Open-loop zeros

If there are no open-loop zeros in the transfer funetion, then the denominator of equation (5.62) is unity. 

Termination points K = oo) The root loei terminate at the open-loop zeros when they exist, otherwise at infinity. 

S open-loop poles — S open-loop zeros =------------------7------1----------------... 

The control strategy for the root-locus diagram shown in Figure 5.24 is called PIDD, because of the additional open-loop zero. The system is unstable between K = 0.17 and K = 1.06, but exhibits good transient response at A" = 10.2 on both complex loci. 

Radius = open-loop poles Centre = (Open-loop zero, 0)... 

Although the solution is the same, the important differenee is with state feedbaek, the elosed-loop poles are plaeed at direetly the required loeations. With root loeus, a eertain amount of trial and error in plaeing open-loop zeros was required to aehieve the desired elosed-loop loeations. 

Note that with this very specific case by choosing x = 1, the open-loop zero introduced by the PI controller cancels one of the open-loop poles of the process function at -1. If we do a root locus plot later, we d see how the root loci change to that of a purely second order system. With respect to this example, the value is not important as long as x > 1/2. 

This is the idea behind the plotting of the closed-loop poles—in other words, construction of root locus plots. Of course, we need mathematical or computational tools when we have more complex systems. An important observation from Example 7.5 is that with simple first and second order systems with no open-loop zeros in the RHP, the closed-loop system is always stable. 

In other words, on a root locus plot, we expect the "trace" of the root loci to begin at the open-loop poles and terminate at the open-loop zeros (if there is one). For real systems, m > n, and n>... 

The system in (e) can become unstable, while a proper addition of an open-loop zero, as in (f), can help stabilize the system (Fig. E7.6c). In (e), the two loci from -1 and -2 approach each other (arrows not shown). They then break away and the closed-loop poles become unstable. The... 

A locus (closed-loop root path) starts at an open-loop pole and either terminates at an open-loop zero or extends to infinity. 

From the perspective of a root locus plot, a phase-lag compensator adds a large open-loop zero... 

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). 

To find the new state feedback gain is a matter of applying Eq. (9-29) and the Ackermann s formula. The hard part is to make an intelligent decision on the choice of closed-loop poles. Following the lead of Example 4.7B, we use root locus plots to help us. With the understanding that we have two open-loop poles at -4 and -5, a reasonable choice of the integral time constant is 1/3 min. With the open-loop zero at -3, the reactor system is always stable, and the dominant closed-loop pole is real and the reactor system will not suffer from excessive oscillation. 

Try an open-loop zero at -1 % to cancel the open-loop pole at -1... 

MATLAB automatically selects a reasonable vector for k, calculates the roots, and plots them. The function rlocus () also adds the open-loop zeros and poles of G(s) to the plot. 

To implement an ideal PD controller, we ll have an additional open-loop zero. Two (of infinite) possibilities are... 

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. 

The major regions for placing the zero are the same, but the interpretation as to the choice of the integral time constant is very different. We now repeat adding the open-loop zeros ... 

You may want to try some sample calculations using a PID controller. One way of thinking we need to add a second open-loop zero. We can limit the number of cases if we assume that the value of the derivative time constant is usually smaller than the integral time constant. 

For a first order function with deadtime, the proportional gain, integral and derivative time constants of an ideal PID controller. Can handle dead-time easily and rigorously. The Nyquist criterion allows the use of open-loop functions in Nyquist or Bode plots to analyze the closed-loop problem. The stability criteria have no use for simple first and second order systems with no positive open-loop zeros. 

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