**Posted Dec 1, 1995**

This article concerns piecewise linear systems and determining if they meet givenHperformance specifications. Such problems occur in control of linear systems with saturation nonlinearities. While one could imagine many mathematically natural piecewise linear systems problems we took care to extract one which does correspond to control of saturated plants. We think many such problems will fall into the category treated here.After a serious compromise (which makes our test for meeting specifications conservative), we arrive at a type of piecewise Riccati inequality. We show how these can be converted to matrix inequalities which are linear in the unknowns. Thus modern methods for optimizing such expressions can be used to obtain solutions.