Loop gain

We have seen in all the mning methods that the product Kp.K should be constant. Once we have established what the controller gain should be, we will need to change the value if there is any change which affects the loop gain. For example if the instmment range of either the PVor MV is changed, will need adjustment. From Equations (2.2) to (2.4) we define 

So if we change the range of the PV or the MV then, to keep the loop gain constant, the 

The same correction would be necessary if we change the MV of the controller, for example changing a primary controller cascaded to a flow controller so that it instead directly manipulates the control valve. This is often a quick fix if the secondary flow transmitter has a problem. We will show in Chapter 6 that adding a ratio-based feedforward can also require recalculation of Kc again because the effective range of the MV may be changed. 

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The response time is equal to the ZC product, where Cis the diode capacitance, or when using an integrating amplifier, the response time is determined by the closed loop gain. 

The speed of the controller is adjusted by the proportional band and reset rate (proportional and integral gains). These parameters also influence the stability of the control loop. All control loops are limited to a gain of less than one at their critical frequency. Higher closed-loop gain will make the loop unstable. 

The current-mode controlled forward converter has one additional consideration there is a double pole at one-half the operating switching frequency. The compensation bandwidth normally does not go this high, but it may cause problems if the closed-loop gain is not sufficiently low enough to attenuate its effects. Its influence on the control-to-output characteristic can be seen in Figure B-14. 

The elosed-loop gain eross-over frequency should be as high as practically possible. This quickens the transient response time of the supply. 

The closed-loop gain at dc should be as high as possible. This has a direct bearing on the output load regulation of the supply. 

The resulting average slope of the closed-loop gain curve should be an average of -20dB/decade. 

The schematic and Bode plot for the single-pole method of compensation are given in Figure B-16. At dc it exhibits the full open-loop gain of the op amp, and its gain drops at -20dB/decade from dc. It also has a constant -270 degree phase shift. Any phase shift contributed by the control-to-output characteristic... 

Next find the closed-loop gain cross-over frequency by deciding how much phase margin you desire in your system. A good value is 45 degrees. Ignoring any effect of the Q of the T-C filter, the gain cross-over point is found from... 

Next determine the maximum elosed-loop gain eross-over frequency. Less than one-fifth of the switehing power supply s operating frequency is a good rule. 

Next, the maximum closed-loop gain cross-over frequency (/(o) is determined and is, once again, no more than one-fifth the minimum switching of the switching power supply ... 

This ean only happen if the open-loop gain eonstant K K is infinite. In praetiee this is not possible and therefore the proportional eontrol system proposed in Figure 4.23 will always produee steady-state errors. These ean be minimized by keeping the open-loop gain eonstant K K as high as possible. 

Maximum value of the open-loop gain constant for the stability of a closed-loop system... 

If a point. vi lies on a loeus, then the value of the open-loop gain eonstant K at that point may be evaluated by using the magnitude eriterion. 

Note that method (b) provides both the erossover value (i.e. the frequeney of oseillation at marginal stability) and the open-loop gain eonstant. 

Value of open-loop gain constant K Applying the magnitude criterion to the above point... 

Gain Margin (GM) The gain margin is the inerease in open-loop gain required when the open-loop phase is —180° to make the elosed-loop system just unstable. 

In general, type zero are unsatisfaetory unless the open-loop gain K ean be raised, without instability, to a suffieiently high value to make 1/(1 + A p) aeeeptably small. Most eontrol systems are type one, the integrator either oeeurring naturally, or deliberately ineluded in the form of integral eontrol aetion, i.e. PI or PID. 

The Nichols chart shown in Figure 6.26 is a rectangular plot of open-loop phase on the x-axis against open-loop modulus (dB) on the jr-axis. M and N contours are superimposed so that open-loop and closed-loop frequency response characteristics can be evaluated simultaneously. Like the Bode diagram, the effect of increasing the open-loop gain constant K is to move the open-loop frequency response locus in the y-direction. The Nichols chart is one of the most useful tools in frequency domain analysis. 

The open-loop transfer function is third-order type 2, and is unstable for all values of open-loop gain K, as can be seen from the Nichols chart in Figure 6.33. From Figure 6.33 it can be seen that the zero modulus crossover occurs at a frequency of 1.9 rad/s, with a phase margin of —21°. A lead compensator should therefore have its maximum phase advance 0m at this frequency. Flowever, inserting the lead compensator in the loop will change (increase) the modulus crossover frequency. 

Figure 6.35 shows the Bode gain and phase for both compensated and uncompensated systems. From Figure 6.35, it can be seen that by reducing the open-loop gain by 5.4dB, the original modulus crossover frequency, where the phase advance is a maximum, can be attained. 

Using MATLAB to design a system, it is possible to superimpose lines of constant ( and ajn on the root locus diagram. It is also possible, using a cursor in the graphics window, to select a point on the locus, and return values for open-loop gain K and closed-loop poles using the command... 

Requires K12K21 = 0. Open-loop gain is the same as the closed-loop gain. The controlled variable (or loop) / is not subject to interaction from other manipulated variables (or other loops). Of course, we know nothing about whether other manipulated variables may interact and affect other controlled variables. Nevertheless, pairing the i-th controlled variable to they-th manipulated variable is desirable. 

The open-loop gain is zero. The manipulated variablej has no effect on the controlled variable i. Of course nij may still influence other... 

No doubt there are interactions from other loops, and from (10-37), some of the process gains must have opposite signs (or act in different directions). When Ay = 0.5, we can interpret that the effect of the interactions is identical to the open-loop gain—recall statement after (10-36). When Ay > 0.5, the interaction is less than the main effect of nij on Cj. However, when Ay < 0.5, the interactive effects predominate and we want to avoid pairing nij with c . 

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