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An alternative to Liapunov's stability method and its application to higher-order systems
COMPUTER METHODS NORTH-HOLLAND
IN APPLIED
MECHANICS
AND ENGINEERING
47 (1984) 299-314
AN ALTERNATIVE TO LIAPUNOV’S STABILITY METHOD AND ITS APPLICATION TO HIGHER-ORDER SYSTEMS H.H.E.
LEIPHOLZ*
Departments of Civil and Mechanical Engineering, Solid Mechanics Division, University of Waterloo, Waterloo, Ontario N2L 361, Canada Received
30 January
1984
This paper is an extension of [l] and should be read in conjunction with that reference. It proves that an alternative to Liapunov’s stability method, which removes the obstacle of having to establish a Liapunov function, can be generalized to cover also those cases in which higher-order differential equation systems are involved.
1. Introduction In a preceding paper [l], a proposal for an alternative to Liapunov’s stability method was made. However, that proposal was restricted in its argumentation and application to secondorder systems of differential equations. In this paper, it will be shown how the new method can be extended for the handling of stability problems posed by higher-order systems of differential equations. Examples, supporting the new technique, will also be presented.
2. Mathematical foundations and stability conditions As already pointed out in [l], the system of differential equations ii =
fi(Xl,
ii=dxi/dt,
X27 . . .
7
X”)
i=l,2
7
Xi(t)
Xi =
,...,
7
(1)
It,
will be assumed to be autonomous, i.e., the functions fi are supposed not to involve time t explicitly. Also, like in [l], the system (1) will be viewed as determining a flow, the i representing a vector field in an n-dimensional space. Again, as described in [l], attention will be paid only to the stability of the origin 0 = (X = 0) of the n-dimensional space, yet without restricting generality. If 0 is to be (asymptotically) stable, one has to require that, for increasing t, the flow tends towards 0. This means, 0 has to be a sink of the flow. *This research 00457825/84/$3.00
was supported
through
@ 1984, Elsevier
NSERC
Science
Grant
Publishers
No. A 7297. B.V. (North-Holland)
300
H.H.E.
Let m be the magnitude
Leipholz,
An alternative
to Liapunov
of the sink. It can be calculated, using the hypersphere
S = C XT= r* = const.,
(2)
with centre in 0. If 0 is the surface of S, dm z=
x-n
(3)
where n is the (outer) normal to 0’. Yet, n = grad S/lgrad Sl
(4)
and grad S = 2Xi,
x: = 2r
Igrad SI = 2Jc I
(5)
Hence.
2
dm -= d0
xi -&Xi.-
-
2r
ci
iiXi/r
Finally, one has for the magnitude C
m=
PROOF.
(6)
of the sink the relationship
i %Xi
,d6.
r
f0
PROPOSITION
.
2.1.
If
(7)
m < 0 for r + O’, the origin 0 is asymptotically stable.
Since S = E i 2iiXi, m = B (,!?/2r) d6. I
But, then, for sufficiently small values of r, (S),0’ - 2mr
and
(S), - 2mr/B.
If for r+ O’, m < 0 holds, the time derivative of the positive definite function S is negative around the origin 0. This fact is sufficient for asymptotic stability of 0 according to Liapunov’s second theorem. COROLLARY 2.2. If m, as calculated by means of (7) should tend towards zero for r -0, could still claim stability of the origin 0.
one
H.H.E.
Leipholz, An alternative to Liapunov
301
This claim can be justified by arguing as before and invoking Liapunov’s first theorem. Let the case of II = 3 be taken under specific consideration. In this case, S is an actual sphere of radius r in the three-dimensional Euclidean space with centre in the origin of the space. Using spherical coordinates, one has x1 = r sin 8 cos C#J , x2 = r x3=
(8)
sin 8 sin 4,
rcos
8,
and d6’ = r’sin 8 de d4.
(9)
With (8) and (9) (7) becomes m =~02-[\~(,*,i)rsinBdB]d~.
(10)
With (1) and (8) (10) can be rewritten to yield 2lT
m=
m
I iI 0
1r
0 If(
sin e cos 4, r sin 8 sin 4, r cos 8)r2sin2tl cos fp
+ f2(r sin e cos 4, r sin e sin 4, r cos 8)r2sin28 sin 4 + f3(r sin e cos 4, r sin 8 sin 4, r cos 8)r’cos 8 sin e] de
d4 .
(11)
As a first stability condition, m
forr+O+,
02)
may be established on the basis of the preceding deliberations. In (12) m is to be calculated using (7). In the case of n = 3, one may use (11) instead of (7). At this point, further stability conditions shall be derived. This is done by interpreting the vector
as a force acting on the image point which traces the trajectory. This trajectory is obtained from the system (1) through integration. Hence, in (13), the ‘components’ fi, i = 1,2, . . . , n, are the functions on the right-hand side of (1). In the case of asymptotic stability, ‘force’ F should ‘drive’ the image point towards the origin. This will indeed happen, if F were a ‘central force’, as shown in Fig. 1.
H.H.E. Leipholz, An alternative to Liapunov
302
orbit
Fig. 1. Vector
Let r = (x: 0 to the point by the nature The vector
F = (fi) interpreted
as a central
force in the case of asymptotic
stability
+ x$ + . . . + x2,)“* = +(x1, x2, . . . , x,) be the radius vector leading from the origin P of the trajectory. In this relationship, &(x1, x2, . . . , x,) is a function prescribed of the trajectory. F is a ‘central force’ if it satisfies the condition (14)
under the assumption
that Re A < 0 holds for the constant A. Hence, the requirement
. . 7X”)-hXi=O,
J(x,,xz,.
that
Reh CO, i= 1,2,. . .,lt,
WI
holds, which follows from (14) yields additional stability conditions, which are expressed terms of system parameters involved in the expressions ,fi(xl, x2, . . . , x,). In a number of cases, fi(Xl,
x2,
* * . ,
X,)= C
UijXj + @i(Xl,X2,.
. .3
xfl)
where the Uij are constants and the @i are nonlinear functions of their arguments. becomes C
(Uij
-
Gijh)Xj
+ @i(Xl,
~2,
. . .,Xn) =
0,
Re A < 0, i = 1,2, . . . , n,
in
(16)
Then, (15)
(17)
where Sij is Kronecker’s symbol. Moreover, assume that the @i are sufficiently smooth functions which vanish at the origin. Then, @i
=
2
(@i,j)x=dj
9
@i, j = d@JdXj
,
(18)
is a first approximation of the @i in the vicinity of 0. Therefore, in order to assess stability in the neighbourhood of 0, one can use (18) in (17) in order to obtain C {[aij + (@i,j)x=o]- &jh}xj = 0 ,
Re A < 0, i = 1, 2, . . . , II .
(19)
H.H.E. Leipholz, An alternative to Liapunov
This is a system of homogeneous, solution, one must demand that
linear, algebraic equations.
303
For it to have a nontrivial
det{[aij+ (@i,j)x=o] - SijA}= 0 ,
(20)
which can be simply written as
(21)
det{& - Sijh} = 0 with izij
=
Uij
(22)
(GDi,j)x=O
+
From (20) one can derive the algebraic equation A” +
al/i”-’ + * * * + U”_lh + a, = 0
(23)
)
where coefficients ai, i = 1,2, . . . , n, are the invariants of the matrix
A=
(6ij)
(24)
.
These invariants are connected
with the it roots of (23) by the relationship
ul=Al+A2+*~~+A,, u2 = AIA2+ AIA3+ **. + A,_lA, ,
(25)
u3= A,A2A3+ AlA2A4+ . * * + A,-2A,-IA, a, = A,A2A;. * *A, .
Moreover, the invariants ai, i = 1,2, . . . , n, can be calculated, using the elements dij of the matrix A, in the following way: Consider the determinant IAl of matrix A. The subdeterminant, obtained from IA\ by cancelling those lines and the rows which include the diagonal elements iiii and tijj, may be called cof cliitijj.For example, let A be a 4 X 4 matrix, that is to say,
624 _ = &if44 - 624642. cof 611633= a22 _ I a42
a44
Let C,,i be the combination
I
without repetition
of the natural numbers
1,2,. . . , n, to the
304
H.H.E.
Leipholz,
An alternative
to Liapunov
class i, where n is the order of the quadratic matrix A. For A given by (26) for example, C,, = 12, 13, 14,23,24,34. Furthermore,
set
cof tin&z + cof 611& + cof Ci,1ti44+ cof
622ii33
+
cof &*Lz44 + cof
ti3364.4
=
c
cof Lid
cd.2
where d,, indicates that one has to operate invariants of (26) are given by al = (-1)’ c cof & = -tr(A)
,
c4.3
u3
with the diagonal
=
of A. Then, the
(-1)3 c cof & ) c4,
az=(-1)2CCOf&,
elements
1
a4=(-1)4xcof&=detA,
c4.2
c4.0
which explicitly reads
u4
a21
622
623
is1
632
ii33
a41
-4 a2
a43
=
fz
=detA.
644
In general terms one has for a matrix of order n the invariants
Ui
=
(-l)i
2 COf C”,n-i
iid.
(27)
Stability conditions are obtained by enforcing for the roots hi of (23) the requirement Re A -=c0. Using (25), one arrives in that way at certain conditions for the invariants ai of matrix A. Then, through (27) one translates these conditions into conditions for the elements iZij of A, which through (22) are related with the property determining quantities of the original problem (16). Thus one has indeed stability conditions for this problem (16). Take a matrix A of order n = 3 as an example. For Re Ai < 0, i = 1,2,3, it follows from (25)
H.H.E.
305
Leipholz, An alternative to Liapunov
that aI > 0, a2 > 0 and a, > 0 must hold. That is identical with tril
det A < 0.
a2>0,
(28)
Again by means of (25) one is also led from the requirement ala2
(29)
> a3,
which is a direct consequence (A,+
Re Ai < 0 to the inequality
h2+
A3)(hIh2+
of the inequality hIA,+
h2h3)=lh2h3
>
which holds true if Re Ai < 0, i = 1,2,3, is satisfied. But, this inequality is indeed (29) in terms of (25). It is not surprising to realize, that with (28) and (29) one has obtained the well-known stability conditions of Hurwitz. Generalizing, one can make the following statement: Conditions (15), (17) respectively, yield stability conditions for a nonlinear system of differential equations (1) in addition to condition in (12). Under special circumstances, one can linearize and replace conditions (15) and (17) by condition (19). This condition leads to Hurwitz’ conditions in terms of quantities (27). Hence, when linearization of (15), (17) respectively, (not necessarily linearization of the actual system (1)) is appropriate, one may add to stability condition (12) the Hurwitz conditions, expressed in terms of the ai, and calculated by means of (27) from the tiij, which are given by (22). Finally, it may be mentioned that if the rank of the ‘reduced’ matrix A = (aij)
WV
is not smaller than IZ,one may use the quantities aij’ instead of the quantities 6ij for the setting up of Hurwitz’ conditions. This possibility is to be welcomed as it simplifies calculations. 3. Application of the theory to examples EXAMPLE 3.1. The first example system of differential equations
can be found in [2, p. 1071. It concerns
the nonlinear
~‘1=-X1-X2+X3+Xlr2=fl,
3i2=x1-2X2+2X3+X2r2=f2, ~3=x1+2x2+x3+x3r2=f3, r2=x:+x;+x;. 'Forthe definition of the aij, see (16).
(31)
306
H.H.E. Leipholz, An alternative to Liapunov
For (31) it is predicted in [2] that the trivial equilibrium position is unstable. It will be shown that the alternative method can be used to confirm this fact. Applying (16) to (31) one realizes that matrix A defined by (30) reads in this case
A=
1
-1 -1 l-2 2 [ 1
1 2’ 1
Since (13) is a system of the third order, conditions (28) and (29) must hold for stability. Using (27) to determine the quantities ai, i = 1,2,3, involved in (28) and (29), one finds a,=-(trA)=3-1=2>0, -1 -2 -1 -1 1 -2 1 2
a,=-detA=-
=-2-4-l-1+2+1=-5.~0,
1 2 =-9<0, 1
These results indicate that conditions (28) and (29) are partly violated. Hence, stability cannot exist as predicted. In view of this fact, there is no need to consider condition (12) in addition. EXAMPLE
3.2. The next example is again from [2, p. 1061. It concerns the nonlinear
system
& = X2- 3X3- X1X;+ 4X&X3 - 4X& = fl , 12 =
-2X1 + 3x3 - X& - 2X&X3 - x2x; = f2 )
(32)
&=2x1-X2-x3=f3, which has been shown in [2] to possess a stable trivial equilibrium confirmed by means of the alternative method. Using (8) and (ll), (32) yields
m=
r +
position. This fact will be
sin e sin 4 - 3r cos e - r3 sin3 e cos 4 sin2 4 4r3sin28 cos e sin 4 cos 4 - 4r3sin e c03e
cos 4]r2sin28 cos C#J
+ [-2r sin e cos f#~ + 3r cos e - r3sin3e cos2~ sin 4 -2r3sin28 cos e sin 4 cos C#J - r3sin e c02e
sin 4]r2sin28 sin 4
+ [2r sin e cos 4 - r sin e sin 4 - r cos 8]r2 cos 8 sin 8
d$J .
(33)
H.H.E.
An easy and elementary
Leipholz, An alternative to Liapunov
307
calculation using (33) yields 28
m
=
-$rr3-is7Fr5<0,
which satisfies condition (12). Next, applying (16) to (32) yields
By means of A and (27) one finds al=
-(-l)=
l>O,
a,=3+6+2=11>0, a3=-[-(2-6)-3(2)]=-[4-6]=2>0, al& -
u3
=
Hence, all conditions
(l)(ll) - 2 = 9 > 0 . (28) and (29) are satisfied. This indeed confirms stability.
EXAMPLE 3.3. The following example is from [2, p. 1391 and deals with the stability of the trivia1 equilibrium position of the differential equation ... x+u~itu;itxf(x)=o.
(34)
It is shown in [2] that stability is warranted
u;>o,
uT>O,
f(Y)'07
if the conditions UTU*2-f(Y)>O
(35)
are being satisfied. In the following, it will be shown that the alternative method yields essentially the same results. First, let (34) be transformed into a system of first-order differential equations which reads Ii, =
&=X3,
x2,
i3=-UTX3-U*2X2_XS(X1).
(36)
Matrix A is found to be 0
A=
1
0 [
0 1
-f(O) -“a; -u’i
if implying that x1 is considered
1 ’
to be taken in the neighbourhood
of the point 0 = (0,0,O).
308
H.H.E.
Conditions
Leipholz,
An alternative
to Liapunov
(28) and (29) are in this case given as
aI=-(trA)=aT>O, a2= a3
aZ>O, =
-(-f(O))
=
f(O)
’
0
,
These conditions are identical with (35) if one restricts oneself with f(xJ to the neighbourhood of the origin. This may be interpreted as a limitation of the domain of attraction of the origin. In order to establish stability definitely, one must show that condition (12) is being satisfied. Using (8) (11) and (36) one finds with f(xr) -f(O) + x$(O) that r sin 8 sin 4]r2sin20 cos 4 + [r cos 8]r2sin28 sin 4
m=L2Y\X
+[-aTrcos8-a*2rsin8sin~-rsin8cos~f(O)
which yields m = -$aTr3. Hence, for a; > 0, which is already supposed to be the case, condition (12) is indeed fulfilled. This warrants stability. EXAMPLE equations
3.4. The next example is given in [3, p. 4711. It involves the system of differential
&
=
-x1
+
i2
=
-9(X3)
i3
=
(y
-
f(X3)
)
(37)
,
1)X, + yxz -
Under the assumption if
kf(x3)
.
that f(0) = 0, Popov shows that the trivial equilibrium
forO
k>O,
for y > 1,
k>y-1.
(38)
The corresponding stability domain is depicted in Fig. 2. Linearizing, one has f(X3)
=
f(O)
+
fl(O)XS
=
position is stable
fl(O)XJ
,
H.H.E. Leipholz, An alternative to Liapunov
Stability
domain
UIIIII
Popov’s
Coordinates Coordinates
for exomple
stability
309
(IV)
domain,
extended stobi Ii ty domain, EI of point P: r = 2.0, k = 0.5, P is in the extended stobili ty domain, of point Q: 7:
2.0,
k= fi - I = 0.414, 0 is on the extended stobllity boundary.
Fig. 2.
which is true due to the assumption reads i1= i*
=
-x1
+
f’(O)x,
-f)(O)&
on f(0). Hence, the linearized system following from (37)
)
)
i3 = (y - l)xr + yxz- kf)(O)X3. matrix A turns out to be
The corresponding -1
0
0 7-l
Conditions
f’(0)
0 Y
-f’(O)
.
-kf’(O) I
(28) and (20) are found to be al=-(trA)=l+kkf’(O)>O, a2 = yf’(0) + kf’(0) - f’(O)(y - l)= (k + l)f’(O) > 0 ) a3 =
-det A = -[- l(f’(O)y)] = f’(O)y > 0,
~1~22-
a3
=
(1 + kf’(O))(k + l)f’(O) - f’(O)y > 0 .
Assume in addition to f(0) = 0 that also f’(0) > 0. Then, one is left with the requirements Y>O,
k+l>O,
l+ kf’(O)>O,
(39)
310
H.H.E. Leipholz, An alternative to Liapunov
In order to simplify the further discussion of these conditions, f63)
=
x:
+
let (40)
x3,
so that f(0) = 0, f’(0) = 1 > 0 as assumed. With f’(0) = 1, (39) changes into k + 1 > y’12.
kt1>0,
Y’O,
(41)
Turning to condition (12), one obtains by means of (18) and (11) 27r
m=
n
I (I( 0
-r [
0
sin 6 cos 4 + f(r cos 8)]r2sin20 cos 4
+ [-f(r
cos 8)]r2sin28 sin C#J + [(y - 1)r sin 8 cm C$
t yr sin e sin 4 - kf(r
cos
e)]r%os
B
sin 8
and -r3sin36
m=
cos24,] -
[kr’f(r
In order to evaluate this integral, approximate f (x3)- x3.Then
cos
t9) cos
8
sin 01
dd
f(x3) in the neighbourhood
m= ~2~(~~((-r3sin3e~~s2~]-(kr3co~2Bsin8])d~}d~
= -$r3(1+
. of the origin by
k).
0
This result satisfies condition (12) if one requires that k + 1 > 0. This is in agreement The relevant conditions are therefore k + 1>
Y>O,
Y”~.
with (41).
(42)
They are to be compared with Popov’s conditions (38). It becomes obvious that conditions (38) do not contradict conditions (42). But it is also clear, that the new conditions (42) do indicate that the actual stability domain in the space of the parameters k and y is larger than the domain predicted by Popov. For the point P, which is located in the extended stability domain with coordinates y = 2.0 and k = 0.5 (see Fig. 2), stability of the trivial solution of (37) for f (x3)= x: t x3 could not be claimed according to Popov. Yet, the alternative method applied in this paper asserts stability. For a verification of this assertion, the system i1=-.q+x33+x3, i2=
-&-
i3=
x1+2X2-&~-&3,
x3
9
H.H.E. L-eipholz, An alternative to Liapunov
which corresponds
311
to point P, has been solved for the initial conditions
Xl(O)= 1.0 )
x*(O) =
1.0 )
x3(0) =
1.0 .
The solution is depicted in Fig. 3. One realizes that there is indeed asymptotic stability of the trivial solution, and the assertion of the existence of a larger stability domain than the one given by Popov has been justified. EXAMPLE 3.5. The last example deals with the rotation [4, PP. 3% 311. Consider the perturbed system
of a gyroscope
as presented
A& = (B - C)x*rfI + (B - C)x2x3 ) Si,=(C-A)
Xlr.0 + (C - &x1x3
C& = (A - @x1x*
)
)
which describes a rotation about the axis of the gyroscope with the moment of inertia Using (8) and (11) one obtains
C-A
+
A-B
r sm 8 cos 4 sin 4 r2cos 8 sin 8
-2.o-
7
1
2.2
-_ Solution
of
I, = -x,+x3’+ x2’-x:-
x3 x3
x3= x,+2x2 for
I7
x,(O) 2 x2(01 = XJO’ =
-1/2x;
-1/2x3
I0 Fig. 3.
1
r2sin 8 cos e cos 4 r2sin2tI sin
ror sin 8 cos 4 + -
[ E +
C-A
-
in
H.H.E. Leipholz, An alternative to Liapunov
312
and finds after a simple calculation that m=O. This result is ambiguous and indicates that one is dealing with a critical case of stability: When m = 0,it means that with respect to the origin, as much ‘substance’ flows in as it flows out. This is of course the case, if the trajectories are closed and cross the sphere around the origin in an even number of times. In that case, one could have stability, yet not an asymptotic one. But, if the trajectories consist of curves open at one end or of pairs of curve branches (open at both ends), an even number of crossings of the sphere around the origin could again occur for a trajectory. Then, m would be zero again and, in spite of this, one would not have stability. Hence, m = 0 is indecisive concerning stability, see Fig. 4. In order to come to a conclusion, one had to turn to conditions (28) and (29). Yet, one finds
,trajectory
(a)
m ‘0,
stability
of Q
,trajectory
(b) m = 0, no stability m = 0 indecisive
with
respect Fig. 4.
to stability
of p
H.H.E. Leipholz, An alternative to Liapunov
313
which shows that A is degenerate. Conditions (28) and (29) cannot be met, which indicates that asymptotic stability is in fact out of question. But, when linearizing (43) one obtains
which leads to x3 =
const.
and
(44)
- -
~
_&CC-A 1 ~-r&l A B
=0.
(45)
The result (44) is compatible with stability. Moreover, x1 and x2 will be periodic functions of time, thus implying stability, if in (45) (46) Condition
(46) is satisfied provided
and also if
B-&-O,
c-A-co,
B>A>C.
Wb)
Inequalities (47) indicate that a small rotation about the axis with the moment of inertia c is periodic if c is either the largest or the smallest of the gyroscope’s moments of inertia. In the sense of Liapunov, one is facing a critical case, and a decision on stability had to be made taking the nonlinearities of the problem into account. In the calculations here, this has already been done by calculating m. It is interesting to note, that the result obtained, i.e. m = 0, left in contrast to Liapunov’s method the decision to the linearized situation. The answer, following from (44), (45) and (46) leads to the well-known fact that a rotation about the extreme axes of the gyroscope is stable.
4. Conclusions In this paper, an alternative to Liapunov’s stability method has been presented. It is based on the flow concept, visualizing trajectories as flow lines, which must tend to the origin in the case of asymptotic stability. Therefore, condition (12) i.e. m < 0,is a necessary condition for asymptotic stability. But, in order to obtain a complete picture of the stability domain in the
314
H.H.E. Leipholz, An alternative to Liapunov
parameter space, one must also consider the condition (15). This condition can be linearized to yield (19) (also (28) (29) in the three-dimensional case), if one is only interested in that part of the domain of attraction which is in the close neighbourhood of the origin. The examples presented confirmed the applicability of the alternative method and lead to two important facts: (i) The alternative method is more powerful than Liapunov’s method in determining the stability domain in the parameter space. (ii) The method can also be used to determine simple stability (not only asymptotic stability) and even instability, if the results are properly interpreted.
References [l] H.H.E. Leipholz, An alternative to Liapunov’s stability method, Comput. Meths. Appl. Mech. Engrg. 43 (3) (1984) 293-313. [2] W. Hahn, Stability of Motion (Springer, New York, 1967). [3] E.P. Popov, Dynamik von Systemen der Automatischen Regelung (Akademie Verlag, Berlin, 1958). [4] N.G. Chetayev, The Stability of Motion (Pergamon Press, Oxford, 1961).
IN APPLIED
MECHANICS
AND ENGINEERING
47 (1984) 299-314
AN ALTERNATIVE TO LIAPUNOV’S STABILITY METHOD AND ITS APPLICATION TO HIGHER-ORDER SYSTEMS H.H.E.
LEIPHOLZ*
Departments of Civil and Mechanical Engineering, Solid Mechanics Division, University of Waterloo, Waterloo, Ontario N2L 361, Canada Received
30 January
1984
This paper is an extension of [l] and should be read in conjunction with that reference. It proves that an alternative to Liapunov’s stability method, which removes the obstacle of having to establish a Liapunov function, can be generalized to cover also those cases in which higher-order differential equation systems are involved.
1. Introduction In a preceding paper [l], a proposal for an alternative to Liapunov’s stability method was made. However, that proposal was restricted in its argumentation and application to secondorder systems of differential equations. In this paper, it will be shown how the new method can be extended for the handling of stability problems posed by higher-order systems of differential equations. Examples, supporting the new technique, will also be presented.
2. Mathematical foundations and stability conditions As already pointed out in [l], the system of differential equations ii =
fi(Xl,
ii=dxi/dt,
X27 . . .
7
X”)
i=l,2
7
Xi(t)
Xi =
,...,
7
(1)
It,
will be assumed to be autonomous, i.e., the functions fi are supposed not to involve time t explicitly. Also, like in [l], the system (1) will be viewed as determining a flow, the i representing a vector field in an n-dimensional space. Again, as described in [l], attention will be paid only to the stability of the origin 0 = (X = 0) of the n-dimensional space, yet without restricting generality. If 0 is to be (asymptotically) stable, one has to require that, for increasing t, the flow tends towards 0. This means, 0 has to be a sink of the flow. *This research 00457825/84/$3.00
was supported
through
@ 1984, Elsevier
NSERC
Science
Grant
Publishers
No. A 7297. B.V. (North-Holland)
300
H.H.E.
Let m be the magnitude
Leipholz,
An alternative
to Liapunov
of the sink. It can be calculated, using the hypersphere
S = C XT= r* = const.,
(2)
with centre in 0. If 0 is the surface of S, dm z=
x-n
(3)
where n is the (outer) normal to 0’. Yet, n = grad S/lgrad Sl
(4)
and grad S = 2Xi,
x: = 2r
Igrad SI = 2Jc I
(5)
Hence.
2
dm -= d0
xi -&Xi.-
-
2r
ci
iiXi/r
Finally, one has for the magnitude C
m=
PROOF.
(6)
of the sink the relationship
i %Xi
,d6.
r
f0
PROPOSITION
.
2.1.
If
(7)
m < 0 for r + O’, the origin 0 is asymptotically stable.
Since S = E i 2iiXi, m = B (,!?/2r) d6. I
But, then, for sufficiently small values of r, (S),0’ - 2mr
and
(S), - 2mr/B.
If for r+ O’, m < 0 holds, the time derivative of the positive definite function S is negative around the origin 0. This fact is sufficient for asymptotic stability of 0 according to Liapunov’s second theorem. COROLLARY 2.2. If m, as calculated by means of (7) should tend towards zero for r -0, could still claim stability of the origin 0.
one
H.H.E.
Leipholz, An alternative to Liapunov
301
This claim can be justified by arguing as before and invoking Liapunov’s first theorem. Let the case of II = 3 be taken under specific consideration. In this case, S is an actual sphere of radius r in the three-dimensional Euclidean space with centre in the origin of the space. Using spherical coordinates, one has x1 = r sin 8 cos C#J , x2 = r x3=
(8)
sin 8 sin 4,
rcos
8,
and d6’ = r’sin 8 de d4.
(9)
With (8) and (9) (7) becomes m =~02-[\~(,*,i)rsinBdB]d~.
(10)
With (1) and (8) (10) can be rewritten to yield 2lT
m=
m
I iI 0
1r
0 If(
sin e cos 4, r sin 8 sin 4, r cos 8)r2sin2tl cos fp
+ f2(r sin e cos 4, r sin e sin 4, r cos 8)r2sin28 sin 4 + f3(r sin e cos 4, r sin 8 sin 4, r cos 8)r’cos 8 sin e] de
d4 .
(11)
As a first stability condition, m
forr+O+,
02)
may be established on the basis of the preceding deliberations. In (12) m is to be calculated using (7). In the case of n = 3, one may use (11) instead of (7). At this point, further stability conditions shall be derived. This is done by interpreting the vector
as a force acting on the image point which traces the trajectory. This trajectory is obtained from the system (1) through integration. Hence, in (13), the ‘components’ fi, i = 1,2, . . . , n, are the functions on the right-hand side of (1). In the case of asymptotic stability, ‘force’ F should ‘drive’ the image point towards the origin. This will indeed happen, if F were a ‘central force’, as shown in Fig. 1.
H.H.E. Leipholz, An alternative to Liapunov
302
orbit
Fig. 1. Vector
Let r = (x: 0 to the point by the nature The vector
F = (fi) interpreted
as a central
force in the case of asymptotic
stability
+ x$ + . . . + x2,)“* = +(x1, x2, . . . , x,) be the radius vector leading from the origin P of the trajectory. In this relationship, &(x1, x2, . . . , x,) is a function prescribed of the trajectory. F is a ‘central force’ if it satisfies the condition (14)
under the assumption
that Re A < 0 holds for the constant A. Hence, the requirement
. . 7X”)-hXi=O,
J(x,,xz,.
that
Reh CO, i= 1,2,. . .,lt,
WI
holds, which follows from (14) yields additional stability conditions, which are expressed terms of system parameters involved in the expressions ,fi(xl, x2, . . . , x,). In a number of cases, fi(Xl,
x2,
* * . ,
X,)= C
UijXj + @i(Xl,X2,.
. .3
xfl)
where the Uij are constants and the @i are nonlinear functions of their arguments. becomes C
(Uij
-
Gijh)Xj
+ @i(Xl,
~2,
. . .,Xn) =
0,
Re A < 0, i = 1,2, . . . , n,
in
(16)
Then, (15)
(17)
where Sij is Kronecker’s symbol. Moreover, assume that the @i are sufficiently smooth functions which vanish at the origin. Then, @i
=
2
(@i,j)x=dj
9
@i, j = d@JdXj
,
(18)
is a first approximation of the @i in the vicinity of 0. Therefore, in order to assess stability in the neighbourhood of 0, one can use (18) in (17) in order to obtain C {[aij + (@i,j)x=o]- &jh}xj = 0 ,
Re A < 0, i = 1, 2, . . . , II .
(19)
H.H.E. Leipholz, An alternative to Liapunov
This is a system of homogeneous, solution, one must demand that
linear, algebraic equations.
303
For it to have a nontrivial
det{[aij+ (@i,j)x=o] - SijA}= 0 ,
(20)
which can be simply written as
(21)
det{& - Sijh} = 0 with izij
=
Uij
(22)
(GDi,j)x=O
+
From (20) one can derive the algebraic equation A” +
al/i”-’ + * * * + U”_lh + a, = 0
(23)
)
where coefficients ai, i = 1,2, . . . , n, are the invariants of the matrix
A=
(6ij)
(24)
.
These invariants are connected
with the it roots of (23) by the relationship
ul=Al+A2+*~~+A,, u2 = AIA2+ AIA3+ **. + A,_lA, ,
(25)
u3= A,A2A3+ AlA2A4+ . * * + A,-2A,-IA, a, = A,A2A;. * *A, .
Moreover, the invariants ai, i = 1,2, . . . , n, can be calculated, using the elements dij of the matrix A, in the following way: Consider the determinant IAl of matrix A. The subdeterminant, obtained from IA\ by cancelling those lines and the rows which include the diagonal elements iiii and tijj, may be called cof cliitijj.For example, let A be a 4 X 4 matrix, that is to say,
624 _ = &if44 - 624642. cof 611633= a22 _ I a42
a44
Let C,,i be the combination
I
without repetition
of the natural numbers
1,2,. . . , n, to the
304
H.H.E.
Leipholz,
An alternative
to Liapunov
class i, where n is the order of the quadratic matrix A. For A given by (26) for example, C,, = 12, 13, 14,23,24,34. Furthermore,
set
cof tin&z + cof 611& + cof Ci,1ti44+ cof
622ii33
+
cof &*Lz44 + cof
ti3364.4
=
c
cof Lid
cd.2
where d,, indicates that one has to operate invariants of (26) are given by al = (-1)’ c cof & = -tr(A)
,
c4.3
u3
with the diagonal
=
of A. Then, the
(-1)3 c cof & ) c4,
az=(-1)2CCOf&,
elements
1
a4=(-1)4xcof&=detA,
c4.2
c4.0
which explicitly reads
u4
a21
622
623
is1
632
ii33
a41
-4 a2
a43
=
fz
=detA.
644
In general terms one has for a matrix of order n the invariants
Ui
=
(-l)i
2 COf C”,n-i
iid.
(27)
Stability conditions are obtained by enforcing for the roots hi of (23) the requirement Re A -=c0. Using (25), one arrives in that way at certain conditions for the invariants ai of matrix A. Then, through (27) one translates these conditions into conditions for the elements iZij of A, which through (22) are related with the property determining quantities of the original problem (16). Thus one has indeed stability conditions for this problem (16). Take a matrix A of order n = 3 as an example. For Re Ai < 0, i = 1,2,3, it follows from (25)
H.H.E.
305
Leipholz, An alternative to Liapunov
that aI > 0, a2 > 0 and a, > 0 must hold. That is identical with tril
det A < 0.
a2>0,
(28)
Again by means of (25) one is also led from the requirement ala2
(29)
> a3,
which is a direct consequence (A,+
Re Ai < 0 to the inequality
h2+
A3)(hIh2+
of the inequality hIA,+
h2h3)=lh2h3
>
which holds true if Re Ai < 0, i = 1,2,3, is satisfied. But, this inequality is indeed (29) in terms of (25). It is not surprising to realize, that with (28) and (29) one has obtained the well-known stability conditions of Hurwitz. Generalizing, one can make the following statement: Conditions (15), (17) respectively, yield stability conditions for a nonlinear system of differential equations (1) in addition to condition in (12). Under special circumstances, one can linearize and replace conditions (15) and (17) by condition (19). This condition leads to Hurwitz’ conditions in terms of quantities (27). Hence, when linearization of (15), (17) respectively, (not necessarily linearization of the actual system (1)) is appropriate, one may add to stability condition (12) the Hurwitz conditions, expressed in terms of the ai, and calculated by means of (27) from the tiij, which are given by (22). Finally, it may be mentioned that if the rank of the ‘reduced’ matrix A = (aij)
WV
is not smaller than IZ,one may use the quantities aij’ instead of the quantities 6ij for the setting up of Hurwitz’ conditions. This possibility is to be welcomed as it simplifies calculations. 3. Application of the theory to examples EXAMPLE 3.1. The first example system of differential equations
can be found in [2, p. 1071. It concerns
the nonlinear
~‘1=-X1-X2+X3+Xlr2=fl,
3i2=x1-2X2+2X3+X2r2=f2, ~3=x1+2x2+x3+x3r2=f3, r2=x:+x;+x;. 'Forthe definition of the aij, see (16).
(31)
306
H.H.E. Leipholz, An alternative to Liapunov
For (31) it is predicted in [2] that the trivial equilibrium position is unstable. It will be shown that the alternative method can be used to confirm this fact. Applying (16) to (31) one realizes that matrix A defined by (30) reads in this case
A=
1
-1 -1 l-2 2 [ 1
1 2’ 1
Since (13) is a system of the third order, conditions (28) and (29) must hold for stability. Using (27) to determine the quantities ai, i = 1,2,3, involved in (28) and (29), one finds a,=-(trA)=3-1=2>0, -1 -2 -1 -1 1 -2 1 2
a,=-detA=-
=-2-4-l-1+2+1=-5.~0,
1 2 =-9<0, 1
These results indicate that conditions (28) and (29) are partly violated. Hence, stability cannot exist as predicted. In view of this fact, there is no need to consider condition (12) in addition. EXAMPLE
3.2. The next example is again from [2, p. 1061. It concerns the nonlinear
system
& = X2- 3X3- X1X;+ 4X&X3 - 4X& = fl , 12 =
-2X1 + 3x3 - X& - 2X&X3 - x2x; = f2 )
(32)
&=2x1-X2-x3=f3, which has been shown in [2] to possess a stable trivial equilibrium confirmed by means of the alternative method. Using (8) and (ll), (32) yields
m=
r +
position. This fact will be
sin e sin 4 - 3r cos e - r3 sin3 e cos 4 sin2 4 4r3sin28 cos e sin 4 cos 4 - 4r3sin e c03e
cos 4]r2sin28 cos C#J
+ [-2r sin e cos f#~ + 3r cos e - r3sin3e cos2~ sin 4 -2r3sin28 cos e sin 4 cos C#J - r3sin e c02e
sin 4]r2sin28 sin 4
+ [2r sin e cos 4 - r sin e sin 4 - r cos 8]r2 cos 8 sin 8
d$J .
(33)
H.H.E.
An easy and elementary
Leipholz, An alternative to Liapunov
307
calculation using (33) yields 28
m
=
-$rr3-is7Fr5<0,
which satisfies condition (12). Next, applying (16) to (32) yields
By means of A and (27) one finds al=
-(-l)=
l>O,
a,=3+6+2=11>0, a3=-[-(2-6)-3(2)]=-[4-6]=2>0, al& -
u3
=
Hence, all conditions
(l)(ll) - 2 = 9 > 0 . (28) and (29) are satisfied. This indeed confirms stability.
EXAMPLE 3.3. The following example is from [2, p. 1391 and deals with the stability of the trivia1 equilibrium position of the differential equation ... x+u~itu;itxf(x)=o.
(34)
It is shown in [2] that stability is warranted
u;>o,
uT>O,
f(Y)'07
if the conditions UTU*2-f(Y)>O
(35)
are being satisfied. In the following, it will be shown that the alternative method yields essentially the same results. First, let (34) be transformed into a system of first-order differential equations which reads Ii, =
&=X3,
x2,
i3=-UTX3-U*2X2_XS(X1).
(36)
Matrix A is found to be 0
A=
1
0 [
0 1
-f(O) -“a; -u’i
if implying that x1 is considered
1 ’
to be taken in the neighbourhood
of the point 0 = (0,0,O).
308
H.H.E.
Conditions
Leipholz,
An alternative
to Liapunov
(28) and (29) are in this case given as
aI=-(trA)=aT>O, a2= a3
aZ>O, =
-(-f(O))
=
f(O)
’
0
,
These conditions are identical with (35) if one restricts oneself with f(xJ to the neighbourhood of the origin. This may be interpreted as a limitation of the domain of attraction of the origin. In order to establish stability definitely, one must show that condition (12) is being satisfied. Using (8) (11) and (36) one finds with f(xr) -f(O) + x$(O) that r sin 8 sin 4]r2sin20 cos 4 + [r cos 8]r2sin28 sin 4
m=L2Y\X
+[-aTrcos8-a*2rsin8sin~-rsin8cos~f(O)
which yields m = -$aTr3. Hence, for a; > 0, which is already supposed to be the case, condition (12) is indeed fulfilled. This warrants stability. EXAMPLE equations
3.4. The next example is given in [3, p. 4711. It involves the system of differential
&
=
-x1
+
i2
=
-9(X3)
i3
=
(y
-
f(X3)
)
(37)
,
1)X, + yxz -
Under the assumption if
kf(x3)
.
that f(0) = 0, Popov shows that the trivial equilibrium
forO
k>O,
for y > 1,
k>y-1.
(38)
The corresponding stability domain is depicted in Fig. 2. Linearizing, one has f(X3)
=
f(O)
+
fl(O)XS
=
position is stable
fl(O)XJ
,
H.H.E. Leipholz, An alternative to Liapunov
Stability
domain
UIIIII
Popov’s
Coordinates Coordinates
for exomple
stability
309
(IV)
domain,
extended stobi Ii ty domain, EI of point P: r = 2.0, k = 0.5, P is in the extended stobili ty domain, of point Q: 7:
2.0,
k= fi - I = 0.414, 0 is on the extended stobllity boundary.
Fig. 2.
which is true due to the assumption reads i1= i*
=
-x1
+
f’(O)x,
-f)(O)&
on f(0). Hence, the linearized system following from (37)
)
)
i3 = (y - l)xr + yxz- kf)(O)X3. matrix A turns out to be
The corresponding -1
0
0 7-l
Conditions
f’(0)
0 Y
-f’(O)
.
-kf’(O) I
(28) and (20) are found to be al=-(trA)=l+kkf’(O)>O, a2 = yf’(0) + kf’(0) - f’(O)(y - l)= (k + l)f’(O) > 0 ) a3 =
-det A = -[- l(f’(O)y)] = f’(O)y > 0,
~1~22-
a3
=
(1 + kf’(O))(k + l)f’(O) - f’(O)y > 0 .
Assume in addition to f(0) = 0 that also f’(0) > 0. Then, one is left with the requirements Y>O,
k+l>O,
l+ kf’(O)>O,
(39)
310
H.H.E. Leipholz, An alternative to Liapunov
In order to simplify the further discussion of these conditions, f63)
=
x:
+
let (40)
x3,
so that f(0) = 0, f’(0) = 1 > 0 as assumed. With f’(0) = 1, (39) changes into k + 1 > y’12.
kt1>0,
Y’O,
(41)
Turning to condition (12), one obtains by means of (18) and (11) 27r
m=
n
I (I( 0
-r [
0
sin 6 cos 4 + f(r cos 8)]r2sin20 cos 4
+ [-f(r
cos 8)]r2sin28 sin C#J + [(y - 1)r sin 8 cm C$
t yr sin e sin 4 - kf(r
cos
e)]r%os
B
sin 8
and -r3sin36
m=
cos24,] -
[kr’f(r
In order to evaluate this integral, approximate f (x3)- x3.Then
cos
t9) cos
8
sin 01
dd
f(x3) in the neighbourhood
m= ~2~(~~((-r3sin3e~~s2~]-(kr3co~2Bsin8])d~}d~
= -$r3(1+
. of the origin by
k).
0
This result satisfies condition (12) if one requires that k + 1 > 0. This is in agreement The relevant conditions are therefore k + 1>
Y>O,
Y”~.
with (41).
(42)
They are to be compared with Popov’s conditions (38). It becomes obvious that conditions (38) do not contradict conditions (42). But it is also clear, that the new conditions (42) do indicate that the actual stability domain in the space of the parameters k and y is larger than the domain predicted by Popov. For the point P, which is located in the extended stability domain with coordinates y = 2.0 and k = 0.5 (see Fig. 2), stability of the trivial solution of (37) for f (x3)= x: t x3 could not be claimed according to Popov. Yet, the alternative method applied in this paper asserts stability. For a verification of this assertion, the system i1=-.q+x33+x3, i2=
-&-
i3=
x1+2X2-&~-&3,
x3
9
H.H.E. L-eipholz, An alternative to Liapunov
which corresponds
311
to point P, has been solved for the initial conditions
Xl(O)= 1.0 )
x*(O) =
1.0 )
x3(0) =
1.0 .
The solution is depicted in Fig. 3. One realizes that there is indeed asymptotic stability of the trivial solution, and the assertion of the existence of a larger stability domain than the one given by Popov has been justified. EXAMPLE 3.5. The last example deals with the rotation [4, PP. 3% 311. Consider the perturbed system
of a gyroscope
as presented
A& = (B - C)x*rfI + (B - C)x2x3 ) Si,=(C-A)
Xlr.0 + (C - &x1x3
C& = (A - @x1x*
)
)
which describes a rotation about the axis of the gyroscope with the moment of inertia Using (8) and (11) one obtains
C-A
+
A-B
r sm 8 cos 4 sin 4 r2cos 8 sin 8
-2.o-
7
1
2.2
-_ Solution
of
I, = -x,+x3’+ x2’-x:-
x3 x3
x3= x,+2x2 for
I7
x,(O) 2 x2(01 = XJO’ =
-1/2x;
-1/2x3
I0 Fig. 3.
1
r2sin 8 cos e cos 4 r2sin2tI sin
ror sin 8 cos 4 + -
[ E +
C-A
-
in
H.H.E. Leipholz, An alternative to Liapunov
312
and finds after a simple calculation that m=O. This result is ambiguous and indicates that one is dealing with a critical case of stability: When m = 0,it means that with respect to the origin, as much ‘substance’ flows in as it flows out. This is of course the case, if the trajectories are closed and cross the sphere around the origin in an even number of times. In that case, one could have stability, yet not an asymptotic one. But, if the trajectories consist of curves open at one end or of pairs of curve branches (open at both ends), an even number of crossings of the sphere around the origin could again occur for a trajectory. Then, m would be zero again and, in spite of this, one would not have stability. Hence, m = 0 is indecisive concerning stability, see Fig. 4. In order to come to a conclusion, one had to turn to conditions (28) and (29). Yet, one finds
,trajectory
(a)
m ‘0,
stability
of Q
,trajectory
(b) m = 0, no stability m = 0 indecisive
with
respect Fig. 4.
to stability
of p
H.H.E. Leipholz, An alternative to Liapunov
313
which shows that A is degenerate. Conditions (28) and (29) cannot be met, which indicates that asymptotic stability is in fact out of question. But, when linearizing (43) one obtains
which leads to x3 =
const.
and
(44)
- -
~
_&CC-A 1 ~-r&l A B
=0.
(45)
The result (44) is compatible with stability. Moreover, x1 and x2 will be periodic functions of time, thus implying stability, if in (45) (46) Condition
(46) is satisfied provided
and also if
B-&-O,
c-A-co,
B>A>C.
Wb)
Inequalities (47) indicate that a small rotation about the axis with the moment of inertia c is periodic if c is either the largest or the smallest of the gyroscope’s moments of inertia. In the sense of Liapunov, one is facing a critical case, and a decision on stability had to be made taking the nonlinearities of the problem into account. In the calculations here, this has already been done by calculating m. It is interesting to note, that the result obtained, i.e. m = 0, left in contrast to Liapunov’s method the decision to the linearized situation. The answer, following from (44), (45) and (46) leads to the well-known fact that a rotation about the extreme axes of the gyroscope is stable.
4. Conclusions In this paper, an alternative to Liapunov’s stability method has been presented. It is based on the flow concept, visualizing trajectories as flow lines, which must tend to the origin in the case of asymptotic stability. Therefore, condition (12) i.e. m < 0,is a necessary condition for asymptotic stability. But, in order to obtain a complete picture of the stability domain in the
314
H.H.E. Leipholz, An alternative to Liapunov
parameter space, one must also consider the condition (15). This condition can be linearized to yield (19) (also (28) (29) in the three-dimensional case), if one is only interested in that part of the domain of attraction which is in the close neighbourhood of the origin. The examples presented confirmed the applicability of the alternative method and lead to two important facts: (i) The alternative method is more powerful than Liapunov’s method in determining the stability domain in the parameter space. (ii) The method can also be used to determine simple stability (not only asymptotic stability) and even instability, if the results are properly interpreted.
References [l] H.H.E. Leipholz, An alternative to Liapunov’s stability method, Comput. Meths. Appl. Mech. Engrg. 43 (3) (1984) 293-313. [2] W. Hahn, Stability of Motion (Springer, New York, 1967). [3] E.P. Popov, Dynamik von Systemen der Automatischen Regelung (Akademie Verlag, Berlin, 1958). [4] N.G. Chetayev, The Stability of Motion (Pergamon Press, Oxford, 1961).