(206f) Optimization Methods for Robust Model Predictive Control

Nominal model predictive control (MPC) is a feedback control scheme that predicts the future states of a dynamic system, using a process model, over a finite time window. Based on the predicted and current states, the control scheme optimizes an objective function to determine an optimal manipulated variable profile, i.e. an optimal control action sequence. The term nominal refers to the assumption that the system is deterministic for the optimization problem. If instead unknown disturbances cause a system to be uncertain, then robust model predictive control (RMPC) is a suitable control scheme. However, practical applications of RMPC have been limited by the optimization problem, which is computationally expensive and intractable for complex systems. While many aspects of nominal MPC for deterministic systems are well understood (Mayne et al., 2000), the presence of uncertain systems, which motivates a robust controller in the form of RMPC, is still a significant challenge that is receiving major attention (Mayne, 2014; Bujarbaruah et al. 2020). Min-max RMPC, which is considered in this study, was first developed by Campo et al. (1987) and minimizes the worst-case performance over all uncertainty realizations. Solving a min-max robust optimization problem is intractable or time consuming and ineffective for online implementation and industrial control applications (Veres et al., 1993; Lee et al. 1997; Kerrigan et al. 2004; Eliasi et al. 2012). Highly complex and inefficient numerical optimization algorithms are generally not suitable for most industrial platforms and computationally simple optimization statements for RMPC have not been formulated (Saltik et al. 2018). Therefore developing a tractable problem for min-max RMPC is an active area of research, and to this date no optimization strategies have been deployed industrially (Yan et al., 2021). The traditional min-max robust optimization problem solved in discrete time for linear process models given an energy bound on the variable to maximize, the disturbance, is also known as disturbance attenuation problem (Basar et al, 1995), which is a constrained min-max optimization. Algorithms have been developed to solve for the suboptimal version of this problem, but iterative or trial-and-error methods are usually required to solve for the optimal solutions to the disturbance attenuation problem, which are possible limitations to the industrial applications of this type of robust controller (Basar et al, 1995; Hespanha, 2017).

In this study, we present an optimization formulation and algorithm to solve for the optimal disturbance attenuation problem. Differently from what is presented in the previous literature, we propose an algorithm which efficiently solve for the optimal solutions without requiring any additional iterative or trial-and-error methods. The proposed algorithm translates the constrained min-max optimization into an equivalent constrained minimization. When there are no constraints on the manipulated variable, then the proposed algorithm is actually solving a constrained scalar minimization and finds analytical solutions. When the manipulated variable is constrained, the proposed algorithm is still able to efficiently find the optimal solutions. We also discuss the claim made by Basar et al, (1995) that in the limit of $x_0 \rightarrow 0$, where $x_0$ is the initial condition for the state of the system, the disturbance attenutaion problem solves the $H_{\infty}$ optimal control problem. Furthermore, we present a case study where the performance of nominal MPC and RMPC, implemented using the proposed optimization algorithm, is compared, as applied to a generic process monitoring control problem. The RMPC simulation is solved efficiently with the proposed algorithm and the numerical simulations suggest that RMPC is a better controller in limiting the influence of disturbances to the state of the system. However, robustness comes with a higher average cost than nominal MPC.

References

Basar, Tamer, and Pierre Bernhard. "$H^{\infty}$-optimal control and related minimax design problems. Systems and Control: Foundations and Applications." (1995).

Bujarbaruah, Monimoy, Ugo Rosolia, Yvonne R. Stürz, Xiaojing Zhang, and Francesco Borrelli. "Robust MPC for linear systems with parametric and additive uncertainty: A novel constraint tightening approach." arXiv preprint arXiv:2007.00930 (2020).

Campo, Peter J., and Manfred Morari. "Robust model predictive control." 1987 American control conference. IEEE, 1987.

Eliasi, H., M. B. Menhaj, and H. Davilu. "Robust nonlinear model predictive control for a PWR nuclear power plant." Progress in Nuclear Energy 54.1 (2012): 177-185.

Hespanha, João P. Noncooperative game theory. Princeton University Press, 2017.

Lee, Jay H., and Zhenghong Yu. "Worst-case formulations of model predictive control for systems with bounded parameters." Automatica 33.5 (1997): 763-781.

Kerrigan, Eric C., and Jan M. Maciejowski. "Feedback min‐max model predictive control using a single linear program: robust stability and the explicit solution." International Journal of Robust and Nonlinear Control: IFAC‐Affiliated Journal 14.4 (2004): 395-413.

Mayne, David Q., James B. Rawlings, Christopher V. Rao, and Pierre OM Scokaert. "Constrained model predictive control: Stability and optimality." Automatica 36, no. 6 (2000): 789-814.

Mayne, David Q. "Model predictive control: Recent developments and future promise." Automatica 50.12 (2014): 2967-2986.

Saltık, M. Bahadır, Leyla Özkan, Jobert HA Ludlage, Siep Weiland, and Paul MJ Van den Hof. "An outlook on robust model predictive control algorithms: Reflections on performance and computational aspects." Journal of Process Control 61 (2018): 77-102.

Veres, Sandor M., and J. P. Norton. "Predictive self-tuning control by parameter bounding and worst-case design." Automatica 29.4 (1993): 911-928.

Yan, Yan, and Longge Zhang. "Robust Model Predictive Control with Almost Zero Online Computation." Mathematics 9.3 (2021): 242.