Explicit model predictive control: A connected-graph approach☆
Introduction
Model Predictive Control (MPC) is a powerful, model-based control paradigm, which enables the optimal control of multivariable, constrained systems (Bahakim and Ricardez Sandoval, 2014, Chawankul et al., 2007, Heidarinejad et al., 2013, Mastragostino et al., 2014, Mhaskar et al., 2005, Nikandrov and Swartz, 2009, Soliman et al., 2008, Trifkovic et al., 2012). At its heart is thereby the idea of a receding horizon approach, where the optimization problem associated with the MPC problem is solved online at each time step (Rawlings & Mayne, 2009). One way to avoid this online computational step is the explicit solution of the MPC problem via its reformulation into an equivalent multi-parametric quadratic programming (mp-QP) problem, which is solved once and offline (Bemporad, Morari et al., 2002, Mayne and Raković, 2002, Pistikopoulos et al., 2015); also used in many applications (see e.g. Axehill, Besselmann, Raimondo, & Morari, 2014; Feller, Johansen, & Olaru, 2013; Kouramas, Faísca, Panos, & Pistikopoulos, 2011; Kouramas, Panos, Faísca, & Pistikopoulos, 2013; Krieger & Pistikopoulos, 2014; Mayne, Raković, & Kerrigan, 2007; Oberdieck & Pistikopoulos, 2015; Pistikopoulos, 2009; Pistikopoulos, 2012; Pistikopoulos & Diangelakis, 2016; Rivotti & Pistikopoulos, 2014a; Rivotti & Pistikopoulos, 2014b; Wen, Ma, & Ydstie, 2009).
Due to the reformulation into a mp-QP problem, the application of explicit MPC is intimately linked to the ability to solve mp-QP problems. The most efficient solution procedures to date can broadly be classified into geometrical and combinatorial approaches. In the geometrical approaches, the region where a certain active set is optimal is used as the basis for the algorithm, with the focus then being laid on exploring the parameter space by moving from one region to another (Baotic, 2002, Bemporad, 2015, Bemporad, Morari et al., 2002, Dua et al., 2002, Spjøtvold et al., 2006, Tøndel et al., 2003). On the other hand, combinatorial approaches are based on the optimal active set itself, and then an evaluation of all feasible combinations of active sets via a suitable branch-and-bound approach (Feller et al., 2013, Gupta et al., 2011, Herceg et al., 2015).
In this paper we use the ability to infer the optimal active set of adjacent critical regions (Tøndel et al., 2003) to show that the solution of mp-QP problems, and thus of explicit MPC problems, is given by a connected graph. This property is used to devise a novel solution algorithm, which uses the concept of a connected graph in a combinatorial setting, where the powerful fathoming criterion of Gupta et al. (2011) is still applicable. The merit of this new algorithm is shown in an extensive computational study featuring test sets of 100 mp-LP and mp-QP problems of various size each, as well as in the explicit model predictive control of a combined heat-and-power (CHP) system, where the scalability of the novel approach is highlighted by considering an increasing control and output horizon.
We denote as the zero matrix of dimension . Let and , then and denote the th element and row of and , respectively, and denotes the element in the -row and th column of . Additionally, denotes the cardinality of the set . Let and be a set. Then the binomial coefficient is denoted as , while denotes the set of all possible sets of cardinality which are subsets of . Lastly, let be a polytope, then denotes the interior of .
Section snippets
From a MPC to a mp-QP problem
The general MPC problem formulation is considered as follows: where are the state variables, and are the control variables and their respective set points, denotes the difference between two consecutive control actions, and are the outputs and their respective set points, are the measured
The connected graph approach for mp-QP problems
The combinatorial approach does not feature the limitations of the geometrical approach such as the necessity of consider facet-to-facet properties and step-size determination. However, as shown in Fig. 1, even a small example requires the evaluation of a potentially large number of candidate active sets, most of which are not optimal. Thus, the key to a more efficient algorithm is either to increase the fathoming efficiency or to reduce the number of candidate active sets. In this paper, we do
Computational study
In this section we display the computational prowess of the newly developed algorithm. To highlight this, we use the test sets ‘POP_ mpLP1’ and ‘POP_ mpQP1’ available at http://paroc.tamu.edu/Software consisting of 100 mp-LP and 100 mp-QP problems, respectively (see Fig. 3 for the problem statistics). All computational experiments are performed on a 4-core machine with an Intel Core i5-4200M CPU at 2.50 GHz and 8 GB of RAM. Furthermore, MATLAB R2014a and IBM ILOG CPLEX Optimization Studio
Conclusions
In this paper we have described a unified approach towards the solution of mp-LP and mp-QP problems. Based on seemingly disjoint results from the different approaches, the connected graph theorem from (Gal & Nedoma, 1972) is extended to the mp-QP case. This enables the formulation of a novel, efficient active-set based algorithm which uses the concept of the connected graph to find all candidate active sets. The efficiency of the new algorithm for mp-QP problems is highlighted in a
Richard Oberdieck received his B.Sc. and M.Sc. degrees in Chemical Engineering from ETH Zurich, Switzerland in 2013. He spent the academic year of 2011/12 at the Centre of Process Systems Engineering, Imperial College London under the supervision of Prof. Pistikopoulos. Since 2013, he is pursuing a Ph.D. degree in Chemical Engineering at Imperial College London in the area of multi-parametric programming and multi-parametric/explicit model-predictive control.
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Richard Oberdieck received his B.Sc. and M.Sc. degrees in Chemical Engineering from ETH Zurich, Switzerland in 2013. He spent the academic year of 2011/12 at the Centre of Process Systems Engineering, Imperial College London under the supervision of Prof. Pistikopoulos. Since 2013, he is pursuing a Ph.D. degree in Chemical Engineering at Imperial College London in the area of multi-parametric programming and multi-parametric/explicit model-predictive control.
Nikolaos A. Diangelakis received his B.Sc. in Chemical Engineering from the National Technical University of Athens in 2011 and his M.Sc. in Advanced Chemical Engineering from Imperial College London in 2012. Since 2013 he is pursuing a Ph.D. degree under the supervision of Prof. E.N. Pistikopoulos in the area of simultaneous design and operational optimization via multi-parametric programming and multi-parametric control.
Efstratios N. Pistikopoulos is a TEES Distinguished Research Professor in the McFerrin Department of Chemical Engineering and Interim Co-Director of the Energy Institute at Texas A&M University, where he moved after having spent 24 years at Imperial College London. He obtained a Diploma in Chemical Engineering from the Aristotle University of Thessaloniki, Greece in 1984 and his Ph.D. in Chemical Engineering from Carnegie Mellon University, USA in 1988. His research interests include the development of theory, algorithms and computational tools for multi-parametric programming and multi-parametric/explicit model-predictive control with applications in process & energy systems, smart manufacturing and personalized health care engineering. He has authored or co-authored over 350 research publications in the area of optimization, control and process systems engineering.
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Financial support from EPSRC (EP/M027856/1, EP/M028240/1), Texas A&M University and Texas A&M Energy Institute is gratefully acknowledged. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Riccardo Scattolini under the direction of Editor Ian R. Petersen.