ROOT LOCUS
The root locus is one of the most powerful techniques used to analyse the stability of a closed- loop system. This technique is also used to design controllers with required time response characteristics. The root locus is a plot of the locus of the roots of the characteristic equation as the gain of the system is varied. The rules of the root locus for discrete-time systems are identical to those for continuous systems. This is because the roots of an equation Q(z) = 0 in the z-plane are the same as the roots of Q(s) = 0 in the s-plane. Even though the rules are the same, the interpretation of the root locus is quite different in the s-plane and the z-plane. For example, a continuous system is stable if the roots are in the left-hand s-plane. A discrete-time system, on the other hand, is stable if the roots are inside the unit circle. The construction and the rules of the root locus for continuous-time systems are described in many textbooks. In this section only the important rules for the construction of the discrete-time root locus are given, with worked examples.
Given the closed-loop system transfer function
we can write the characteristic equation as 1 + kF (z) = 0, and the root locus can then be plotted as k is varied. The rules for constructing the root locus can be summarized as follows:
1. The locus starts on the poles of F (z) and terminate on the zeros of F (z).
2. The root locus is symmetrical about the real axis.
3. The root locus includes all points on the real axis to the left of an odd number of poles and zeros.
4. If F (z) has zeros at infinity, the root locus will have asymptotes as k → ∞. The number of asymptotes is equal to the number of poles n p , minus the number of zeros nz . The angles of the asymptotes are given by
At this point one part of the locus moves towards the zero at z = −0.717 and the other moves towards the zero at −∞.
Figure 8.4 shows the root locus with the unit circle drawn on the same axis. The system will become marginally stable when the locus is on the unit circle. The value of k at these points can be found either from Jury’s test or by using the Routh–Hurwitz criterion.
Using Jury’s test, the characteristic equation is
Example 8.9
For Example 8.8, calculate the value of k for which the damping factor is ζ = 0.7.
Solution
In Figure 8.5 the root locus of the system is redrawn with the lines of constant damping factor and constant natural frequency.
From the figure, the roots when ζ = 0.7 are read as s1,2 = 0.61 ± j 0.25 (see Figure 8.6). The value of k can now be calculated as
which gives k = 0.12 and k = 2.08. The root locus of the system is shown in Figure 8.7. It is clear from this plot that the system is always stable since all poles are inside the unit circle for all values of k.
Lines of constant damping factor and constant angular frequency are plotted on the same axis in Figure 8.8.
Assuming that T = 1 s, ωn > 0.6 if the roots are on the left-hand side of the constant angular frequency line ωn = 0.2π/ T . The damping factor will be greater than 0.6 if the roots are below the constant damping ratio line ζ = 0.6. A point satisfying these properties has been chosen
