Файл: Microcontroller based applied digital control (D. Ibrahim, 2006).pdf

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z-plane

Figure 7.8 Time response of z-plane pole locations

Equation (7.5) describes a logarithmic spiral in the z-plane as shown in Figure 7.9(b). The spiral starts from z = 1 when ω = 0. Figure 7.10 shows the lines of constant damping ratio in the z-plane for various values of ζ .

7.4.2 Undamped Natural Frequency

As shown in Figure 7.11, the locus of constant undamped natural frequency in the s-plane is a circle with radius ωn . From this figure, we can write

ω2 + σ 2 = ωn2 or σ = ωn2 − ω2. (7.6)

Line of

constant ζ

Line of constant ωn

ωn

The locus of constant ωn in the z-plane is given by (7.7) and is shown in Figure 7.10 as the vertical lines. Notice that the curves are given for values of ωn ranging from ωn = π/10T to

ωn = π/ T .

Notice that the loci of constant damping ratio and the loci of undamped natural frequency are usually shown on the same graph.

7.5DAMPING RATIO AND UNDAMPED NATURAL FREQUENCY USING FORMULAE

In Section 7.4 above we saw how to find the damping ratio and the undamped natural frequency of a system using a graphical technique. Here, we will derive equations for calculating the damping ratio and the undamped natural frequency.

The damping ratio and the natural frequency of a system in the z-plane can be determined if we first of all consider a second-order system in the s-plane:

ω2

G(s) = n . (7.8) s2 + 2ζ ωn s + ωn2

The poles of this system are at

s1,2 = −ζ ωn ± j ωn 1 − ζ 2. (7.9)

We can now find the equivalent z-plane poles by making the substitution z = es T , i.e.

z = es T = e−ζ ωn T ± ωn T 1 − ζ 2, (7.10)

Find the damping ratio and the undamped natural frequency. Assume that T = 1 s.

Solution

We need to find the poles of the closed-loop transfer function. The system characteristic equation is 1 + G(z) = 0,

i.e.

z2 − z + 0.632 = (z − 0.5 − j 0.618)(z − 0.5 + j 0.618) = 0,

EXERCISES 183

Figure 7.12 Finding ζ and ωn graphically

Example 7.3

Find the damping ratio and the undamped natural frequency for Example 7.2 using the graphical method.

Solution

The characteristic equation of the system is found to be

z2 − z + 0.632 = (z − 0.5 − j 0.618)(z − 0.5 + j 0.618) = 0

and the poles of the closed-loop system are at

z1,2 = 0.5 ± j 0.618.

Figure 7.12 shows the loci of the constant damping ratio and the loci of the undamped natural frequency with the poles of the closed-loop system marked with an × on the graph. From the graph we can read the damping ratio as 0.25 and the undamped natural frequency as

ωn = 0.29π = 0.91.

T

7.6 EXERCISES

1.Find the damping ratio and the undamped natural frequency of the sampled data systems whose characteristic equations are given below

(a)z2 − z + 2 = 0

(b)z2 − 1 = 0

(c)z2 − z + 1 = 0

(d)z2 − 0.81 = 0