Describing functions for stability analysis of integral pulse frequency modulated unity feedback closed loop syste

Fitzgerald, J. F.

Monopoli, R. V.

English

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A DESCRIBING FUNCTION ANALYSIS I N  CLOSED LOOP SYSTEMS 
WITH PULSE FREQUENCY MODULATORS 
* 
R.  V .  Moliopoli J .  F. F i t zge ra ld  
Univers i ty  of Massachusetts PI Bell Telephone Laborator ies ,  Inc.  
Amher s t , Mas s ac.huset t s Whippany, New Jersey 
Abstract  - Equivalent c i r c u i t s  are presented  f o r  i n t e g r a l  and sigma pulse  
frequency modulators (IPFM and CPFM). Both conta in  l i n c a r  elements and a 
n o n l i n e a r i t y  with s t e p  d i s c o n t i n u i t i e s  and h y s t e r e s i s .  
i s  developed f o r  t h i s  non l inea r i ty ,  and appl ied i n  t h e  usual  manner t o  t h e  
s t a b i l i t y  a n a l y s i s  of a u n i t y  feedback closed loop system with t h e  modulator 
A desc r ib ing  func t ion  
i n  t h e  forward pa th .  The t r a n s f e r  func t ion  f o r  t h e  l i n e a r  p a r t  of t h e  system 
is assumed t o  have a s i n g l e  pole  a t  t h e  o r i g i n .  
i n d i c a t e  t h a t  t h i s  a n a l y s i s  g ives  b e t t e r  p red ic t ed  va lues  f o r  l i m i t  cyc le  
Analog computer r e s u l t s  
per iods  than  t h e  quas i -descr ib ing  func t ion  a n a l y s i s  prev ious ly  repor ted .  
I .  INTRODUCTION 
The app l i ca t ion  of  pu lse  frequency modulation i n  feedback systems i s  
of i n t e r e s t  i n  p re sen t  day technology [l-51. In  t h i s  paper ,  an equivz len t  
c i r c u i t  and a descr ib ing  func t ion  a n a l y s i s  based on t h i s  equivalent  i s  
presented f o r  a certain c l a s s  of these  systems. 
Pulse  frequency modulation (PFM) i s  t h e  coding of  information i n t o  t h e  
in te rva l  between pu l ses .  
is  t o  i n t e g r a t e  t h e  input  s i g n a l  and dec ide  on t h e  emission of a pulse  whenever 
t h e  i n t e g r a l  reaches a c e r t a i n  l e v e l .  
A common scheme f o r  implementing t h e  coding process  
This  is  known as i n t e g r a l  pu lse  
frequency modulation (IPFM) and has been t h e  sub jec t  of considerable  s tudy [3-51. 
A more general  PFM scheme was introduced i n  [lj and is  known as sigma pulse  
* 
Research upported i n  p a r t  by NASA under gran t  NGR-22-010-018. 
- 
' .  H l L  3600 
-e/- 96 7K3 K * G f  
https://ntrs.nasa.gov/search.jsp?R=19680025122 2020-03-12T07:01:15+00:00Z
- 2 -  
frequency modulation (CPFN) . 
over IPFM fecdback systems, improved s t a b i l i t y  and ease of  phys ica l  imple- 
CPFM feedback systems have two b a s i c  advantages 
mentation. 
A frequency dependent s inuso ida l  descr ib ing  func t ion  was proposed by 
L i [ 3 ]  f o r  IPFM. However, it cannot be appl ied  t o  CPFM feedback systems [SI. 
Pav l id i s  and Ju ry  presented a quasi-descr ibing func t ion ,  based on a square 
wave inpu t ,  f o r  IPFM and t h e  subc lass  of CPFM known as Neural pu lse  frequency 
modulation (NPFEI). I n  t h i s  paper an IPFM equiva len t  c i r c u i t  proposed by 
Meyer [4] i s  used as t h e  basi.s t o  develop an equiva len t  c i r c u i t  f o r  NPFM. 
This  equiva len t  conta ins  l i n e a r  elements and a non l inea r i ty  with s t e p  discon- 
t i n u i t i e s  and h y s t e r e s i s .  
descr ib ing  func t ion  techniques convenient t o  apply t o  both IPFP.1 and NPFFl 
feedback systems. 
Use of t h i s  equiva len t  makes conventional s inusoida l  
The t r a n s f e r  func t ion  f o r  t h e  l i n e a r  p a r t  of t h e  system is 
assumed t o  have one pole  a t  the o r i g i n .  
Analog computer r e s u l t s  i n d i c a t e  t h a t  t h e  approach prcsented here  gives  
b e t t e r  pred ic ted  values  f o r  l i m i t  cycle  per iods  than does t h e  quasi-descr ibing 
func t ion .  
11. DEFINITIONS AND EQUIVALENT MODELS FOR IPFEl AND CPFbl 
If t h e  on-time of a pulse  i s  very s h o r t  compared with t h e  response time 
of t h e  system, t h e  pu l ses  can be assumed t o  be impulses, and IPFM can be  
descr ibed by t h e  following equat ions [l ] 
* =  x - ry . d t  
Y = S P ( P )  S i lP l  - r) (2) 
where x denotes t h e  input ,  y t he  output ,  p t h e  value o f  t h e  i n t e g r a t o r  ou tput ,  
and r t h e  pulse  t r i g g e r i n g  and i n t e g r a t o r  r e s e t  l e v e l .  Note t h a t  i f  an impulse 
- 3 -  
occurs  at t = to, then p(to-) = r and p( to+)  = 0. 
If a continuous func t ion  o f  p, g(p) t h a t  is odd and monotonically 
inc reas ing  is added t o  t h e  l e f t  hand s i d e  of (1) it  leads  t o  t h e  equat ions 
descr ib ing  CPFM [l], i .e.  
If g(p) = cp where c i s  a p o s i t i v e  cons tan t  equat ions (3)  and (4) de f ine  NPF’M. 
An a l t e r n a t e  way of desc r ib ing  these  two PFM processes  i s  developed i n  
[ G I .  Based on t h e  equiva len t  c i r c u i t  f o r  IPFfil proposed by Myer [4] it i s  
shown t h a t  t h e  input  output  r e l a t i o n s h i p  of equat ions (1) and (2) can be 
represented  i n  block diagram form as shown i n  Fig.  l ( a )  with N as sliom i n  
Fig.  l ( b ) .  Furthermore t h e  block diagram i n  Pig.  l ( a )  may be manipulated t o  
produce t h e  CPFM equiva len t  and the  NPFM equiva len t  shown i n  F i g s .  2(a)  and 
2(b) r e s p e c t i v e l y  [ 6 ] .  These equiva len ts  a r e  s i g n i f i c a n t  f o r  two reasons.  
F i r s t ,  f o r  l a rge  q ( t ) ,  N may be reasonably approximated as a l i n e a r  element 
with u n i t y  ga in .  This l a rge  s i g n a l  l i n e a r  approximation lead t o  good p red ic t ed  
r e s u l t s  f o r  t h e  t r a n s i e n t  response t o  l a rge  s t e p  inpu t s  [GI. Furthermore, t h e  
nonl inear  element N admits a s inuso ida l  desc r ib ing  func t ion  which can be r e a d i l y  
der ived as opposed t o  a descr ib ing  func t ion  which assumes t h a t  x ( t ) ’ i s  
s inuso ida l  [ 33 . 
111. DERIVATION OF THE DESCRIBING FUNCTION 
In  t h e  descr ib ing  func t ion  method t h e  s inuso ida l  input  t o  t h e  nonl inear  
element i s  usua l ly  assumed t o  have zero average va lue .  This  assumption is not  
very  use fu l  f o r  NPPN systems, however, s i n c e  t h e  output  of  t h e  nonl inear  element 
can e x h i b i t  only an even number of pu l ses  per h a l f  cyc le  i f  dr iven  by such an 
s 
- 4 -  
i npu t .  
p e r  ha l f  cyc le  and a l l  o t h e r  cases  where t h e  observed l i m i t  cyc le  conta ins  an 
This  e l imina tes  from cons idera t ion  t h e  very important case of one pu l se  
odd number of pu l se s  p e r  h a l f  cycle .  
is developed with q ( t )  assumed t o  be of t h e  form 
For t h i s  reason t h e  .descr ibing ' funct ion 
q ( t )  = A s i n  w t + b 
where A > 0 and 0 < - b < - 1 /2 .  For t h i s  choice of  q ( t ) ,  t h e  following p r o p e r t i e s  
of f ( q )  are evident  by inspec t ion  of f i g u r e  l ( b ) :  
f ( q  + M) = f ( q )  + FI; for  Ff an i n t e g e r  
Because of t hese  p r o p e r t i e s ,  a l l  u s e f u l  f i rs t  harmonics of f ( q )  may be computcd 
with A and b r e s t r i c t e d  as i n  (5).  
For b = 0 only an even number of pu l ses  p e r  h a l f  cyc le  can occur ,  and f o r  
b = 1 / 2 ,  only an odd number can occur. I n  t h e  former case  2n pu l ses  a r e  produced 
i f  n < A < n + 1 ( f o r  n = 0,  1, 2 ,  . . . I  and i n  t h e  l a t t e r  2n + 1 pu l ses  are 
produced f o r  n + 1 / 2  < - A < n + 3/2 ( fo r  n = 0, 1, 2 ,  3 ,  ...). 
pulses  p e r  h a l f  cyc le  which occur f o r  var ious  combinations o f  A and b may be odd 
- 
The number of 
o r  even. If n < A + b < n + 1 ( f o r  n = 1, 2,  3 ,  . . . I  then  2n - 1 pu l ses  occurs f o r  
A - b < n and 2n pu l ses  occur f o r  A - b > n .  
- 
These r e s u l t s  are indica ted  - 
graph ica l ly  i n  Fig.  4 which shows the A-b plane separa ted  i n t o  t r i a n g u l a r  regions 
by s t r a i g h t  l i n e s .  For a given p a i r  (A,b), t h e  number of pu lses  p e r  h a l f  cyc le  
is  uniquely spec i f i ed .  
u se fu l  i n  t h e  graphica l  r ep resen ta t ion  of t h e  negat ive inverse descr ib ing  func t ion .  
The boundaries f o r  t hese  t r i a n g u l a r  regions w i l l  prove 
I n  computing t h e  descr ib ing  func t ion ,  l e t  w t  = 0 .  Since f (A s i n  0 + b) 
+ + 
changes only when A s i n  (0 +) + b = m and A s i n  (0 -) + b # m f o r  m = 0,  - 1, - 2 ,  
..., it may be represented  by t h e  fol lowing s e r i e s  of  u n i t  s t e p  func t ions :  
- 5 -  
n 
f(A s i n  0 + b) = C 
m= 1 
[u(0 - s i n  -1 ((m-b)/A)) - u(0 - II + sin-'((m-l-b)/A))] 
j - c[u(0 - II - sin-' ((m+b)/A))- u(0  - 2II + sin-'((m-l+b)/A))I (8) 
m= 1 
where j equals  t h e  g r e a t e s t  i n t e g e r  i n  A - b 
The f i r s t  harmonic of f (A s i n  0 + b) w i l l  be denoted by € (A s i n  0 + b) and 1 
i n  phasor form by y1 = a1 + j B 1  where 
-a 2n 
- f f ( A  s i n  0 + b) cos OdO a1 - K O  
1 2n 
B1 = - 1 f ( A  s i n  0 + b) s i n  OdO 
n o  
Performing the  i n t e g r a t i o n  on (9) and (10) y i e l d s  
1 1 
a i n  =-- ((2n-f)/A + { K } )  
The bracketed term, { 1 ,  does not  appear i f  j = n-1. 
- ((n-l+b)/*)*' +J1 - ((n-b)/A)2> 
1 I n  order t o  ob ta in  t h e  descr ib ing  func t ion ,  one need only normalize y 
with r e spec t  t o  A; thus ,  
N = Y p  (13) 
I V .  APPLICATION OF THE DESCRIBING FUNCTION TO NPFbi FEEDBACK SYSTEhIS 
An NPFM feedback system i s  shown i n  F i g .  3 ( a ) .  A modified form of t h i s  
system's block diagram shown i n  Fig.  3(b) is seen t o  be s u i t a b l e  f o r  descr ib ing  
func t ion  ana lys i s .  The loop t r a n s f e r  €unction, €I(s), i s  
-6- 
(l/rc)sP(s) -1 
SIC + 1 
H ( s )  = 
where P(s), the plant transfer function, is assumed t o  have one pole at the 
origin. 
the equation 
Limit cycle amplitudes and frequencies are determined by solving 
H ( j w )  = - 1/N (15) 
graphically using either a polar plot or a magnitude-phase plot. The 
negative inverse describing function, -l/N, is plotted in Fig. 5 f o r  n = 1 
and 2, i . e .  for one and two pulses per half cycle. Since no stable limit 
cycles with more than two pulses per half cycle were observed experimentally, 
no other characteristics are shown. 
It is t o  be noted that the -1/N plot is not a single curve. Instcad, 
each triangular region in Fig. 4 leads to a unique bounded region in Fig. 5. 
Thus, for n = 1 and n = 2, -1/N may be anywhere in the regions indicated in 
Fig. 5. Thus, a range of predicted values of limit cycle frequencies is 
determined by the two intersections where the M ( j w )  
each region. 
below. 
plot enters and leaves 
This is indicated in the results included in Tables I and I1 
Fitzgerald Pavlidis 
T 
3.7 - 3.9 
T = predicted period, T = experimental period P e 
Table I 
-7- 
A = predic ted  amplitude, A = experimental  amplitude P e 
Table I1 
In Fig.  5, (15) is  solved g raph ica l ly  f o r  fou r  examples. The curves 
labeled I ,  11, 111, and I V  are those of H ( j w )  f o r  examples 1, 2 ,  3, and 4 
r e spec t ive ly .  Examples 1 and 2 a r e  v i r t u a l l y  t h e  same examples used i n  
[l] with t h e  exception t h a t  K and r have been sca l ed  t o  s u i t  a TR-10 analog 
computer. 
Ik levant  da t a  f o r  t hese  examples i s  as fol lows:  
K 3-s 
(2s+ l )  (s+l) P(s) = __ II(s) = Example 1: s ( s + l )  ’ 
K = 2,  r = 1, c = 0.5 ( K  = l o ,  r = 5 ,  c = 0.5 i n  [ l ] )  
Example 2: 
Example:3: 
Example 4: 
Same as Example 1 except K = 4 
7- s 
( K  = 20 i n  [ l ] ) .  H(s) = 
( 2 s + l ) ( s  + I )  
2 
K(5s +1) -s + 9s + 1 
P ( s )  = -p; H(S) = 
2 
s(s + s + 1) (s + 1)  (s + s + 1)  
K = 2 , r - c - l .  
‘ 2  + 14 s + 1 K(5s+l) -3s 
s ( s + l )  (s + 1) 3 
P ( s )  = ____T_ ; H(s) = (1/3) 
, 
K = 4/3, r = c = 1 
As seen from Tables I and 11, experimental  per iods  f a l l  wi th in  t h e  
-8 - 
range of  p red ic t ed  per iods  f o r  a l l  examples except 4. Predic ted  and 
experimental  amplitudes were i n  b e t t e r  o r  worse agreement as q ( t )  was 
more or less s inuso ida l .  
are b e t t e r  than  those  obtained using t h e  quas i  descr ib ing  func t ion .  
comparison of p red ic t ed  amplitude is  made with r e s u l t s  i n  [ l ]  s ince  
d i f f e r e n t  s i g n a l s  are involved. No explanat ion i s  of fe red  f o r  d i s -  
crepancies  i n  experimental  per iods  presented  here  and i n  [l] s ince  none 
i s  known t o  t h e  authors .  
Examples 1 and 2 show t h a t  pred ic ted  per iods  
No 
V Conclusions 
An equiva len t  c i r c u i t  has  been developed f o r  CPFN and NPFM which 
allows convenient app l i ca t ion  of conventional descr ib ing  func t ion  techniques 
t o  a class of c losed  loop systems containing these  types  of modulators. 
Predicted r e s u l t s  f o r  l i m i t  cyc l e  c h a r a c t e r i s t i c s  are i n  q u i t e  good agree- 
ment with experimental  f o r  t h e  examples presented .  
f i l t e r i n g  i s  not  provided by H C s ) ,  then p red ic t ed  and experimental  r e s u l t s  
If adequate low pass 
are not  always i n  such good agreement. 
with the  quas i  descr ib ing  func t ion  method shows t h i s  approach b e t t e r  f o r  
t h e  two examples considered. 
A comparison of t h i s  technique 
- 9 -  
REFERENCES 
1. Pav l id i s ,  T. and E. I .  Ju ry ,  "Analysis of a New Class of Pulse 
Frequency Modulated Feedback Systems", I .E . F  . E .  Trans. on *Auto- 
matic Control ,  V . ,  AC-10, Jan .  1965, pp. 35-43. 
Pavl id is ,  T . ,  Pulse Frequency Nodulated Feedback Systc~s, - - ~--- - Ph.D. 
Di s se r t a t ion ,  Universi ty  of CalifGrXia, Berkeley, June 1964. 
2. 
3.  L i ,  C. C . ,  I n t eg ra l  Pulse  Frequency Modulated.Feedback - Systems, 
Ph. D .  D i s se r t a t ion ,  Northwestern Universi ty ,  June 1961. 
4 .  Meyer, A.  U., Pulse Frequency Flodulation and i t s  Effects i n  
Feedback Control Systems, Ph.D. Di s se r t a t ion ,  Northwestern 
Universi ty ,  Aug. 1961. 
5. L i .  C .  C .  and R.  1V. Jones,  " In tegra l  Pulse Frequency Modulated 
Feedback Systems", 2nd I .F .A .C .  Congress, Basilc, Switzerland, 
1963, pp. 217/1 - 417/8. 
6. F i tzgera ld ,  J .  F . ,  Frequency Domain Techniques fo r  Pulse  Frequency 
Modulated Feedback Systems, 11. S .  Thesis ,  Universi ty  of Massachusetts, 
Ju ly ,  1967. 
Fig. l(a) 
3 
2 
1 
-3 -2 -1 
_I 
3 1  2 3  
-1 
-2 
-3 
P i g .  l ( b )  
Fig. 3(a) 
Fig .  3(b) 
b 
F i g .  4 
3 0  
.. 
- 
. .  - 
. .  . .- 
Fig. 5 
ANGLE 
LIST OF FIGURES -- 
Fig. l ( a )  Equivalent C i r c u i t  fo r  IPFM. 
Fig. l ( b )  The Nonlinear i ty ,  N. 
Fig.  2(a) Equivalent C i r c u i t  f o r  CPFM. 
Fig.  2(b) Equivalent C i r c u i t  fo r  NPFM. 
Fig.  3 (a) Feedback System with MPFM Equivalent.  
Fig. 3(b) Reduced Form of Fig. 3 ( a ) .  
F i g .  4 The Input Signal  Parameter (A,b) Plane. 
Fig.  5 Magnitude vs .  Phase f o r  -1/N and Linear Part. 


A describing function analysis in closed loop systems with pulse frequency modulators

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