(363u) Tuning MPC through System-Level Parameterization and Inverse Optimization

Model predictive control (MPC) is the state-of-the-art strategy for multivariable constrained control. To obtain satisfactory control performance, it is necessary to correctly and automatically tune the controller parameters, although such automatic tuning can be challenging due to the implicitness of the relations between the parameters and performance. Through the research in the past decades, many guidelines and heuristics have been proposed for MPC tuning [1]. However, it should be noted that there are some common difficulties in the existing approaches: (i) analytical characterization of quantitative features (such as overshoot and settling time) cannot be easily derived except for simple dynamics; (ii) the number of parameters to be tuned is usually high and a low-dimensional simplification or separation into individual loops may not be desirable; (iii) direct formulation as dynamic optimization problems can be computationally prohibitive; and (iv) the users may give diverse and multiple specifications on the variables, and these requirements may frequently change on need.

Motivated by these considerations, this work proposes a generic and systematic tuning method for the MPC of linear dynamical systems represented by state-space models.

1. The approach adopts the so-called system-level parameterization (SLP), which refers to the representation of controllers using matrices that characterize the closed-loop responses of states and inputs to the exogenous disturbances and initial conditions [2]. User specifications, such as the (i) magnitudes of initial responses to impulse inputs, (ii) overshoot, (iii) robustness with respect to the model uncertainties, (iv) closed-loop time constants, and (v) presence/absence/directions of responses of given states to given inputs, can all be interpreted as constraints or cost terms to be minimized. In this way, given the system model and user specifications, the feasible response matrices that best satisfy the need can be sought.

2. The MPC parameters are then tuned in such a way that the resulting controller best matches the above obtained response matrices. This is considered as an inverse optimization problem, which refers to the problem of finding the objective function according to given optimal solution [3]. Specifically, the violation to the optimality conditions of the MPC formulation (in the sense of the squared Frobenius norm) is used as the objective function to be minimized, and the MPC parameters and the corresponding actual response matrices are used as decision variables. While conceptually connected to the classical inverse linear quadratic regulator problem [4] and the controller matching approach of [5], the proposed approach here does not need to assume the prior knowledge of a well-performing controller (which should be the result rather than the prerequisite of controller tuning).

The proposed method is computationally efficient, since the constituent optimization problems are both convex quadratic programs. For demonstration, a case study on a benchmark heavy oil fractionator unit [6] is conducted, with comparison to some present approaches in the literature.

References

1. Alhajeri, M., & Soroush, M. (2020). Tuning guidelines for model predictive control. Ind. Eng. Chem. Res., 59, 4177-4191.

2. Wang, Y.-S., Matni, N., & Doyle, J. C. (2019). A system-level approach to controller synthesis. IEEE Trans. Autom. Control, 64, 4079-4093.

3. Ahuja, R. K., & Orlin, J. B. (2001). Inverse optimization. Oper. Res., 49, 771-783.

4. Jameson, A., & Kreindler, E. (1973). Inverse problem of linear optimal control. SIAM Control, 11, 1-19.

5. Di Cairano, S., & Bemporad, A. (2009). Model predictive control tuning by controller matching. IEEE Trans. Autom. Control, 55, 185-190.

6. Prett, D., & Morari, M. (1987). Shell Process Control Workshop. Butterworth.