Parameter tuning of second-order sliding mode controllers for linear plants with dynamic actuators

Abstract

Tuning of second-order sliding mode control (2-SMC) algorithms in linear systems with dynamic actuators is considered. By means of a describing function (DF) approach, it is investigated how the parameters of a 2-SMC algorithm (the so-called “generalized sub-optimal” algorithm) affect the frequency and the magnitude of the limit cycles that occur when the overall relative degree of the plant plus the actuator is three or more. Explicit formulas are given that allow for setting the parameters of the algorithm to obtain a periodic solution with the prescribed characteristics. By means of simulation examples, we show that the estimated chattering parameters are in good agreement with the actual ones. We also show that the proposed design procedure can also be useful when the local linearization of a nonlinear dynamics is sufficiently accurate.

Introduction

Sliding mode control (SMC) is a popular approach to control system design under heavy uncertainty, which remains one of the main subjects of modern control theory. SMC is precise, rather insensitive to disturbances (Utkin, 1992, Utkin et al., 1999), and usually very simple to implement.

The main drawbacks of classical first-order Sliding Modes (1-SM) are principally related to the so-called chattering effect (Boiko, 2003, Fridman, 2001, Utkin et al., 1999). The main cause of chattering has been identified as the presence of unmodelled parasitic dynamics in the switching devices (Bondarev et al., 1985).

Three main approaches to counteract the chattering phenomenon in SMC systems were proposed in the mid-eighties: the use of a continuous approximation of the relay (e.g. the saturation function (Burton and Zinober, 1986)), the use of an asymptotic state-observer to confine chattering in the observer dynamics bypassing the plant (Bondarev et al., 1985), the use of higher-order sliding mode control algorithms (Emelyanov et al., 1986).

The higher-order sliding mode approach has been actively developed over the last two decades not only for chattering attenuation but also for the robust control of uncertain systems with relative degree two and higher (see Bartolini et al., 2003; Bartolini et al., 2001; Fridman and Levant, 2002, Levant, 1993, Levant, 2003, Orlov et al., 2003, Sira-Ramirez, 2002 and references therein).

The main drawbacks of the continuous approximations and of the observed-based approach are the deterioration of accuracy and system robustness, respectively. In recent papers (Boiko and Fridman, 2004, Boiko et al., 2004; Boiko et al., 2004a) it has been shown that even the second-order sliding mode (2-SM) algorithms suffer from chattering if parasitic dynamics are present increasing the system relative degree.

In this note we consider the generalized sub-optimal control algorithm (Bartolini et al., 2001, Bartolini et al., 2003), which is the generalization of a 2-SM control algorithm derived from the time-optimal control law of a double integrator (Bartolini et al., 1997). We analyze the dependence of the frequency and the amplitude of chattering from the tuning parameters of the algorithm when it applied to dynamical systems with relative degree three and higher. The capability of affecting the frequency of the residual steady state oscillations may be useful, for example, to keep it far enough from the resonant frequencies of the plant.

There are two main approaches to chattering analysis: the time-domain analysis of the system dynamics or the use of frequency-domain techniques.

Analysis of the magnitude of the oscillations based on singularly perturbed relay systems was developed in Utkin (1992) and Fridman (2002). Poincare maps and LMIs have been suggested to investigate the existence and stability of periodic solutions in relay systems (Di Bernardo et al., 2001, Goncalves et al., 2001 and the reference therein). A special decomposition of Poincare maps allowing for analyzing chattering in 1-SM control systems has been proposed in Fridman (2001). Preliminary results regarding the time-domain analysis of chattering in 2-SMC systems were presented in Boiko et al. (2004a).

When linear plants are considered several frequency-domain techniques can be used to assess the existence and stability of periodic solutions. The Tsypkin locus method (Tsypkin, 1984) provides exact values of the amplitude and frequency of chattering. The recently proposed “Locus of a Perturbed Relay System” (LPRS) method (Boiko, 2005) gives the exact values of chattering frequency and amplitudes and also allows to perform some robustness analysis (Boiko, 2003). All these approaches require quite tedious computations. Therefore the application of the approximate analysis method based on the Describing Function (DF) technique could be useful whenever the low-pass filtering condition is satisfied (Atherton, 1975). The DF method has already been used to estimate the frequency and the amplitude of the periodic motions in the 1-SMC systems (Shtessel and Lee, 1996, Zhilcov, 1974). The results obtained via the use of exact frequency-domain techniques feature a satisfactory correspondence with those obtained via the approximate DF method (Boiko, 2003).

In the present paper a parametric relay representation of the generalized sub-optimal 2-SMC algorithm (Bartolini et al., 2003) is given. The “Twisting” algorithm (Levant, 1993), the “sub-optimal” algorithm (Bartolini et al., 1997) and even the classical relay (Utkin, 1992) can be obtained as particular cases. Such representation is exploited for analysis and design purposes in the frequency domain in order to provide effective tuning rules for chattering attenuation. We assume that the cascade actuator-plant dynamics is a low-pass filter. We also assume that the steady-state oscillations are periodic, symmetric and with zero mean. The analysis results give the designer useful tuning rules to set the controller parameters so as the chattering effect is counteracted.

This paper is organized as follows: in Section 2 we formulate the problem under investigation and detail the class of controlled plants we are concerned with. Section 3 contains the main results, namely a DF analysis of the considered class of control systems and the derivation of tuning rules for setting the parameters of the 2-SMC algorithm. In Section 4 the proposed tuning procedure is applied and verified by means of computer simulations. The estimated chattering parameters (amplitude and frequency of the periodic solution) obtained via the DF analysis are shown to be in good agreement with the simulated system's behaviour. Section 5 provides concluding remarks.