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A simple observer for distributed systems: Application to a heat exchanger. (English) Zbl 0924.93006

The authors adapt results of a previous paper [C. Z. Xu, P. Ligarius, and J. P. Gauthier, Comput. Math. Appl. 29, No. 7, 13-21 (1995; Zbl 0829.93006)], dealing with an observer for the general infinite-dimensional dissipative bilinear distributed system

$\stackrel{˙}{x}\left(t\right)={A}_{u}\left(t\right)x\left(t\right)+Bx\left(t\right)+b,\phantom{\rule{1.em}{0ex}}x\left(0\right)={x}_{0}\in H,\phantom{\rule{1.em}{0ex}}y\left(t\right)={〈x\left(t\right),c〉}_{H},\phantom{\rule{2.em}{0ex}}\left(1\right)$

over a separable Hilbert space $H$ with scalar product ${〈·,·〉}_{H},$ to prove the convergence of the observer. The following candidate observer

$\stackrel{˙}{\stackrel{^}{x}}\left(t\right)={A}_{u}\left(t\right)\stackrel{^}{x}\left(t\right)+B\stackrel{^}{x}\left(t\right)+b-r\left({〈\stackrel{^}{x}\left(t\right),c〉}_{H}-y\left(t\right)\right)c,\phantom{\rule{1.em}{0ex}}\stackrel{^}{x}\left(0\right)={\stackrel{^}{x}}_{0}\in H,\phantom{\rule{1.em}{0ex}}r>0,$

for the system (1) is proposed. All the solutions of the systems considered throughout the paper are in the weak sense. Using the class of regularly persistent inputs which preserve some uniform observability, with respect to time, the convergence property of a very simple Luenberger-like observer is proved: Given a regularly persistent positive input $u\in {ℒ}^{\infty }\left[0,\infty \right),$ the estimation error $\epsilon \left(t\right)$ converges weakly to zero in $H$ when time goes to infinity. This result means, roughly speaking, that regularly persistent inputs are sufficiently rich to preserve an asymptotic “amount of observability” on some bounded intervals ensuring the convergence of the observer. The proposed observer is applied for a heat exchanger process. Some numerical simulations are presented to illustrate the convergence property of the designed observer.

##### MSC:
 93B07 Observability 93C25 Control systems in abstract spaces 93C95 Applications of control theory