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

Given the output feedback system with uncertain measurement:

$\stackrel{˙}{x}\in f\left(x\left(t\right),V\left(y\left(t\right)\right),\phantom{\rule{1.em}{0ex}}x\in {ℝ}^{n},\phantom{\rule{2.em}{0ex}}y\in g\left(x\left(t\right),K\right),\phantom{\rule{1.em}{0ex}}y\in {ℝ}^{m},\phantom{\rule{4pt}{0ex}}k\subset {ℝ}^{m},$

one says that the mapping $V\left(·\right):{ℝ}^{m}\to U\subset {ℝ}^{r}$ is a viable regulation map with respect to the viability constraint $x\left(t\right)\in X\subset {ℝ}^{n}$, $t\in \left[0,+\infty \right)$ if for every $x\left(0\right)\in X$, $x\left(t\right)\in X$, $t\in \left[0,+\infty \right)$. One proves that if there is $\epsilon >0$ and a constant $\rho$ such that for every $y\in {Y}_{\epsilon }$

$\underset{u\in U}{min}\underset{x\in {E}_{\epsilon }\left(y\right)}{max}\underset{z\in {𝒫}_{X}\left(x\right)}{min}〈f\left(x,u\right)-\rho \left(x-z\right),x-z〉\le 0$

the viable regulation map is given by:

$V\left(y\right)=\left\{\begin{array}{cc}\left\{u\in U;\phantom{\rule{4pt}{0ex}}\underset{x\in {E}_{\epsilon }\left(y\right)}{max}\left[\underset{z\in {𝒫}_{X}\left(x\right)}{min}〈f\left(x,u\right),x-z〉-\rho \phantom{\rule{4.pt}{0ex}}\text{dist}{\left(x,X\right)}^{2}\right]\le 0\right\},\phantom{\rule{4pt}{0ex}}\hfill & y\in {Y}_{\epsilon },\hfill \\ U,\hfill & y\notin {Y}_{\epsilon },\hfill \end{array}\right\$

where ${Y}_{\epsilon }=g\left(X+\epsilon ℬ,K\right)$, ${E}_{\epsilon }\left(y\right)=\left\{x\in X+\epsilon ℬ$; $y\in g\left(x,K\right)\right\}$, with $y\in {Y}_{\epsilon }$, and ${𝒫}_{X}\left(x\right)=\left\{{x}^{\text{'}}\in X$; $|x-{x}^{\text{'}}|=\text{dist}\left(x,X\right)\right\}$, with $x\in X$, and $ℬ$ is the unit ball in ${ℝ}^{n}$. Two examples illustrate the obtained results.

Reviewer: M.Voicu (Iaşi)
##### MSC:
 49N35 Optimal feedback synthesis 93B52 Feedback control
##### Keywords:
output feedback system; uncertain measurement
##### References:
 [1] Aubin, J.-P.:Viability Theory Birkhäuser, Boston, 1991. [2] Aubin, J.-P. and Frankowska, H.:Set-Valued Analysis Birkhäuser, Boston, 1990. [3] Clarke, F.:Optimization and Nonsmooth Analysis, Wiley, New York, 1983. [4] Guseinov, K., Subotin, A., and Ushakov, V., Derivatives for multivalued mappings with applications to game-theoretical problems of control,Problems Control Inform. Theory 14 (1985), 155-167. [5] Krasovskii, N.N. and Subbotin, A.I.:Positional Differential Games Nauka, Moscow, 1974; English transl.,Game-Theoretical Control Problems, Springer-Verlag, New York, 1988. [6] Kurzhanski, A. and Nikonov, O., On adaptive processes in problems of guaranteed control,Izv. Akad. Nauk SSSR Techn. Kibernet. (4), 1986 (in Russian). [7] Kurzhanski, A.B. and Nikonov, O.I.: On adaptive processes in problems of guaranteed control, inProc. 9th World Congress of IFAC, Vol. 5, Budapest, 1984, pp. 176-180. [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.