PI-controller design for constrained linear systems using positive invariance and bilinear programming

We address the step reference tracking for mono-variable systems subject to state and control constraints via PI-control law with a feedforward term. We determine amplitude limits for the step reference that guarantees the regional closed-loop stability and constraints fulfillment, using invariance and contractivity properties of polyhedral sets, such that the Internal Model Principle remains valid in the polyhedral region. The first main result is based on the positive-invariance property applied to an extended state-space representation that takes the reference model into account, leading to more complex conditions than the ones based on the $\Delta$-invariance property. Based on these two complementary results, a set of bilinear programming problems is proposed to find solutions for the associated constrained tracking problem. A numerical example illustrates the proposal.