(16b) The Inherent Robustness of Model Predictive Control: Discrete and Infrequent Disturbances

To ensure successful industrial implementation, control algorithms must be robust to disturbances. In robust or stochastic MPC formulations, disturbances are explicitly addressed in optimization problem. However, characterizing the set and probability distribution for a disturbance is difficult and solving a stochastic or min-max optimization problem is computationally expensive. Fortunately, the inherent robustness of nominal MPC, i.e., the robustness obtained solely through feedback, is often sufficient in industrial applications.

Typically, inherent robustness is characterized by robust asymptotic (or exponential) stability of the closed-loop system (Allan et al., 2017). However, this definition applies only for disturbances that can be characterized as "sufficiently small". This characterization is reasonable for most process control applications in which disturbances arise from measure noise, modeling errors, and small perturbations. Furthermore, these results extend to MPC implementations with integrality constraints on the input, i.e., discrete-valued decisions, and discontinuous optimal cost functions (Allan et al., 2017).

Theoretical advances and improvements in optimization algorithms have led to a growing interest in applying MPC to higher-level planning and scheduling problems that include many discrete-valued decision variables (Risbeck et al., 2019). In these scheduling problems, however, relevant disturbances such as equipment breakdowns and delays are not small. These disturbances are discrete-valued and typically large, i.e., the effect of the disturbance often exceeds the effect of the control action. Thus, constructing a bound for the worst deterministic performance possible, e.g., the entire facility is broken, results in an excessively conservative bound that offers little insight into the true behavior of the underlying system. In fact, we may even consider disturbances that are sufficiently large such that they preclude any deterministic upper bound on the closed-loop system. Although the main application of these results is to address disturbances in planning and scheduling applications of MPC, these theoretical results also apply to large and infrequent disturbances such as faults or communication failures that occur in other MPC applications.

Although discrete-valued and large, we exploit the infrequent nature of these disturbances, e.g., the probability that we experience a breakdown is small, to define a stochastic form of robustness. To define this stochastic form of robustness, we leverage the concept of stochastic stability for nonlinear systems, originally developed by Kushner (1967). Analogous to input-to-state stability (ISS) for deterministic systems, stochastic input-to-state stability (SISS) has been used to characterize the robustness of stochastic nonlinear systems (Krstic and Deng, 1998). Using these results as a foundation, we define the term robust exponential stability in expectation (RESiE) for (large) discrete and infrequent disturbances and discuss the implications of this definition through a motivating example.

Although we use the term 'large' to classify the disturbance of interest, we cannot permit any type of disturbance and nonlinear system. Specifically, we require that the effect of the disturbance, defined as the distance between the state evolution with and without the disturbance, is bounded by an affine function of the current state of the system. This bound permits additive disturbances as well as certain types of multiplicative disturbances. In addition, we require the same basic stability assumptions used to ensure that MPC is exponentially stable for the nominal closed-loop system. Subject to these assumptions, and provided that the optimization problem remains feasible, we can establish that MPC is RESiE to discrete and infrequent disturbances. We conclude with an example to illustrate the implications of this analysis and its application to planning and scheduling problems.

Works Cited
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Allan, D.A., Bates, C.N., Risbeck, M.J., and Rawlings, J.B. (2017). On the inherent robustness of optimal and suboptimal nonlinear MPC. Sys. Cont. Let., 106, 68-78. doi:10.1016/j.sysconle.2017.03.005.

M. J. Risbeck, C. T. Maravelias, and J. B. Rawlings. (2019). Unification of closed-loop scheduling and control: State-space formulations, terminal constraints, and nominal theoretical properties. Comput. Chem. Eng., 129:106496.

Krstic, M. and Deng, H. (1998). Stabilization of nonlinear uncertain systems. Springer-Verlag.

Kushner, H.J. (1967). Stochastic Stability and Control, volume 33 of Mathematics in Science and Engineering. Academic Press, New York.