29th Benelux Meeting
on
Systems and Control
March 30 – April 1, 2010
Heeze, The Netherlands
Issues on global stabilization of linear systems subject to actuator
saturation
Tao Yang
School of Electrical Engineering and Computer Science,
Washington State University,
Pullman, WA 99164-2752, U.S.A.
E-mail: tyang1@eecs.wsu.edu
Anton A. Stoorvogel
Department of Electrical Engineering, Mathematics and Computer Science,
University of Twente,
P.O. Box 217, Enschede, The Netherlands
E-mail: A.A.Stoorvogel@utwente.nl
Ali Saberi
School of Electrical Engineering and Computer Science,
Washington State University,
Pullman, WA 99164-2752, U.S.A.
E-mail: saberi@eecs.wsu.edu
1 Abstract
Linear systems subject to actuator saturation are ubiquitous and have been the subject of extensive study. Internal stabi-lization for this class of systems has a long history. Fuller’s paper [?] established that a chain of intergrators with order greater or equal to three cannot be globally stabilized by any saturating linear static state feedback law with only one input channel. Sontag and Sussmann [?] established that, global stabilization of continuous-time linear systems with bounded input can be achieved if and only if the linear sys-tem in the absence of actuator saturation is stabilizable and critically unstable (equivalently, asymptotically null
con-trollable with bounded control(ANCBC)). In general, this
requires a nonlinear feedback laws. However, for certain cases, the global stabilization can be achieved by linear con-trol laws. More precisely, the paper [?] noted that systems which are asymptotically null controllable with bounded in-puts can be globally stabilized by linear static state feedback control laws if all non-zero eigenvalues on the imaginary axis are semi-simple (geometric and algebraic multiplicities are equal) while zero is allowed to be an eigenvalue whose Jordan blocks can be at most of size 2×2 (which are associ-ated with double integrators). The quoted paper [?] does not give a full proof of this result. In this paper we prove this result. Moreover, this proof is constructive.
Another issue is that in the literature, there is this general belief that if there are Jordan blocks of size greater or equal to three associated to an eigenvalue in zero then we need nonlinear controllers. This is a misconception. Such a
mis-conception is possibly based on a misreading of the result of [?]. One should emphasize that the beautiful result of Fuller does not claim anything beyond static state feedback con-trollers for chains of integrators. In this paper we illustrate this issue by showing that a triple-integrator with multiple inputs subject to actuator saturation can be globally stabi-lized by a linear static state feedback. This is clearly a first step towards a better understanding when nonlinear static state feedbacks are needed.
Two general open problems are still unresolved; (1) under what conditions one can utilize a linear static state feedback control law to globally stabilize a linear system subject to actuator saturation?, and (2) under what conditions one can utilize a linear dynamic state feedback control law to glob-ally stabilize a linear system subject to actuator saturation?
References
[1] A.T. FULLER, “In-the-large stability of relay and sat-urating control systems with linear controller”, Int. J. Contr., 10(4), 1969, pp. 457–480.
[2] E.D. SONTAG AND H.J. SUSSMANN, “Nonlinear output feedback design for linear systems with saturating controls”, in Proc. 29th CDC, Honolulu, 1990, pp. 3414– 3416.
[3] F. TYAN AND D.S. BERNSTEIN, “Global stabiliza-tion of systems containing a double integrator using a satu-rated linear controller”, Int. J. Robust & Nonlinear Control, 9(15), 1999, pp. 1143–1156.
Book of Abstracts 29th Benelux Meeting on Systems and Control
