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An Efficient Mixed-Integer Formulation and an Iterative Method for Optimal Control of Switched Systems Under Dwell Time Constraints

Ramin Abbasi-Esfeden, Armin Nurkanovic, Moritz Diehl, Panagiotis Patrinos, Jan Swevers

TL;DR

This work tackles optimal control of switched systems with discrete inputs under MDT constraints by formulating a MINLP that decomposes into Sequence Optimization (SO) and Switching Time Optimization (STO). A master sequence $\bar{v}$ and a binary vector $b \in \{0,1\}^{\bar{M}}$ restrict SO to subsequences, yielding a discretization-size–independent number of binaries and enabling scalable solvers. An Iterative Switching Time Optimization (ISTO) algorithm solves STO and SO heuristically by softly enforcing MDT via a slack $e$ and a quadratic penalty, then pruning collapsed modes and repeating, with convergence under MPCC-LICQ-type conditions. Numerical experiments on four switched systems (DTS, LVF, VDP, TRJ) show that the MINLP formulation with ISTO achieves near-optimal performance with significantly reduced computation time compared to full MILP approaches and the Combinatorial Integral Approximation (CIA). The proposed framework thus provides a practical, scalable method for high-fidelity optimal control of switched systems under dwell-time constraints.

Abstract

This paper presents an efficient Mixed-Integer Nonlinear Programming (MINLP) formulation for systems with discrete control inputs under dwell time constraints. By viewing such systems as a switched system, the problem is decomposed into a Sequence Optimization (SO) and a Switching Time Optimization (STO) -- the former providing the sequence of the switched system, and the latter calculating the optimal switching times. By limiting the feasible set of SO to subsequences of a master sequence, this formulation requires a small number of binary variables, independent of the number of time discretization nodes. This enables the proposed formulation to provide solutions efficiently, even for large numbers of time discretization nodes. To provide even faster solutions, an iterative algorithm is introduced to heuristically solve STO and SO. The proposed approaches are then showcased on four different switched systems and results demonstrate the efficiency of the MINLP formulation and the iterative algorithm.

An Efficient Mixed-Integer Formulation and an Iterative Method for Optimal Control of Switched Systems Under Dwell Time Constraints

TL;DR

This work tackles optimal control of switched systems with discrete inputs under MDT constraints by formulating a MINLP that decomposes into Sequence Optimization (SO) and Switching Time Optimization (STO). A master sequence and a binary vector restrict SO to subsequences, yielding a discretization-size–independent number of binaries and enabling scalable solvers. An Iterative Switching Time Optimization (ISTO) algorithm solves STO and SO heuristically by softly enforcing MDT via a slack and a quadratic penalty, then pruning collapsed modes and repeating, with convergence under MPCC-LICQ-type conditions. Numerical experiments on four switched systems (DTS, LVF, VDP, TRJ) show that the MINLP formulation with ISTO achieves near-optimal performance with significantly reduced computation time compared to full MILP approaches and the Combinatorial Integral Approximation (CIA). The proposed framework thus provides a practical, scalable method for high-fidelity optimal control of switched systems under dwell-time constraints.

Abstract

This paper presents an efficient Mixed-Integer Nonlinear Programming (MINLP) formulation for systems with discrete control inputs under dwell time constraints. By viewing such systems as a switched system, the problem is decomposed into a Sequence Optimization (SO) and a Switching Time Optimization (STO) -- the former providing the sequence of the switched system, and the latter calculating the optimal switching times. By limiting the feasible set of SO to subsequences of a master sequence, this formulation requires a small number of binary variables, independent of the number of time discretization nodes. This enables the proposed formulation to provide solutions efficiently, even for large numbers of time discretization nodes. To provide even faster solutions, an iterative algorithm is introduced to heuristically solve STO and SO. The proposed approaches are then showcased on four different switched systems and results demonstrate the efficiency of the MINLP formulation and the iterative algorithm.
Paper Structure (8 sections, 4 theorems, 24 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 8 sections, 4 theorems, 24 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. PROBLEM FORMULATION
  3. Discrete Input Systems
  4. Minimum Dwell Time Constraints
  5. ITERATIVE SWITCHING TIME Optimization
  6. ISTO Algorithm
  7. NUMERICAL RESULTS
  8. CONCLUSIONS

Key Result

theorem 1

Inequalities so:condition:ineq imply $z_I = 1$ is equivalent to so:conditions.

Figures (1)

  • Figure 1: From left to right: the Double Tank problem, the Lotka Volterra Fishing binary problem, the Van Der Pol oscillator, and the Particle Trajectory problem.

Theorems & Definitions (12)

  • definition 1
  • theorem 1
  • proof
  • theorem 2
  • proof
  • definition 2
  • lemma 1
  • proof
  • theorem 3
  • proof
  • ...and 2 more