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The relation between the spatial weighting function of a manipulated variable and the locations of the zeros is discussed for some class of first order partial differential equation system which are sometimes found in a single tube heat exchanger and many other industrial processes. When the spatial weighting function is constant and the gain is high, the closed loop poles may become unstable, since the zeros exist in the neighborhood of the imaginary axis. It is found that the modification of the form of a spatial weighting function is desirable so as to move the zeros to the left on the complex plane. When the spatial weighting function is linear and quadratic, the locations of the zeros move to the left on the complex plane as the magnitude of the spatial weighting function in the outlet is specified larger than that in the inlet. When the spatial weighting function is piecewise constant, some unmovable zeros exist in the neighborhood of the imaginary axis even if the magnitude of the outlet weighting is large. It is confirmed that the continuous gradient of the spatial weighting function plays an important role in moving the zeros to the left on the complex plane.

keisoku jidō seigyo gakkai ronbunshū/keisoku jido seigyo gakkai ronbunshu

On the Relation between the Spatial Weighting Function of Manipulated Variable and the Positions of Zeros in Some System of First Order Partial Differential Equation