Leader-Follower Controls in Systems with Two Controllers

The classical control theory deals with dynamical systems that are controlled by one controller designed to ensure maximum performance of the system over a specific interval of time. More recently, there has been considerable interest in systems that are controlled by more than one controller. In this paper, we consider control systems that are controlled by two independent controllers whose roles and objectives in terms of the performance of the system are generally different. More specifically, we consider systems where one controller has an advantage over the other in that it has the capability of designing and implementing its control first, before the other controller. With such control hierarchy this controller is designated as leader and the other as follower. To take advantage of its primary role, the leader control is designed by anticipating and taking into consideration the follower's control. The follower becomes the sole controller in the system after the leader's control has been implemented. In this paper, we describe such systems and derive in detail the controls of both, the leader and follower. We consider an economic control system that models the behavior of two firms in a dynamic market where one firm is more powerful than the other and as a result assumes the role of a market leader. We derive the general conditions that govern the existence of the leader and follower's control functions. We then treat in detail the case where the system dynamics are linear and productions costs are quadratic. Finally, we illustrate the results with a numerical example.