Gain-scheduling Distributed MPC for Polytopic Uncertain Systems
subject to Actuator Saturation
Langwen Zhang
*
, Jingcheng Wang
*
, Kang Li
**
*Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and
Information Processing, Ministry of Education of China, Shanghai 200240
(e-mail: zhang.langwen@163.com; jcwang@sjtu.edu.cn)
**School of Electronics, Electrical Engineering and Computer Science Queen’s University Belfast, UK
(e-mail: k.li@qub.ac.uk)
Abstract: In this paper, we present a gain-scheduling distributed model predictive control (MPC)
algorithm for polytopic uncertain systems subject to actuator saturation. A large-scale system is
decomposed into subsystems and sub-controllers are designed independently. An invariant set condition is
provided and a min-max distributed MPC strategy is proposed based on the invariant set. The distributed
MPC controller is determined by solving a linear matrix inequality (LMI) optimization problem. An
iterative algorithm is provided to coordinate the sub-controllers. A numerical example is carried out to
demonstrate the effectiveness of the proposed algorithm.
1. INTRODUCTION
Model predictive control technique has been adopted in a
wide variety of application areas (Qin & Badgwell, 2003). At
each time interval, a MPC algorithm optimizes a cost
function, which is associated with the predicted future states
and inputs. Only the first input in the optimal sequence is
injected into the processes, and the procedure is repeated at
subsequent control intervals. The optimal solution depends
on a linear dynamic model of the plant, respects all output
and input constraints, and minimizes a performance index.
An impressive review of theoretical results on the closed-
loop behavior of MPC algorithms is provided in (Mayne,
Rawlings, Rao, & Scokaert, 2000).
The closed-loop system performance can be severely
degraded when the actuator is saturated. Many researchers
have focused on the actuator saturation problem (Bernstein &
Michel, 1995; Cao & Lin, 2005; Casavola, Giannelli, &
Mosca, 2000; Casavola & Mosca, 2007; Hu, Lin, & Chen,
2002; D. Li, Xi, & Lin, 2011; Wu, Lin, & Zheng, 2007).
Usually, there are two approaches to deal with actuator
saturation problem: designing low gain control laws, which
can avoid saturation, or estimating the domain of attraction
(Cao & Lin, 2005). The latter one is common with its
advantage of less conservative. A new centralized MPC
algorithm was developed for LPV systems subject to input
saturation in (Cao & Lin, 2005). (Cao & Lin, 2003) discussed
the stability of discrete-time systems with actuator saturation
by a saturation-dependent Lyapunov function approach.
In recent years, there has been significant interest in the study
of LPV systems (Cao & Lin, 2005; Cao, Lin, & Shamash,
2002; Yun, Choi, & Park, 2010). LPV systems are systems
that depend on measurable time-varying parameters. The
measurement of these parameters provides real-time
information on the variations of the plant characteristics (Cao
& Lin, 2005). Hence, it is desirable to design controllers that
are scheduled based on the real-time information on the
variations of the plant characteristics. Gain-scheduling
approach is very popular approaches to LPV control systems
design and has been successfully applied in process control
(Cao, et al., 2002; Gao & Budman, 2005; Jungers & Castelan,
2011; Leith & Leithead, 2000; Rugh & Shamma, 2000; Yin,
Shi, & Liu, 2011).
The motivation of this paper is to reduce the conservatism by
gain-scheduling approach in distributed MPC design. The
Lyapunov function we consider is saturation-dependent,
which captures the real-time information on the severity of
saturation and thus leads to a less conservative MPC design
(Cao & Lin, 2003). The proposed distributed controller is
determined by solving an LMI optimization problem.
The paper is organized as follows. In section 2, problem
formulations are presented. Gain-scheduling distributed MPC
strategy is presented in section 3. Numerical example is
carried out in section 4 to demonstrate the effectiveness of
the proposed strategy. The paper is concluded in section 5.
2. PROBLEM FORMULATIONS
Consider a discrete-time linear input-saturated system:
(1) ()
kAkxkBkuk
(1)
where
n
R is the system state;
m
uR is the control input;
and B are system matrices of appropriate dimensions;
The function
():
mm
RR
is the standard saturation
function of appropriate dimensions defined as
1
T
m
uu u
in which
i
uk
min 1,
ii
ign u k u k . It is assumed that
k and