Deadtime, transfer function

As shown in Fig. 12.17, the deadtime transfer function has a flat L at 0 dB curve for all frequencies, but the phase angle drops off to minus infinity. The phase angle is down to —180° when the frequency is n/D. So the bigger the deadtime, the lower the frequency at which the phase angle drops off rapidly. 

Fitting Dynamic Models to E erimental Data In developing empirical transfer functions, it is necessary to identify model parameters from experimental data. There are a number of approaches to process identification that have been pubhshed. The simplest approach involves introducing a step test into the process and recording the response of the process, as illustrated in Fig. 8-21. The i s in the figure represent the recorded data. For purposes of illustration, the process under study will be assumed to be first order with deadtime and have the transfer func tion ... 

It is not always possible to achieve perfect feedforward control. If the transfer function has a deadtime that is larger than the deadtime in the G, ... 

Draw a block diagram of a process that has two manipulated variable inputs (Mi and M]) that each affect the output (2T). A feedback controller Si is used to control X by manipulating Mi since the transfer function between Mj and X has a small time constant and smaU deadtime. 

G. GENERAL TRANSFER FUNCTIONS IN SERIES. The historical reason for the widespread use of Bode plots is that, before the use of computers, they made it possible to handle complex processes fairly easily. A complex transfer function can be broken down into its simple elements leads, lags, gains, deadtimes, etc. Then each of these is plotted on the same Bode plots. Finally the total complex transfer function is obtained by adding the individual log modulus curves and the individual phase curves at each value of frequency. 

Figure 13.20 shows the closedloop servo transfer function Bode plots for P and PI controllers with the ZN settings for a deadtime of 0.5 min. The effect of... 

A process has an openloop transfer function that is approximately a pure deadtime of D minutes. A proportional-derivative controller is to be used with a value of a equal to 0.1. What is the optimum value of the derivative lime constant Note that part of this problem involves defining what you mean by optimum. 

J2. A process with an openloop transfer function consisting of a steadystate gain, deadtime, and liist-ordcr lag is to be controlled by a PI controller. The deadtime (O) is onc-fiTth the magnitude of the time constant (t). 

Once the log modulus curve has been adequately fitted by an approximate transfer function G(J ), the phase angle of G( a) is compared with the experimental phase-angle curve. The difference is usually the contribution of deadtime. The procedure is illustrated in Fig. 14.2. 

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. 

B. DEADTIME. The deadtime D in the transfer function can be easily read off the initial part of the ATV test. It is simply the time it takes x to start responding to the initial change in m. 

These transfer functions have either one or two unknown parameters since we know the gain and the deadtime D. 

In Chap. 15 we reviewed a tittle matrix mathematics and notation. Now that the tools are available, we will apply them in this chapter to the analysis of multivariable processes. Our primary concern is with closedloop systems. Given a process with its matrix of openloop transfer functions, we want to be able to see the effects of using various feedback controllers. Therefore we must be able to find out if the entire closedloop multivariable system is stable. And if it is stable, we want to know how stable it is. The last question considers the robustness of the controller, i.e., the tolerance of the controller to changes in parameters. If the system becomes unstable for small changes in process gains, time constants, or deadtimes, the controller is not robust. 

The procedure has been tested primarily on realistic distillation column models. This choice was deliberate because most industrial processes have similar gain, deadtime, and lag transfer functions. Undoubtedly some pathological transfer functions can be found that the procedure cannot handle. But we are interested in a practical engineering tool, not elegant, rigorous all-inclusive mathematical theorems. 

C EVALUATE PROCESS TRANSFER FUNCTION MATRIX C GAIN KP(I), DEADTIME D(I), LEAD TAU(1,I,J)... 

The basic idea in multivariable IMC is the same as in single-loop IMC. The ideal controller would be the inverse of the plant transfer function matrix. This would give perfect control. However, the inverse of the plant transfer function matrix is not physically realizable because of deadtimes, higher-order denominators than numerators, and RHP zeros (which would give an openloop unstable controller). 

Example 19.6. The chromatographic system studied in Example 18.9 had a first-order lag openloop process transfer function and a deadtime of one sampling period. The closedloop characteristic equation was [see Eq. (18.100)]... 

Example 19.12. The first-order system with a one sampling-period deadtime considered in Example 19.3 had the continuous openloop transfer function... 

Example 203. If we have a first-order lag process with a deadtime that is equal to one sampling period, the process transfer function becomes... 

Example 20.10. Suppose we add a one-sampling period deadtime to the first-order system. The openloop system transfer function becomes... 

Design a minimal prototype sampled-data controller for a process with an openloop process transfer function that is a pure deadtime. 

It is important to remember that a deadtime or several lags must be inserted in most control loops in order to mn a relay-feedback test. To have an ultimate gain, the process must have a phase angle that drops below —180°. Many of the models in Aspen Dynamics have only a first-order transfer function between the controller variable and the manipulated variable. In the CSTR temperature controller example, the controlled variable is reactor temperature and the manipulated variable is medium temperature. The phase angle of a first-order process goes to only —90°, so there is no ultimate gain. The relay-feedback test will fail without the deadtime element inserted in the loop. 

Thus, the transfer function between output and input variables for a pure deadtime process is e ... 

It is not always possible to achieve perfect feedforward control. If the Gm(s) transfer function has a deadtime that is larger than the deadtime in the Gl(s) transfer function, the feedforward controller will be physically unrealizable because it requires predictive action. Also, if the Gm(s) transfer function is of higher order than the Gi( s) transfer function, the feedforward controller will be physically unrealizable [see Eq. (9.28)]. 

This case illustrates that the desired closedloop relationship cannot be chosen arbitrarily. You cannot make a jumbo jet behave like a jet fighter, or a garbage truck drive like a Ferrari We must select the desired response so that the controller is physically realizable. In this case all we need to do is modify the specified closedloop servo transfer function S(,t) to include the deadtime. 

Combining the transfer functions for deadtime and first-order lag gives... 

Many chemical engineering systems can be modeled by a transfer function involving a first-order lag with deadtime. Let us consider a typical transfer function ... 

We can often capture the feedback-control-relevant dynamics of a process by assuming that the openioop process transfer function Gm(s) relating the controlled and the manipulated variables is a simple gain, a deadtime, and a pure integrator in series. 

Therefore, we should pair variables that are related through low-order transfer functions having large steady-state gains, small time constants, and small deadtimes. A number of dynamically poor pairings can be eliminated by inspection. 

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