By K. J. Astrom, T. Hagglund
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Ia) that rn rer: This formula 24 will now be illustrated G'(0) -C(0) by a few examples. rG) that the average residence time is Tor:f:Y' q The average residence time is thus the ratio of the volume and the flow through the tank. b is G(" ) : e-' L. rG) that the average residence time is To, : L. The average residence time is thus equal to the time delay. 11-AvnRAGE RnsnnNcn TrnappoR cascADED sysrpnrs A system that is the cascade combination of two stable linear systems with transfer functions Gr(r) and G2(s) has the transfer function c(') - Gr(s)G2(s).
46) The fact that the time delay L can be estimated in two ways can be used to asses if a process can be modeled by an FOTD model. The selection of the pulse time To can be determined automatically, for example, as the time when the output has changed a specified amount. The method can be applied to SOTD models, but the formulas are more complicated. ,1. stepresponses determine the dynamics is short because it is not necessary to wait for steady state as for the bump test. The disadvantages of the method are that it is difficult to determine times when the extrema occurs accurately, and the estimate of the gain is poor because the excitation of the pulse is mainly in the highfrequency regime.
81 step responses for a large batch of systems that are normalized to give the same average resiclence time. ) The figure shows that all step responses are close for t - Tor. 08. 63 where the step response has reached 68 percent of its final value. 3i. The static gain Ko is simply determined from the steady-state values of the signals before and afier the siep change' The apparent time delay Z is given by the point where the steepest tangent intersects the steady-state level before the step change.
Advanced PID Control by K. J. Astrom, T. Hagglund