Technical communiqueChattering-free discrete-time sliding mode control☆
Introduction
As a popular nonlinear control method, sliding mode control (SMC) has been widely studied in the theoretical research community (Utkin, 1992) and successfully applied in industry (Perruquetti & Barbot, 2002). The main reason is due to its many advantages, such as simple design idea and good robustness (Yu & Xu, 2002). Since more and more modern control systems are implemented by computers, the study of SMC in the discrete-time domain, i.e., discrete-time SMC, has been an important topic in the SMC literature (Yu, Wang, & Li, 2012).
The main difference between discrete-time SMC (DSMC) and continuous-time SMC (CSMC) lies in that the switching frequency of DSMC is limited, which leads to the celebrated invariance property of CSMC systems no longer holds (Drazenovic, 1969). In this case, the study of DSMC has been paid attention by many researchers and can be divided into two directions. One direction is to follow the design idea of CSMC and the switching term is still preserved, e.g., the design of DSMC law directly based on discrete-time systems (Gao et al., 1995, Qu et al., 2014) (usually called reaching-law-based DSMC), the study of discretization effect on continuous-time SMC systems (Galias and Yu, 2007, Li et al., 2014, Wang et al., 2009, Xia and Zinober, 2006, Yu et al., 2008). The other direction is based on the equivalent control for discrete-time system, which is called equivalent-control-based DSMC (Su et al., 2000, Utkin, 1994).
In the reaching-law-based DSMC, since the switching term is still employed, the chattering problem will be inevitable. In the equivalent control-based DSMC, although there is no switching term, it will generate an over-large control effort since there is no reaching process. Actually, in practice, due to the existence of disturbances, no matter which kinds of DSMC methods are employed, the sliding mode state cannot be precisely kept at zero. In such a case, the central issue is how to guarantee a smaller boundary layer for the sliding mode motion. In this regard, some improved DSMC methods have been proposed, such as disturbance observer-based DSMC (Su et al., 2000), discrete-time integral SMC (Abidi, Xu, & Yu, 2007), etc. However, these improved DSMC methods belong to the equivalent-control-based DSMC.
In this paper, we provide a new DSMC design method which is based on non-smooth control. The advantage of non-smooth control lies in its good performances such as better robustness (Bhat & Bernstein, 2000). In this new DSMC, to avoid the chatting phenomenon and the generation of overly large control action, a non-smooth term (continuous function) is employed instead of the switching term and a reaching process is added. Under the proposed DSMC, a rigorous theoretic analysis shows that the same accuracy for the sliding mode motion can be obtained as that of the equivalent-control-based DSMC, who provides a higher precision than that of the traditional reaching-law-based DSMC. Finally, an example is provided to demonstrate the potential of the proposed method.
Section snippets
System description
As in Wang et al. (2009) and Su et al. (2000), consider the following single-input continuous-time system with matched disturbances: where , and are the state, input, and disturbance, and are constant matrices of appropriate dimensions. The matching condition implies that . The disturbance is assumed to be smooth and bounded.
Assume the control law is digitally implemented through a zero-order-holder (ZOH), i.e., over the
New chattering free discrete-time SMC
How to guarantee not only a smaller boundary layer for sliding mode motion but also avoidance of the chattering phenomenon will be an interesting problem. In this section, we will employ a non-smooth control method to design a non-smooth control-based DSMC, which is a chatter-free DSMC method.
Theorem 3.1 For the discrete-time system (2) under Assumption 2.1, if the discrete-time sliding mode surface is chosen as (3), the disturbance estimation is defined as (6), and DSMC law is chosen as:
Conclusions
A new chattering-free DSMC method based on non-smooth control has been presented. It has shown that there is no chattering phenomenon and no need for an overly large control effort in the proposed DSMC. By synthesizing the benefit of non-smooth control, the high precision for sliding mode motion can be kept under the proposed DSMC, who provides a new choice for the design of DSMC law.
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This work is supported in part by the Natural Science Foundation of China (61304007, 61374053, 61473080) and the Australian Research Council (DP130104765). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Keqin Gu under the direction of Editor André L. Tits.