The double description method for the approximation of explicit MPC control laws

Abstract

Abstract-A standard model predictive controller (MPC) can be written as a parametric optimization problem whose solution is a piecewise affine (PWA) map from the measured state to the optimal control input. The primary limitation of this optimal 'explicit solution' is that the complexity can grow quickly with problem size, and so in this paper we seek to compute approximate explicit control laws that can tradeoff complexity for approximation error. This computation is accomplished in a two-phase process: First, inner and outer polyhedral approximations of the the convex cost function of the parametric problem are computed with an algorithm based on an extension to the classic double-description method; a convex hull approach. The proposed method has two main advantages from a control point of view: it is an incremental approach, meaning that an approximation of any specified complexity can be produced and it operates on implicitly-defined convex sets, meaning that the optimal solution of the parametric problem is not required. In the second phase of the algorithm, a feasible approximate control law is computed that has the cost function derived in the first phase. For this purpose, a new interpolation method is introduced based on recent work on barycentric interpolation. The resulting control law is continuous, although non-linear and defined over a non-simplical polytopic partition of the state space. The non-simplical nature of the partition generates significantly simpler approximate control laws than current competing methods, as demonstrated on computational examples.

A Multiresolution Approximation Method for Fast Explicit Model Predictive Control

“…Additionally, the interpolation of the data at the vertices by barycentric coordinates is always guaranteed to be inside the convex hull of the points being interpolated [23]. As a result of this property, the authors in [10] were able to show that an approximate MPC law constructed via interpolation over polytopes by barycentric coordinates guaranteed feasibility, stability, and performance bounds.…”

“…for all x ∈ R. Proof: Follows as consequence ofû(x) expressed as an interpolation by barycentric coordinates [10], [26]. Lemma 2 leads us to the following result.…”

“…In the same spirit as [10], we now show that the function approximation by adaptive hierarchical basis function expansion introduced in Section II-C generates a grid (of hypercubes) spanned by an interpolation by barycentric coordinates. Given a nodal function approximation space V d l and a d-dimensional space Ω d , consider the set of level l hypercubic regions R l,j of width h l whose centers are given by x (l+1,j) ∈ Ω d , j ∈ I l,j where…”

“…The second portion of the proof can be found in [10], [26] and is briefly outlined as follows. Note that due to the barycentric interpolation it holds thatx i (x 0 ) =…”

“…Then, a feasible control lawũ(x) is computed such thatJ is a Lyapunov function for the resulting closed-loop system. The computation of a feasible control lawũ which preserves the stability and performance of the cost function approximation can be done using the techniques discussed in [4], [6], [7], [9], [10].…”