Abstract— Lyapunov’s direct method is useful technique especially for nonlinear systems. Lyapunov function is energy-like function. This function may draw conclusions about the stability of system without solving the set of non linear equations. Discovering a Lyapunov function for particular system is really a hard job which requires experience and physical insight. Stability problem is a real problem for any control system which requires appropriate control law for a given plant. This paper presents Lyapunov’s local stability for uni-axial vehicle by designing appropriate control law. After the designing of control law candidate Lyapunov function becomes a real Lyapunov function. Index Terms— Nonlinear control, Lyapunov’s direct method, , Local stability, State-space. I. INTRODUCTION Nonlinear control is one of the biggest challenges in modern control theory. Nonlinear processes are difficult to control because there can be so many variations of the nonlinear behavior. The second method of Lyapunov also know as Lyapunov s direct method [1] , is becoming increasingly recognized as having great potentiality, both for resolving nonlinear stability and performance problems. Lyapunov’s direct method is now being widely used for designing stable controllers for various fields like Power system stabilizing controllers [2], Adaptive Control of a MEMS Gyroscope [3], replicator system with entropy-like applications [4], Neuro stabilizing control for power systems [5], Suppression Control of Rotor Oscillation for Stepping Motor [6], Application to induction machines [7]. In order to find the stability characteristics of the system we have to solve the system equations. Sometimes especially for nonlinear systems it is hard to solve system equations. Lyapunov’s theory purpose a way to analyze stability of the system without necessarily solving the system equations [1]. Lyapunov’s direct method is a useful technique for determining the stability of non linear system [1]. An energy-like scalar function is generated through this method. Stability of the system can be examined, through variations of that function [8]. There is no effective approach to find the lyapunov function; it requires experience, physical insight and trial-error approach. Candidate lyapunov function has to fulfill certain conditions in order to become a valid lyapunov Ahmed Farid ,Tehseen Zia, Fahad Maqbool, Saad Razzaq, fahad shahbaz, kashif irfan are with Department of Computer Sciences & IT ,University of Sargodha ,Pakistan e-mail ahmedfarid80@yahoo.com,:msaadrazzaq@yahoo.com,fahadmaqbool@yah oo.com, ,fahadjee@yahoo.com,kashif_irfan31@hotmail.com function i.e. it should be zero at equilibrium and positive elsewhere (positive definite function)[8][9]. In many control problems the task is to find an appropriate control law for a given plant [8]. Stable control systems can be designed by using lyapunov’s direct method. In this paper we have designed a control law for uni-axial vehicle by hypothesizing a lyapunov fuction candidate. Later in the paper it is showed that this candidate function is a real lyapunov function. Section 2 presents lyapunov’s applications in various fields of stability in sense of lyapunov and conditions for real lyapunov function. In the section 3, dynamics of uni-axial vehicle are described. Section 4 contains the hypothetical lyapunov function, control law and lyapunov’s local stability proof for uni-axial vehicle. In the fifth section simulations and results are shown and last section concludes our work.Procedure for Paper Submission. II. RELATED WORK Lyapunov’s direct method is applied to design various controllers, their brief description is provided in this section. M. Januszewski, J. Machowski and J.W. Bialek presented in [2] an approach to improve damping of power swings. This is done by using the unified power flow controller (UPFC). Available signals of real and reactive power are used and then state-variable strategy has been derived. This state-variable control provides damping independent of operating conditions. Robert P. Leland has explained [3] two adaptive controllers for a vibrational MEMS gyroscope. One based on low frequency model and other based on the full gyroscope model. Good transient response is obtained force-to-rebalance and automatic gain control loops using The Lyapunov function used is critical in obtaining a good transient response, especially for the force-to-rebalance and automatic gain control loops. Yuri A. Pykh has shown in [4] that there exists entropy-like lyapunov function. For replicator systems known entropy measures may be obtain from entropy-like Lyapunov function. Hirata, Atsushi and Nishigaito used [5] Particle swarm optimization to find set of connecting weights matrix of NN. Probability of agents entrapped into local optima by improving algorithm is reduced. Senjyu, Nakahama and Uezato achieved [6] the suppression control of the rotor oscillation by excitation control using Lyapunov function. Because of this the rotor oscillation was effectively vanished by the simple excitation sequence and quick tracking control of rotor position is becomes possible. Ludvigsen, Ortega, Albertos and Egeland studied in [A] that, due to technological or information transmission considerations, fast switching is not possible. Predictive Control Law for UNI-Axial Vehicle Using Lyapunov Analysis Ahmed Farid, Tehseen Zia, Fahad Maqbool, Saad Razzaq, Fahad Shahbaz & Kashif Irfan Proceedings of the World Congress on Engineering 2008 Vol II WCE 2008, July 2 - 4, 2008, London, U.K. ISBN:978-988-17012-3-7 WCE 2008