A nonlinear programming approach for the sliding mode control design

We treat the sliding mode control problem by formulating it as a two phase problem consisting of reaching and sliding phases. We show that such a problem can be formulated as bicriteria nonlinear programming problem by associating each of these phases with an appropriate objective function and constraints. We then scalarize this problem by taking weighted sum of these objective functions. We show that by solving a sequence of such formulated nonlinear programming problems it is possible to obtain sliding mode controller feedback coefficients which yield a competitive performance throughout the control. We solve the nonlinear programming problems so constructed by using the modified subgradient method which does not require any convexity and differentiability assumptions. We illustrate validity of our approach by generating a sliding mode control input function for stabilization of an inverted pendulum.