(199a) Fast and Large-Scale Model Predictive Control Using Neural Networks

Fast and large-scale model predictive control using neural networks
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Pratyush Kumar, James B. Rawlings

Department of Chemical Engineering, University of California - Santa Barbara

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Model predictive control (MPC) is a feedback control technology
that uses a dynamic model of the plant to forecast the internal plant states
and solves an optimization problem in real-time to determine
the manipulated control inputs. The use of a plant model
allows MPC to handle multivariable processes, and the use of the
online optimization allows MPC to naturally handle physical constraints and
decide the control inputs by optimizing the desired objectives (e.g, a tracking error or an economic cost).
These features led to a widespread adoption of MPC in
process industries (Qin and Badgwell, 2003).

Real-time optimization in MPC is challenging
on large-scale processes, which have limited available time for online computation.
The improvements in optimization algorithms and software over the years certainly help to
mitigate this challenge. But there is still a need to develop faster methods to make
MPC applicable on such large-scale processes. For linear
plant models and quadratic or linear objectives, the MPC control
law is known to be a piecewise affine function of the system state
defined on polytopic partitions of the state-space
(Bemporad et al., 2002; Seron et al. 2000;Borreli et al., 2017; Rawlings et al., 2017, Chapter 7).
The number of the polytopic state-space regions which
define the piecewise affine control law increases exponentially
with increase in the system size and the length of the prediction horizon in MPC.
Deep neural networks that use rectified linear units (ReLU) as
their activation function also represent piecewise affine functions
defined on polytopic partitions of the input-space of the network. The
number of the polytopic input-space regions represented by such deep networks grows exponentially
with the addition of hidden layers in the network (Montufar et al., 2014).
This property makes deep neural networks a
reasonable choice of functions required to approximate the MPC control law (Karg and Lucia, 2018;Cavagnari et al., 1999).
We demonstrate this neural network approximation procedure
and the efficacy of this approach via several case studies. The success of this approximate neural network
controller can enable MPC applications on processes which are out of reach
with the current optimization solvers.

To start, we demonstrate the neural network approximation
procedure on a simple two dimensional linear system with one control input
and constraints on the control input. The approximation
procedure goes as follows. We first sample a set of initial states inside the
entire feasible region of states for which the MPC controller is known
to be stabilizing. We then
solve quadratic programs offline for each of those states and collect
the pairs (x, u) for the neural network training dataset. Finally, we
train the neural network on the collected training dataset
using a standard optimization algorithm used
in the machine learning literature. Once the neural
network is trained, we compare the state-space regions which define
the neural network control law
to the state-space regions which define the optimal control law.
We then perform closed-loop simulations using the approximated neural
network controller and demonstrate that the approximated controller
is as good as the optimal MPC controller. We also point out that even on
small-scale problems like this example, the standard machine learning procedure to train neural
networks can get stuck in bad local solutions. This behavior necessitates
stability analysis of the approximate neural network controller after
the training phase.

Finally, we move on to challenging large-scale problems.
We discuss one problem in which solving the MPC optimization problem
in real-time can be challenging, thereby requiring faster alternative techniques
than online optimization.
We train a neural network to mimic the MPC controller
using an approximation procedure similar to the one discussed previously.
A major challenge associated with large-scale problems
is the inability to tractably sample states for the
neural network training dataset with sufficient density coverage inside the feasible region. Hence,
we sample only the states that are frequently encountered during the plant
operation due to the desired setpoint changes and the anticipated disturbances.
We report the data requirements to obtain a good approximation of the optimal controller
and the average speed-up times achieved
by the use of a neural network compared to state-of-the-art optimization solvers.

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References:

Bemporad, Alberto, et al. "The explicit linear quadratic regulator for constrained systems." Automatica 38.1 (2002): 3-20.

Borrelli, Francesco, Alberto Bemporad, and Manfred Morari. Predictive control for linear and hybrid systems. Cambridge University Press, 2017.

Cavagnari, L., Lalo Magni, and Riccardo Scattolini. "Neural network implementation of nonlinear receding-horizon control." Neural computing & applications 8.1 (1999): 86-92.

Karg, Benjamin, and Sergio Lucia. "Efficient representation and approximation of model predictive control laws via deep learning." arXiv preprint arXiv:1806.10644 (2018).

Montufar, Guido F., et al. "On the number of linear regions of deep neural networks." Advances in neural information processing systems. 2014.

Qin, S. Joe, and Thomas A. Badgwell. "A survey of industrial model predictive control technology." Control engineering practice 11.7 (2003): 733-764.

Rawlings, James Blake, David Q. Mayne, and Moritz Diehl. Model Predictive Control: Theory, Computation, and Design. Nob Hill Publishing, 2017.

Seron, Maria M., José A. De Doná, and Graham C. Goodwin. "Global analytical model predictive control with input constraints." Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No. 00CH37187). Vol. 1. IEEE, 2000.