Elsevier

Automatica

Volume 76, February 2017, Pages 103-112
Automatica

Explicit model predictive control: A connected-graph approach

https://doi.org/10.1016/j.automatica.2016.10.005Get rights and content

Abstract

The ability to solve model predictive control (MPC) problems of linear time-invariant systems explicitly and offline via multi-parametric quadratic programming (mp-QP) has become a widely used methodology. The most efficient approaches used to solve the underlying mp-QP problem are either based on combinatorial considerations, which scale unfavorably with the number of constraints, or geometrical considerations, which require heuristic tuning of the step-size and correct identification of the active set. In this paper, we describe a novel algorithm which unifies these two types of approaches by showing that the solution of a mp-QP problem is given by a connected graph, where the nodes correspond to the different optimal active sets over the parameter space. Using an extensive computational study, as well as the explicit MPC solution of a combined heat and power system, the merits of the proposed algorithm are clearly highlighted.

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 0n×m as the zero matrix of dimension n×m. Let aRn and ARn×n, then ak and Ak denote the kth element and row of a and A, respectively, and Akj denotes the element in the k-row and jth column of A. Additionally, |a| denotes the cardinality of the set a. Let n,kR and p be a set. Then the binomial coefficient is denoted as (nk), while (pk) denotes the set of all possible sets of cardinality k which are subsets of p. Lastly, let P be a polytope, then int(P) denotes the interior of P.

Section snippets

From a MPC to a mp-QP problem

The general MPC problem formulation is considered as follows: min.uxNTPxN+k=1N1(xkTQkxk+(ykykR)TQRk(ykykR))+k=0M1((ukukR)TRk(ukukR)+ΔukTR1kΔuk)s.t.xk+1=Axk+Buk+Cdkyk=Dxk+Euk+euminukumaxΔuminΔukΔumaxxminxkxmaxyminykymax, where xk are the state variables, uk and ukR are the control variables and their respective set points, Δuk denotes the difference between two consecutive control actions, yk and ykR are the outputs and their respective set points, dk 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.

References (47)

  • K.I. Kouramas et al.

    Explicit/multi-parametric model predictive control (MPC) of linear discrete-time systems by dynamic and multi-parametric programming

    Automatica

    (2011)
  • K.I. Kouramas et al.

    An algorithm for robust explicit/multi-parametric model predictive control

    Automatica

    (2013)
  • A. Krieger et al.

    Model predictive control of anesthesia under uncertainty

    Computers & Chemical Engineering

    (2014)
  • R. Mastragostino et al.

    Robust decision making for hybrid process supply chain systems via model predictive control

    Computers & Chemical Engineering

    (2014)
  • P. Mhaskar et al.

    Robust hybrid predictive control of nonlinear systems

    Automatica

    (2005)
  • A. Nikandrov et al.

    Sensitivity analysis of LP-MPC cascade control systems

    Journal of Process Control

    (2009)
  • R. Oberdieck et al.

    Explicit hybrid model-predictive control: The exact solution

    Automatica

    (2015)
  • R. Oberdieck et al.

    Parallel computing in multi-parametric programming

  • P. Patrinos et al.

    A new algorithm for solving convex parametric quadratic programs based on graphical derivatives of solution mappings

    Automatica

    (2010)
  • E.N. Pistikopoulos

    From multi-parametric programming theory to MPC-on-a-chip multi-scale systems applications

    Computers & Chemical Engineering

    (2012)
  • E.N. Pistikopoulos et al.

    Towards the integration of process design, control and scheduling: Are we getting closer?

    Computers & Chemical Engineering

    (2016)
  • E.N. Pistikopoulos et al.

    PAROC - an integrated framework and software platform for the optimization and advanced model-based control of process systems

    Chemical Engineering Science

    (2015)
  • P. Rivotti et al.

    Constrained dynamic programming of mixed-integer linear problems by multi-parametric programming

    Computers & Chemical Engineering

    (2014)
  • Cited by (0)

    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.

    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.

    View full text