A hybrid system is a dynamical model that describes the coupled evolution of both continuous-valued variables and discrete patterns. A prime example of such a system is a power electronic circuit, where the semiconductor transistors behave as ideal switches whose switching actions effectively change the circuit topology (i.e., the discrete pattern) that in turn defines the dynamics of currents and voltages (i.e., the continuous variables) and hence the switching actions. There have been two disparate paths to analyzing and designing hybrid systems. One path is to focus on the discrete patterns and achieve scalable, high-level analysis and synthesis. The other path is to pay attention to the dynamics of continuous variables and guarantee low-level properties such as stability and transient performance. The research objective of this proposal is to bridge these approaches by enabling a synergy between the discrete pattern based and continuous variable based approaches. The theory and algorithms developed in course of this work will be applied to digital control of power electronic circuits in order to overcome the scalability and stability issues suffered by existing approaches to power electronics design. The PIs envision that a successful completion of the project will establish a new paradigm in the analysis and design of hybrid systems, and thus contribute to the needs of modern society, such as microgrids and embedded generation, where power electronic circuits are integral parts. The research will be integrated into educational programs through student mentoring and development of courses and laboratory equipment. The PIs will make a special effort to recruit women and minority students. These broader-impact programs will help innovate science and engineering education and prepare for next-generation scientists and engineers.