Lead-lag element

Lag-Lead Compensation The transfer function of a lag-lead element is ... 

What is a lag-lead element, and why is it considered to be a versatile component for feedforward control ... 

The controller given by eq. (21.15a) is called a lag-lead element because fis + 1 introduces phase lead and the l/(as + 1) adds phase lag. a and fi are adjustable parameters for the controller. For the set point element GSp(s), eq. (21.15b) indicates that we should use a lead element. 

The lag-lead element is the most commonly used in dynamic feedforward control. It is quite versatile because the two adjustable parameters a and fi allow its use as a lead element (when fi is much larger than a) or as a lag element (when a is much larger than fi). Finally, lag-lead elements can be bought easily and they are not expensive as are special-purpose analog computational devices. 

Consider the feedforward control of a distillation column. What kind of dynamic feedforward element will be needed lag-lead, lag only, lead only, gain only Give a rather qualitative explanation. 

Development of the steady-state model for an evaporator involves material and energy balance. A relationship between feed composition and product composition is also required. The process dynamics require that a lead-lag dynamic element be incorporated in the system to compensate for any dynamic imbalance. In some applications evaporators are operated with waste steam, in which case the feed rate is proportionally adjusted to the available steam, making the feed the manipulated variable and steam the load variable. Generally, steam is the manipulated variable. The final consideration Is feedback trim. As a... 

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

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). 

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. 

This energy difference may be interpreted in terms of two elliptical trajectories separated by Ae and with a phase lag between the leading and following edges of the element, ApAq that moves along the fuzzy trajectory. The two edges remain separated in time by a fixed amount A and define the elements A and B at -q0 and p respectively. 

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 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,. 

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. 

It is also possible to use an internal standard to correct for sample transport effects, instrumental drift and short-term noise, if a simultaneous multi-element detector is used. Simultaneous detection is necessary because the analyte and internal standard signals must be in-phase for effective correction. If a sequential instrument is used there will be a time lag between acquisition of the analyte signal and the internal standard signal, during which time short-term fluctuations in the signals will render the correction inaccurate, and could even lead to a degradation in precision. The element used as the internal standard should have similar chemical behaviour as the analyte of interest and the emission line should have similar excitation energy and should be the same species, i.e. ion or atom line, as the analyte emission line. 

The output of the element represented by equation 7.155 lags the input. However, the destabilising effect of this additional lag is more than offset by an associated decrease in amplitude ratio. This decrease is more pronounced as the difference between r, and tj is increased. Lag compensators can be designed to produce different total open-loop stability specifications (e.g. in terms of allowable gain margin, phase margin, etc.) in a manner similar to that for lead compensators. 

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