Zeroes and Poles

Let us try to connect the dots now. Both the first- and second-order filters we have discussed gave us poles. That is because they both had s in the denominators of their transfer functions — if s takes on specific values, it can force the denominator to become zero, and the transfer function then becomes infinite, and we get a pole by definition. The values of 5 at which the denominator becomes zero are the resonant (or break) frequencies, that is, the locations of the poles. For example, a hypothetical transfer function 1/s will give us a pole at zero frequency (the pole-at-zero we talked about earlier). 

Note that the gain, which is the magnitude of the transfer function (calculated by putting s = jco), won t necessarily be infinite at the pole location. For example, in the case of the RC filter, we know that the gain is in fact always less than or equal to unity, despite a pole being present at the break frequency. 

Zeros are anti-poles in many senses. For one, their presence is indicated by both the gain and the phase increasing with frequency — opposite to a pole. Further, zeros also cancel poles if they happen to fall at the same frequency location. 

One double-zero at f=0 One double-pole at - wg (fwo single poles very close to each other) 

Now we can generalize our approach. A network transfer function can be described as a ratio of two polynomials  

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Where Ui denotes input number i and there is an implied summation over all the inputs in the expression above A, Bj, C, D, and F are polynomials in the shift operator (z or q). The general structure is defined by giving the time delays nk and the orders of the polynomials (i.e., the number of poles and zeros of the dynamic models trom u to y, as well as of the noise model from e to y). Note that A(q) corresponds to poles that are common between the dynamic model and the noise model (useful if noise enters system close to the input). Likewise Fj(q) determines the poles that are unique for the dynamics from input number i and D(q) the poles that are unique for the noise N(t). 

Calculations of mutual locations of poles and zeros for these TF models allow to trace dynamics of moving of the parameters (poles and zeros) under increasing loads. Their location regarding to the unit circle could be used for prediction of stability of the system (material behavior) or the process stationary state (absence of AE burst ) [7]. 

Figure 6. Location of poles and zeros for visco-elasto-plastic material (left) and brittle material (right) under loading close to fiacture.

Locating the compensating poles and zeros in the error amplifier... 

Assign the loeation of the error amplifier eompensating pole and zero. To eompensate for the light-load output filter pole with a zero,... 

Using the proeedure given in Appendix B, the loeation of the pole and zero within the error amplifier s eompensation network are... 

The phase boost is proportional to the separation of the zero-pole pair in the error amplifier, but that is of seeondary nature sinee the error amplifier s pole and zero were plaeed to eompensate the worst ease zero and pole in the eontrol-to-output eharaeteristies. The aetual loeation of the zero eaused by the ESR, whieh amounts to the ehoiee of a vendor and part number, will affeet the amount of exeess phase the supply will exhibit. So the designer may have to reloeate the eompensating pole if there is any possibility that the exeess phase will fall to less that 30 degrees (-330 degree lag). 

The more complicated methods of compensation, such as this, allow the designer much more control over the final closed-loop bode response of the system. The poles and zeros can be located independently of one another. Once their frequencies are chosen, the corresponding component values can be easily determined by the step-by-step procedure below. The zero and pole pairs can be kept together in pairs, or can be separated. The high-frequency pole pair appear to yield better results if they are separated and placed as below. The zero pair are usually kept together, but can be separated and placed either side of the output filter pole s corner frequency to help minimize the gain effects of the Q of the T-C filter (refer to Figure B-23). 

The method performed above with the plaeement of the poles and zeroes will yield a minimum value for the exeess phase of 45 degrees, whieh is satisfaetory. If other pole and zero loeations are attempted, then loeate the maximum phase lag point of the L-C filter at the geometrie mean frequency between/ez2 and/epi. This will guarantee the best phase performance. The amount of phase boost of the compensation design will be... 

Root-locus analysis 5.3.1 System poles and zeros... 

Figure 5.12 shows veetors from open-loop poles and zeros to a trial point. vi. From Figure 5.12 and equation (5.57), for. vi to lie on a loeus, then... 

For Example 5.7, if. vi lies on a loeus, then the pole and zero magnitudes are shown in Figure 5.13. From Figure 5.13 and equation (5.61), the value of the open-loop gain eonstant K at position. vi is... 

Root locus locations on real axis. A point on the real axis is part of the loei if the sum of the number of open-loop poles and zeros to the right of the point eoneerned is odd. 

A compensator, or controller, placed in the forward path of a control system will modify the shape of the loci if it contains additional poles and zeros. Characteristics of conventional compensators are given in Table 5.2. 

All the features about poles and zeros can be obtained from this simpler equation. 

Suppose that x2 is associated with the slower pole (x2 > X ), we now require xf = x2 such that the pole and zero cancel each other. The result is a PI controller ... 

On the real axis, a root locus only exists to the left of an odd number of real poles and zeros. (The explanation of this point is on our Web Support.)... 

A transfer function can be written in terms of its poles and zeros. For example,... 

Double check that we can recover the poles and zeros with... 

We may not need to use them, but it is good to know that there are functions that help us extract the polynomials, or poles and zeros back from an object. For example ... 

Optional reading In the initial learning stage, it can be a bad habit to rely on MATLAB too much. Hence the following tutorial goes the slow way in making root locus plots, which hopefully may make us more aware of how the loci relate to pole and zero positions. The first thing, of course, is to identify the open-loop poles. 

Save loci to array "r" first % Now use plot() to do the dots % hold the plot to add goodies % pzmap() draws the open-loop poles % and zeros... 

The default is a proportional controller, but the K block in Fig. M6.1 can easily be changed to become a PI, PD, or PID controller. The change can be accomplished in different ways. One is to retrieve the compensator-editing window by clicking on the K block or by using the Tools pull-down menu. The other is to use the set of arrow tools in the root locus window to add or move open-loop poles and zeros associated with the compensator. 

We will ignore the values of any gains. We focus only on the probable open-loop pole and zero positions introduced by a process or by a controller, or in other words, the shape of the root locus plots. 

One final point should be made about transfer functions. The steadystate gain Kp for alt the transfer functions derived in the examples was obtained by expressing the transfer function in terms of time constants instead of in terms of poles and zeros. For the general system of Eq. (9.91) this would be... 

G(i, is a ratio of polynomials in s that can be factored into poles and zeros. 

Thus, we should expect the magnitude plot to start at -40 dB and finish at 0 dB. The next question is, where are the poles and zeros Let... 

EXERCISE 5-5 Plot the Bode phase and magnitude plots for frequencies from 1 Hz to 1 MHz. Use the cursors to find the frequencies of the poles and zeros. 

The following equations are used to predict the pole and zero locations of the feedback loop. The output filter causes a double pole at... 

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