Smoothed Piecewise Linear Lyapunov Function for the First Order Dynamical Systems

The paper deals with the mathematical backgrounds of design piecewise single-variable Lyapunov functions. These backgrounds are based on the study of the main features of generalized Lyapunov function and using them to analyze the solution of dynamical programming problem. This analysis shows that the optimal control signal, which is supplied to the control plant, depends on the partial derivative of the Lyapunov function. This fact allows us to consider the Lyapunov function as a control signal integral over the perturbed motion variable. If the control signal is produced by the sliding mode controller, our approach defines the Lyapunov function as a non-quadratic absolute value function. Since this function is not differentiable in the origin, it is hard enough to use it while the optimal controller is being designed with optimal theory methods. We avoid this problem by applying a smoothing procedure, which is based on considering a small neighborhood near the fracture point. In this neighborhood, we replace the piecewise linear function with a polynomial one. Factors of the polynomial are defined to have the same function values and its derivatives in the boundary points. In our paper, we show that our approach can be extended to the case of a controller which produces a multilevel signal. Signal of such form causes occurring the piecewise linear Lyapunov function, which can be smoothed by using polynomials. Since such a piecewise function can have a lot of branches, we offer to consider it as two functions, which are defined in their intervals.