An optimal finite-diemensional modeling in heat conduction and diffusion equations with partially known eigenstructure

An optimal finite-dimensional modeling technique is presented for a standard class of distributed parameter systems for heat and diffusion equations. A finite-dimensional nominal model with minimum error bounds in frequency domain is established for spectral systems with partially known eigenvalues and eigenfunctions. The result is derived from a completely characterized geometric figure upon complex plane, of all the frequency responses of the systems that have (i) a finite number of given time constants T_i's and modal coefficients k_i's, (ii) an upper bound ρ to the infinite sum of the absolute values of all the modal coefficients k_i's, (iii) an upper bound T to the unknown T_i's, and (iv) a given dc gain G(0). Discussions are made on how each parameter mentioned above makes contribution to bounding error or uncertainty, and we stress that steady state analysis for dc input is used effectively in reduced order modeling and bounding errors. The feasibility of the presented scheme is demonstrated by a simple example of heat conduction in ideal copper rod.