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(33.2)
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S(t,P) = -1 i S(t + P,P)
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0 < t < PI2 PI2 < t < P all t
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is the undelayed unit square wave of period P shown in Fig. 33.3.
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k=Asin at = - co
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FIGURE 33-2 Result of application of sinusoidal error signal to relay with dead zone.
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FIGURE 33-3 The unit square wave S(t, P).
As is well known from Fourier series analysis, (see Churchill and Brown, 1986), S(t, 27rl0.1) may be expanded in a series of sine waves to give S t 2 = 4 sinot + !jsin3wt + +.sin5wr (h) ,( Hence, by Eq. (33.2) M(t) = 4 sin wt - sin- e +;sin(3wt A) 7T I ( - 3sin- ;) =t- e-e)
(33.4)
(33.5)
+lsin 5wt -5sin- 2 + **a 5 ( A) 1 According to Eq. (33.5), M(t) contains a fundamental and odd harmonics. Let us consider what happens to these components of M as they pass around the control loop. Assuming that o is sufficiently large, the harmonics are much more heavily attenuated by the two first-order elements than is the fundamental, because the harmonic frequencies are higher. For example, if w is 9 t-ad/mm, the relative attenuation of the fundamental and third harmonic between M and B is expressed by the quotient
Since the initial amplitude of the third harmonic in M(t) is one-third of the fundamental, it is clear that the amplitude of the third harmonic will be less than 4 percent of the amplitude of the fundamental in B(t). The amplitudes of the higher harmonics will be even less. To all intents and purposes, B(t) is sinusoidal and, hence, so is e(t). Furthermore, the presence of harmonics in M(t) may be ignored, and the approximation
THE,
DESCRIBING FUNCTION
TECHNIQUE
509 (33.6)
M(t) = $ sin wc - sin- 2 ( 1
is acceptable because the higher harmonics are filtered out by the rest of the loop. In order for a limit cycle to be maintained, it is necessary that B(t) = -e(t) = -Asinot (33.7)
However, if M(t) is given by Eq. (33.6), B(t) can be calculated by frequency response. The AR between M and B is
and the phase difference between B and M is &B - 4M = - tan- w - tan- : (33.9)
According to Eq. (33.7), the overall amplitude ratio between B and E must be unity and the overall. phase lag 180 . Also, according to Eq. (33.6), the AR between E and M is 4/7rA and the phase lag is sin- (O.Ol/A). Combining these facts results in 4 1 1 ~J1+7;;-jzJiTqzp= (33.10) 0 -sin- _ tan-lw - tan-12 = -180 Equations (33.10) are a system of two equations in the unknowns A and o . Trialand-error solution yields A = 0.03 w = 9 rad/min in excellent agreement with the results of the phase-plane method presented in Chap. 32. The reason for the accuracy of these results is the high attenuation of harmonics provided by the linear elements in the loop. The labor saving of this method over the phase-plane method is apparent. Of more direct interest is the amplitude of the signal C in the limit cycle. This may now be estimated by frequency response to be ICI= J + ($0.03) = 0.14 The true result derived by the phase-plane method is 0.17, so that an error of 18 percent is attributed to the neglect of harmonics in C. The reason for the decreased accuracy in the amplitude of C over that in the amplitude of E is that only one of the linear elements has acted on the squarewave output of the relay before it reaches C. Hence, the harmonics are not fully attenuated in C and the
NONLINEAR CONTROL
signal C will be less sinusoidal than E. However, for engineering purposes the . error in C is probably not excessive.
THE DESCRIBING FUNCTION
Because the basic technique of harmonic analysis often yields accurate results with modest effort, it is profitable to systematize the procedure. To do this, a describing function is defined for the nonlinear loop element. This function assumes a sinusoidal input to the nonlinearity and gives the AR and phase lag of the fundamental in the output. Thus, for the relay considered in the last section, the describing function is defined by
where N is used as the symbol for the describing function. In general, the loop diagram for a relay control system appears as in Fig. 33.4. As shown previously, the necessary condition for a limit cycle, ignoring harmonics, is (33.11)
i$N + i$G,H = -180
As in the case of the relay, the magnitude and angle of N in general depend on the amplitude A of the input to N. The magnitude and angle of G# depend on o. Equations (33.11) can be rewritten (33.12)
&G,H(o) = -180 - &N(A)
Equations (33.12) can be solved graphically on a gain-phase plot. This is a plot of the log of AR versus phase, as shown in Fig. 33.5 for the case treated in the previous section. The linear elements are plotted as IG#l versus &GpH, with o plotted as a parameter on the curve. The relay is plotted as l/l N(A) I versus -180 - &N(A), with A plotted as a parameter on the curve. According to Eqs. (33.12), a limit cycle occurs at the intersection of the two curves, where the amplitude and frequency can be read from the parametric labeling of A and o.
+!z gT HGm33-4
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