Linear Matrix Inequality Techniques for the Optimization of Interval Observers for Spatially Distributed Heating Systems

Interval observers provide the possibility to estimate guaranteed enclosures for the state variables of a dynamic system that are compatible on the one hand with a predefined mathematical model in which uncertain but bounded parameters may be included. On the other hand, they allow for a correction of the state estimates by a Luenberger-like observer where bounded tolerances of the measured system outputs are taken into consideration. Especially for cooperative system models, these interval observers can be implemented in a straightforward manner. Then, two separate bounding systems (one for the lower and one for the upper bounds of the respective state variables) have to be defined. In previous work, an offline parameter identification scheme was interfaced with a fundamental interval observer for a class of distributed heating systems. There, preserving the property of cooperativity by the Luenberger-like observer and guaranteeing asymptotic stability of the error dynamics was in focus. In addition, the current paper aims at optimizing the observer gains in such a way that the widths of the resulting state estimates can be influenced in a systematic manner. For that purpose, linear matrix inequality techniques are employed which aim at the minimization of a suitable $H_{\infty}$ norm. Experimental state estimation results for a lab-scale distributed heating system conclude this contribution.