Global dynamic optimization of linear hybrid systems

Hybrid systems, which exhibit both discrete state and continuous state behavior, have become the modeling framework of choice for a wide variety of applications that require detailed dynamic models with embedded discontinuities. Economic, environmental, safety and quality considerations in these applications, such as the automated design of safe operating procedures and the formal verification of embedded systems, strongly motivate the development of algorithms and tools for the global optimization of hybrid systems. For safety critical tasks such as formal verification, it is crucial that the global solution be found within epsilon optimality. Recently, a deterministic algorithm has been proposed for the global optimization of linear time varying hybrid systems in the continuous time domain, provided the transition times and the sequence of modes are fixed. This work is extended by presenting a method for determining the optimal mode sequence when the transition times are fixed. A reformulation of the problem by introducing binary decision variables while retaining the linearity of the underlying dynamic system is proposed. This allows recently developed convexity theory for linear time varying continuous systems to be employed to construct a convex relaxation of the resulting mixed-integer dynamic optimization problem, which enables the global solution to be found in a finite number of iterations using nonconvex outer approximation.