Optimal sliding mode control for nonlinear systems with uncertainties

A nonlinear sliding mode in an optimal fashion is designed for nonlinear systems affected by uncertainties. A quadratic performance index is given and an optimal nonlinear switching manifold is obtained. The switching manifold obtained consists of analytic terms and a compensation term which is the limit of the adjoint vector sequence. The analytic terms can be found by solving a Riccati matrix equation and a matrix equation. The compensation term can be obtained from an iterative formula of adjoint vectors. Based on the reaching law approach for uncertain systems, a control input that forces the system´s state to reach the nonlinear sliding surface in finite time is obtained. A disturbance observer is constructed to make the control input physically realizable. The stability of the nonlinear sliding mode is analysed. Simulation results are employed to test the effect of the proposed design algorithm.