Pole at the origin

It ean be seen in Figure 5.17 that the pole at the origin and the zero at. v = —1 dominate the response. With the eomplex loei, ( = 0.7 gives K a value of 15. ITowever, this value of K oeeurs at —0.74 on the dominant real loeus. The time response shown in Figure 5.20 shows the dominant first-order response with the oseillatory seeond-order response superimposed. The settling time is 3.9 seeonds, whieh is outside of the speeifieation. 

In order to eneirele any poles or zeros of F s) that lie in the right-hand side of the. v-plane, a Nyquist eontour is eonstrueted as shown in Figure 6.16. To avoid poles at the origin, a small semieirele of radius e, where e 0, is ineluded. 

Under normal circumstances, we would pick a x which we deem appropriate. Now if we pick x to be identical to xp, the zero of the controller function cancels the pole of the process function. We are left with only one open-loop pole at the origin. Eq. (6-29), when x = xp, is reduced to... 

We also see another common definition—bounded input bounded output (BIBO) stability A system is BIBO stable if the output response is bounded for any bounded input. One illustration of this definition is to consider a hypothetical situation with a closed-loop pole at the origin. In such a case, we know that if we apply an impulse input or a rectangular pulse input, the response remains bounded. However, if we apply a step input, which is bounded, the response is a ramp, which has no upper bound. For this reason, we cannot accept any control system that has closed-loop poles lying on the imaginary axis. They must be in the LHP. 1... 

Do not be confused by the function of integral control its pole at the origin is an open-loop pole. This point should clear up when we get to the root locus section. 

The shape of the root locus plot resembles that of a PI controller, except of course we do not have an open-loop pole at the origin anymore. The root loci approach one another from -xp and -p0, then break away from the real axis to form a circle which breaks in to the left of the open-loop zero at -z0. One locus approaches negative infinity and the other toward -z0. One may design the controller with an approach similar to that in Example 7.7 (p. 7-16). 

Integral control will add an open-loop pole at the origin. Again, we have two regions where we can put the open-loop zero ... 

Finally, let s take a look at the probable root loci of a system with an ideal PID controller, which introduces one open-loop pole at the origin and two open-loop zeros. For illustration, we will not use the integral and derivative time constants explicitly, but only refer to the two zeros that the controller may introduce. We will also use zpk () to generate the transfer functions. 

This system has a pole at the origin. We pick a contour in the s plane that goes counterclockwise around the origin, excluding the pole from the area enclosed by the contour. As shown in Fig. 13.4b, the contour is a semicircle of radius r. And Fo is made to approach zero. 

Connection formula pertaining to a first-order transition pole at the origin... 

Now we assume that in a certain region of the complex 2-plane around a first-order transition pole at the origin, i.e., a first-order pole of Q2(z) at the origin, we have... 

Note that the integrator has a single-pole — at zero frequency . Therefore, we will often refer to it as the pole-at-zero stage or section of the compensation network. This pole is more commonly called the pole at the origin or the dominant pole. 

This is one of the fundamental results of contour integration and will find widespread applications the point here being that the enclosure of a simple pole at the origin always yields 2tt/. 

Now, if n is a positive integer, or zero, the above is obviously in accordance with Cauchy s First theorem, since then fis) = s" is analytic for all finite values of s. However, if n becomes a negative integer, the s" is clearly not analytic at the point s = 0. Nonetheless, the previous result indicates the closed integral vanishes even in this case, provided only that n — 1. Thus, only the simple pole at the origin produces a finite result, when the origin is enclosed by a closed contour. 

We have seen in Section 9.5 that simple poles or nth order poles at the origin are removable type singularities, so that if f(s) contains a singularity at the origin, say a pole of order N, then it can be removed and the new function so generated will be analytic, even at the origin... 

Control engineers refer to the values of s that are roots of the denominator polynomial as the poles of transfer function G s). Sometimes it is useful to plot the roots (poles) and to discuss process response characteristics in terms of root locations in the complex s plane. In Fig. 6.1 the ordinate expresses the imaginary part of each root the abscissa expresses the real part. Figure 6.1 is based on Eq. 6-2 and indicates the presence of four poles an integrating element (pole at the origin), one real pole (at and a pair of complex... 

In this example, integration of the input introduces a pole at the origin (the term in the denominator), an important point that will be discussed later. 

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