Output Reach Control Problem with Applications to Motion Planning for Robotic Systems

As electronic devices become more embedded into everyday products, an increasing number of consumer devices are being equipped with control systems. These control systems are sometimes required to achieve complex specifications such as safety constraints or switching between distinct tasks. Among a wide range of techniques to deal with the control design of complex systems, one is to partition the state space into polytopic regions, and solve a control problem on each region such that the collective behaviour of the hybrid system solves the original complex control specification. In this dissertation we consider such a methodology where the control problem on each polytopic region is called the {\em Reach Control Problem} (RCP). The RCP is to find a state feedback to make the closed-loop trajectories of an affine control system defined on a polytope reach and exit a prescribed facet of the polytope in finite time. There is already an extensive literature on the RCP. This dissertation extends this literature by considering the {\em Output Reach Control Problem} (ORCP). The ORCP is to control the output trajectory of the control system on a simplex in the output space, in contrast to control the entire state trajectory. We also extend the applications of the RCP by developing a unified framework for control with complex specifications. This dissertation contains four distinct contributions. The first contribution provides an extension of classical linear regulator theory to affine systems and exosystems. This extension then provides the basis to develop a framework for the output reach control problem with disturbances. The second contribution leverages existing literature on viability theory to convert the ORCP in the output space on a simplex to a RCP in the full state space on a polytope. The third contribution is to propose a unified framework for solving control problems with complex specifications. Our framework is targeted at robotic systems. Finally, our fourth contribution is a set of so called {\em motion primitives} which comprise one of the layers of our aforementioned framework. These motion primitives are based on our approach to the ORCP.Ph.D