Lead-lag

Fig. 14. Examples of feedforward (EE) controls having feedback (EB) trim, where L/L = lead/lag element ...

The multiloop controller contains a variety of func tion blocks (for example, PID, totalizer, lead/lag compensator, ratio control, alarm, sequencer, and Boolean) that can be soft-wired together to form complex control strategies. The multiloop controller, as part of a DCS, communicates with other controllers and man/machine interface (MMI) devices also on the DCS network. 

Two vessels arranged in series, a lead/lag arrangement, will allow the SulfaTreat material to be used more efficiently with no interruption in unit service and greater process reliability. The first vessel, the lead ... 

Thermal fatigue is a form of metal failure whereby fracturing occurs under repeated cycles of thermally induced stress. These cycles make take place, for example, because of stop-start, lead-lag, or peak-load boiler firing arrangements. 

For real physical processes, the orders of polynomials are such that n > m. A simple explanation is to look at a so-called lead-lag element when n = m and y(L + y = x(L + x. The LHS, which is the dynamic model, must have enough complexity to reflect the change of the forcing on the RHS. Thus if the forcing includes a rate of change, the model must have the same capability too. 

The inherent dynamics is governed by the poles, but the zeros can impart finer "fingerprint" features by modifying the coefficients of each term in the time domain solution. That was the point which we tried to make with the examples in Section 2.5 (p. 2-10). Two common illustrations on the effects of zeros are the lead-lag element and the sum of two functions in parallel. 

The so-called lead-lag element is a semi-proper function with a first order lead divided by a first order lag ... 

Figure 3.5. Time response of a lead-lag element with x = 2 s. The curves from top to bottom are plotted with xz = 4,3,2, 1,-1, -2, and —4 s, respectively.

With MATLAB, try do a unit step response of a lead-lag element in as in Eq. (3-49). 

Example 4.4 Derive the state space representation of the lead-lag transfer function... 

We may note that the coefficient D is not zero, meaning that with a lead-lag element, an input can have instantaneous effect on the output. Thus while the state variable x has zero initial condition, it is not necessarily so with the output y. This analysis explains the mystery with the inverse transform of this transfer function in Eq. (3-49) on page 3-15. 

In practice, we cannot build a pneumatic device or a passive circuit which provides ideal derivative action. Commercial (real ) PD controllers are designed on the basis of a lead-lag element ... 

In effect, we are adding a very large real pole to the derivative transfer function. Later, after learning root locus and frequency response analysis, we can make more rational explanations, including why the function is called a lead-lag element. We ll see that this is a nice strategy which is preferable to using the ideal PD controller. 

This configuration is also referred to as interacting PID, series PID, or rate-before-reset. To eliminate derivative kick, the derivative lead-lag element is implemented on the measured (controlled) variable in the feedback loop. 

The closed-loop system remains first order and the function is that of a lead-lag element. We can rewrite the closed-loop transfer function as... 

X Example 8.13. Derive the magnitude and phase lag of the transfer functions of phase-lead and phase-lag compensators. In many electromechanical control systems, the controller Gc is built with relatively simple R-C circuits and takes the form of a lead-lag element ... 

The consequence is that most simple implementation of a feedforward controller, especially with off-the-shelf hardware, is a lead-lag element with a gain ... 

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. 

When we tune the feedforward controller, we may take, as a first approximation, xFLD as the sum of the time constants xm and x v. Analogous to the "real" derivative control function, we can choose the lag time constant to be a tenth smaller, xFLG = 0.1 xFLD. If the dynamics of the measurement device is extremely fast, Gm = KmL, and if we have cascade control, the time constant x v is also small, and we may not need the lead-lag element in the feedforward controller. Just the use of the steady state compensator Kpp may suffice. In any event, the feedforward controller must be tuned with computer simulations, and subsequently, field tests. 

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. 

Lead iodide, 14 785-786 Lead-lag systems, 14 408 Lead lanthanum zirconate titanate (PLZT), 5 583 22 713... 

This is called a lead-lag element and contains a first-order lag and a first-order lead. See Table 9.1 for some commonly used transfer function elements. 

The transfer function of a real PID controller, as opposed to an ideal one, is the PI transfer function with a lead-lag element placed in series. 

The lead-lag unit is called a derivative unit, and its step response is sketched in Fig. 9.10. For a unit step change in the input, the output jumps to /a and then decays at a rate that depends on z. So the derivative unit approximates an ideal derivative. It is physically realizable since the order of its numerator polynomial is the same as the order of its denominator polynomial. 

The feedforward controller contains a stcadyslate gain and dynamic terms. For this system the dynamic element is a first-order lead-lag. The unit step reaponae of this lead-lag is an initial change to a value that is (—followed by an exponential rise or decay to the final steadystate value... 

Figure 11.5a shows a typical implementation of feedforward controller. A distillation column provides the specific example. Steam flow to the reboiler is ratioed to the feed flow rate. The feedforward controller gain is set in the ratio device. The dynamic elements of the feedforward controller are provided by the lead-lag unit. 

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. 

The Bode plot for the lead-lag element is sketched in Fig. 13.13c. It contributes positive phase-angle advance over a range of frequencies between 1/tj, and l/arj,. 

The lead-lag clement can move the G, B curve away from the (—1,0) point and improve stability. When the derivative setting on a PID controller is tuned, the location of the phase-angle advance is shifted so that it occurs near the critical (— 1,0) point. 

Sampled-data controllers can be designed in the same way continuous controllers are designed. Root locus plots in the z plane or frequency-response plots are made with various types of >(z) s (different orders of M and N and different values of the a, and 6, coefficients). This is the same as using different combinations of lead-lag elements in continuous systems. 

There are several other methods of achieving stability in potentiostatic circuits. A capacitor may be added between the counter and reference electrodes to reduce phase shift in the critical frequency region. Some caution must be exercised since a low-resistance reference electrode then becomes the counterelectrode at high frequencies. A particularly interesting method is known as input lead-lag compensation a series RC is connected between the input terminals of the control amplifier, and a second resistor is connected between the noninverting input and common. This form of compensation has minimum effect on the slew rate of the control amplifier. Further details can be found in the book by Stout and Kaufman listed in the bibliography. 

Gcomp Transfer function of lead, lag or lag-lead compensator — —... 

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