Reference signal shaping for closed-loop systems with causality constraints

A reference signal shaping problem formulated as a convex optimization problem is presented for the design of the reference signal in a closed-loop discrete-time lineartime-invariant system, with the purpose that internal control signals and system output are bounded within constraints. A causal solution endures the reference profiles changes only after a system output is required to change. The proposed solution allows us to compute a causal or noncausal reference profile, by adding a time-dependent signal constraint. Feasibility and existence of a reference profile is verified with a linear programming (LP) problem, while an optimal reference profile for the closed-loop system is obtained via a quadratic program (QP) problem. A mass-spring-damper system paired with a PID controller is the illustrative example for closed-loop reference shaping. To evaluate the proposed design, the closed-system response for both causal and noncausal reference profiles are compared.