LMI-Based Design of State Feedback Controller for Lipschitzian Nonlinear Systems

State-feedback controller design is crucial for dynamical systems when direct measurement of all state variables is possible. This paper deals with such design problem for Lipchsitzian nonlinear systems within the framework of Linear Matrix Inequalities (LMIs). We present first Lipchsitzian nonlinearity conditions, namely the classical, the one-sided, the quasi-one-sided and the quadratic inner-bounded Lipschitz conditions. Moreover, we establish LMI-based stability conditions using several forms of the Lipchsitzian conditions in order to demonstrate and consequently reduce their conservatism feature. Our main goal is to show how linear state-feedback laws can be formulated to stabilize nonlinear Lipchsitzian systems and, at the same time, maximize the limits on non-linearity that the system can tolerate without becoming unstable. Furthermore, in order to overcome the problem of high size of the feedback gain, we consider the case of input saturation where an additional LMI is added. Finally, an illustrative example, namely the 1-DOF inverted pendulum, is given to compare between the different Lipchsitzian conditions and hence to show their conservativeness.