Abstract
Tailored Mixed-Integer Optimal Control policies for real-world applications usually have to avoid very short successive changes of the active integer control. Minimum dwell time (MDT) constraints express this requirement and can be included into the combinatorial integral approximation decomposition, which solves mixed-integer optimal control problems (MIOCPs) to epsilon -optimality by solving one continuous nonlinear program and one mixed-integer linear program (MILP). Within this work, we analyze the integrality gap of MIOCPs under MDT constraints by providing tight upper bounds on the MILP subproblem. We suggest different rounding schemes for constructing MDT feasible control solutions, e.g., we propose a modification of Sum Up Rounding. A numerical study supplements the theoretical results and compares objective values of integer feasible and relaxed solutions.
Highlights
We consider mixed-integer optimal control problems (MIOCPs) on the fixed and finite time horizon T := [t0, t f ] ⊂ R of the following form inf x,v Φ (x(t f ))s. t. x(t) = f(x(t), v(t)) for t ∈ T, (1.1) x(t0) = x0, (1.2) v ∈ VMDT ⊂ V . (1.3)The differential states x ∈ W 1,∞(T, Rnx ) with fixed initial values x0 ∈ Rnx are governed by the right-hand side ordinary differential equation (ODE) function f : Rnx × {v1, . . . , vnω } → Rnx, which is assumed to be continuous in the first argument
As this study does not focus on the nonlinear program (NLP) formulation, i.e., how a is achieved, and since we propose to solve (CIA-UD) by means of tailored rounding heuristics or a branch and bound algorithm and not with a standard mixed-integer linear program (MILP) solver exploiting extended formulations, we would benefit neither numerically nor theoretically from alternative MILP formulations and we skip the presentation of these
We introduced in the dwell time next forced rounding (DNFR) scheme the min down mode χD = 1 and are going to deduce in the sequel an alternative upper bound compared to the one obtained by DNFR with χD = 0
Summary
We consider mixed-integer optimal control problems (MIOCPs) on the fixed and finite time horizon T := [t0, t f ] ⊂ R of the following form inf x,v. Vnω } is assumed to be measurable and is further restricted by minimum dwell time (MDT) constraints represented by the subset VMDT. Vnω } ⊂ Rnv so that the discrete control function v : T → {v1, . We stress that this function takes values out of a finite set with cardinality nω ∈ N and exclude the trivial case nω = 1. We minimize Φ ∈ C0(Rnx , R) over the end state, which in turn depends on the discrete control function v
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