A Multiresolution Approximation Method for Fast Explicit Model Predictive Control

Abstract

Abstract-A model predictive control law is given by the solution to a parametric optimization problem that can be precomputed offline and provides an explicit map from state to control input. In this paper, an algorithm is introduced based on wavelet multiresolution analysis that returns a low complexity explicit model predictive control law built on a hierarchy of second order interpolets. The resulting interpolation is shown to be everywhere feasible and continuous. Further, tests to confirm stability and to compute a bound on the performance loss are introduced. Since the controller approximation is built on a grid hierarchy, convergence to a stabilizing control law is guaranteed and the evaluation of the control law in real-time systems is naturally fast and runs in a bounded logarithmic time. Two examples are provided; A two-dimensional example with an evaluation speed of 31 ns and a four-dimensional example with an evaluation speed of 119 ns.

Integration of real-time and stability constraints via hybrid polytopic partitions

“…Partitions obtained by hierarchical subdivision were studied in [6], [18] in the context of approximate explicit MPC. The BHS partition is obtained by partitioning X through a single hyperplane, i.e., [1,n] .…”

“…These issues lead to complex optimization problems in online MPC or highly-complex controller partitions in explicit MPC. To reduce the computational power of the control hardware required by MPC algorithms sub-optimal on-line MPC solutions and approximations of explicit MPC solutions were recently proposed, see, e.g., [2], [5], [18] and the references therein. Nevertheless, guaranteeing stability for suboptimal and approximative solutions is generally difficult and often requires an a posteriori test.…”

A parametric branch and bound approach to suboptimal explicit hybrid MPC

“…Even though this potentially allows for fast online applications, the complexity of the solution that is sought for might make the process intractable already at the point of offline computation of the explicit solution. This situation motivated the development of a variety of different approximate explicit MPC schemes, see, e.g., Summers, Jones, Lygeros, and Morari (2011). In the case of hybrid systems and a quadratic cost function, the computation of explicit MPC control laws boils down to solving a parametric mixed-integer quadratic program.…”

Analysis and Control of Flywheel Energy Storage Systems

“…In the past three decades the theory of optimal control has been well developed in nearly all aspects, such as stability, nonlinearity, and robustness (Summers et al, 2011;Rawlings et al, 2008;Mayne, et al, 2000). It is known that multivariable constrained control problems in state-space can be effectively handled using Linear Quadratic Gaussian (LQG).…”