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Sufficient conditions for viability under imperfect measurement. (English) Zbl 0802.49029

Given the output feedback system with uncertain measurement:

x ˙f(x(t),V(y(t)),x n ,yg(x(t),K),y m ,k m ,

one says that the mapping V(·): m U r is a viable regulation map with respect to the viability constraint x(t)X n , t[0,+) if for every x(0)X, x(t)X, t[0,+). One proves that if there is ε>0 and a constant ρ such that for every yY ε

min uU max xE ε (y) min z𝒫 X (x) f(x,u)-ρ(x-z),x-z0

the viable regulation map is given by:

V(y)={uU;max xE ε (y) [min z𝒫 X (x) f(x,u),x-z-ρdist(x,X) 2 ]0},yY ε ,U,yY ε ,

where Y ε =g(X+ε,K), E ε (y)={xX+ε; yg(x,K)}, with yY ε , and 𝒫 X (x)={x ' X; |x-x ' |=dist(x,X)}, with xX, and is the unit ball in n . Two examples illustrate the obtained results.

Reviewer: M.Voicu (Iaşi)
MSC:
49N35Optimal feedback synthesis
93B52Feedback control
References:
[1]Aubin, J.-P.:Viability Theory Birkhäuser, Boston, 1991.
[2]Aubin, J.-P. and Frankowska, H.:Set-Valued Analysis Birkhäuser, Boston, 1990.
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[8]Veliov, V.: Guaranteed control of uncertain systems: Funnel equations and existence of regulation maps, Working Paper WP-92-62, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1992.