#### Informal Systems Seminar (ISS), Centre for Intelligent Machines (CIM) and Groupe d'Etudes et de Recherche en Analyse des Decisions (GERAD)

### The Large Gain Theorem and its Applications

Ryan James Caverly

Department of Aerospace Engineering , University of Michigan

October 7, 2016 at 11:00 AM

McConnell Engineering Room 437

The Small Gain Theorem is a well-known stability result that it widely used in the field of robust control. However, the Small Gain Theorem cannot be directly used to assess the stability of a feedback interconnection that involves open-loop unstable systems. The Large Gain Theorem is a recently developed and little-known stability result that uses the concept of minimum gain, rather than maximum gain, to assess the stability of a feedback interconnection that may be open-loop stable or unstable. In this talk, the concepts of minimum gain and the Large Gain Theorem are introduced and comparisons are made to the Small Gain Theorem using input-output theory and a Nyquist stability criterion interpretation. Applications of the Large Gain Theorem are also presented, including the design of robust feedback controllers.

Controllers designed with the Large Gain Theorem are capable of robustly stabilizing systems with uncertainty that has large gain or is even unstable. This is a feat that can be difficult, or even impossible, for any controller relying on the Small Gain Theorem, such as an H-infinity controller, to achieve without significant reformulation of the problem.

Controllers designed with the Large Gain Theorem are capable of robustly stabilizing systems with uncertainty that has large gain or is even unstable. This is a feat that can be difficult, or even impossible, for any controller relying on the Small Gain Theorem, such as an H-infinity controller, to achieve without significant reformulation of the problem.