# More Safe Optimal Input Signals for Parameter Estimation of Linear Systems with Spatio-Temporal Dynamics

Ewaryst Rafajłowicz^{*} and Wojciech Rafajłowicz

Our starting point is the ascertainment that D-optimal input signals recently considered by the same authors \cite{MODA13} can be too dangerous for applying them to real life system identification. The reason is that they grow too fast in time, while their spatial behaviour is quite satisfactory. In order to obtain more safe input signals, but still leading to a good estimation accuracy of parameter estimates, we propose a quality criterion that is a mixture of D-optimality and a penalty for too fast growth of input signals in time. Then the problem of finding an optimal input signal for estimating parameters of systems with spatio-temporal dynamics, described by linear partial differential equations (PDE), that admit the modal decomposition is stated. Related problems of algorithms for estimating parameters in PDE's and selecting allocation of sensors are not discussed (we refer the reader to \cite{Banks}, \cite{Ucinski} and \cite{Patan}). Our derivations are parallel to those in \cite{MODA13} up to a certain point only, since we obtain different optimality conditions in the form of an integral equation. We also briefly discuss a numerical algorithm for its solution and briefly discuss the selection of input signal when a feedback is present, since -- as it was demonstrated in \cite{Hjal} for systems described by ODE's -- the presence of feedback can be beneficial. \begin{thebibliography}{99} \bibitem{Banks} Banks H. T., Kunisch K., Estimation Techniques for Distributed Parameter Systems. Birkhauser, Boston, 1989. \bibitem{Hjal} Hjalmarsson H., Gevers M., De Bruyne F., For model-based control design, closed-loop identification gives better performance. Automatica, Vol. 32, pp 1659-1673, 1996. \bibitem{MODA13} Rafajlowicz E. and Rafajlowicz W. A variational approach to optimal input signals for parameter estimation in systems with spatio-temporal dynamics, Proceedings of Conference Model Oriented Data Analysis MODA 2013, Lagow, Poland, June 2013. \bibitem{Patan} Patan M., Optimal sensor network scheduling in identification of distributed parameter systems, Springer-Verlag, Lecture Notes in Control and Information Sciences, 2012. \bibitem{Ucinski} Ucinski D., Optimal Measurement Methods for Distributed Parameter System Identification. CRC Press, London, New York, 2005. \end{thebibliography}

Mathematics Subject Classification: 62K05 34H05

Keywords: optimal control; system identification; spatio-temporal system

Minisymposion: Optimal Design of Spatio-Temporal Networks and Systems