Lag-lead compensation

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. 

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. 

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. 

On the other hand, conventional control approaches also rely on models, but they are usually not built into the controller itself. Instead the models form the basis of simulations and other analysis methods that guide in the selection of control loops and suggest tuning constants for the relatively simple controllers normally employed [PI, PID, I-only. P-only, lead-lag compensation, etc. (P = proportional, PI = proportional-integral, PID = proportional-integral-derivative)]. Conventional control approaches attempt to build the smarts into the system (the process and the controllers.) rather than only use complex control algorithms. 

Given a process whose dynamics consist of first-order lags, = 3 min and T, = 1 min For conditions of rand q, both at 50 percent, what is the integrated area per unit load change with an uncompensated forward loop What is the Integrated area with lead-lag compensation, if the lead time is 2.5 min and the lag time is 1 min ... 

Design experience with flooded reboilers is limited but indicates that typical time constants are of the order of 2—5 minutes. Simulation studies show that substantial improvement in response speed may be achieved by lead-lag compensations with transfer functions such as ... 

The pressure at the turbine inlet is kept constant by regulating the opening of the turbine control valves. The same logic as used in BWRs is adopted. It is shown in Fig. 4.17. The opening is proportional to the deviation of the pressure from the setpoint with lead-lag compensation. The turbine control valve opening ratio is calculated from the following equations. 

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 ... 

Here, z0 and p0 are just two positive numbers. There are obviously two possibilities case (a) z0 > po, and case (b) z0 < p0. Sketch the magnitude and phase lag plots of Gc for both cases. Identify which case is the phase-lead and which case is the phase-lag compensation. What types of classical controllers may phase-lead and phase-lag compensations resemble ... 

Example 8.14. Designing phase-lead and phase-lag compensators. Consider a simple unity feedback loop with characteristic equation 1 + GCGP = 0 and with a first order process... 

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. 

Can sense as the basis of phase-lead and phase-lag compensator design... 

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. 

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

For noninteracting control loops with zero dead time, the integral setting (minutes per repeat) is about 50% and the derivative, about 18% of the period of oscillation (P). As dead time rises, these percentages drop. If the dead time reaches 50% of the time constant, I = 40%, D = 16%, and if dead time equals the time constant, I = 33% and D = 13%. When tuning the feedforward control loops, one has to separately consider the steady-state portion of the heat transfer process (flow times temperature difference) and its dynamic compensation. The dynamic compensation of the steady-state model by a lead/lag element is necessary, because the response is not instantaneous but affected by both the dead time and the time constant of the process. 

Figure 15.56 shows the effect of the ratio of x,/x,j on the dynamic response of a lead/lag element. When 1 /1 is greater than 1, overcompensation is used. That is, when the process responds faster to the disturbance than to the controller output, larger than steady-state changes in the controller output are required to compensate for dynamic mismatch. On the other hand, when x,/x, ... 

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... 

We should also remember that the tuning has been based on the assumption that the process is first order plus deadtime. It is theoretically possible to implement a second order equivalent of the lead-lag algorithm but this would require the identification of second order models for the DV and MV, and the calculation of additional tuning constants. It is unlikely therefore to be practical. It would be easier to fine tune the dynamic compensation. This also takes account of any abnormalities in the way in which the DCS vendor may have coded the lead-lag algorithm. 

Figure 8.10 shows the first of the decouplers. When PIDi takes corrective action, the decoupler applies dynamic compensation to the change in output (AOPi) and makes a change to MV2 that counteracts the disturbance that the change in MVi would otherwise cause to PV2- Dynamic compensation is provided by a deadtime/lead-lag algorithm. 

We apply dynamic compensation in the form of a deadtime/lead-lag algorithm. This is tuned in exactly the same way as described in Chapter 6 covering bias feedforward. By performing open loop steps on the MV we obtain the dynamics of both the inferential and... 

If the analyser is discontinuous and its sample interval greater than the time it takes the process to reach steady state, then it may not show significant lag. As T2 should not be set to zero (because of the effect on the TUTl ratio) then it is wise only to include the deadtime compensation - by removing the lead-lag or setting T equal to T2. 

Dynamic compensation is likely to be necessary to ensure that the reflux and steam flows are adjusted at the right time. The method for tuning these deadtime/lead-lag algorithms is described in Chapter 6. Part of this procedure involves steptesting the DV, in this case feed rate, to obtain the dynamic response of the PV, in this case tray temperature. This can present a problem on some columns. 

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