A Contractive Sliding- mode MPC Algorithm for Nonlinear Discrete- time Systems
Meng Zhao
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Bao-Cang Ding
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College of Automation, Chongqing University
, Chongqing 400044,
China
This paper investigates a sliding-mode model predictive control (MPC) algorithm with auxiliary contractive sliding vector constraint for constrained nonlinear discrete-time systems. By adding contractive constraint into the optimization problem in regular sliding-mode MPC algorithm, the value of the sliding vector is decreased to zero asymptotically, which means that the system state is driven into a vicinity of sliding surface with a certain width. Then, the system state moves along the sliding surface to the equilibrium point within the vicinity. By applying the proposed algorithm, the stability of the closed-loop system is guaranteed. A numerical example of a continuous stirred tank reactor (CSTR) system is given to verify the feasibility and efiectiveness of the proposed method.
1 Introduction
Model predictive control (MPC), also called moving
horizon control (MHC) and receding horizon control (RHC), is
the most attractive control strategy for systems with input
and state constraints. The current control action of MPC
is obtained by solving a flnite horizon optimization problem
at each sampling time, and the flrst one is applied to the
plant. At the next sampling time, the same procedure is
repeated.
Linear MPC (LMPC) is a control scheme for linear
systems, which has been studied extensively[1]. However, most
of the practical systems have nonlinearities. Hence, the
nonlinear MPC (NMPC) algorithms should be applied
instead of LMPC strategies in order to get the high quality
of control performance. Because of the inherent
diculties in analyzing nonlinear control systems, NMPC theory
is far from perfect and many challenges still exist, such as
stability, robustness, computational burden, etc.[25]
The major diculty of NMPC is guaranteeing the
closedloop stability. In order to guarantee the closed-loop
stability, various stability constraints have been proposed. The
simplest approach is to add a terminal equality constraint
into the optimization problem[6]. It requires that the state
exactly converges to zero in flnite steps. It is
conservative, and the optimization problem may become infeasible.
For relaxation, the terminal inequality constraint is applied,
where terminal state is enforced to a region which includes
equilibrium point in its interior, instead of a point
(equilibrium point). By combining the terminal cost function
with terminal inequality constraint, Chen and Allgower[7]
proposed a quasi-inflnite NMPC strategy, which can get the
inflnite horizon control performance by minimizing the
upper bound of inflnite horizon cost functions. Oliveira and
Morari[8] proposed the contractive constraint, which adds
a terminal contractive constraint in the optimization
problem to guarantee the system stability. In order to prove
the closed-loop stability, a block optimization strategy is
Manuscript received August 2, 2012; revised September 25, 2012
This work was supported by Fundamental Research Funds for the
Central Universities (Nos. CDJXS10170008 and CDJXS10171101).
adopted. Xie[9] presented the flrst state contractive NMPC
algorithm, in which the contractive constraints are enforced
on the one-step ahead predicted state. Sun et al.[10]
presented another contractive NMPC algorithm, which adopts
a time-varying implementation horizon conflrmed by
solving an appropriate optimization problem.
As an important branch of variable structure
control[1113], sliding mode control (SMC) is
characterized by switching the control law during the evolution
of the state, and enforcing the states to the predeflned
asymptotic stable sliding surface. We call the control
algorithm which combines MPC with SMC the sliding-mode
MPC (SM-MPC). Parte et al.[14] designed a generalized
predictive control (GPC) method based on sliding mode
controller. Xiao et al.[15] addresed a similar approach,
where the model algorithm control (MAC) is used. Zhou
et al.[16] presented an SM-MPC algorithm for systems with
state space model, which takes sliding vector as a new
variable, and stabilizes it by dual-mode MPC.
Inspired by [16], this paper takes sliding vector as a new
variable, and stabilizes it by MPC algorithm with extra
contractive sliding vector constraint for constrained
nonlinear systems. This makes the sliding vector contract to zero
step by step. It implies that the system state implicitly
satisfles the reaching condition. The proposed algorithm
improves the overall feasibility, and avoids the switching
between inner mode controller and outer mode controller.
The closed-loop stability is guaranteed if asymptotic stable
sliding mode is predesigned.
This paper is organized as follows. Section 2 describes
the problem to be studied. Section 3 presents the new
contractive SM-MPC. The stability is discussed and proved in
Section 4, which is mainly inspired by the method in [8]. In
Section 5, we apply the proposed algorithm to a pract (...truncated)