# 9 Systems Oscillations Nonlinear Sampled Data In-Books Pdf

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The material presented in Sec 9 2 is readily applicable to the study of. limit cycles in two level relay systems but these systems are of such impor. tance that it seems worth exploiting the simpler approach which is possible in. The configuration of the system is shown in Fig 9 1 1 The two level. relay is shown as having possible hysteresis A zero order hold is considered. to follow the sampling switch A great many systems do employ a zero. order hold or simple clamp which clamps a sampled signal t o a constant over. the following sampling period This analysis is not however limited to such. systems If the actual system does not include a hold or uses a higher. ordered hold the transfer function of that hold is included in the linear part. as shown in Fig 9 1 1 along with the reciprocal of the zero order hold. transfer function The linear part may include any number of continuous. and discrete linear elements, T H E DESCRIBING F U N C T I O N. One has free choice in deciding how much of the system to characterize with a. describing function so long as the nonlinear part is included The analysis. of this system is most like the analysis of continuous systems considered. heretofore if one chooses to represent the effect of the nonlinearity the. sampling switch and the zero order hold by a describing function To this. end x t is taken to be a sinusoid unbiased to begin with and the funda. mental harmonic component of z t is calculated The frequencies we shall. consider according to the discussion of the preceding section are whole. fractions of the sampling frequency o Moreover we shall center attention. on the even fractions 9 t Q since these are the limit cycle modes one. might expect to see in the very common case in which the linear part of the. system includes a pole at the origin an integration In that case z t must. be an unbiased function in any steady state limit cycle with no input to the. Figure 9 1 1 Two level relay system configurarion, 466 O S C I L L A T I O N S IN N O N L I N E A R SAMPLED DATA SYSTEMS. system The drive signal into the linear part will then consist of a periodic. cycle which includes an equal number of sampling periods of plus and minus. drive The only arrangement of these periods of plus and minus drive which. is consistent with the sinusoid assumed as the input to the nonlinearity is. n positive drive periods followed by n negative drive periods in the case of a. cycle with period 2nTs T being the sampling period Such a cycle will be. termed an n n mode, The input and output waveforms for the 2 2 mode are shown in Fig. 9 1 2 x t is a sinusoid with period 4 T s and z t is a square wave with that. period The output of the hold z t is shown lagging the output of the. nonlinearity y t because y t is not in phase with the sampling points. The lag between the zero crossing of y t and the next sampling point is not. known a priori evidently it can take any value between 0 and T in time or 0. and n n in phase angle The amplitude of the fundamental harmonic of. z t is 4 n D and the phase lag of that component relative to x t is. sin 6 A y where p is the sampling lag The describing function for. the chain of elements nonlinearity sampling switch and hold is then. This expression holds for an n n mode of any order. T H E LINEAR PART, The remainder of the system the linear part as shown in Fig 9 1 1 is now. characterized by its steady state sinusoidal response at the frequency 1 2 n o. Sampling points, Figure 9 1 2 Signal wai eformsfor the 2 2 mode.
L I M I T CYCLES IN SAMPLED T W O L E V E L RELAY SYSTEMS 467. If this is a continuous linear operator the only requirement for applicability. of describing function theory is that it attenuate the higher harmonics of. z t sufficiently to return essentially the fundamental sinusoid to x t If. this includes samplers and discrete linear elements a more restrictive condi. tion is placed on them Consider the linear elements in Fig 9 1 3 In this. example L and L represent continuous linear filters whereas D represents. a discrete linear filter It may be mechanized as a pulsed analog filter or. perhaps as a digital computer solving a linear difference equation L andlor. L may include data holds We wish to find the sinusoidal component of. frequency w in the steady state output of this chain when the input is. periodic with that frequency, C jw L j o V jw, jw D jw W jo 9 1 2. and W j w 2 Ll j w Iw lZ j w Zw, From this expression one can see that if z t were just a sinusoid of frequency. w w none of the complementary components of w t would have the. frequency w The only component of c t which would have the frequency. w is that due to the I 0 term in the sum of Eq 9 1 3 Thus the sinusoidal. response function representing the fundamental transfer through the chain. of elements in Fig 9 1 3 would be just l T L jw D jw L jw. This result is complicated however by the fact that in the system under. study z t is a periodic function which includes harmonics in addition to the. fundamental component With o a n even fraction of w some of the odd. harmonics of z t are modulated to additional components of w t at the. frequency w The harmonics which contribute to the fundamental com. ponent after sampling are those with frequencies equal to Iw Lt w for all. integers I The effect of any significant contributors could be included by. calculating the harmonics of z t passing them through L jkw k is the. order of the harmonic and adding the term in Eq 9 1 3 However for. ponents are generated not only by the nonlinear part but by the sampling operations as well Consider the system configuration of Fig 9 0 la In the study of steady state oscillations in this system the nonlinearity input being the output of the continuous linear filter may reasonably be taken as a

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