Robust L-2-gain control for nonlinear systems with projection dynamics andinput constraints: an example from traffic control

We formulate the L-2-gain control problem for a general nonlinear, state-sp ace system with projection dynamics in the state evolution and hard constra ints on the set of admissible inputs. We develop specific results for an ex ample motivated by a traffic signal control problem. A state-feedback contr ol with the desired properties is found in terms of the solution of an asso ciated Hamilton-Jacobi-Isaacs equation (the storage function or value funct ion of the associated game) and the critical point of the associated Hamilt onian function. Discontinuities in the resulting control as a function of t he 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 t he closed-loop dynamics must be interpreted as a differential set inclusion in the sense of Filippov. Construction of the storage function is via a ge neralized stable invariant manifold for the flow of a discontinuous Hamilto nian vector-field, which again must be interpreted in the sense of Filippov . For the traffic control model example, the storage function is constructe d explicitly. The control resulting from this analysis for the traffic cont rol example is a mathematically idealized averaged control which is not imm ediately implementable; implementation issues for traffic problems will be discussed elsewhere. (C) 1999 Elsevier Science Ltd. All rights reserved.