H∞ sub-optimization in dynamic back-stepping multiple surface control

This paper considers the optimization problem for dynamic multiple surface control of nonlinear systems in strict feedback form with additive uncertainties. Back-stepping combined with multiple surfaces sliding mode control is the control design method. Integral filters are used to estimate the derivative of the composite reference state at each step to avoid explosion of the number of terms. The problem can be described as H_∞ sub-optimization in s- space where s is the coordinates determined by sliding functions s = (s₁, ..., s_n). Here the sub-optimization is treated as a whole with respect to s instead of each state respectively. These sliding gains can be determined by solving a set of triangularly coupled algebraic inequalities, which is easy to implement. This is a partial optimization in the sense that optimization is not with respect to integral filter gains. A pertinent third order example is given