A Comparison of the Embedding Method to Multi-Parametric Programming,   Mixed-Integer Programming, Gradient-Descent, and Hybrid Minimum Principle   Based Methods
  In recent years, the embedding approach for solving switched optimal control problems has been developed in a series of papers. However, the embedding approach, which advantageously converts the hybrid optimal control problem to a classical nonlinear optimization, has not been extensively compared to alternative solution approaches. The goal of this paper is thus to compare the embedding approach to multi-parametric programming, mixed-integer programming (e.g., CPLEX), and gradient-descent based methods in the context of five recently published examples: a spring-mass system, moving-target tracking for a mobile robot, two-tank filling, DC-DC boost converter, and skid-steered vehicle. A sixth example, an autonomous switched 11-region linear system, is used to compare a hybrid minimum principle method and traditional numerical programming. For a given performance index for each case, cost and solution times are presented. It is shown that there are numerical advantages of the embedding approach: lower performance index cost (except in some instances when autonomous switches are present), generally faster solution time, and convergence to a solution when other methods may fail. In addition, the embedding method requires no ad hoc assumptions (e.g., predetermined mode sequences) or specialized control models. Theoretical advantages of the embedding approach over the other methods are also described: guaranteed existence of a solution under mild conditions, convexity of the embedded hybrid optimization problem (under the customary conditions on the performance index), solvability with traditional techniques (e.g., sequential quadratic programming) avoiding the combinatorial complexity in the number of modes/discrete variables of mixed-integer programming, applicability to affine nonlinear systems, and no need to explicitly assign discrete/mode variables to autonomous switches. 
Richard Meyer and Milo\v{s} \v{Z}efran and Raymond A. DeCarlo
math.OC cs.SY