Elsevier

Journal of Process Control

Volume 100, April 2021, Pages 80-92
Journal of Process Control

Robust model predictive control via multi-scenario reference trajectory optimization with closed-loop prediction

https://doi.org/10.1016/j.jprocont.2021.02.006Get rights and content

Highlights

  • The paper presents a novel two-layer robust MPC paradigm.

  • An upper level optimization problem computes set-point trajectories for nominal MPC.

  • Utilizes predicted closed-loop response of plant scenarios under constrained MPC.

  • Performance is demonstrated through application to linear and nonlinear case studies.

Abstract

This paper presents a two-layer control structure to address parameter uncertainty within a plant. The lower layer is formulated as a nominal MPC that computes control actions to regulate the underlying plant, and the upper layer computes optimal set-point trajectories for the lower level to track. The upper layer is formulated as an optimization problem that takes into account the closed-loop behavior of uncertain plant scenarios under the action of nominal MPC. The upper layer facilitates the lower layer in avoiding constraint violations and producing less conservative control actions by assigning time-varying set-point trajectories to the nominal MPC. The benefits of this approach are illustrated through application to linear single-input–single-output transfer function case studies, and a nonlinear multi-input multi-output evaporator process.

Introduction

Model predictive control (MPC) has been widely adopted as an advanced, multivariable control strategy across several industrial sectors [1]. The control inputs are generated by an optimization problem that predicts the future behavior of the system based on a dynamic plant model, and optimizes a performance criterion that may involve set-point tracking, input move suppression, plant economics, or combinations thereof. The inputs corresponding to the first control interval are implemented, and at the end of the sample time new measurements are used to provide estimates of the plant states and/or disturbances, and the control calculation process repeated.

However, control performance can degrade due to plant model inaccuracies as a result of process parameter uncertainty, even though state or output feedback can provide some assistance for disturbance rejection and modeling discrepancies. The inaccurate prediction based on the available model may still not reflect the correct dynamics of the real plant, resulting in potential stability or feasibility problems. This has spurred the development of robust MPC paradigms to incorporate modeling uncertainties into the formulation to address the issues of feasibility, stability and constraint satisfaction.

In the paragraphs below, we review key approaches that have been proposed for incorporation of uncertainty within the MPC formulation. This is followed by a brief discussion of reference governor and dynamic real-time optimization (DRTO) schemes, since the robust MPC formulation proposed in this work combines concepts utilized in scenario-based robust MPC within reference governor and DRTO paradigms. A key feature of the proposed scheme is that it can be implemented as a separate module, with uncertainty accounted for in the calculation of time-varying set-point trajectories that are provided to a standard MPC system. This novel approach is outlined in the final paragraph of this section and described in detail in the sections that follow.

A key factor in robust MPC is open-loop versus-closed loop prediction. Open-loop prediction does not consider the effect of future feedback action on the propagation of uncertainty over the considered time horizon. Closed-loop prediction takes future control action into account which generally leads to less conservative control. Various approaches differ in the mechanism for achieving this. Other differentiating characteristics between the robust MPC approaches include worst-case versus expected performance and the characterization of uncertainty.

Bemporad and Morari [2] provide a comprehensive review of robust MPC. Since then, work in this area has continued to grow, with many recent contributions following a scenario tree paradigm. The paragraphs below briefly review some key approaches that have been proposed. This is followed by a discussion of a two-layer reference governor control paradigm which we utilize in our robust MPC approach, as well as the related two-layered dynamic real-time optimization (DRTO) architecture.

Campo and Morari [3] propose a robust MPC formulation which seeks to minimize the maximum deviation of the outputs from their set-points over a set of uncertain parameters, with constraint satisfaction enforced for all parameter realizations within the uncertainty set. The effect of feedback on the predicted response is not considered. Kothare et al. [4] consider minimization of a worst-case quadratic performance index under the assumption of a constant state feedback control law. The problem is posed as a linear matrix inequality (LMI) corresponding to minimization of an upper bound on the robust performance objective. Lee and Yu [5] investigate both open- and closed-loop robust MPC formulations, with the latter based on dynamic programming. They consider both time-invariant and time-varying uncertainty and also propose suboptimal algorithms for the closed-loop case. Kouvaritakis et al. [6] consider a polytopic set of linear time-invariant or time-varying state space systems, and use a fixed state-feedback law with a free variable to introduce extra degrees of freedom to provide a larger invariant set and to avoid constraint violations during the transient behavior. The feedback gain matrix may be determined off-line, with the controller bias term calculated online. Cuzzola et al. [7] consider polytopic uncertainty and improve the robust MPC algorithm of [4] by utilizing multiple Lyapunov functions corresponding to uncertainty parameter vertices rather than a single one.

Wan and Kothare [8] propose an LMI-based robust MPC strategy in which a sequence of state feedback matrices is computed off-line and employed online via a lookup table and bisection calculation. The control algorithm includes state estimation for output feedback. Bemporad et al. [9] utilize a multiparametric MPC (mp-MPC) formulation in a min–max robust MPC scheme. They consider a linear performance objective and both open- and closed-loop formulations. The mp-MPC formulation permits online implementation through evaluation of a piecewise affine function at each control execution. Sakizlis et al. [10] propose a robust mp-MPC scheme that considers a quadratic nominal or expected performance objective.

Tube-based MPC is a robust MPC paradigm that involves determination of a nominal reference trajectory and a local ancillary controller that maintains the perturbed plant trajectories within a tubed centered around the nominal trajectory. A linear ancillary controller proposed for linear systems [11] is replaced in [12] by an ancillary model predictive controller for nonlinear systems. A comprehensive description of tube-based MPC is given in [13].

A number of recent contributions follow a scenario-based robust MPC approach. Munoz de la Peña et al. [14] generate a scenario tree by considering at each control interval the response of the process to a set of uncertain parameter realizations. The control optimization problem takes the form of a multistage stochastic programming problem in which the expected value of a performance objective is optimized. The scenarios emanating from the same node should respond to the same input; a condition enforced through the imposition of non-anticipativity constraints on the input. Lucia et al. [15] propose a similar strategy for nonlinear MPC in which they utilize implicit Euler integration. They also propose branching over a limited horizon, referred to as the robust horizon, as a means to limit the problem size, and apply the method to a semi-batch polymerization case study. Lucia et al. [16] utilize an economic objective function and compare the performance of scenario-based robust nonlinear MPC (NMPC), an open-loop NMPC formulation, and an NMPC formulation involving optimization over feedback policies. Mastragostino et al. [17] apply a two-stage stochastic programming approach in robust MPC of supply chain systems and compare its performance to that of nominal MPC and an open-loop robust MPC approach. They include in their formulation transportation delays as well as binary production scheduling decisions. Martí et al. [18] investigate decomposition strategies for improving the computational efficiency of multi-stage stochastic optimization problems resulting from the robust MPC formulation. Krishnamoorthy et al. [19] propose a primal decomposition algorithm for scenario-based MPC, which unlike previously proposed approaches, ensures feasibility of non-anticipativity constraints throughout the iterations. They demonstrate its performance under conditions in which the control input is applied after a fixed number of iterations of the decomposition algorithm. Velarde et al. [20] compare the performance of open-loop multi-scenario, closed-loop multi-scenario and chance constrained MPC applied to a renewable hydrogen-based microgrid simulation case study. Puschke and Mitsos [21] consider two different techniques for selection of the scenarios used in multi-stage economic MPC. One considers worst-case scenarios identified upon discretization of the uncertain parameter set, while the other is a sensitivity-based heuristic approach. Holtorf et al. [22] propose a scenario-based approach to robust nonlinear MPC in which the scenario tree is adaptively generated based on a constraint sensitivity analysis. A least-squares parameter estimation step is included in which parameter estimates and their confidence regions are used to reduce the uncertainty range over the prediction horizon.

The robust MPC algorithm proposed in this article involves use of a set-point trajectory as additional optimization degrees of freedom in the control input calculation. We review briefly two related paradigms — reference or command governors, and dynamic real time optimization.

A reference or command governor is an “add-on” to a previously designed unconstrained feedback control system that is used to modify the reference or set-point, primarily to avoid violation of state and/or input constraints. The configuration of a reference governor feedback system is illustrated in Fig. 1. Systems of this type are proposed in [23] and [24] for linear stable feedback control systems. The method is extended in [25] to uncertain linear systems in which additive uncertainty to impulse and step response models of the primary feedback system are considered. Extensions to nonlinear systems are proposed in [26] and [27], with the latter considering alternative approaches of direct use of the nonlinear model versus an uncertain linear time-varying system whose dynamic evolution contains that of the nonlinear system. A comprehensive review of the reference or command governor approach, including extensions and applications, is given in [28].

Dynamic real-time optimization (DRTO) is a supervisory system that accounts for plant dynamics and which provides set-point trajectories, typically economics based, to an underlying feedback control system [29], [30]. DRTO and reference governor schemes are therefore similar in the way in which they interact with a regulatory feedback system. Of particular relevance to the present study is the DRTO approach of Jamaludin and Swartz [31] in which the predicted response of the plant under the action of constrained MPC is utilized. Earlier DRTO schemes that utilize only the plant model do not account for the action of MPC on the predicted plant response, and have been shown in [31] to yield suboptimal performance, particularly in the presence of non-minimum phase characteristics where the MPC is detuned for stability and robustness. The presence of embedded MPC subproblems in the DRTO optimization gives rise to a multilevel optimization problem which is transformed into a single-level mathematical program with complementarity constraints (MPCC) by replacing the MPC optimization subproblems by their first-order optimality conditions. Approximation of the closed-loop dynamics is described in [32], and application of the approach to the coordination of distributed MPC systems is proposed in [33], [34]. Closed-loop plant dynamics under the action of MPC are also considered in the control-cognizant scheduling approaches proposed in Dias et al. [35], Pattison et al. [36] and Caspari et al. [37]. In [35] the closed-loop response is generated via simulation, whereas [36] and [37] utilize low-order scale-bridging models identified using closed-loop simulation data.

In this article we propose a robust MPC scheme that utilizes both concepts of a reference governor or DRTO scheme and consideration of multiple uncertainty scenarios. The reference trajectory is generated by an optimization problem which fully considers the closed-loop behavior of the plant under the action of constrained MPC, as proposed in the DRTO formulation of Jamaludin and Swartz [31], and is assigned to the MPC at each sample time when the plant measurement becomes available. Within this optimization problem, the dynamics of multiple uncertain plant scenarios under nominal input-constrained MPC are considered in parallel. Only one set of reference trajectories acts as the decision variables (degrees of freedom) for the MPCs associated with the plant scenarios to track. In this way, the effects of input saturation, constraint violation and target tracking can be simultaneously addressed within the formulation without complicating the lower-level control strategy, which can remain as a simple quadratic programming formulation.

Section snippets

The reference trajectory optimization formulation

The conceptual design of the two-layer architecture for the proposed robust MPC approach is shown in Fig. 2. The upper layer includes different scenarios generated by considering discrete realizations of uncertain parameters. Thus, each scenario has its own dynamic model to generate outputs specific to that scenario. MPC optimization subproblems are embedded along the closed-loop prediction horizon to generate the control actions for the plant scenarios, whose outputs provide disturbance

Case studies

In this section we demonstrate the performance of the proposed robust MPC approach on a linear, single-input–single-output (SISO) system without dead time, a SISO system with dead time, and a nonlinear multi-input multi-output (MIMO) case study. The performance is also compared against that obtained using nominal MPC.

In the first case study, a set-point hold of SPH=2 is used, with the default value of 1 used for the other cases. Although the robust MPC formulation is considerably more complex

Conclusion

This work presents a novel control approach for addressing uncertainty inherent in process parameters in dynamic systems. This approach adopts a two-layer structure where the lower-layer takes the form of a nominal MPC that considers only one particular case of process dynamics and the upper layer computes the optimal set-point trajectories for the lower layer. This upper layer considers discrete scenarios over the uncertainty range and generates closed-loop predictions of plants corresponding

CRediT authorship contribution statement

Hao Li: Conceptualization, Methodology, Software, Investigation, Writing - original draft, Visualization. Christopher L.E. Swartz: Supervision, Conceptualization, Methodology, Visualization, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

Support through an Imperial Oil University Research Award and the McMaster Advanced Control Consortium (MACC) is gratefully acknowledged.

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