A control theory for complex specifications

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Control theory is a discipline bridging mathematics and engineering and its function is to supply the theories, frameworks, principles, and algorithms for analysis and control synthesis problems arising in a spectrum of application disciplines. Its clients include energy systems, biomedical engineering, mechatronics, robotics, aerospace systems, nanotechnology, transportation systems, process control, manufacturing, biology, ecology, economics, and other areas. Control theory has focused over the last 200 years on problems of regulation and tracking. Common examples of regulation are maintaining a constant temperature in a building or maintaining a desired speed in a vehicle cruise control. Optimizing a solar cell performance by rotating its surface to continually face the sun during daylight hours is an example of tracking. Tracking and regulation are simple control system specifications. As control systems become more integrated within high-end engineering systems as well as consumer products, they are expected to achieve richer specifications that intertwine regulation and tracking with safety requirements, startup procedures, logic rules, and so forth. Good examples of complex specifications are the startup of a chemical or nuclear reactor, requirements for an automated parking system, or transition between different gaits in a bipedal robot. Because of the historical emphasis on regulation and tracking, even the simplest control problems with non-traditional specifications have no known solution method. Our research program concerns the development of a control theory for complex specifications. This is a multistep endeavor involving both the discovery of fundamental mathematical structures within a control system that determine if it can achieve a complex specification, as well as the development of concrete design methods that can be used by practitioners in diverse application fields.