Implicit vs explicit MPC — Similarities, differences, and a path owards a unified method

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The challenge of implementing Model Predictive Control (MPC) on embedded hardware is to obtain the optimal solution while taking into account limited resources of the implementation hardware. Traditionally, the optimal solution is obtained either by an iterative numerical procedure (referred to as implicit MPC), or by evaluating the explicit representation of the MPC feedback law, which is obtained off-line using parametric programming (referred to as explicit MPC). Both approaches have their pros and cons. Implicit MPC requires more computational resources, but is able to handle large systems. Explicit MPC, on the other hand, requires less online computation, but the off-line construction of the feedback law scales badly with increasing dimensionality of the problem. Moreover, the memory footprint of the explicit solutions can easily violate limits of the available memory storage. In this tutorial talk we first highlight the similarities between implicit and explicit MPC. Specifically, we point out that explicit MPC is nothing but an active set method with a complete pre-factorization of all locally optimal Karush-Kuhn-Tucker systems. This observation allows to address the two main shortcomings of explicit MPC. First, we show how to obtain the explicit solution even for systems with a large number of states, provided the number of optimization variables is modest. Second, and more importantly, we illustrate that the memory footprint of such solutions can be continuously scaled to meet the hardware limits with implicit MPC being one endpoint of such a scale. In particular, we show that explicit MPC solutions can be characterized without the need to store the associated critical regions. As an outcome we obtain an embedded MPC algorithm which can abolish the maximal amount of on-line computation in exchange for memory storage.