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G(z) = KS
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where b = emT . Using the transformation given by Eq. (24.4), we obtain for 1 + G(z) = 0 1+ K(l- b) = 0 w+l b w - l or
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1 + KU b)(w 1) = o
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w + 1 - b(w - 1)
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Rearranging this result in polynomial form for applying the Routh test gives
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[(K + l)(l - b)]w + [(l + b) - K(l - b)] = 0
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STABILITY
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The Routh array for this expression becomes
R o w 1t e r m
(K + l)(l - b) (1 + b) - K(l - b)
Since b = emT/r is always positive and less than one and K is positive, the first element in the array is positive. For stability, the Routh test requires that all elements of the first column be positive. Therefore, (1 + b) - K(l - b) > 0
1+ b > K(l- b) or l+b KCl - b
for stability
This is the same result given by Rq. (23.37), which was obtained from an expression for c(nT>. We shall now use the root locus method on the same example. For this simple problem, there is only one pole of the open-loop transfer function, G(z), which is located at b as shown in Fig. 24.5. The root locus consists of one branch that moves from the pole at b along the real axis to the left. The intersection of this branch with the unit circle at z = - 1 gives us the stability boundary. Using the magnitude criterion of root locus construction gives
KU-b) -= 1
1% - Pll We can obtain the value of K at the stability boundary by solving for K, thus K ~ hi-Pll _ l+b l - b l - b
z-plane
FIGURE
24-5
Root locus plot for Example 24.1.
SAMPLED-DATAcoNTRoLSYSTEMS
Since the root locus branch moves to the left with increase of K, we see that l+b Kcl-b for stability
For this simple example, the root locus method is easier, for one does not need to use a transformation and the root locus diagram is very simple. However, for higherorder systems, the apparent advantage of the root locus method over the Routh test is lost. To appreciate the details of applying the stability criterion to sampled-data systems, the reader is encouraged to work a few of the more complex problems at the end of this chapter. Other methods for determining stability of sampled-data systems include the Schur-Cohn criterion and the Jury criterion (see Jury, 1964 and Tou, 1959). The Jury criterion is a simplification of the Schur-Cohn criterion. These methods, which can be applied directly to the characteristic equation written in the z-domain, can detect roots outside the unit circle of the z-plane. Since these methods require the evaluation of high-order determinants, they are limited to simple systems.
SUMMARY
The presence of sampling in a control system contributes to instability. The criterion for stability of a sampled-data system requires that the roots of the characteristic equation, 1 + G(z) = 0, fall within the unit circle of the complex z-plane.
Based on this criterion, two methods were developed to determine stability: (a)
the modified Routh test and (b) the root locus method. To use the Routh test, one
must first apply the bilinear transformation, which maps the inside of the unit circle into the left half of the w-plane. The usual rules of the Routh test am then applied to the transformed characteristic equation. Using the root lqcus method is simpler, for one simply constructs the root locus diagram from the poles and zeroes of the open-loop transfer function G(z). When a branch of the root locus diagram crosses the unit circle, the system becomes unstable. It is of interest to note that systems having transport lag can be analyzed easily for stability in sampled-data systems by either the modified Routh test or the root locus method; this was not the case for continuous systems having transport lag.
PROBLEMS
24.1. For the system shown in Fig. P23.5, determine the ultimate gain by use of the Routh test and by use of the root locus method. 24.2. For the control system shown in Fig. P24.2, determine (a) an expression for C(z) when a unit-step change occurs in U, R remaining 0. (b) the stability criteria for the control system. (c) plot the continuous response c(t) for at least a period of time equal to 2T. Obtain this information from basic knowledge of first-order systems. Note: Clamping is not present in this system, for them is no zero-order hold in the block diagram.
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