1274 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 7, JULY 2003 [15] S. S. Stankovic ´ and D. D. ˇ Siljak, “Contractibility of overlapping decen- tralized control,” Syst. Control Lett., vol. 44, pp. 189–200, 2001. [16] D. D. ˇ Siljak, Decentralized Control of Complex Systems. New York: Academic, 1991. [17] D. D. ˇ Siljak, S. M. Mladenovic ´, and S. S. Stankovic ´, “Overlapping de- centralized observation and control of a platoon of vehicles,” in Proc. Amer. Control Conf., San Diego, CA, 1999, pp. 4522–4526. Semiglobal Stabilization and Output Regulation of Singular Linear Systems With Input Saturation Weiyao Lan and Jie Huang Abstract—The semiglobal stabilization problem and output regulation problem of singular linear systems subject to input saturation are addressed. A reduced-order normal system is obtained by a standard coordinate transformation. It is further shown that the controller that solves the stabilization (output regulation) problem of the reduced-order normal systems also solves the stabilization (output regulation) problem of the original singular systems. Index Terms—Input saturation, output regulation, singular system, sta- bilization. I. INTRODUCTION Singular systems arise in a variety of practical systems such as net- works, circuits, power systems, and so on [10]. Since the late 1970s, there has been extensive study on singular systems. However, singular systems subject to input saturation are hardly studied. In this note, we will consider two important control problems for singular systems sub- ject to input saturation, namely, the output feedback semiglobal sta- bilization problem and the output feedback semiglobal output regula- tion problem. For normal linear systems with saturation actuators, the semiglobal stabilization problem and the semiglobal output regulation problem have been thoroughly studied in quite a few papers ([12]–[14]) and two monographs [15] and [16]. In particular, it was shown in [12] that one can semiglobally stabilize a linear system subject to input sat- uration using linear feedback control laws if the system is asymptot- ically null controllable with bounded controls. Also, solvability con- ditions for the output regulation problem of linear systems subject to input saturation was established in [14]. In this note, under some standard assumptions, by employing the fa- miliar system equivalence technique, we will convert a singular system into a reduced-order normal system. Further, we will show that the re- duced-order normal systems satisfy the solvability conditions of the semiglobal stabilization (output regulation) problem. Thus, a normal output feedback controller can be constructed explicitly to solve the semiglobal stabilization (output regulation) problem for the reduced- order normal systems with input saturation. Finally, we will show that Manuscript received October 30, 2002; revised March 4, 2003. Recom- mended by Associate Editor Z. Lin. This work was supported in part by the Hong Kong Research Grant Council under Grant CUHK 4316/02E. W. Lan is with the Department of Automation and Computer-Aided Engi- neering, The Chinese University of Hong Kong, Hong Kong. J. Huang is with the Department of Automation and Computer-Aided Engi- neering, The Chinese University of Hong Kong, Hong Kong. He is also with the School of Automation Science and Engineering, South China University of Technology (e-mail: jhuang@acae.cuhk.edu.hk). Digital Object Identifier 10.1109/TAC.2003.814276 the same controller also solves the semiglobal stabilization (output reg- ulation) problem for the original singular systems with input saturation. It should be noted that the output regulation problem for linear sin- gular systems without input saturation is studied in [2] and [11]. The output regulation problem for singular nonlinear systems is formulated and solved in [8]. However, technically, this note is more relevant to [12]–[16]. Notations: For an -dimensional vector , , and . For a piecewise continuous bounded function , , and for , . This note is organized as follows. In Section II, we formulate the stabilization problem and output regulation problem of the singular linear systems with input saturation. Section III solves the stabiliza- tion problem for the singular systems with input saturation, while the output regulation problem is considered in Section IV. An example for semiglobal output regulation problem is given in Section V. Finally, Section VI concludes this note. II. PROBLEM STATEMENT AND PRELIMINARIES A singular linear system subject to input saturation is described as follows: (1) where is the system state, the control input, and the measurable output. , , , and are constant matrices. Without loss of generality, we assume that is singular and ( , ) is regular, i.e, . is a vector-valued saturation function defined as (2) where if if if We will consider the normal output feedback control law of the form (3) where for some integer , , , , , and are constant matrices to be determined. A. Semiglobal Stabilization Problem Given any compact sets containing the origin and containing the origin, find an output feedback control law (3) such that for all , the solution of the closed-loop system consisting of (1) and (3) exists for all and there exist and such that 0018-9286/03$17.00 © 2003 IEEE