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Thus, the system is stable for 0.285 < K < 2.

Example 8.5

The characteristic equation of a system is given by

F (z) = z3 − 2z2 + 1.4z − 0.1 = 0.

Determine the stability of the system.

192 System Stability

8.3 ROUTH–HURWITZ CRITERION

The stability of a sampled data system can be analysed by transforming the system characteristic equation into the s-plane and then applying the well-known Routh–Hurwitz criterion.

A bilinear transformation is usually used to transform the left-hand s-plane into the interior of the unit circle in the z-plane. For this transformation, z is replaced by

Given the characteristic equation in w,

F (w) = bn wn + bn−1wn−1 + . . . + b1w + b0 = 0,

then the Routh–Hurwitz array is formed as follows:

The first two rows are obtained from the equation directly and the other rows are calculated as follows:

c1 = bn−1bn−2 − bn bn−3 ,

bn−1

c2 = bn−1bn−4 − bn bn−5 , bn−1

c3 = bn−1bn−6 − bn bn−7 ,

bn−1

d1 = c1bn−3 − bn−1c2 , c1

. . . .

The Routh–Hurwitz criterion states that the number of roots of the characteristic equation in the right hand s-plane is equal to the number of sign changes of the coefficients in the first column of the array. Thus, for a stable system all coefficients in the first column must have the same sign.

Example 8.6

The characteristic equation of a sampled data system is given by

z2 − z + 0.7 = 0.

Determine the stability of the system using the Routh–Hurwitz criterion.

Solution

Transforming the characteristic equation into the w-plane gives

Figure 8.2 Closed-loop system

or

2.7w2 + 0.6w + 0.7 = 0.

Forming the Routh–Hurwitz array,

w2 2.7 0.7

w1 0.6 0 w0 0.7

there are no sign changes in the first column and thus the system is stable.

Example 8.7

The block diagram of a sampled data system is shown in Figure 8.2. Use the Routh–Hurwitz criterion to determine the value of K for which the system is stable. Assume that K > 0 and T = 1 s.

or

z2 − z(1.368 − 0.368K ) + 0.368 + 0.264K = 0.

194 System Stability

Transforming into the w-plane gives

The system is stable if there is no sign change in the first column. Thus, for stability,

1.264 − 0.528K > 0

or

K < 2.4.

8.4 ROOT LOCUS

The root locus is one of the most powerful techniques used to analyse the stability of a closedloop 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

G(z)

,

1 + GH(z)

we can write the characteristic equation as 1 + k F (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

180r

θ = , where r = ±1, ±3, ±5, . . . .

n p − nz

The asymptotes intersect the real axis at σ , where

p − z

5. The breakaway points on the real axis of the root locus are at the roots of

dF(z) = 0.

dz

6.If a point is on the root locus, the value of k is given by

1 + kF(z) = 0 or k = − 1 .

F (z)

Example 8.8

A closed-loop system has the characteristic equation

1 + GH(z) = 1 + K 0.368(z + 0.717) = 0. (z − 1)(z − 0.368)

Draw the root locus and hence determine the stability of the system.

Solution

Applying the rules:

1. The above equation is in the form 1 + kF(z) = 0, where

F (z) = 0.368(z + 0.717) . (z − 1)(z − 0.368)

The system has two poles at z = 1 and at z = 0.368. There are two zeros, one at z = −0.717 and the other at minus infinity. The locus will start at the two poles and terminate at the two zeros.

2.The section on the real axis between z = 0.368 and z = 1 is on the locus. Similarly, the section on the real axis between z = −∞ and z = −0.717 is on the locus.

3. Since n p − nz = 1, there is one asymptote and the angle of this asymptote is

axis intersection point of the asymptotes.