PII: S0005 – 1098(98)00102 – 2 Automatica, Vol. 34, No. 12, pp. 1547—1557, 1998  1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0005-1098/98 $—see front matter Brief Paper Simultaneous External and Internal Stabilization for Continuous and Discrete-Time Critically Unstable Linear Systems with Saturating Actuators* PING HOU,- ALI SABERI,- ZONGLI LIN‡ and PEDDAPULLAIAH SANNUTI Key Words—¸  stability; internal stability; simultaneous external and internal stabilization; saturation. Abstract—The issues arising in hybrid or simultaneous external as well as internal stabilization of linear systems with saturating actuators are considered. Four different stabilization problems are studied. Roughly, these problems are (1) simultaneous semi- global external as well as semi-global internal stabilization, (2) simultaneous semi-global external as well as global internal stabilization, (3) simultaneous global external as well as semi- global internal stabilization, and (4) simultaneous global ex- ternal as well as global internal stabilization. As evident from the literature, the requirement of internal stabilization alone either in the global or semi-global sense demands that the linear part of the given system be (a) stabilizable, and (b) has all its poles in the closed left half complex plane for continuous-time systems while it has all its poles inside and/or on the unit circle for discrete- time systems. This implies that the posed simultaneous stabiliz- ation problems are solvable at best only under the conditions (a) and (b). Under such conditions, we construct here explicit state as well as measurement feedback controllers for all the four problems in the case of continuous-time systems, and for the problems (1), (2) and (4) in the case of discrete-time systems. The design methodologies used to construct appropriate feedback laws are based on by now familiar low-gain and low-and-high gain design concepts or certain scheduled versions of them.  1998 Elsevier Science Ltd. All rights reserved. 1. Introduction Recently, there has been a renewed interest in the study of linear systems subject to actuator saturation, presumably owing to a wide recognition of inherent limitations of physical actuators (see, for a recent survey, Bernstein and Michel, 1995). Several types of studies on stabilizing such systems have already been done. These include, global, semi-global, and local internal or external stabilization, and global, semi-global, and local simul- taneous internal as well as external stabilization. The internal stability is also known as state-space stability. On the other hand, the external stability (see, Liu et al., 1996) is also known as the finite gain ¸  or l  stability, and is often discussed while studying the input—output behavior of systems. As is well known, the traditional global (or local) framework of analysis and design *Received 27 May1997; revised 8 January 1998; received in final form 19 May 1998. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Kenko Uchida under the direction of editor Tamer Bas,ar. Corresponding author Profes- sor Peddapullaiah Sannuti. Tel. # 1 732-445-3127; Fax# 1 732- 445-5313; E-mail sannuti@ece.rutgers.edu. - School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA 99164-2752, U.S.A. ‡ Department of Electrical Engineering, University of Vir- ginia, Charlottesville, VA 22903, U.S.A.  Department of Electrical and Computer Engineering, Rut- gers University, 94 Brett Road, Piscataway, NJ 08854-8058, U.S.A. requires methods and design techniques that are valid globally (or respectively locally). In contrast with this, as already pro- posed in a number of papers (see Lin and Saberi, 1993a, b, 1995a, b; Saberi et al., 1996; Lin, 1996a, 1997; Lin et al., 1996; Teel, 1992), the semi-global framework of analysis and design deals with methods and design techniques that are valid in an a priori given (arbitrarily large) region of operation. For both continuous- and discrete-time linear systems subject to actuator saturation, our interest here is to examine the issues that arise in local, semi-global, and global hybrid stabilization, which we label as simultaneous external as well as internal stabilization. A typical problem studied here is that of finding a controller C such that the operator (d  , d  ) >(y  , y  ) defined by the standard systems interconnection y  "P(d  #y  ), and y  "C(d  #y  ), (1) where P denotes the input/output behavior of the given open- loop system, is well-posed and is externally-stable, while C achieves simultaneously the internal stability of the closed- loop system. Different such problems arise depending upon the type of external and the internal stability required. We are concerned here with problems that arise when the external as well as the internal stabilization is required in either a global or a semi-global sense. The concept of either external or internal stabilization in a global sense is well understood in the literature. Also, as we mentioned earlier, the concept of internal stabilization in a semi-global sense has already been discussed in a number of papers. Semi-global stabilization requires local asymptotic stability of the equilibrium point of the closed-loop system with a critical requirement that the domain of attraction include a priori given bounded set. On the other hand, the concept of external stabilization in a semi-global sense was first introduced in Lin et al. (1995) and has recently been generalized in Hou et al. (1997). Providing ¸  -[or l  -] external stabilization in a semi-global sense is to provide ¸  -[or l  -] external stability for the closed-loop system while the external inputs are re- stricted to be in a closed ¸  -[or l  -] ball centered at the origin with an arbitrarily prescribed radius D. Thus, we are concerned here with four types of simultaneous stabilization problems: (1) Simultaneous ¸  -[or l  -] semi-global finite gain ¸  -[or l  -] sta- bilization and semi-global asymptotic stabilization (SG   /SG), (2) Simultaneous ¸  -[or l  -] semi-global finite gain ¸  -[or l  -] stabilization and global asymptotic stabilization (SG   /G) , (3) Simultaneous global finite gain ¸  -[or l  -] stabilization and semi-global asymptotic stabilization (G  /SG), and (4) Simulta- neous global ¸  -[or l  -] stabilization and global asymptotic stabilization (G  /G). These four problems are rigorously defined in Section 2. Our concern here is to solve these four problems for both continuous- and discrete-time systems, and for any p 3[1, R ] and q 3[1, R ] via either by the feedback of state (i.e. y  is the state of the plant P and d  "0) or by the measure- ment feedback. It is well understood in the literature that a lin- ear system with saturating actuators cannot be internally stabil- ized either in a global or semi-global sense if the linear part of the given system does not have all its poles in the closed left half 1547