Dead time function

The dead time function is also called the time delay, transport lag, translated, or time shift function (Fig. 2.3). It is defined such that an original function f(t) is "shifted" in time to, and no matter what f(t) is, its value is set to zero for t < to- This time delay function can be written as ... 

Example 3.2 Using the first order Pade approximation, plot the unit step response of the first order with dead time function ... 

It is clear that as n increases, the response, as commented in Example 2.9, becomes slower. If we ignore the data at small times, it appears that the curves might be approximated with first order with dead time functions. We shall do this exercise in the Review Problems. 

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... 

The summation to estimate the dead time is over all the other time constants (/ = 3, 4, etc ). This idea can be extended to the approximation of a first order with dead time function. 

The choice of the time constant and dead time is meant as an illustration. The fit will not be particularly good in this example because there is no one single dominant pole in the fifth order function with a pole repeated five times. A first order with dead time function will never provide a perfect fit. 

The real time data (the process reaction curve) in most processing unit operations take the form of a sigmoidal curve, which is fitted to a first order with dead time function (Fig. 6.2) 1... 

Using the first order with dead time function, we can go ahead and determine the controller settings with empirical tuning relations. The most common ones are the Ziegler-Nichols relations. In process unit operation applications, we can also use the Cohen and Coon or the Ciancone and Marlin relations. These relations are listed in the Table of Tuning Relations (Table 6.1). 

The open-loop test response fitted to a first order with dead time function GPRC can be applied to other tuning relations. One such possibility is a set of relations derived from the minimization of error integrals. Here, we just provide the basic idea behind the use of error integrals. 

As far as we are concerned, using the error integral criteria is just another empirical method. We are simply using the results of minimization obtained by other people, not to mention that the first order with dead time function is from an open-loop test. 

Even though this result is based on what we say is a process function, we could apply (E6-4) as if the derivation is for the first order with dead time function GPRC obtained from an open-loop step test. 

With the IMC tuning setting in Example 5.7B, the resulting time response plot is (very nicely) slightly underdamped even though the derivation in Example 6.4 predicates on a system response without oscillations. Part of the reason lies in the approximation of the dead time function, and part of the reason is due to how the system time constant was chosen. Generally, it is important to double check our IMC settings with simulations. 

Empirical tuning with open-loop step test Measure open-loop step response, the so-called process reaction curve. Fit data to first order with dead-time function. 

When the system has dead time, we must make an approximation, such as the Pade approximation, on the exponential dead time function before we can apply the Routh-Hurwitz criterion. The result is hence only an estimate. Direct substitution allows us to solve for the ultimate gain and ultimate frequency exactly. The next example illustrates this point. 

Example 8.5. What are the Bode and Nyquist plots of a dead time function G(s) = e-6s ... 

When co = ir/0. ZG(jco) = -k. On the polar plot, the dead time function is a unit circle. 

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.)... 

From Example 8.5, we know that the magnitude of the dead time function is 1. Combining also with the results in Example 8.2, the magnitude and phase angle of G(jco) are... 

The very first step is to find the ultimate gain. Following Example 8.6 (p. 8-11), we can add easily the extra phase lag due to the dead time function ... 

This is roughly how we did it. All the simulations are performed with Simulink. First, we use Gj j and G22 as the first order with dead time functions and apply them to the ITAE tuning... 

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