462 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. Sampled data systems have come into practical importance for a variety. of reasons The earliest of these had primarily to do with economy in the. design and use of equipment Many problems in impedance matching or. power level matching can be avoided if critical components are isolated. disconnected most of the time and the connection made only briefly a t. periodic intervals to read out a sample of the signal The possibility of time. sharing one component among several systems also gives rise to a sampled. form of signal processing A major increase in interest in sampled data. systems was caused by the development of radar systems during the 1940s. Most radars provide information only in the form of periodic samples either. because of a periodic scanning process or because of pulsed transmission of. the microwave energy A more recent surge of interest has been due to the. increasing utilization of digital computers as controllers in feedback systems. In some areas of application especially aerospace guidance and control the. use of discrete data processors is often a practical necessity Thus many. system engineers find themselves concerned almost exclusively with the. design of sampled data systems And as with continuous systems these. systems may be designed with or otherwise may suffer from a number of. important nonlinear effects, T H E EFFECTS O F SAMPLING. In this chapter as in most of the preceding material attention is centered on. systems which can be reduced to single loop configurations having a single. nonlinear part separated from the linear part The linear part in this case. may include any number of continuous linear elements and discrete or. pulsed linear elements The ordering of these elements around the loop is. of some consequence to the application of describing function theory. because in this case higher frequency and possibly lower frequency com. 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. sinusoid for the purpose of quasi linearization The output of the non. linearity y t then contains harmonic components at the fundamental and. higher harmonic frequencies On a two sided frequency scale the harmonic. components of y t would in general have the frequencies k w k 0. 1 2 where w is the frequency of the input sinusoid The sampling. operation modulates y t with the frequencies h l w I 0 1 2 where. o is the frequency of closure of the sampling switch l Thus y t contains. The reader who needs a basic treatment of the description of the sampling operation. the transfer of sampled signals through linear systems and z transform theory is directed. to any one of a number of texts on the subject Among them are Jury Ref 7 Kuo Ref. 15 Raggazzini and Franklin Ref 22 and Tou Ref 27, INTRODUCTION 463. Figure 9 0 1 Nonlinear sampled data system confgurations N static nonlinear elemenr. H data hold D discrete linear element L continuous linear element Asterisks. denote sampled signals, harmonic components with frequencies f k w f Zw These frequencies. which appear in the loop determine to a large extent the possibility of. successful application of describing function theory. 1 If the frequency ratio w w is irrational y t is aperiodic It contains. a harmonic component with frequency o in fact that component is just. 1 T times the fundamental component of the nonlinearity output This. may be seen from the familiar expression for the transform of y t. If w is not rationally related to o the only term in this sum with frequency. w is the primary term for I 0 Thus if describing function theory can be. applied at all in the case of irrational frequencies the describing function. relating x t to y t is just 1 T times the ordinary single sinusoid input. describing function for the nonlinearity which relates x t to y t The. question of applicability is raised because y t in this case may very well. contain harmonic components with frequencies lower than w These. components cannot be discarded on the basis of the filter hypothesis The. low frequency components in y t are due to higher harmonics of y t. which lie close to lo and thus are modulated to frequencies near zero. Describing function theory would then seem to be applicable only if w is. so small that the effect of the sampling on the operation of the system is. 2 If the frequency ratio w w is rational o ws m n y t is periodic. with a frequency which is an integral multiple of wlm Again y t may. 464 OSCILLATIONS IN N O N L I N E A R SAMPLED DATA SYSTEMS. contain harmonic components with lower frequencies than w and essentially. the same comments about applicability of describing function theory made in. regard to irrational frequencies are pertinent in this case. 3 If the frequency ratio w w is a whole fraction w o Iln y t is. periodic with frequency o It contains no harmonic component with. frequency lower than o except possibly for a dc component if o 40. With the possible effects of a dc component taken into consideration these. are the right conditions for applicability of describing function theory and. the remainder of the chapter is restricted to this case Fortunately this. includes a most important class of problems since the limit cycles which are. most commonly observed in sampled nonlinear systems have periods which. are whole multiples of the sampling period It is important to observe that. the component of frequency w in y t is not just l T stimes the corresponding. component of y t Thus it would be quite incorrect to employ a describing. function which relates x t to y t and ignores the higher harmonics of a. signal which is being sampled It is essential that the describing function. characterize directly the relation between x t and y t. As another illustration consider the system configuration of Fig 9 0 lb. In this case the higher harmonics in y t are attenuated by the continuous. linear filter before being sampled Thus the modulating effect of the sampler. on these harmonics may be of little importance The greater question in this. case is whether the input to the nonlinearity can be assumed a sinusoid. The hold does not provide very complete filtering of the high frequency. content of the sampled signal Thus unless there were additional filtering. in the position of the hold it might be necessary to characterize the transfer. from z t to y t by a describing function a task which promises to be. The following sections deal with the determination of and stability of. limit cycle modes in sampled nonlinear systems where the limit cycles tested. have periods which are whole multiples of the sampling period These are. not the only limit cycles which may be possible in such systems but experience. with both real and simulated systems has shown these to be by far the most. commonly occurring modes This does not exhaust the usefulness of. describing function theory in its application to sampled nonlinear systems. But other applications such as the study of forced sinusoidal response must. be considered carefully in each individual case because of the possibility of. lower frequency components as discussed above under irrationally related. frequencies and the possible existence of limit cycle modes in addition to. the forced response A very important special case system which can be. dealt with by a simple extension of previous techniques is treated in the. following section Then in the next we turn to the study of limit cycles in. more general systems, L I M I T CYCLES IN SAMPLED T W O L E V E L RELAY SYSTEMS 465. 9 1 L I M I T CYCLES IN SAMPLED TWO LEVEL, RELAY SYSTEMS. 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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