Robust \(L_2\)-gain control for nonlinear systems with projection dynamics and input constraints: An example from traffic control. (English) Zbl 0946.93015

Ball, Joseph A.; Day, Martin V.; Yu, Tungsheng; Kachroo, Pushkin

Automatica 35, No. 3, 429-444 (1999).

The main aim of this paper consists in solving a Hamilton-Jacobi equation related to an \(\mathbb{L}_2\)-gain control problem through construction of a stable “invariant manifold” for the Hamiltonian flow. Because in general this “invariant manifold” is not regular the stability has to be understood in the sense of Filippov’s solutions to ODEs.
Another difficulty in this topic is the presence of constraints. So the original dynamics of the system is replaced by a “projected” one on the set of constraints (assumed to be convex). This method presents the state of the new dynamics to escape the constraint set.
This idea is, in general, of great use in viability theory and is also related to Krasovski solutions of a dynamical system with disturbance. This method of projection together with the Hamilton-Jacobi equation allows to define a set-valued feedback attenuating the \(\mathbb{L}^2\)-gain.
Also a construction of a storage function derived from the Hamiltonian system (the solutions of which are taken in Filippov’s sense) is given.
One main motivation of this very interesting paper is a traffic signal control problem the solution of which is discussed in this work.

Reviewer: Marc Quincampoix (Brest)

MSC:

93B36	\(H^\infty\)-control
93C10	Nonlinear systems in control theory
93D15	Stabilization of systems by feedback
93C95	Application models in control theory
34A60	Ordinary differential inclusions

Keywords:

\(H^\infty\)-control; invariant manifold; \(\mathbb{L}_2\)-gain control problem; Filippov’s solutions; constraints; method of projection; Hamilton-Jacobi equation; set-valued feedback; storage function; traffic signal control problem