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A compensation technique for certain nonlinear autonomous systems
Brief Communication A Compensation Technique for Certain Nonlinear Autonomous Systems b_jJ CHARLES
J.
MONIER
Nicholls State College Thibodaux, Louisiana
1. Introduction
In this paper a method of compensation* is introduced to improve the stability of certain nonlinear systems by the addition of zero networks. Improving system stability implies increasing the region of operation for which stability can be predicted. In linear systems compensation techniques are extremely well developed and many types of system performance can be altered by synthesizing the appropriate network. In nonlinear systems, compensation techniques have not been developed to any large extent.
ZZ. Systems
Considered
Consider the autonomous system in Fig. 1 where D E d/dt, b,‘s are constants, and f represents a nonlinearity such that f is a continuous function
I
Dm+bm_,Dm-‘+...e..... .....+b,
f(Y)
f
y(t)
Nonlinearity FIG.
1. mth Order uncompensated
system.
of y. The differential equation describing this system is &n-l
dtm-lY++b,$y+boy+fh.d =0. ~y+b,-1 Liapunov’s Second Method can be employed system described by Eq. 1. [See (l).]
to predict
stability
for the
* The process of synthesizing a network that must be added to a system in order to improve the system performance.
319
Brief Communication Now consider the autonomous describing this system is
system in Fig. 2. The differential equation
&n-l ~Y+Lldtnr_1V+
. . . +b,&+b,y
+&&)+a
nz- 1$&l/)+
..* + a1 $m
+ G&f)
=
0.
It is considerably more difficult to determine the stability of the system described by Eq. 2 since terms containing the time derivative of the nonlinearity are present. Dm+om_, Dm-’
+.. . . . . . . . .._.._ +
0,
>
Dm+bm_, Dm-‘+ .......... ......._..+
v(t)
b,
I/
f(Y)
f
FIa.
2. mth Order compensated
y(t)
Nonlineorify
system.
In this paper, the system given by Eq. 1 is compensated and the resulting system is described by an equation of the form of Eq. 2. By determining the region of stability for the system described by Eq. 1 it is possible to predict the region of stability for the system described by Eq. 2 under certain conditions, Further, it is possible to improve the stability of system 1 by choosing the appropriate compensation network. Theorem If m = 1, Eq. 2 reduces to
$Y+boY+~f(Y)+a,f(y) =0. It is obvious that if one lets y = w-f(y), Eq. 1. That is, $w+b,w+g(w)
then Eq. 3 reduces to the form of
= 0,
where g(w) = (a,, - b,)f(y). This leads to the hypothesis that the transformation y = w-f(y) reduces an mth order system of the form of Eq. 2 to the form of Eq. 1. This hypothesis is proved in Theorem 1 with certain restrictions which greatly reduce the practical use of the theorem but lead to interesting theoretical conclusions. Consider Eq. 2 where ai = bi for i = 1,2,3, . . , , m - 1.
320
Journal of The Franklin Institute
Brief Theorem equivalent
1. If the transformation equation in w is
w = y-f(y)
Communication
is introduced,
then an
dm-1
. ..+b.$w+b,w+g(w)
$&+b,-lslw+
where g(w) = (a, - bJf(y). Proof. The proof is by mathematical
induction.
$dw+&f@)+~,f(y) Let w = y-f(y).
= 0,
(5)
For m = 1,
=a
(6)
Then $w+b,w+g(w)
= 0,
(7)
where g(w) = (a,,- b,)f(y). Therefore, the theorem is true for m = 1. Assume that the theorem is true for m = k. Then the differential equation dk
dk +b,Y+@f(Y)+ak-1-f(Y)+...
dk-1
k ldtk_1y+.‘.
@Y+b-
dk-1
d fQf(Y) has an equivalent
representation
d”w+b dtk
dk-lw+ k _ l&k-l
+%f(Y)
= 0
(8)
in w as follows: . ..+b.$w+b,w+g(w)
= 0,
(9)
where g(w) = (ao-b&f(y). Now consider m = k + 1. This results in dk+l ajY+bk$Y+e..+b,Y+
Sf
(Y)
+ak$f(y)+~~*+aI$f(y)+a,f(y)
= OS
By hypothesis a, = b, for i = 1,2,3, . . ., k. Therefore, ok = b,, ak_l = bk+ and a, = b,. It follows that dk+l cf(y)+bk[$ dtk+ly+dtk+l
y+...+$y+$f(y)+...+zf(y)]
(10) . . .,
=o.
(11)
= 0,
(12)
Using Eqs. 8 and 9, it follows that dk+l
dk+l
-f(y)+bk
dtk+lY+&k+l
where
$w+bk_l~w+...+;w+h(w)] [
k
bo h(w) =z-c 1 [
Vol.288.No. 4, October1969
f(y).
321
Brief Colnmunication Since y t-f (y) = w, dkfl
d”
dk-1
+b,w+g(w)
---+fk~W+bk-l~+... dtkfl
= o,
(13)
where g(w) = b,h(w) = (a,- b,)f (y). The theorem is therefore true for m = k + 1 and this completes tion process and the theorem is proved.
III.
Total
Gain
the induc-
of a Nonlinearity
Let the total gain of a function f (y) be the real function gt(f, y) defined by gdf, y) = f(y) Y *
(14)
This is illustrated in Fig. 3. The region of stability for the systems discussed is given in terms of the bounds on the total gain of the nonlinearity.
“S
=-
Y
f(Y, 1 yI
FIG. 3. Nonlinearity.
Results Consider the system in Fig. 1 which is described as follows: @n-l
;mY+b,-~d~lY+..~+b,Y+f(Y)
= 0.
(15)
Assume that the region of stability for this system is N<‘fo
(16)
*
That is, the system is stable as long as the total gain remains within the bounds given by Eq. 16. Now suppose that a zero network is added to the
322
Journalof The Franklin
Institute
Brief Communication system, as shown in Fig. 4. By Theorem 1, the system in Fig. 4 can be represented by the following equivalent equation in w: dm-1
+b,w+g(w)
~w+bm-~F~w+... where g(w) = (a,-
b,)f(y).
= 0,
(17)
S ince Eq. 17 is in the same form as Eq. 16, the 1
I
om+b,,,_,Dm-I+...+b 0
Dm+b,_,Dm-'~..-+a,
-
=
y(t)
FIG. 4. rnth Order compens&ed system. region of stability for Eq. 17 is given by the following: N< g(w)
(18)
and g(w) = (a,, - b,)f(y). N < (ao -b,)f(Y) Y+f(Y)
Upon dividing the numerator
< j,,f
(19)
-
and denominator
This results in
of
(a,-b,)f(Y) Y +.f(y) by y, the result is N < (a,-b,) [f(Y)/Yl 1+ [f(Y)/Yl
< J,J *
(20)
It is further recalled that the f (y)/y is the total gain of the nonlinearity g,(f, y). Equation 20 then becomes N < (% - ‘0) gt(f, Y) < M * 1 +st(f,y) Therefore,
(21)
if the system in Fig. 1 is stable for (22)
N < s,(.L Y) -=zM then the system in Fig. 4 will be stable for N<
Vol. 288, No. 4, October
1969
(a,-b,)g,(f,Y) 1+ %(f, Y)
(23)
323
Brief Co~nmunicution Example Consider the system given in Fig. 5. The estimated region of stability based on a particular choice of a Liapunov function for this system is 0< where
f (y) is
a continuous
SAfP Y)< 4.4,
function
(24)
such that gt(f, y) > 0.
I D3 + 2.5D2 + 2D
+
y(t)
>
-5
f
Nonlinearity FIG. 8. Third-order uncompenseted
I D3 +
,
2*5D*+ 20
system.
f.?+ 2*5D2+ 2 D + a,,
.
‘r
y(t)
+*5 \I
/\
f Nonlinrarlty
FIG. 6. Third-order
Now consider the compensated system in Fig. 6 is stable for o
<
compensated
system given in Fig. 6. From Eq. 23, the
(a, - O-5)gdf, y) < 4.4 1 +gAf,y)
This results in
system.
(25)
*
4.4 g;< a,-4.9’ ___
(26)
g,ao.
(27)
Equation 27 results from the restriction on the nonlinearity. From Eqs. 26 and 27 it is seen that a, must obey the following relationship, otherwise Eqs. 26 and 27 have no significance, a, > 4.9.
324
(28)
Journal
of The Franklin
Institute
Brief Communication From Eq. 27, it is seen that the upper bound on gr is increased if a, is selected sufficiently close to the quantity 4.9. For example, let a, = 5.5. Then the system in Fig. 6 is stable for 0 < gt(f, y) < 7.1. It is also worth noting that if the quantity a, is continually increased from 5.9, then the stability region for the system in Fig. 6 is continually decreased.
IV.
Conclusions
A method is presented to improve the stability of a certain class of nonlinear autonomous systems. The method involves the synthesis of an appropriate zero network which when added to the system increases the region of operation for which stability can be predicted. It is well known that zero networks cannot be realized exactly. However, zero networks can be very closely approximated and it is feasible to consider their use in the approximate form (2). The use of zero networks to compensate linear systems is well documented (3).
References (1) E. J. Lefferts, “A Guide of the Applications of the Liapunov Direct Method to Flight Control Systems”, N.A.S.A.CR-209,Wash., D.C., 1965. (2) L. J. Fairchild, “Ten Circuits for Differentiation on Analog Computers”, Control Engineering, pp. 38-40, Feb. 1965. (3) J. J. D’Azzo and C. H. Houpis, “Feedback Control System Analysis and Synthesis”, pp. 275-276, McGraw-Hill Co., New York, 1960. (4) C. J. Monier, “Stability Investigation of Certain Nonlinear Autonomous Systems”, Ph.D. Dissertation, Mississippi State University, June 1966.
Vol. 288, No. 4, October
1969
325
J.
MONIER
Nicholls State College Thibodaux, Louisiana
1. Introduction
In this paper a method of compensation* is introduced to improve the stability of certain nonlinear systems by the addition of zero networks. Improving system stability implies increasing the region of operation for which stability can be predicted. In linear systems compensation techniques are extremely well developed and many types of system performance can be altered by synthesizing the appropriate network. In nonlinear systems, compensation techniques have not been developed to any large extent.
ZZ. Systems
Considered
Consider the autonomous system in Fig. 1 where D E d/dt, b,‘s are constants, and f represents a nonlinearity such that f is a continuous function
I
Dm+bm_,Dm-‘+...e..... .....+b,
f(Y)
f
y(t)
Nonlinearity FIG.
1. mth Order uncompensated
system.
of y. The differential equation describing this system is &n-l
dtm-lY++b,$y+boy+fh.d =0. ~y+b,-1 Liapunov’s Second Method can be employed system described by Eq. 1. [See (l).]
to predict
stability
for the
* The process of synthesizing a network that must be added to a system in order to improve the system performance.
319
Brief Communication Now consider the autonomous describing this system is
system in Fig. 2. The differential equation
&n-l ~Y+Lldtnr_1V+
. . . +b,&+b,y
+&&)+a
nz- 1$&l/)+
..* + a1 $m
+ G&f)
=
0.
It is considerably more difficult to determine the stability of the system described by Eq. 2 since terms containing the time derivative of the nonlinearity are present. Dm+om_, Dm-’
+.. . . . . . . . .._.._ +
0,
>
Dm+bm_, Dm-‘+ .......... ......._..+
v(t)
b,
I/
f(Y)
f
FIa.
2. mth Order compensated
y(t)
Nonlineorify
system.
In this paper, the system given by Eq. 1 is compensated and the resulting system is described by an equation of the form of Eq. 2. By determining the region of stability for the system described by Eq. 1 it is possible to predict the region of stability for the system described by Eq. 2 under certain conditions, Further, it is possible to improve the stability of system 1 by choosing the appropriate compensation network. Theorem If m = 1, Eq. 2 reduces to
$Y+boY+~f(Y)+a,f(y) =0. It is obvious that if one lets y = w-f(y), Eq. 1. That is, $w+b,w+g(w)
then Eq. 3 reduces to the form of
= 0,
where g(w) = (a,, - b,)f(y). This leads to the hypothesis that the transformation y = w-f(y) reduces an mth order system of the form of Eq. 2 to the form of Eq. 1. This hypothesis is proved in Theorem 1 with certain restrictions which greatly reduce the practical use of the theorem but lead to interesting theoretical conclusions. Consider Eq. 2 where ai = bi for i = 1,2,3, . . , , m - 1.
320
Journal of The Franklin Institute
Brief Theorem equivalent
1. If the transformation equation in w is
w = y-f(y)
Communication
is introduced,
then an
dm-1
. ..+b.$w+b,w+g(w)
$&+b,-lslw+
where g(w) = (a, - bJf(y). Proof. The proof is by mathematical
induction.
$dw+&f@)+~,f(y) Let w = y-f(y).
= 0,
(5)
For m = 1,
=a
(6)
Then $w+b,w+g(w)
= 0,
(7)
where g(w) = (a,,- b,)f(y). Therefore, the theorem is true for m = 1. Assume that the theorem is true for m = k. Then the differential equation dk
dk +b,Y+@f(Y)+ak-1-f(Y)+...
dk-1
k ldtk_1y+.‘.
@Y+b-
dk-1
d fQf(Y) has an equivalent
representation
d”w+b dtk
dk-lw+ k _ l&k-l
+%f(Y)
= 0
(8)
in w as follows: . ..+b.$w+b,w+g(w)
= 0,
(9)
where g(w) = (ao-b&f(y). Now consider m = k + 1. This results in dk+l ajY+bk$Y+e..+b,Y+
Sf
(Y)
+ak$f(y)+~~*+aI$f(y)+a,f(y)
= OS
By hypothesis a, = b, for i = 1,2,3, . . ., k. Therefore, ok = b,, ak_l = bk+ and a, = b,. It follows that dk+l cf(y)+bk[$ dtk+ly+dtk+l
y+...+$y+$f(y)+...+zf(y)]
(10) . . .,
=o.
(11)
= 0,
(12)
Using Eqs. 8 and 9, it follows that dk+l
dk+l
-f(y)+bk
dtk+lY+&k+l
where
$w+bk_l~w+...+;w+h(w)] [
k
bo h(w) =z-c 1 [
Vol.288.No. 4, October1969
f(y).
321
Brief Colnmunication Since y t-f (y) = w, dkfl
d”
dk-1
+b,w+g(w)
---+fk~W+bk-l~+... dtkfl
= o,
(13)
where g(w) = b,h(w) = (a,- b,)f (y). The theorem is therefore true for m = k + 1 and this completes tion process and the theorem is proved.
III.
Total
Gain
the induc-
of a Nonlinearity
Let the total gain of a function f (y) be the real function gt(f, y) defined by gdf, y) = f(y) Y *
(14)
This is illustrated in Fig. 3. The region of stability for the systems discussed is given in terms of the bounds on the total gain of the nonlinearity.
“S
=-
Y
f(Y, 1 yI
FIG. 3. Nonlinearity.
Results Consider the system in Fig. 1 which is described as follows: @n-l
;mY+b,-~d~lY+..~+b,Y+f(Y)
= 0.
(15)
Assume that the region of stability for this system is N<‘fo
(16)
*
That is, the system is stable as long as the total gain remains within the bounds given by Eq. 16. Now suppose that a zero network is added to the
322
Journalof The Franklin
Institute
Brief Communication system, as shown in Fig. 4. By Theorem 1, the system in Fig. 4 can be represented by the following equivalent equation in w: dm-1
+b,w+g(w)
~w+bm-~F~w+... where g(w) = (a,-
b,)f(y).
= 0,
(17)
S ince Eq. 17 is in the same form as Eq. 16, the 1
I
om+b,,,_,Dm-I+...+b 0
Dm+b,_,Dm-'~..-+a,
-
=
y(t)
FIG. 4. rnth Order compens&ed system. region of stability for Eq. 17 is given by the following: N< g(w)
(18)
and g(w) = (a,, - b,)f(y). N < (ao -b,)f(Y) Y+f(Y)
Upon dividing the numerator
< j,,f
(19)
-
and denominator
This results in
of
(a,-b,)f(Y) Y +.f(y) by y, the result is N < (a,-b,) [f(Y)/Yl 1+ [f(Y)/Yl
< J,J *
(20)
It is further recalled that the f (y)/y is the total gain of the nonlinearity g,(f, y). Equation 20 then becomes N < (% - ‘0) gt(f, Y) < M * 1 +st(f,y) Therefore,
(21)
if the system in Fig. 1 is stable for (22)
N < s,(.L Y) -=zM then the system in Fig. 4 will be stable for N<
Vol. 288, No. 4, October
1969
(a,-b,)g,(f,Y) 1+ %(f, Y)
(23)
323
Brief Co~nmunicution Example Consider the system given in Fig. 5. The estimated region of stability based on a particular choice of a Liapunov function for this system is 0< where
f (y) is
a continuous
SAfP Y)< 4.4,
function
(24)
such that gt(f, y) > 0.
I D3 + 2.5D2 + 2D
+
y(t)
>
-5
f
Nonlinearity FIG. 8. Third-order uncompenseted
I D3 +
,
2*5D*+ 20
system.
f.?+ 2*5D2+ 2 D + a,,
.
‘r
y(t)
+*5 \I
/\
f Nonlinrarlty
FIG. 6. Third-order
Now consider the compensated system in Fig. 6 is stable for o
<
compensated
system given in Fig. 6. From Eq. 23, the
(a, - O-5)gdf, y) < 4.4 1 +gAf,y)
This results in
system.
(25)
*
4.4 g;< a,-4.9’ ___
(26)
g,ao.
(27)
Equation 27 results from the restriction on the nonlinearity. From Eqs. 26 and 27 it is seen that a, must obey the following relationship, otherwise Eqs. 26 and 27 have no significance, a, > 4.9.
324
(28)
Journal
of The Franklin
Institute
Brief Communication From Eq. 27, it is seen that the upper bound on gr is increased if a, is selected sufficiently close to the quantity 4.9. For example, let a, = 5.5. Then the system in Fig. 6 is stable for 0 < gt(f, y) < 7.1. It is also worth noting that if the quantity a, is continually increased from 5.9, then the stability region for the system in Fig. 6 is continually decreased.
IV.
Conclusions
A method is presented to improve the stability of a certain class of nonlinear autonomous systems. The method involves the synthesis of an appropriate zero network which when added to the system increases the region of operation for which stability can be predicted. It is well known that zero networks cannot be realized exactly. However, zero networks can be very closely approximated and it is feasible to consider their use in the approximate form (2). The use of zero networks to compensate linear systems is well documented (3).
References (1) E. J. Lefferts, “A Guide of the Applications of the Liapunov Direct Method to Flight Control Systems”, N.A.S.A.CR-209,Wash., D.C., 1965. (2) L. J. Fairchild, “Ten Circuits for Differentiation on Analog Computers”, Control Engineering, pp. 38-40, Feb. 1965. (3) J. J. D’Azzo and C. H. Houpis, “Feedback Control System Analysis and Synthesis”, pp. 275-276, McGraw-Hill Co., New York, 1960. (4) C. J. Monier, “Stability Investigation of Certain Nonlinear Autonomous Systems”, Ph.D. Dissertation, Mississippi State University, June 1966.
Vol. 288, No. 4, October
1969
325