Towards Hybrid Optimal Control

Abstract

In this article a general class of hybrid optimal control problems with continuous and discrete state variables and control inputs is defined. After a brief review of conventional optimal control, major novel challenges resulting from the hybrid nature are discussed. Some application problems are comparatively easy to solve because of the fixed or known sequence of discrete events; however, if the number and the sequence of discrete phases is not known a priori, the solution must then be found among a combinatorial number of possible sequence candidates. The article presents several preliminary approaches to the (numerical) solution of hybrid optimal control problems by hybrid dynamic programming, by decomposition using branch-and-bound, or fixing transversality conditions to obtain suboptimal solutions. The last two methods rely on the capabilities of the direct collocation method DIRCOL to solving multi-phase optimal control problems robustly and efficiently. Results obtained by the proposed methods are presented in two examples: an underactuated robotic system with a holding brake as the discrete component, and a hybrid, motorized traveling salesman problem.

Review of Mixed‐Integer Nonlinear and Generalized Disjunctive Programming Methods

“…Start-up and shut-down operations [262], batch systems [263], and grade transitions [258,259] are some of the main applications of optimization of hybrid systems. Several hybrid formulations have been developed and implemented in recent years [264][265][266][267][268][269][270][271]. A comprehensive review on modelling, simulation, sensitivity analysis and optimization of hybrid systems is provided by Barton and…”

Nonlinear Hybrid Dynamical Systems: Modeling, Optimal Control, and Applications

“…This solution strategy is applied to solve the cooperative multi-arm transport problem in Sect. 5.1, see also [19,18,20]. Disadvantages of this approach are the possibly high number of multi-point boundary value problems to be solved and the inherent suboptimality of the obtained solution.…”

“…The solution method for mixed-binary optimal control problems (MBOCP) using a combination of sparse direct collocation and branch-and-bound was first presented in [44] and further investigated in [19,47,48]. Given certain assumptions, the HOCP may be transformed into a MBOCP with a simple transformation of its discrete variables.…”

“…To mention some of the many applications: walking pattern and trajectory planning [26], mobile robot path planning [29], optimal payload (weight) lifting, and acrobatics [2,34], etc. Numerical algorithms designed for hybrid optimal control problems (HOCPs) with variable structure, nonlinear differential equations have recently been published [15,19,28,41]. These efforts were applied to low-dimensional illustrative problems, yet the results presented here demonstrate that numerical methods do exist which are promising for dealing with realistic, higherdimensional system models.…”

“…The key to numerically solving HOCPs seems to be the combination of efficient numerical solvers -such as direct collocation -for optimal control problems together with (heuristical) approaches to reduce the combinatorial complexity of the discrete event aspect in HOCPs [19,47,48,44]. This paper presents numerical solution techniques for HOCPs with applications in mechatronics and robotics.…”