Model predictive controller tuning via eigenvalue placement

This paper presents an analytical study of the effect of model predictive control (MPC) tunable parameters over a wide range, on the closed-loop performance quantified in terms of the location(s) of closed-loop eigenvalue(s) of a large set of widely common, single-input single-output, linear plants whose constraints are inactive. Symbolic manipulation capabilities of MATHEMATICA are used to obtain analytical expressions describing the dependence of closed-loop eigenvalues on the tunable parameters. This work is first to investigate how MPC tuning-parameters affect the location of the eigenvalues of the closed-loop system of a plant in the discrete-time setting. It is to provide theoretical basis/justification for many of the existing qualitative MPC tuning rules and propose new tuning guidelines for MPC. For example, as the prediction horizon is increased while other tunable parameters remain constant, a subset of the closed-loop eigenvalues (poles) move non-monotonically towards the open-loop eigenvalues (poles) of the plant. If a prediction horizon much longer than the reference-trajectory time-constant is used, the value of reference-trajectory time-constant has little effect on the closed- loop performance. As the weights on the magnitude or the rate of change of the manipulated input are increased, the closed- loop eigenvalues move towards the open-loop eigenvalues. As the control horizon is increased from one, the dominant eigenvalue of the closed-loop system initially moves towards the origin and then away from the origin to a location that does not change with a further increase in the control horizon.