by a state feedback control law, the full information of the dominant state is not always available. For this reason, we establish a separation principle so that the state of the observer may be used in lieu of the dominant state of the original plant, which facilitates the implementation of the stabilizing feedback control law derived. The model-reduction, observer design, stabilization, and separation principle derived in this paper for discrete-time linear control systems have important applications in practice. This paper is organized as follows. In Section 2, we derive the reduced-order model of a given discrete-time linear system. In Section 3, we derive necessary and sufficient conditions for exponential observer for the reduced-order plant. In Section 4, we derive necessary and sufficient conditions for the reduced-order plant to be stabilizable by a state feedback control law, and we also present a separation principle. 2, R E D U C E D O R D E R M O D E L O F T H E L I N E A R S Y S T E M Consider a discrete-time linear time-invariant system given by x(k + 1) = Ax(k) + Bu(k) , y (k) = c x (k) , (1) where x E •n is the state, y E R p, the output, and u E ]~m, the input of linear plant (1). First, we suppose that we have performed an identification of the dominant and nondominant states of the given linear system using the modal approach as in [9]. Without loss of generality, we may assume that