A Contractive Sliding- mode MPC Algorithm for Nonlinear Discrete- time Systems

International Journal of Automation and Computing, Apr 2013

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 effectiveness of the proposed method.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

http://link.springer.com/content/pdf/10.1007%2Fs11633-013-0709-x.pdf

A Contractive Sliding- mode MPC Algorithm for Nonlinear Discrete- time Systems

Meng Zhao 0 Bao-Cang Ding 0 0 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)


This is a preview of a remote PDF: http://link.springer.com/content/pdf/10.1007%2Fs11633-013-0709-x.pdf

Meng Zhao, Bao-Cang Ding. A Contractive Sliding- mode MPC Algorithm for Nonlinear Discrete- time Systems, International Journal of Automation and Computing, 2013, pp. 167-172, Volume 10, Issue 2, DOI: 10.1007/s11633-013-0709-x