Root locus diagram

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.

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. 

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... 

The root locus diagram can be used to provide a quick estimate of the transient response of the closed-loop system. The roots closest to the imaginary axis correspond to the slowest response modes. If the two closest roots are a complex conjugate pair (as in Fig. 11.28), then the closed-loop system can be approximated by an underdamped second-order system as follows. 

Figure 11.27 Root locus diagram for third-order system.

Consider the root locus diagram in Fig. 11.27 for the third-order s5Tstem of Example 11.13. For Kc = 10, determine values of 5 and t that can be used to characterize the transient response approximately. 

The utility of root locus diagrams has been illustrated by the third-order system of Examples 11.13 and 11.14. The major disadvantage of root locus analysis is that time delays cannot be handled conveniently, and they require iterative solution of the nonlinear and nonra-tional characteristic equation. Nor is it easy to display simultaneous changes in more than one parameter (e.g., controller parameters Kc and t/). For this reason, the root locus technique has not found much use as a design tool in process control. 

Root locus diagrams can be quickly generated by using a hand calculator or a computer with root-finding techniques such as are provided in MATLAB. 

The root locus diagrams of Section 11.5 (e.g.. Fig. 11.27) show how the roots of the characteristic equation change as controller gain Kc changes. By definition, the roots of the characteristic equation are the numerical values of the complex variables that satisfy Eq. 14-53. Thus, each point on the root locus also satisfies (14-54), which is a rearrangement of (14-53) ... 

Figure J.2 Root locus diagram for a process with three poles and no zeroes.

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