A parametric branch and bound approach to suboptimal explicit hybrid MPC

Abstract

a b s t r a c tIn this article we present a parametric branch and bound algorithm for computation of optimal and suboptimal solutions to parametric mixed-integer quadratic programs and parametric mixed-integer linear programs. The algorithm returns an optimal or suboptimal parametric solution with the level of suboptimality requested by the user. An interesting application of the proposed parametric branch and bound procedure is suboptimal explicit MPC for hybrid systems, where the introduced user-defined suboptimality tolerance reduces the storage requirements and the online computational effort, or even enables the computation of a suboptimal MPC controller in cases where the computation of the optimal MPC controller would be intractable. Moreover, stability of the system in closed loop with the suboptimal controller can be guaranteed a priori.

Explicit hybrid model-predictive control: The exact solution

“…Furthermore, it is assumed that (1c) uniquely determines δ k and z k , implying that x k+1 is also uniquely determined (Axehill et al, 2014).…”

“…In the next step, the solution of this mp-QP was compared to the current best upper bound. While Oberdieck et al (2014) applied McCormick relaxations to increase the fathoming efficiency of the comparison procedure between the solution with the upper bound, Axehill et al (2014) did not apply a comparison procedure for the nonconvex objective functions but rather compared solutions over the entire critical region. Additionally, Almér and Morari (2013) proposed an algorithm for a special class of mp-MIQP problems found in a certain type of hybrid MPC problems.…”

“…This combination was then fixed resulting in a multiparametric quadratic programming (mp-QP) subproblem. Recently, Axehill, Besselmann, Raimondo, and Morari (2014) and Oberdieck, Wittmann-Hohlbein, and Pistikopoulos (2014) independent of each other proposed a branch-and-bound procedure which uses a binary search tree to successively fix the integer variables. At each node of the search tree, the remaining integer variables were relaxed to bounded continuous variables, and the resulting mp-QP problem was solved.…”

Model-based multi-parametric programming strategies towards the integration of design, control and operational optimization

“…While di erent algorithms are available to solve hybrid mp-MPC problems [14,48,98,257], their applicability is still limited due to the high computational burden associated with the solution of multi-parametric mixed-integer programming (mp-MIP) problems.…”

“…The parameter dependence of the objective function can be avoided using the , 96, 201, 257, 294, 359] 2-norm Hybrid system [7,14,98,152,257] *:…”