Dead time transfer function

The dead time transfer function has to be handled differently in classical control, and well use the Pade approximation for this purpose. 

Now, we switch gears and look into the dead time transfer function approximation. To do a Pade approximation, we can use the MATLAB function 3... 

Distance-Velocity Lag (Dead-Time Element) The dead-time element, commonly called a distance-velocity lag, is often encountered in process systems. For example, if a temperature-measuring element is located downstream from a heat exchanger, a time delay occurs before the heated fluid leaving the exchanger arrives at the temperature measurement point. If some element of a system produces a dead-time of 0 time units, then an input to that unit,/(t), will be reproduced at the output a.s f t — 0). The transfer function for a pure dead-time element is shown in Fig. 8-17, and the transient response of the element is shown in Fig. 8-18. 

The positive results obtained at production scale give us confidence in the validity of our approach. Derivation of a simple scaling factor enabled us to conduct a series of experiments in a small pilot plant which would have been expensive and time-consuming on a production scale. Time series analysis not only provided us with estimates of the process gain, dead time and the process time constants, but also yielded an empirical transfer function which is process-specific, not one based on... 

Let say we have a high order transfer function that has been factored into partial fractions. If there is a large enough difference in the time constants of individual terms, we may try to throw away the small time scale terms and retain the ones with dominant poles (large time constants). This is our reduced-order model approximation. From Fig. E3.3, we also need to add a time delay in this approximation. The extreme of this idea is to use a first order with dead time function. It obviously cannot do an adequate job in many circumstances. Nevertheless, this simple... 

Plot the unit step response using just the first and second order Pade approximation in Eqs. (3.30) and (3-31). Try also the step response of a first order function with dead time as in Example 3.2. Note that while the approximation to the exponential function itself is not that good, the approximation to the entire transfer function is not as bad, as long as td x. How do you plot the exact solution in MATLAB ... 

The important point is that the phase lag of the dead time function increases without bound with respect to frequency. This is what is called a nonminimum phase system, as opposed to the first and second transfer functions which are minimum phase systems. Formally, a minimum phase system is one which has no dead time and has neither poles nor zeros in the RHP. (See Review Problems.)... 

With frequency response analysis, we can derive a general relative stability criterion. The result is apphcable to systems with dead time. The analysis of the closed-loop system can be reduced to using only the open-loop transfer functions in the computation. 

This is the steady state compensator. The lead-lag element with lead time constant xFLD and lag time constant XpLG is the dynamic compensator. Any dead time in the transfer functions in (10-7) is omitted in this implementation. 

If the load transfer function in Eq. (10-7) had also been approximated as a first order function with dead time, say, of the form KLe tds/(xps+l), the feedforward controller would appear as... 

In this illustration, we do not have to detune the SISO controller settings. The interaction does not appear to be severely detrimental mainly because we have used the conservative ITAE settings. It would not be the case if we had tried Cohen-Coon relations. The decouplers also do not appear to be particularly effective. They reduce the oscillation, but also slow down the system response. The main reason is that the lead-lag compensators do not factor in the dead times in all the transfer functions. 

Bode plots for transfer functions with dead time... 

To handle dead time, all we need is a simple modification using the fact that the time delay transfer function has magnitude 1 and phase angle - tdoo. We need one single statement... 

Thus the transfer function between output and input variables for a pure dead time process is as sketched in Fig. 9.6. 

This method can be extended to evaluate models with more parameters and with diHerent kinds of transfer functions (e.g., underdamped second-order lag) by using hysteresis in the relay feedback or by inserting an additional dead time in the loop to produce a limit cycle with a different frequency. The two autotune tests give four equations so four parameters can be evaluated. 

In the unity feedback system shown in Fig. 7.59 (set-point following case), it is assumed that all the dead time is contained within the process and is represented by the transfer function G3(j), where ... 

It can be seen from equation 7.144 that implementation of the Smith predictor assumes knowledge of the transfer functions (i.e. the dynamics) of the process (including the dead time) and the final control element. These are unlikely to be... 

The estimator assumes a simple linear model for the process. For instance, a first-order system with dead time may be employed for which the transfer function may be written as ... 

Fitting Dynamic Models to Experimental Data In developing empiricaTtransfer functions, it is necessary to identify model parameters from experimental data. There are a number of approaches to process identification that have been published. 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 xs in the figure represent the recorded data. For purposes of illustration, the process under study will be assumed to be first-order with dead time and have the transfer function... 

In some situations where one or more of the latex properties are measured either directly or indirectly through their correlation with surrogate variables and where extreme nonlinearities such as the periodic generation of polymer particles does not occur, one can use much simpler modehng and control techniques. Linear transfer function-type models can he identified directly from the plant reactor data. Conventional control devices such as PID controllers or PID controllers with dead-time compensation can then be designed. If process data is also used to identify... 

Construct the Bode diagram and Nyquist plot of a first-order system with dead time, having a transfer function... 

In the absence of dead time a closed-loop system may become unstable if its open-loop transfer function is of third order or higher. 

We see from Example 19.1 that as the dead time of an open-loop transfer function increases, the following two undesirable effects take place ... 

Figure 19.3 Dead-time compensation with inaccurate knowledge of process transfer function and dead time.

Let us introduce perfect dead-time compensation. This is possible if the true transfer function of the process is known. Then the... 

The transfer function of a first-order process with dead time is... 

A given process has unknown detailed dynamics that is, it exhibits overdamped open-loop behavior but its exact order and parameter values are poorly known. From the process reaction curve (see Section 16.5) we have approximated its transfer function by the following first-order system with dead time ... 

Consider a first-order system with a dead time t between the input f(t) and the output y(t). We can represent such system by a series of two systems as shown in Figure 12.2a (i.e., a first-order system in series with a dead time). For the first-order system we have the following transfer function ... 

Let us assume now that the dead time for the process has not been estimated accurately and that its true value is 0.15 instead of 0.1. Then the open-loop transfer function is given by... 

The dead-time compensator predicts the delayed effect that the manipulated variable will have on the process output. This prediction led to the term Smith predictor and it is possible only if we have a model for the dynamics of the process (transfer function, dead time). 

---