Robust L2-gain control for nonlinear systems with projection dynamics and input constraints:: an example from traffic control

We formulate the L(2)-gain control problem for a general nonlinear, state-space system with projection dynamics in the state evolution and hard constraints on the set of admissible inputs. We develop specific results for an example motivated by a traffic signal control problem. A state-feedback control with the desired properties is found in terms of the solution of an associated Hamilton-Jacobi-Isaacs equation (the storage function or value function of the associated game) and the critical point of the associated Hamiltonian function. Discontinuities in the resulting control as a function of the state and due to the boundary projection in the system dynamics lead to hybrid features of the closed-loop system, specifically jumps of the system description between two or more continuous-time models. Trajectories for the closed-loop dynamics must be interpreted as a differential set inclusion in the sense of Filippov. Construction of the storage function is via a generalized stable invariant manifold for the flow of a discontinuous Hamiltonian vector-field, which again must be interpreted in the sense of Filippov. For the traffic control model example, the storage function is constructed explicitly. The control resulting from this analysis for the traffic control example is a mathematically idealized averaged control which is not immediately implementable; implementation issues for traffic problems will be discussed elsewhere. (C) 1999 Elsevier Science Ltd. All rights reserved.