On regulator design for spatial temperature distribution using finite-dimensional modeling of heat equations

A regulator problem for a heat conduction system, of which the eigenstructure is just partially known, is formulated to design a stabilizing controller that keeps a performance index less than a prescribed value. The index is made of the spatio integral of the squared deviation from reference temperature distribution. It is shown that through characterizing frequency response from input to temperature at each spatial point, a distributed parameter system with nominal model and additive uncertainty weight, both of which are real rational, is reconstructed using knowledge of the eigenstructure. A main result claims that the formulated problem is reduced to a standard mixed H₂/H_∞ one for a linear finite dimensional time-invariant system. Numerical study demonstrates feasibility of the proposed design scheme.