Equations for investigating control signal by continuous and discontinuous sign functions based on synthesis technique using Liapunov direct metho

Monopoli, R. V.

English

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(ACCESSION NUMBER1 0. 
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E LPAOESI 
BY: 
iZesearch A s s i s t a n t  
David ---%I- P. Lindo 
A ppm ved by : 
P r i n c i p a l  Inves tigstor 
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3ynthcsis techniques be8ed on Liapunov's direct nethod of t en  recpire m-1 
signals which are si@ h c t l o n s .  ' - 5  The arguamt of the sign b c t i o n  i n  each 
c m e  i s  a l i n e a r  conbination of system atates. 
that the des ired sa lut im m y  not &et for  equations which have sign Pur,7m*,ic,n8 
as forcing h c t i o n s ,  authors c w t o ~ r i l y  replace the d i a c c m t h u s  8I.m functio!~ 
by a continuow !'Unction, ueuully the saturation function defined as 
Because Flugt-Lotz R 6 has shown 
I for a'>t/k 
k dfor  -?/k 5 $5 l/k sat k 8 -  
-1  rm Yc-i/k 
n 
- 
i
where k > o  i s  a constant. 
This avoids the theoretical d i f f icul ty  discussed by Fluge-htz, but, b 
e 
SUXJ cases, leads to anothcx. concedng  the validity of conclusions reached re!- 
garding s t a b i l i t y  i n  the region where the saturation function ha8 a snaller 
magnitude than the theoretically specified sign function. This problen is in- 
vestigated here in relation to the synthesis technique enp1o-g a nodel 
reference 2 for control of single inpt, @ingle output, linear tin+-- plants 
I 
With WePe-8 m y b g  h 8n DMner within horn, fbdh baumfs, 
The clam of plants considered can b e  deucribed by the equation 
ud kT - [O, 0, 0, ... bnf . - 
I: 
i 
-2- 
hr0#1. The denants of the notrix a R ( t )  are the variations of p h t  
parmeters itbm their desired values which are the elanemtS cf the A ahtrix. 
:; 
1' 
1 
r .  
! ' 
i 
i 
I '  
t 
I 
1. 
A(t) 1 0 
refesnce which is 
An error for the mystan is defined by 
T A P + P A - - Q  
a 
-3- 
According ta two theorem give3 i n  3eference 8, the  following s ta tenent  
can be made concerning ( 5 ) .  If n 
then ( 5 )  i s  asynp to t i ca l ly  a table ,  i.e,. e+O as t+w. 
d i f f i c u l t y  posed by Fluge-htz  to contend with, so ( 9 )  nust be replaoed by 
But there  i s  the  
It 
n 
Now it no longer can be argued t h a t  ( 5 )  is asynpto t ica l ly  stable in the region 
where 
since u as given by (10) nay not have s u f f i c i e n t  nagnitude t o  guarantee that 
V <  0 i n  t h i s  region. 
fallowing discussion considers this question. 
that u as given by (10) is being ccnsidered, not that of (9).  
3 
What, then, can be said regarding s t a b i l l t p ?  The 
It i s  t o  be understood throughoub 
?he tern -2 P b, u i n  (7) i s  always negative because of (10). Therefore, 
neglecting this tern,  V s a t i s f i e s  the i nequa l i ty  
n n 
i-1 i l l  
The variables xdi and ei a r e  introduced into the  square bracket  of (12) by 
adding and subt rac t ing  This 
renoves t h e  components of 
A( t )  x+ within t h e  square brackets  of (7). 
fron t h e  problen. The reason for doing this i s  
explained below, 
-. r : - i s  bounded, then 30 are the cmponents of I& s ince  a stable 
. 
3 t *'a u wii, cause 5 to be w i t k i r .  t h e  
< l/k . Fqr r. -= 2, t h e  si tuatior.  is as 
-1  
t 
2 e 
.- . .. 
1 
I 
! 
i 
# I  
i 
I 
i 
i 
Figure 1. Cross Habched Area Indicating 
----+-e, 
= l/k 
0 
a 
Wjhere V Xay Be Positive 
1 .  R. W. a s s ,  d i scu : s i ? r  cf a ;xi>er by A .  M. Letov, m. Heidelberq 
C o n f .  311 Autorlatic Cfmtrcl ( ItBcgelungstechnik: Moderne Theorien und 
i b s e  Verwendbarkeit" hy R. Cldenbourg, ?+iunich, 1357) pp. 209-210. 
L. P. Srayscn, "Design v i a  Ly~;~unov's Second Xethod," Proc. Fourth 
J o i n t  Cmference on Autonatic Control, pp. 589-595; 1963. 
G. W. Johnson, "Synthesis of Ccntrol Systsms with Stability Constraints ;  
via t h e  Dl rqc t  Eethod of L y a p ~ o v , ~ ~  IEEE Trans. on Automatic Control, 
2. 
3 .  
'101. /IC-9, pp. 27G-273; July,  1964. 
4. R .  Y. Monovli ,  "??;e Direct Kcthod of Liapunov Applied to t h e  Dc3igr-1 
of C m t r o l l e r s  f o r  a Class of I c:;linear and Time-Varying Processes," 
resewch reportcd under RASA Research Grant No. XSE308-63; August, 1963. 
R. V. ;h.foncpoli, I1Controller Dcsign for  h'onlinear and Time Varying Plants , "  
& A X  Contrzctor Report Kisk CR-152; January, 1965. 
I. Fluape-btz ,  "Xscontinuous A u b n a t i c  Control" (bpok) Princeton 
University Press, Pr i rxe tc r  , Kew Jer9ey; 1953. 
7. W. Hahr, "Theory and Application of Liapunov's Direct  Method," 
5. 
!I 
6 .  
(book) 
Prentics-kd1, Tnc., JbglewQod Cliffs, I'ew Jersey; 1963. 
L. P. Graysc:', tlTwo Thoorms on t h e  Seccnc! Nethod," (cor res~ondence)  
- IEE3E Trane. 
8. 
Autmatic Cor,trol, Vel. kC-9, p.587; October, 1964. 
I 


Comments on synthesis techniques employing the direct method

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