(309f) Strategies Towards the Robust Multi-Parametric Control of Continuous-Time Systems

Strategies
towards the robust multi-parametric control of continuous-time
systems

Muxin
Sun^a,b,
Mario E. Villanueva^b Benoît Chachuat^a,
Efstratios N. Pistikopoulos^b

^aCentre
for Process Systems Engineering, Department of Chemical Engineering,
Imperial College London, London, UK

^bArtie
McFerrin Department of Chemical Engineering, Texas A&M
University, College Station, TX, USA.

Keywords:
multi-parametric
programming,NCO-tracking,robust optimal control

The
need to improve performance and reduce economic costs of industrial
processes has led to an increased interest in real-time model-based
optimal control strategies. These strategies need to be able to deal
with fast dynamics and hard process constraints, such as operational
restrictions and safety requirements. Moreover, the presence of
uncertainty in the form of process disturbances or model mismatch can
lead to suboptimal performance and constraint violation. Model
predictive control (MPC) [1] computes a feedback control law by
repeatedly solving an optimization problem on a receding horizon,
which is often a rather computationally demanding task and might
cause serious delays and/or suboptimal performance.

In
the multi-parametric programming paradigm [1], the optimization
problems are solved off-line, which results in an a priori explicit
mapping of the solutions as a function of measurable quantities, thus
avoiding the need for repeatedly solving optimization problems
online. The case of continuous-time dynamic systems calls for the
solution of multi-parametric infinite-dimensional programs, namely
multi-parametric dynamic optimization (mp-DO). Another approach for
reducing the on-line computationally burden is to enforce the
necessary conditions for optimality (NCO). In NCO tracking [2,3],
feedback control laws are derived in order to track the NCOs along
each arc of the nominal optimal solution, effectively converting an
optimal control problem into a self-optimizing feedback control
problem. However, NCO tracking often relies on the assumption that
the switching structure of the optimal solution remains unchanged,
which may not be the case when the process disturbances or the model
mismatch are significant.

The
implementation of classical MPC controllers is based on a
certainty-equivalence principle, whereby the system is optimized as
if neither process disturbances nor model mismatch were present. Due
to its inherent ability to reject such disturbances, these
controllers exhibit a certain degree of robustness. However, certain
constraints may become violated when large disturbances occur, since
this is not taken into account when optimizing system trajectories.
One way of dealing with this problem involves solving a robust
counterpart of the optimal control problem [4], i.e. to guarantee
feasibility of the system assuming the worst-case realization of the
uncertainty.

This
paper explores strategies for robust multi-parametric control of
continuous-time systems. First, we present a methodology for the
development of so-called mp-NCO-tracking controllers (see Figure 1),
which unifies mp-DO and NCO-tracking for linear dynamic systems [5].
From an NCO-tracking perspective, the use of mp-DO provides a means
for relaxing the constant switching structure assumption by
constructing critical regions that correspond to a unique switching
structure of the optimal control trajectories; whereas from a mp-DO
perspective, NCO-tracking leads to a reduction in the number of
critical regions compared to the use of discretization methods. Next,
we present a robust-counterpart formulation of the mp-DO problem in
order for the mp-NCO-tracking controllers to be robust towards given
process disturbances or model mismatch that are not measured on-line.
This robustification involves backing-off the terminal and path
constraints based on rigorous set-propagation techniques [6]. The
methodology is illustrated with numerical case studies drawn from
chemical engineering, including the control of an FCC unit as shown
in Figure 2.

Figure
1: Framework of multi-parametric NCO-tracking

(a)
mp-DO (b) discretized
mp-QP

Figure
2: Critical regions in the controllers of an FCC unit

References

1. E. N. Pistikopoulos. From multi-parametric programming theory to MPC-on-a-chip multi-scale systems applications. Computer and Chemical Engineering, 47:57-66, 2012.

2. J. V. Kadam, M. Schlegel, B. Srinivasan, D. Bonvin and W. Marquardt. Dynamic optimization in the presence of uncertainty: From off-line nominal solution to measurement-based implementation. Journal of Process Control, 17:389-398, 2007.

3. D. Bonvin and B. Srinivasan. On the role of the necessary conditions of optimality in structuring dynamic real-time optimization schemes. Computers and Chemical Engineering, 51:172-180, 2013.

4. B. Houska, F. Logist, J. V. Impe and M. Diehl. Robust optimization of nonlinear dynamic systems with application to a jacketed tubular reactor. Journal of Process Control, 22(6):1152-1160, 2012.

5. M. Sun. , B. Chachuat and E. N. Pistikopoulos, Design of multi-parametric NCO tracking controllers for linear dynamic systems. Computer and Chemical Engineering, in press (DOI: 10.1016/j.compchemeng.2016.04.038).

6. B. Chachuat, B. Houska, R. Paulen, N. Peric, J. Rajyaguru, M.E. Villanueva, "Set-theoretic approaches in analysis, estimation and control of nonlinear systems," IFAC-PapersOnLine, 48(8):981-995, 2015.