Exact Cost Performance Analysis of Piecewise Affine Systems

A method for the exact cost performance analysis of autonomous discrete-time piecewise affine systems is presented. Cost performance refers to the average trajectory cost over the entire region of attraction of the origin. The trajectory cost is based on the infinite-horizon cost. First, a reverse reachability algorithm which constructs the explicit piecewise quadratic trajectory cost function over the entire region of attraction of the origin is presented. Then, an explicit expression for the integral of a quadratic function over a simplex of arbitrary dimension is derived. Thus, the piecewise quadratic trajectory cost function can be integrated exactly and efficiently in order to determine the cost performance of the system as a whole. This alleviates the need to perform a large number of simulations.