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On the Effects of Tunable Parameters of Model Predictive Control on the Locations of Closed-Loop Eigenvalues

Jorge L. Garriga
Masoud Soroush
H.M. Soroush
Journal / Anthology

Industrial & Engineering Chemistry Research
Year: 2010
Volume: 49
Issue: 17
Page range: 7951-7956

This paper presents an analytical study of the effects of model predictive control (MPC) tunable parameters on the closed-loop performance quantified in terms of the location(s) of closed-loop eigenvalue(s) of several common, single-input single-output, linear plants with inactive constraints. 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 locations of the eigenvalues of the closed-loop system of a plant in the discrete-time setting. It provides theoretical basis/justification for several existing qualitative MPC tuning rules and proposes new tuning guidelines. For example, as the prediction horizon is increased while other tunable parameters remain constant, a subset of the closed-loop eigenvalues (poles) move toward the open-loop eigenvalues (poles) of the plant, if the plant is asymptotically stable. If a prediction horizon much longer than the reference-trajectory time constant is used, the value of the 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 toward the open-loop eigenvalues. As the control horizon is increased from one, the dominant eigenvalue of the closed-loop system initially moves toward the origin and then away from the origin to a location that does not change with a further increase in the control horizon.