Dissipativity, Stability, and Connections: Progress in Complexity

This article reviews the development of dissipativity theory, with application to stability analysis from the seminal 1972 articles by Jan Willems toward both basic generalizations and specific results for classes of complex systems. After a brief historical background for stability analysis, a theory by Hill and Moylan is reviewed, which includes a very general input–output-to-state–space dissipativity connection, a framework for abstract stability results, and aspects not often addressed (that is, nonquadratic supply rates and multiple equilibria). Then, some extensions for testing dissipativity and stability for other forms of complexity in switching, impulsive dynamics, and network systems are presented, including connections to the input-to-state stability approach, with some suggestions for further work.