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By K. J. Astrom, T. Hagglund
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Extra info for Advanced PID Control
The average residence time of the system is rt o y t, -- T t The frequency response of the system is G(ia : ' ' t )+ L a '(l r - r-'"). The transfer function is zero for aT - 2nn. 25. An interesting property of this transfer function is that arsG(ia): -+. foratT 12n. A ContinuousStirredTankReactor Consider a continuous-time stirred tank reactor where the reaction A --+ R takes place. The reaction is exothermic, and reaction heat is removed by a coolant. The system is modeled by mass and energy balances.
It can be determined experimentally by sending sinusoids of different frequencies through the system. This may, however, be time consuming. It can also be determined from other signals. The Bode Plot The Bode plot is another graphical representation of the transfer function. 7. The amplitude or gain curve shows the amplitude ratio a(o) - lG(irlo)l as a function of the frequency a. The phase curve shows the phase g@) - arg G(ia) as a function of the frequency ar. The frequency is given in logarithmic scales on both curves, either in rad/s or Hz.
The Wiener model is a good representation for a system with a nonlinear sensor, for example, a pH electrode. If the process is nonlinear the dynamics are varying with the operating conditions. Ideally, the controller should be tuned with respect to these variations. A conservative approach is to tune the controller for the worst case and accept degraded performance at other operating conditions. Another approach is to find a measurable variable that is well correlated with the process nonlinearity.