Root Loci

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

Fig. 5.7 Root-locus diagram for a first-order system. Roots of characteristic equation...

Fig. 5.9 Root locus diagram for a second-order system.

Root locus asymptotes. For large values of k the root loei are asymptotie to straight lines, with angles given by... 

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. 

Find the asymptotes and angles of departure and hence sketch the root locus diagram. Locate a point on the complex locus that corresponds to a damping ratio of 0.25 and hence find... 

The root locus method provides a very powerful tool for control system design. The objective is to shape the loci so that closed-loop poles can be placed in the. v-plane at positions that produce a transient response that meets a given performance specification. It should be noted that a root locus diagram does not provide information relating to steady-state response, so that steady-state errors may go undetected, unless checked by other means, i.e. time response. 

The root locus diagram is shown in Figure 5.19. In this case the real locus occurs between. v = —5 and —3 and the complex dominant loci breakaway at rrh = —1-15. Since these loci are further to the right than the previous option, the transient response will be slower. The compensator gain that corresponds to ( = 0.7 is K = 5.3. The resulting time response is shown in Figure 5.20, where the overshoot is 5.3% and the settling time is 3.1 seconds. 

The resulting root-locus diagram is shown in Figure 5.24. 

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. 

As with the continuous systems described in Chapter 5, the root locus of a discrete system is a plot of the locus of the roots of the characteristic equation... 

Sketch the root locus diagram for Example 7.4, shown in Figure 7.14. Determine the breakaway points, the value of K for marginal stability and the unit circle crossover. 

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... 

This produces the (pole-zero cancellation) root locus plot shown in Figure 5.18. When run, exampSlO.m allows the user to select the value of K that corresponds to ( = 0.7, and then uses this selected value to plot the step response. The text that appears in the command window is... 

Case study Example 5.11 uses root locus to design a ship roll stabilization system. The script file exampSll.m considers a combined PD and PID (PIDD) controller of the form... 

Example 7.6 constructs the root-locus in the z-plane for the digital control system in Example 7.4 (Figure 7.14). 

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