Adaptive fuzzy non-singular TSM control for a class of MIMO nonlinear systems

An adaptive fuzzy non-singular terminal sliding mode(TSM) control scheme is proposed for a class of uncertain multi-input multi-output(MIMO) nonlinear systems with unknown control direction. The control gain matrix is decomposed into the sum of a symmetric matrix and a skew symmetric matrix, the properties of symmetric matrix are exploited to design adaptive fuzzy controller, and a Nussbaum-type function is incorporated in the controller to resolve unknown control direction issue. This method has two prominent advantages: it does not require the priori knowledge of control direction and only three parameters need to be updated on-line for this MIMO systems. It is proved that all the signals in the resulting closed-loop system are bounded and that the tracking errors converge to a small residual set in finite-time. Simulation result demonstrate the effectiveness of the proposed scheme.

1 Introduction

In the past two decades, by using the universal approximation theorem [1], adaptive fuzzy control schemes have been developed for uncertain MIMO nonlinear systems [2]–[7]. Since the approximation error may deteriorate the tracking performance, in [3], [4], the adaptive fuzzy controller is augmented by a robust controller to improve the dynamic performance. However, the key assumption in these schemes is that the control direction is known a priori. When this assumption is not satisfied, incorporating the Nussbaum function into the controller is an effective technique to solve this problem [3], [4]. The stability of these underlying control systems can be guaranteed in the Lyapunov sense. However, the finite-time convergence of tracking errors can not be achieved. TSM control is an effective scheme to achieve finite-time stabilization, several methods have been proposed in [8]–[12] including Non-singular TSM, fast TSM, continuous non-singular TSM. In [13], the continuous non-singular TSM scheme in [11] has been extended to a class of MIMO nonlinear systems. However, the limitation in this scheme is that the control direction is known a priori.