Pop – the Parametric Optimization Toolbox

Recent years have seen an increased interest in multi-parametric programming (mp-P), in large part due to the ever increasing number of areas where mp-P can be applied such as bilevel programming, reactive scheduling and decentralized control. This in turn has led to significant theoretical advances in fields such as multi-parametric mixed-integer programming, multi-parametric moving horizon estimation and global mp-P [1]. For the solution of the underlying mp-P problems, currently only one solver package is openly available, namely the MPT toolbox [2]. Albeit being very complete and providing a wide array of capabilities, the MPT toolbox is computationally limiting when larger problems are considered. Additionally, as it has its own class and object definitions, software interoperability becomes a challenging process especially during the closed-loop validation of the derived controllers.

In this contribution, we describe POP, the Parametric Optimization toolbox, a new, powerful toolbox for the solution of multi-parametric programming problems. For continuous mp-P problems, POP uses a geometrical approach with a variable step-size exploration strategy combined with a novel algorithm for the removal of redundant constraints. For multi-parametric mixed-integer problems a newly developed generalized framework is employed, featuring an efficient comparison procedure that enables the handling of nonconvex critical regions. Additionally, POP allows seamless formulation and solution of explicit control or scheduling problems, either via concatenation or multi-parametric dynamic programming. These capabilities naturally incorporate the ability to handle hybrid systems through the implementation of state-of-the-art multi-parametric programming algorithms. In order to facilitate the analysis and benchmarking of the developed algorithms, POP features a powerful random problem generator for all problems under consideration. It allows for the generation of problems of arbitrary dimensions, where key aspects of the algorithm can be adjusted using several parameters.

Here we examine various benchmark test sets, featuring for each class of problem 250 randomly generated mp-P problems of diverse characteristics. The simulation results are used to highlight the solution capabilities of POP and compare its performance with the MPT toolbox. They also provide valuable insight into future research directions for the improvement of the computational performance.

The applicability and software interoperability of POP is demonstrated via the explicit optimal control of the multicolumn solvent-gradient purification (MCSGP) process, which falls in the challenging class of continuous nonlinear periodic systems [3]. Using the newly developed PAROC framework (http://www.paroc-platform.co.uk/Software/POP/) [1], a high-fidelity, validated model of the system was implemented in gPROMS® ModelBuilder v4.0 [4] and reduced to a linear state-space representation employing system identification techniques. The state-space model considers 4 states, 1 input, 3 output as well as measured disturbances with a sampling time of 6s. The corresponding explicit MPC problem for different control and output horizons is formulated and solved via the POP toolbox (see Table 1).

Table 1: Detailed characteristics of the explicit optimal control of the MCSGP system.

Output Horizon	Control Horizon	Number of Parameters	Number of constraints	Number of optimization variables
10	2	40	84	2
	4		88	4
	6		92	6
	8		96	8
12	2	46	100	2
	4		104	4
	6		108	6
	8		112	8
14	2	52	116	2
	4		120	4
	6		124	6
	8		128	8

The derived controllers are validated in-silico utilizing seamless software interoperability between gPROMS® ModelBuilder and MATLAB. The closed-loop simulation results indicate successful reference tracking of the output variables and disturbance rejection.

Providing a novel and powerful toolbox for the solution of multi-parametric programming problems, this work presents POP, the Parametric Optimization toolbox. Using a newly developed random problem generator, a large number of numerical studies shows both the versatility as well as the computational efficiency of POP. This in turn allows for the development of explicit MPC controllers for very complex systems such as the MCSGP process.

Acknowledgements

This work is the result of many contributions over the last two decades. In particular, we would like to acknowledge Nikolaos A. Bozinis and Martina Wittmann-Hohlbein, who have been pivotal in developing previous versions of POP [5,6].

References

1.            Pistikopoulos, E.N., et al., PAROC - an Integrated Framework and Software Platform for the Optimization and Advanced Model-Based Control of Process Systems. Chemical Engineering Science, 2015, in print.

2.            Kvasnica, M., et al., Multi-Parametric Toolbox (MPT), in Hybrid Systems: Computation and Control, R. Alur and G.J. Pappas, Editors. 2004, Springer Berlin Heidelberg. p. 448–462.

3.            Aumann, L. and M. Morbidelli, A continuous multicolumn countercurrent solvent gradient purification (MCSGP) process. Biotechnology and Bioengineering, 2007. 98(5): p. 1043–1055.

4.            Process Systems Enterprise, gPROMS, www.psenterprise.com/gproms, 1997-2014.

5.            Bozinis, N.A., et al. POP, alpha version, 2000.

6.            Wittmann-Hohlbein, M., et al. POP, beta version, 2013.