Automated control design for a piecewise-affine approximation of a class of nonlinear systems

This paper establishes a framework to automate the generation of a piecewise-affine approximation for a class of nonlinear systems. The results enable a completely automated procedure for designing an output feedback controller using a method described previously. To perform the piecewise-affine approximation it is assumed that a uniform rectangular grid is provided for the domain of the nonlinearity. Given this grid, there are three basic steps of the algorithm presented in this paper. First, a simplicial partition of the grid is found that provides a partition of the entire state space into polytopic cells. Then, the piecewise-affine approximation of the nonlinearity within each cell of the partition is derived. Finally, a polytopic description of the cells and a parametric description of the boundaries between cells is created. Each cell of the partition will have a closed loop equilibrium at a point that must be specified. This paper presents an optimization objective that can be used to automate the selection of these equilibrium points. All steps in this process axe completely automated, which represents a key step in the development of control synthesis tools for piecewise-affine approximations of nonlinear systems.