J Control Theory Appl 2010 8 (2) 245–248
DOI 10.1007/s11768-010-7261-9
Stability analysis for linear discrete-time systems
subject to actuator saturation
Yongmei MA
1,2
, Guanghong YANG
2,3
(1.College of Science, Yanshan University, Qinhuangdao Hebei 066004, China;
2.College of Information Science and Engineering, Northeastern University, Shenyang Liaoning 110004, China;
3.Key Laboratory of Integrated Automation of Process Industry (Ministry of Education),
Northeastern University, Shenyang Liaoning 110004, China)
Abstract: In this paper, stability of discrete-time linear systems subject to actuator saturation is analyzed by combin-
ing the saturation-dependent Lyapunov function method with Finsler’s lemma. New stability test conditions are proposed
in the enlarged space containing both the state and its time difference which allow extra degree of freedom and lead to
less conservative estimation of the domain of attraction. Furthermore, based on this result, a useful lemma and an iterative
LMI-based optimization algorithm are also developed to maximize an estimation of domain of attraction. A numerical
example illustrates the effectiveness of the proposed methods.
Keywords: Linear systems; Actuator saturation; Estimation of domain of attraction; LMIs
1 Introduction
Saturations are probably the most commonly encountered
nonlinearity in control engineering because of the physi-
cal impossibility of applying unlimited control signals. It
is well known that the input saturation is a source of per-
formance degeneration, limit cycles, different equilibrium
points, and even instability. Hence, the study of saturation
systems has received growing attention, and various meth-
ods have been developed on the analysis and design of satu-
rating control laws and other performance analyses: the es-
timation of the domain of attraction, disturbance tolerance,
L
2
gain analysis, etc. (see, for example, [1]∼ [10] and ref-
erences therein).
One of the most relevant approaches to the analysis of sat-
urated systems is based on a novel polytopic model of the
saturation nonlinearity which was proposed in [6]. Based
on that, stability/performance analysis and controller de-
sign have been studied by developing various Lyapunov
functions, for example, quadratic Lyapunov function [6];
saturation-dependent Lyapunov function [2] (parameter-
dependent Lyapunov function), [11] (for linear discrete-
time system); convex hull quadratic Lyapunov function
and max quadratic Lyapunov function [12] (for linear
continuous-time system). The advantages of using the poly-
topic model have been shown in [13], etc. However, all the
existing results were obtained only in the state space by us-
ing the Lyapunov function approach solely. Obviously, in
this case, the degree of freedom is restricted within narrow
limits, compared with an enlarged space.
The objective of this paper is to investigate the estima-
tion problem of domain of attraction under a given feedback
law for discrete-time linear systems subject to actuator sat-
urations by combining the saturation-dependent Lyapunov
function method with Finsler’s Lemma. Firstly, new stabil-
ity test conditions are proposed in the enlarged space con-
taining both the state and its time difference which allow
extra degree of freedom and lead to less conservative es-
timate of the domain of attraction. Furthermore, based on
this result, an important lemma and an iterative LMI-based
optimization algorithm are also developed to maximize the
estimation of domain of attraction. The effectiveness of the
proposed methods is illustrated by numerical examples.
The paper is organized as follows. Section 2 gives the
problem under consideration. In Section 3, for an estima-
tion of domain of attraction, new stability test conditions,
an useful lemma and an iterative LMI-based optimization
algorithm are presented, respectively. Numerical examples
and simulation results are given to show the effectiveness of
the proposed methods in Section 4. Conclusions are made
in Section 5.
Notation For a vector υ ∈ R
n
, we denote the stan-
dard multivariable saturation function as σ(υ)=[σ(υ
1
)
σ(υ
2
) ···σ(υ
n
)]
T
, where σ(υ
i
) = sign(υ
i
)min{1, |υ
i
|},
denotes the transpose of the off-diagonal element of a ma-
trix. I(0) represents the identity(null) matrix of appropri-
ate dimension. For two integers k
1
,k
2
,k
1
<k
2
, [k
1
,k
2
]=
{k
1
,k
1
+1, ··· ,k
2
}.
2 Problem statement
Consider a discrete-time linear system subject to input
saturation
x(k +1)=Ax(k)+Bσ(u(k)), (1)
Received 6 December 2007; 14 November 2008.
This work was partly supported by Program for New Century Excellent Talents in University (No.NCET-04-0283), the Funds for Creative Research
Groups of China (No.60521003), Program for Changjiang Scholars and Innovative Research Team in University (No.IRT0421), the State Key Program
of National Natural Science of China (No.60534010), the Funds of National Science of China (No.60674021), the Funds of PhD program of MOE,
China (No.20060145019), and the 111 Project (No.B08015).
c
South China University of Technology and Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2010