Optimal Sliding Manifold Design for Nonlinear Systems Based on Sensitivity Approach

The problem of designing nonlinear sliding manifolds with a quadratic performance index for a class of nonlinear systems is considered. By applying optimal quadratic control theory to constructing sliding manifolds, nonlinear two-point boundary value (TPBV) problems are induced. Then a novel sensitivity approach is introduced to transform the original problems into solving a series of nonhomogeneous linear TPBV recursive formulas. By finite terms recursion, we obtain approximate solutions of optimal sliding manifolds. The procedure to get the suboptimal sliding manifolds is presented in detail. Employing the reaching law approach, we select a control input that can force the system states to reach the nonlinear sliding manifold in finite time and have low chattering in the sliding motion. An example is given for illustration.