First-Order Lag With Deadtime

Many chemical engineering systems can be modeled by this type of transfer function. Let us consider a typical transfer function 

We will look at several values of deadtime D. For all cases the values of K, and Tj, will be set equal to unity. Other values of simply modify the frequency and time scales. Other values of modify the controller gain. 

The Bode plot of is given in Fig. 13.20 for D = 0.5. The ultimate gain is 3.9 (11.6 dB), and the ultimate frequency is 3.7 radians per minute. The ZN controller settings for P and PI controllers and the corresponding phase and gain margins and log moduli are shown in Table 13.2 for several values of deadtime D. Also shown are the values for a proportional controller that give +2-dB maximum closedloop log modulus. 

Opealoop and closedloop plots for dcadtime with lag process. 

Notice in Fig. 13.20 that the curve for the P controller does not approach 0 dB at low frequencies. This shows that there is a steadystate offset with a proportional controller. The curve for the PI controller does go to 0 dB at low frequencies because the integrator drives the closedloop servo transfer function to unity (i.e., no offset). 

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Three-CSTR Process / 11.4.2 First-Order Lag with Deadtime / 11.4.3 Openloop-Unstahle... 

There are two options as to what type of approximate dynamics to assume CSTR or PFR. The former uses a first-order lag with the appropriate time constant based on the vessel residence time. The latter used a deadtime. 

Dynamic responses can be divided into the categories of selfregulating and non-self-regulating. A self-regulating response has inherent negative feedback and will always reach a new steady-state in response to an input change. Self-regulating response dynamics can be approximated with a combination of a deadtime and a first-order lag with an appropriate time constant. 

A run-away response continues to change at an increasing rate due to inherent positive feedback. The response is exponential and may be thought of as a first-order lag with a negative time constant. Run-away response dynamics may be approximated with a combination of deadtime, a first-order lag, and a second, longer lag with a negative time constant. 

Perform a step test on the three-heated-lank process and fit a first-order lag plus deadtime model to the response curve. Calculate the ultimate gain and the ultimate frequency from the transfer function and compare with the results from Problem 16.1. 

Simulate several first-order lag plus deadtime processes on a digital computer with a relay feedback. Compare the ultimate gains and frequencies obtained by the autOtune method with the real values of and Ku obtained from the transfer functions. 

Example 19.3. Consider the first-order lag process with a deadtime of one sampling... 

The final phase angle (at coJ2) for the first-order lag process with no deadtime was —180°. For the process with a deadtime of one sampling period, it was —360°. If we had a deadtime that was equal to two sampling periods, the final phase angle would be —540°. Every multiple of the sampling period subtracts 180° from the final phase angle. 

Suppose we have a first-order lag process with deadtime. 

Nyquist plot for deadtime with first-order lag. 

A process with an openloop transfer function consisting of a steady-state gain, deadtime, and first-order lag is to be controlled by a PI controller. The deadtime (D) is one-fifth the magnitude of the time constant (t ). 

Once the test has been performed and the ultimate gain and ultimate frequency have been determined, we may simply use it to calculate Ziegler-Nichols settings. Alternatively, it is possible to use this information, along with other easily determined data, to calculate approximate transfer functions. The idea is to pick some simple forms of transfer functions (gains, deadtime, first- or second-order lags) and find the parameter values that fit the ATV results. 

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