Robust optimality of linear saturated control in uncertain linear network flows

Abstract

Abstract-We propose a novel approach that, given a linear saturated feedback control policy, asks for the objective function that makes robust optimal such a policy. The approach is specialized to a linear network flow system with unknown but bounded demand and politopic bounds on controlled flows. All results are derived via the Hamilton-Jacobi-Isaacs and viscosity theory.

Objective function design for robust optimality of linear control under state-constraints and uncertainty

“…The way to prove the equality between V andṼ used in [3] is to consider respectively the Hamilton-JacobiBellman and the Hamilton-Jacobi-Isaacs equations that they must respectively solve in the viscosity sense (see Crandall et al [14] and Bardi and Capuzzo Dolcetta [4]). Such equations are respectively, neglecting for the moment possible boundary conditions, V (ζ) + H(ζ, ∇V (ζ)) = 0, (0.11)…”

“…V (ζ) +H(ζ, ∇Ṽ (ζ)) = 0, (0.12) where ∇ is the gradient, and the Hamiltonians H andH are defined, for all ζ ∈ R n , p ∈ R n , as In [3] we actually exhibit a function φ solving both (0.11)-(0.12), and, using uniqueness results for (0.11)-(0.12) we conclude φ = V =Ṽ . However, apart from a simpler one-dimensional case, it seems suitable to split the problem into two different problems: one outside the target T , giving it a "minimum-time" feature, and one inside the target, giving it a linear-quadratic feature.…”

“…For the second case, some restriction on the behavior of the time-dependent disturbance w(·) is forced by choosing some suitable forcing terms in g. In the present work instead, we address the problem inside the target only, for which the saturated control is then linear and which presents state-constraints. Moreover we relax some of the forcing terms considered in [3]. To this end, as explained next, it seems necessary to introduce a discontinuity (switching term) in the objective function g, and this fact may lead to some problem for the uniqueness of the Hamilton-Jacobi equations.…”

“…In the recent work [3], the authors study the following problem: find an objective function (ζ, μ, ω) → g(ζ, μ, ω) such that the feedback linear saturated control u(t) = sat(−kz(t)) = (u 1 (t), . .…”

“…In [3] the problem is firstly addressed giving a suitable meaning to the robust optimality of the linear saturated control. This is done by a "differential game" approach in the sense of lower value of the game (see Elliot and Kalton [16]).…”