Suboptimal control of linear systems by augmentation with application to freeway traffic regulation

Abstract

The development of on-ramp metering control strategies for a freeway can be aided by the well-developed theory of linear optimal control. However, an adequate model of an appropriate form must be available. Recently developed variable models appear to be best suited to this purpose. In these models the freeway is sectioned into lengths of about \frac{1}{2} mi and traffic is described in terms of the aggregate variables section speed, section density, and volume. The model is used to formulate a linear regulator problem with quadratic cost for the purpose of determining a feedback control rule which returns the freeway to nominal conditions after a disturbance. The solution of the regulator problem involves considerable computation difficulty when the order of the system is large. In a large class of applications the optimal control law has a local structure which can be exploited in a computationally efficient manner to yield a suboptimal control with little loss of system performance. A new technique, the augmentation method, is presented for constructing a suboptimal control from a set of optimal controls for low-order sub-systems. New results are presented which provide a rational and elegant measure for the additional cost associated with an arbitrary stabilizing suboptimal control. Additional results pertain especially to the augmentation procedure. The models and suboptimal control scheme are applied to a specific detailed design for an 8-mi segment of the San Diego Freeway in Los Angeles. Simulations are shown illustrating the effectiveness of such traffic-responsive controls in clearing up congestion due to traffic incidents.