TY - JOUR

T1 - Controlled functional differential equations

T2 - Approximate and exact asymptotic tracking with prescribed transient performance

AU - Ryan, Eugene P.

AU - Sangwin, Chris J.

AU - Townsend, Philip

PY - 2009/10

Y1 - 2009/10

N2 - A tracking problem is considered in the context of a class S of multi-input, multi-output, nonlinear systems modelled by controlled functional differential equations. The class contains, as a prototype, all finite-dimensional, linear, ℳ-input, ℳ-output, minimum-phase systems with sign-definite "high-frequency gain". The first control objective is tracking of reference signals r by the output y of any system in S: given λ ≥ 0, construct a feedback strategy which ensures that, for every r (assumed bounded with essentially bounded derivative) and every system of class S, the tracking error e = y - r is such that, in the case λ > 0, lim supt→∞ ∥e(t)∥ < λ or, in the case λ = 0, limt→∞ ∥e(t)∥ = 0. The second objective is guaranteed output transient performance: the error is required to evolve within a prescribed performance funnel ℱ (determined by a function). For suitably chosen functions α, ν and θ, both objectives are achieved via a control structure of the form u(t) = -ν(κ(t)) θ(e(t)) with κ(t) = α((t)∥e(t)∥), whilst maintaining boundedness of the control and gain functions u and κ. In the case λ = 0, the feedback strategy may be discontinuous: to accommodate this feature, a unifying framework of differential inclusions is adopted in the analysis of the general case λ ≥ 0.

AB - A tracking problem is considered in the context of a class S of multi-input, multi-output, nonlinear systems modelled by controlled functional differential equations. The class contains, as a prototype, all finite-dimensional, linear, ℳ-input, ℳ-output, minimum-phase systems with sign-definite "high-frequency gain". The first control objective is tracking of reference signals r by the output y of any system in S: given λ ≥ 0, construct a feedback strategy which ensures that, for every r (assumed bounded with essentially bounded derivative) and every system of class S, the tracking error e = y - r is such that, in the case λ > 0, lim supt→∞ ∥e(t)∥ < λ or, in the case λ = 0, limt→∞ ∥e(t)∥ = 0. The second objective is guaranteed output transient performance: the error is required to evolve within a prescribed performance funnel ℱ (determined by a function). For suitably chosen functions α, ν and θ, both objectives are achieved via a control structure of the form u(t) = -ν(κ(t)) θ(e(t)) with κ(t) = α((t)∥e(t)∥), whilst maintaining boundedness of the control and gain functions u and κ. In the case λ = 0, the feedback strategy may be discontinuous: to accommodate this feature, a unifying framework of differential inclusions is adopted in the analysis of the general case λ ≥ 0.

KW - Approximate tracking

KW - Asymptotic tracking

KW - Functional differential inclusions

KW - Transient behaviour

UR - http://www.scopus.com/inward/record.url?scp=72649090688&partnerID=8YFLogxK

U2 - 10.1051/cocv:2008045

DO - 10.1051/cocv:2008045

M3 - Article

AN - SCOPUS:72649090688

VL - 15

SP - 745

EP - 762

JO - ESAIM: Control, Optimisation and Calculus of Variations

JF - ESAIM: Control, Optimisation and Calculus of Variations

SN - 1292-8119

IS - 4

ER -