Model predictive control (MPC) has been implemented widely in the process industries since its introduction in the late 1970's. Among factors contributing to the success of MPC are (i) the ability of MPC to handle constraints on manipulated, state and controlled variables in a systematic way during the design and implementation of the controller and (ii) its enormous flexibility that allows the use of models, objective functions and constraint functionalities in a variety of forms [1]. The theory of MPC has received increasingly more attention for the past decade, especially from researchers outside process systems engineering.

A typical model predictive controller has a great number of tunable parameters. These parameters include the prediction horizon(s), control horizon(s), model horizon, output penalties, input magnitude and rate of change penalties, and reference trajectory time constant(s). For the past three decades, a qualitative understanding of the effect of these controller tunable parameters on the stability and performance of the closed-loop system has been gained. This understanding together with closed-loop simulation studies has made possible the wide-spread implementation of MPC. It is known that the effect of these parameters on the closed-loop performance is often not monotonic. There have been several studies on MPC tuning, including [2, 3, 4, 5, 6]. Shridhar and Cooper [2] derived an analytical expression that computes move suppression coefficients as a function of process model parameters, other MPC design parameters, and partitioned block condition numbers of the system matrix. Their tuning method is applicable to unconstrained multivariable processes, including non-square systems. Wojsznis et al. [3] developed a heuristic approach to tuning MPC. They proposed setting penalties on control moves as a function of process dead time as the primary factor, with some correction from process gain. Trierweiler and Farinab [4] also presented a tuning strategy for MPC based on the attainable performance of a system and its degree of directionality.

This paper presents an analytical study of the effect of the 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 processes whose constraints are inactive. The symbolic manipulation capability of MATHEMATICA is used to obtain analytical expressions describing the dependence of closed-loop eigenvalues on the tunable parameters. Simple qualitative and quantitative rules for setting the tunable parameters are developed. This work provides theoretical basis/justification for many of the existing qualitative MPC tuning rules and proposes new tuning guidelines for MPC. For example, as the prediction horizons are increased while other tunable parameters remain constant, a subset of the closed-loop eigenvalues (poles) move non-monotonically toward the open-loop eigenvalues (poles) of the process.

References

[1] Allgower, F., Zheng, A. Nonlinear Model Predictive Control, Progress in Systems and Control Theory series, Vol. 26, Birkhauser-Verlag, Birkhauser Verlag, Basel, 2000

[2] Shridar, R., Cooper, D.J. A tuning strategy for unconstrained multivariable model predictive control. Ind. Eng. Chem. Res., 37(10), 4003–4016, 1998

[3] Wojsznis, W., Gudaz, J., Blevins, T., Mehta, A. Practical approach to tuning MPC. ISA transactions (ISA trans.), 42(1), 149–162, 2003

[4] Trierweiler, J.O., Farinab, L.A. RPN tuning strategy for model predictive control. J. of Process Control, 13(7), 591–598, 2003

[5] Lee, J.H., Yu, Z. H. Tuning of model predictive controllers for robust performance. Computers & Chemical Engineering, 18(1), 15–37, 1994

[6] Qin, J., Badgwell, T.A. A survey of industrial model predictive control technology. Control Engineering Practice, 11, 733–764, 2003