Sliding Mode Observers for Rotational Robotics Structures

Abstract

The problem of controlling uncertain dynamical systems subject to external disturbances has been an issue of significant interest over the past several years. Most systems that we encounter in practice are subjected to various uncertainties such as nonlinearities, actuator faults parameter changes etc. Many of the proposed control strategies suppose that the state variables are available; this fact is not always true in practice, so the state vector must be estimated for use in the control laws. In the past, several types of observers have been designed for the reconstruction of state variables: Kalman filter (Kalman, 1976), adaptive observers (Gevers & Bastin, 1986), high gain observers (Gauthier et al., 1992), sliding mode observers (SMO) (Utkin, 1992; Walcott & Zak, 1986; Edwards & Spurgeon, 1994) and so on see (Thein & Misawa, 1995) for some comparisons. Depending upon the particular application, all these observers can be used with suitable results. Sliding mode observers differ from more traditional observers in that there is a non-linear discontinuous term injected into the observer depending on the output estimation error. These observers are known to be much more robust than Luenberger observers, as the discontinuous term enables the observer to reject disturbances (Tan & Edwards, 2000). The observers based on the variable structure systems theory and sliding mode concept can be classified in two categories (Xiong & Saif, 2000): 1) the equivalent control based methods and 2) sliding mode observers based on the method of Lyapunov. The analysis of these two types of SMO (Edwards & Spurgeon, 1994; Xiong & Saif, 2000) shows that there exist some differences in terms of robustness properties. From practical point of view, the selection of the switched gain for the equivalent control based SMO is difficult (in order to obtain a sliding mode without excessive chattering) (Edwards & Spurgeon, 1994). Also, there exists bounded estimation error for bounded modelling errors (the estimation will not be accurate when uncertainties are presented) (Xiong & Saif, 2000). The Lyapunov based SMO (the so-called Walcott-Zak observer) provides exact estimation for certain class of nonlinear systems under existence of certain class of uncertainties. However, the difficulty in finding the design and gain matrices is the main drawback of this observer. Consider the effect of adding a negative output feedback term to each equation of the Utkin observer. This results in a new error system. The addition of a Luenberger type gain matrix, feeding back the output error, yields the potential to provide robustness against certain classes of uncertainty.