Optimal Control of Hybrid Systems, with Application to Vehicle Dynamics
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Hybrid dynamical systems are common throughout the physical and computer world, and they consist of dynamical systems that contain both continuous time and discrete time dynamics. Examples of this type of system include thermostat controlled systems, multi-geared transmission based systems, and embedded computer systems. Sometimes, complicated non-linear continuous time systems can be simplified by breaking them up into a set of less complicated continuous systems connected through discrete interactions (referred to as system hybridization). One example is modeling of vehicle dynamics with complicated tire-to-ground interaction by using a tire slipping or no slip model. When the hybrid system is to be a controlled dynamical system, a limited number of tools exist in the literature to synthesize feedback control solutions in an optimal way. The purpose of this dissertation is to develop necessary and sufficient conditions for finding optimal feedback control solutions for a class of hybrid problems that applies to a variety of engineering problems. The necessary and sufficient conditions are developed by decomposing the hybrid problem into a series of non-hybrid optimal feedback control problems that are coupled together with the appropriate boundary conditions. The conditions are developed by using a method similar to Bellman's Dynamic Programming Principle. The solution for the non-hybrid optimal control problem that contains the final state is found and then propagated backwards in time until the solution is generated for every node of the hybrid problem. In order to demonstrate the application of the necessary and sufficient conditions, two hybrid optimal control problems are analyzed. The first problem is a theoretical problem that demonstrates the complexity associated with hybrid systems and the application of the hybrid analysis tools. Through the control problem, a solution is found for the feedback control that minimizes the time to the origin problem for a hybrid system that is a combination of two standard optimal control problems found in the literature; the double integrator system and a harmonic oscillator. Through the second problem, optimal feedback control is found for the drag racing and hot-rodding control problems for any initial reachable state of the system and a hybrid model of a vehicle system with tire-to-ground interaction.