Economic Model Predictive Control for Closed-Loop Scheduling: Robustness to Disturbances

Robert D. McAllister

James B. Rawlings

Abstract

Chemical production scheduling is an inherently dynamic process.
As a schedule is executed, additional information becomes available (e.g.,
updated prices, new demands, task delays) and the remaining schedule should
be updated, and ideally improved, by taking this information into account.
In closed-loop scheduling, this problem is addressed by optimizing the schedule
over a fixed horizon, implementing this schedule for the first time interval,
moving the horizon forward, and then re-optimizng the schedule based on any
new information observed. However, the dynamic behavior of closed-loop scheduling,
or any rescheduling approach, has remained difficult to interpret or quantify
for a general scheduling problem. This difficulty is exacerbated by the
presence of disturbances. To overcome this issue, the dynamics of production scheduling can be represented
in a state-space formulation and, therefore, closed-loop scheduling
can be formulated as an economic model predictive control (MPC) problem.
This formulation allows us to draw upon the
insights, tools, and theoretical results developed in the MPC
community for the dynamic behavior of a control algorithm. Previously,
this framework has provided insights and theoretical guarantees
for the dynamic behavior of nominal closed-loop scheduling. Specifically,
it was demonstrated that constraining the individual optimizations to
terminate along a reference trajectory ensures that, in the nominal case,
the closed-loop cost is asymptotically no worse than that of the reference trajectory.

In this work, we extend these nominal guarantees to
the dynamic behavior of closed-loop scheduling subject to disturbances.
In particular, we investigate the inherent robustness of closed-loop scheduling
and determine performance guarantees for the closed-loop trajectory.
These inherent robustness properties are derived for nominal economic MPC design, i.e., economic MPC
that only considers uncertainty through feedback, and do not rely on stochastic optimization or
robust control techniques.

For persistent, sufficiently small, additive disturbances,
tracking MPC is known to be inherently robust even when discrete
decision variables are present. While this class of disturbances is relevant to many control
problems, scheduling problems must also consider discrete-valued disturbances
such as task delays and unit breakdowns, i.e., infrequent disturbances of a
fixed and not small size. As a generalization of these discrete-value disturbances, we consider
a class of bounded, additive disturbances that have a maximum probability of occurring
between each schedule update. For this class of disturbances, we show that tracking MPC is
robustly asymptotically stable in expectation. In addition, we extend the robustness
results of tracking MPC to economic MPC for dissipative systems and provide bounds for
expectation of performance for the perturbed closed-loop trajectory. Through several small
examples, we demonstrate the insights provided by these theoretical results for
closed-loop scheduling subject to both persistent, sufficient small disturbances
and infrequent, non-small disturbances.

Works Cited

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4. Allan, D. A., Bates, C. N., Risbeck, M. J., Rawlings, J. B., 2017.
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