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Scenario Optimization for MPC

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Handbook of Model Predictive Control

Part of the book series: Control Engineering ((CONTRENGIN))

Abstract

In many control problems, disturbances are a fundamental ingredient and in stochastic Model Predictive Control (MPC) they are accounted for by considering an average cost and probabilistic constraints, where a violation of the constraints is accepted provided that the probability of this to happen is kept below a given threshold. This results in a so-called chance-constrained optimization, which however is known for being very hard to deal with. In this chapter, we describe a scheme to approximately solve stochastic MPC using the scenario approach to stochastic optimization. In the scenario approach the probabilistic constraints are replaced by a finite number of constraints, each one corresponding to a realization of the disturbance. Considering a finite sample of realizations makes the problem computationally tractable while the link to the original chance-constrained problem is established by a rigorous theory. With this approach, along with computational tractability, one gains the important advantage that no assumptions on the disturbance, such as boundedness, independence or Gaussianity, are required.

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Notes

  1. 1.

    Often, the total number of parameters is reduced as compared to (4) by imposing internal relation among parameters. This is further discussed in Section 4. When all θ i, j are set to zero, one obtains a classical setup where optimization is directly performed on the control actions.

  2. 2.

    In [12], a mild assumption of existence and uniqueness of the solution (Assumption 1 in [12]) is made which we do not report here for conciseness of presentation. Moreover, paper [12] considers linear cost functions but the extension to generic convex functions is straightforward.

  3. 3.

    Violation of the removed constraints must hold almost surely with respect to the scenario realizations and for any N, see [29] for a broad discussion.

  4. 4.

    In a standard LQG setting, this would require generating M independent Gaussian noise terms for each realization. In the scenario approach, however, there is no limitation on the noise structure and the noise can, e.g., be generated by an ARMA (Auto-Regressive Moving-Average system) or by any other model.

  5. 5.

    Algorithm 1 is a greedy removal algorithm which is here introduced because it can be implemented at relatively low computational cost. Other alternatives exist, and the paper [13] offers an ample discussion on this matter. Algorithm 1 comes to termination provided that at each step an active constraint can be found whose elimination leads to a cost improvement. This is a very mild condition.

  6. 6.

    The value \(\varepsilon = 0.08\) was computed from (7) by bisection instead of using the explicit formula in Theorem 3.

  7. 7.

    It is perhaps worth mentioning that it is possible to obtain better evaluations of constraints satisfaction by using the results of the recent contribution [14]. Specifically, from Theorem 2 of [14], it can be proven for the present setup that, with high confidence 1 − 10−6, it holds that

    $$\displaystyle{\mathbb{P}\left \{\sup _{i=0,\ldots,4}\|u_{\tau +i}\|_{\infty }\leq 1.8\ \ \mbox{ and}\ \ \sup _{i=1,\ldots,5}\|Cx_{\tau +i}\|_{\infty }\leq 1.8\right \} \geq 1 -\varepsilon (s_{N}^{{\ast}}),}$$

    where \(\varepsilon (\cdot )\) is a function defined over the integers given in the paper and s N ∗ is the number of the so-called “support constraints” that have been found in the problem at hand. In other words, ɛ(s N ∗) is not a-priori determined and it is a-posteriori tuned to the number of support constraints. The interested reader is referred to [14] for a more-in-depth discussion. In the present simulation, it turned out that the number of support constraints was 34, resulting in \(1 -\varepsilon (34) = 0.949\).

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Authors and Affiliations

  1. Università di Brescia - Dipartimento di Ingegneria dell’Informazione, via Branze 38, 25123, Brescia, Italy

    Marco C. Campi

  2. Politecnico di Milano - Dipartimento di Elettronica, Informazione e Bioingegneria, piazza L. da Vinci 32, 20133, Milan, Italy

    Simone Garatti & Maria Prandini

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  1. Marco C. Campi
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  2. Simone Garatti
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Correspondence to Marco C. Campi .

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  1. Independent Researcher, London, UK

    Saša V. Raković

  2. Department of Electrical and Computer Engineering, University of Maryland, College Park, MD, USA

    William S. Levine

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Campi, M.C., Garatti, S., Prandini, M. (2019). Scenario Optimization for MPC. In: Raković, S., Levine, W. (eds) Handbook of Model Predictive Control. Control Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-77489-3_19

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  • DOI: https://doi.org/10.1007/978-3-319-77489-3_19

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