(169f) Pop – the Parametric Optimization Toolbox

Authors: 
Oberdieck, R., Texas A&M University
Diangelakis, N. A., Imperial College
Papathanasiou, M. M., Imperial College London
Nascu, I., Texas A&M University
Sun, M., Imperial College London
Avraamidou, S., Artie McFerrin Department of Chemical Engineering, Texas A&M University
Pistikopoulos, E. N., Texas A&M Energy Institute, Texas A&M University

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.