Optimal control of constrained hybrid dynamical systems: Theory, computation and applications
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Hybrid dynamical systems arise in a number of application areas such as power converters, autopilots, manufacturing, process control, hybrid cars, mobile and humanoid robotics etc., to name a few and as such the optimal control of these systems has been an area of active research. These systems are characterized by two components: subsystems (modes) with continuous or discrete dynamics and a switching law which determines which of these subsystems is active at a given time. While in theory, we can switch infinitely many times between different modes in a finite amount of time, physical systems need to spend some minimum time in a mode before they can switch to another mode due to mechanical reasons, power constraints, information delays, stability considerations etc and must spend some minimum amount of time in a mode before they can switch to another mode. This minimum time is known as the dwell time, a term first used in the context of stability of hybrid systems, and the optimal control of hybrid systems under these constraints is the main focus of this thesis. The presence of the dwell time constraints raises interesting theoretical and computational questions which are addressed in thesis. We consider the general hybrid optimal control problem subject to dwell time constraints thereby establishing necessary conditions for optimality and develop numerical schemes to compute solutions to these problems and prove their convergence. Any physical system that switches is subject to dwell time constraints, small or large, and thus amenable to our framework. To demonstrate, however, the generality and thus wide applicability of our results, we consider the application to an interesting problem in Precision Agriculture, namely the problem of optimal pesticide scheduling and present a case study to demonstrate the application of our methodology. In this thesis, we also consider a class of constrained hybrid optimal control problems inspired by problems in power aware mobile robotic networks that are subject to various constraints on inputs and states. In particular, we consider the problem of jointly minimizing motion and communication energy in power aware mobile robotic networks required to perform various co-ordinated tasks such as the transmission of given amount of data to a remote base station under time and resource constraints and where the robot decision variables are acceleration (continuous), for controlling the motion of the robot and spectral efficiency (discrete), catering to data transmission requirements. Framing this co-optimization problem as a constrained hybrid optimal control problem in the general setting and subsequently solving it using efficient algorithms is another main topic of this thesis. This problem, like any other hybrid optimal control problem, is also subject to dwell time constraints, signifying the importance of the dwell time problem addressed in this thesis. We present numerous application scenarios to demonstrate the utility of our framework. Finally, we propose a multiple shooting based gradient descent techniques to solve a class of complex optimal and hybrid control problems with large time horizons, which otherwise are hard to solve due to numerical problems arising from instability issues associated with the state or co-state equation. The two point boundary problem resulting from solving the optimal or hybrid optimal control problem is transformed into an equivalent optimal control problem over extended states comprising of the original state equation and the costate equation and then solved. Again, the results here are general and we demonstrate the effectiveness of our method by considering its application to solving large multi-agent co-optimization problem in power aware mobile robotic networks.
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