Stabilization of Uncertain Systems using Backstepping and Lyapunov Redesign Muhammad Nizam Kamarudin 1 Faculty of Electrical Engineering Universiti Teknikal Malaysia Melaka Malacca, Malaysia Email: 1 nizamkamarudin@utem.edu.my Abdul Rashid Husain 2 , Mohamad Noh Ahmad 3 Faculty of Electrical Engineering Universiti Teknologi Malaysia Johore, Malaysia Email: 2 rashid@fke.utm.my, 3 noh@fke.utm.my Abstract This article presents stabilization method for uncertain system using backstepping technique and Lyapunov redesign. The design begins by obtaining stabilizing function for unperturbed system using control Lyapunov function. As such, the stabilizing function guarantees asymptotic stability in the sense of Lyapunov. The control Lyapunov function is re-used in the re-design phase whereby the nonlinear robust function is then augmented with pre-designed stabilizing function for robustness toward uncertainties. Lyapunov redesign is used in designing overall robust stabilizing function which guarantees asymptotic stability toward uncertainties and also toward any perturbed states within asymptotic stability region. I. INTRODUCTION Backstepping is one of control technique used for nonlinear systems. Many current works on backstepping reveal that the technique increasingly widespread. Backstepping technique also has been used for adaptive control in [1], for position control of electro-hydraulic actuator in [2], for wheeled mobile robot in [3], for spacecraft attitude control in [4-6], for quadrotor in [4-8] and for ship course control in [9]. Backstepping has also been integrated with artificial intelligence methods to improve robustness such as in fuzzy system [10] and artificial neural network [11]. Bakcstepping technique is a flexible concept as compared with its counterpart such as sliding mode control [12]. Often the discontinuous control law offered by sliding mode technique require fast switching and robust only during sliding phase, backstepping offers smooth and proper control law. The flexibility of this technique give advantages to designer when some special criteria of the control law is emphasized. For instance, backstepping can avoid elimination of some useful nonlinear terms. The control law also can be easily be bounded to respect system admissible set of inputs. Moreover in backstepping, an advanced mathematics can be employed in the design process such that the most feasible and less complex control law is obtained. This may require comparing squares or Young’s inequality [13]. In this paper, with consideration of system like     with   , we discus on how the recursive method of backstepping technique can be applied with Lyapunov redesign to achieve robust control law for system with . Backstepping and Lyapunov redesign has been used for Ball and Beam system and two wheeled mobile inverted pendulum in [14] and for uncertain nonlinear system with mixed matched mismatched uncertainties in [13, 15]. In our case, with the appearance of uncertainties cum external disturbance, we design the feedback stabilizing function for unperturbed nominal system based-on positive definite, smooth and proper function such that the derivative of such function about system under control is negative definite. The guaranteed asymptotic stability of the closed- loop unperturbed system is obtained easily as long as the necessity and sufficiency of Lyapunov theorem is fulfilled. The trivial part in the design phase may occur during Lyapunov redesign when the augmented control law is about to design. The augmented control law may be a saturation-like function [15], nonlinear damping function [13] or others. II. PROBLEM FORMULATION Our design procedure is based on the following class of system [12]:           where   is the state,   is the control input and          is the lumped uncertainty which matched to the control input.   and   is the system characteristic and input matrix with full rank . For design purpose, let consider    !   " # with vector and " are smooth, that is $  $ for nonlinear case. System (1) is affine in control with matched uncertainties. Thus the control design begins with stabilization of unperturbed nominal subsystem using Lypaunov, hence back step for higher dimension and redesigned for List of Accepted Papers 4th IGCESH, 2013 List of Accepted Papers 4th IGCESH, 2013 160