Robust Control of Nonlinear Systems: Sliding mode and Non-monotonic Approaches

Abstract

The field of robust controller design for nonlinear systems has been an important area of research in the control community over the past few decades. Lyapunov based control design methods have developed into an established class of procedures for robust control design. This dissertation initially applies the standard Lyapunov function approach to design an adaptive sliding mode controller (SMC). Next, the non-monotonic Lyapunov approach, which is a variant of the classical Lyapunov approach, is utilised to design a robust controller for discrete-time nonlinear systems represented by a Takagi-Sugeno (T-S) fuzzy model. An adaptive scheme of designing SMC for the affine class of multiple input multiple output (MIMO) nonlinear systems with norm bounded uncertainties is proposed during the first stage of the research. The proposed adaptive SMC does not require a priori knowledge of the bounds of uncertainties and therefore offers significant advantages over non-adaptive schemes of SMC design. Next, this controller is developed to be fault tolerant for actuator faults. The actuator fault is represented by a multiplicative factor of the control signal which reflects the loss of actuator effectiveness. The design of the controller is based on the assumption that the maximum loss level of actuator effectiveness is known. The effectiveness of the proposed controllers is demonstrated considering a two-link robot manipulator. The stability analysis and stabilisation of discrete-time nonlinear systems described by T-S fuzzy models are investigated and new design procedures are proposed in the second stage of this thesis. The stability analysis of T-S fuzzy models and subsequently the controller design are often carried out through the Lyapunov method. This leads to some inherent inequalities in which the nonlinear membership functions are excluded from the final analysis and synthesis conditions. As a result the stability conditions, which are often constructed in the form of linear matrix inequalities (LMIs), become only sufficient (conservative). To reduce this conservatism, the assumption of monotonic decrease of Lyapunov function has been relaxed giving a new approach to the control design known as the non-monotonic Lyapunov approach, where the Lyapunov function is no longer required to decrease in each successive time step (i.e., Vt+1 < Vt ). Instead, small local growth is allowed but ultimately it converges to zero. In recent years, various researchers have used the k-samples variations approach (i.e., Vt+k <Vt ) as a special form of non-monotonic Lyapunov approach. In this thesis, a new and more general from of non-monotonic Lyapunov function is proposed for stability and stabilisation of T-S fuzzy systems. Consequently, the conservatism is further reduced. There has been a lack of positive results in output feedback control for discrete-time uncertain T-S fuzzy systems using the non-monotonic Lyapunov approach, which is more practical than state feedback control. In this thesis, a robust output feedback controller for discrete-time uncertain T-S fuzzy systems is proposed and sufficient conditions for the existence of the controller are derived in terms of LMIs. Then, an H∞ robust state-feedback controller, which is often designed to minimise the ratio of the controlled output energy to the disturbance energy, is investigated for uncertain T--S fuzzy systems using a non-monotonic Lyapunov function. Finally, sufficient conditions for the existence of the robust H∞ controller are derived in terms of LMIs. It is well-known in the control community that in practical control systems measurement of all system states is not always possible and sometimes expensive. So, output feedback control is preferred. This situation provided the motivation to design a robust H∞ output feedback controller. The last stage of the research focuses on filtering issues to estimate the system state information for nonlinear systems represented by a T-S fuzzy model in the presence of disturbance and system uncertainties. A robust H∞ filter is designed following a non-monotonic Lyapunov approach to reduce the conservatism of existing filter design methods and improving the H∞ performance index.