Mixed-integer optimal control problems with switching costs: a shortest path approach

Abstract

We investigate an extension of Mixed-Integer Optimal Control Problems by adding switching costs, which enables the penalization of chattering and extends current modeling capabilities. The decomposition approach, consisting of solving a partial outer convexification to obtain a relaxed solution and using rounding schemes to obtain a discrete-valued control can still be applied, but the rounding turns out to be difficult in the presence of switching costs or switching constraints as the underlying problem is an Integer Program. We therefore reformulate the rounding problem into a shortest path problem on a parameterized family of directed acyclic graphs (DAGs). Solving the shortest path problem then allows to minimize switching costs and still maintain approximability with respect to the tunable DAG parameter $$\theta $$ θ . We provide a proof of a runtime bound on equidistant rounding grids, where the bound is linear in time discretization granularity and polynomial in $$\theta $$ θ . The efficacy of our approach is demonstrated by a comparison with an integer programming approach on a benchmark problem.

The integrated control deviation of mixed‐integer optimal control problems with vanishing constraints

“…Setting α t,i = 1 M for all i ∈ [M ], t ∈ [N ] now leads to a deviation of at least ∆ 1 − 1 M at grid point M − 1. Finally we note that a binary control ω ∈ {0, 1} N ×M adhering to Corollary 2.7 can be computed efficiently with the combinatorial approach presented in [2].…”

“…We show that this bound is tight for the problem classes of MIOCPs and MIOCPs with VCs by construction of a counterexample. Finally we note that the deviation adhering binary control can be computed efficiently by the combinatorial approach presented in [2].…”

Application of feedback-corrected Optimal Scheduling for Reducing the Energy Consumption of a Mixing Process in Foundry

“…Theoretical work on model predictive control for switching process variables and average dwelltimes mostly bases on [8]. Other recent approaches handle switching processes with rounding strategies [9] or use stochastic approaches for energy efficient switching [10].…”

Sequential Linear Integer Programming for Integer Optimal Control with Total Variation Regularization

“…High oscillations hamper meaningful interpretations and implementations of the computed control functions in practice and correspond to high values of the total variation. In particular, if a fractional-valued control v is approximated by discrete-valued ones v n in the weak- * topology of L ∞ (v n * v), the corresponding sequence of total variation terms tends to infinity; that is, we have TV(v n ) → ∞ regardless of the value of TV(v) (see also [3]).…”

“…Therefore, minimization and constraining of total variation terms have been included in the approximation step of the combinatorial integral approximation decomposition in recent articles [2,3,33]. Furthermore, the convex relaxation of the multibang regularizer can be integrated into the combinatorial integral decomposition approach as well; see [24].…”