Geometrical properties in state space of linear differential equations with periodic coefficients

GEm.1ETRICAL PROFERTIES IN STATE SPACE, OF LINEAR DIFFERENTlAL EQUATIONS WITH FERIODIC COEFFICIENTS

by P. C. Hughes

•

GEOME4RICAL PROPERTIES IN STATE SPACE OF LINEAR DIFFERENI'IAL EQUATIONS WITH PERIqDIC COEFFICIENTS

by P. C. Hw;hes

Manuscript received March,

1968

ACKNOWLEDGEMENT

The author is grateful to Professor B. Etkin for his encouragement and interest in the progress of this investigation.

This research was sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force under AFOSR Grant Nos. AF-AFOSR-222-66 and AFOSR-68-1490.

SUMMARY

In an attempt to more completely visualize the solution vector to a system of linear differential equations wi th periodic coefficients, geometrical properties in state-time space are derived in detail for second order systems. The solution vector is found to be on a sprface which is periodic in time and whose cross-section is elliptic. The connection between these surfaces and

Liapunov stability is pointed out. An example is discussed in the area of satelli te attitude dynamics.

1. II.

TABLE OF CONTENTS

INTRODUCTION

OCCURRENCE OF LINEAR VARIABLE-COEFFICIENT SYSTEMS IN AEROSPACE FLIGHT DYNAMICS

1 2

2.1 The Two Major Sources of Linear-Variable-Coefficient System 2

2.2 Examples of Linear Variable-Coefficient Systems

4

2.2.1 Example 1: Dumbbell ~ibrations in an Elliptic Orbit

2.2.2 Example 2: Flapping Behaviour of Helicopter Rotors

2.2.3 Example 3: Motion of Aerodynamically Responsive

Sate-lli tes

2.2.4 Example 4: Satellite Attitude Stability

2.2.5 Example 5: Aircraft Attitude Stability

III.

LINEAR DIFFERENTIAL EQUATIONS: GEOMETRICAL VISUALIZATION

3.1 Linear Invariant Systems

3.1.1 Linear Invariant Homogeneous Systems

3.1.2 Linear Invariant Nonhomogeneous Case

3.1.3 Geometrical Visualization for Second Order Systems

3.-2 Linear Periodic Systems

3.2.1 Linear Periodic Homogeneous Systems

3.2.2 Linear Periodic Nonhomogeneous Case

3.2.3 Geometrical Visualization for Second Order Systems

IV. RELATIONSHIP TO LIAPUNOV' S SECOND ME\THOD

V.

EXAMPLE: DUMBBELL LIBRA~IONS IN AN ELLIPTIC ORBIT VIo CONCLUDING REMARKS

REFERENCES APPENDIX FIGURES

4

4

5

5

6

7

8

8

10 11 14 14

16

16

19

20 22 23 ; ,," I ; ...

1. INTRODUC'.I1IQN

i

Modern mathematics may be considered to have begul'l approximately with the seventeenth century (Ref. 1), and with Descartes (1596-1650). The chief con-tribution made by Descartes is well known to be his marriage of geome-pry to

algebra. In his own words, ti • • In this way, I believed that I could bbrrow all that was best both in geome~ry and in algebra, and correct all the defects of the one by help of the othertl (R.ef. 2). And although i t is probable that geometry benefited more from the tools of algebra than conversely, it has never ceased to be true that symbols of any variety take on a more friendly aspect when viewed with the assistance of some visual device. There seems to be somethinginnate in the processes of human thought which can be brought to bear on a given situation if there is a geometrical interpretation available. Anyo~e who has unwisely attempted to teach vectors without tlarrowstl, set theory without Venp. diagrams, complex variables without Argand diagrams, or calculus without tlsldpes tl and.

ti areas tl will be quickly apprised of this tendency. The finest mathematical minds the world can produce are ~ot without this property. Turnbull (Ref. 3)

remarks that the work of Descartes and his contemporariestl ...• consolidated a position which made the differential calc~lus the iuevita~le discovery of Newton and Leibmz'~

Sawyer (Ref. 4) describes a fictional but fascinating experience in which he tries to explain the principles of mathematics to an ti angel tl who is intelligent but has had no physical experie~ce whatever. Needless to say,

ex-~reme difficulty was encountered on those aspects which touched on geometry. Óne conclusion of this tale must be that geometrical intuition, which is so

invaluable an aid to meditations on analysis and algebra, is nourished by physical experience.

I~ view of all this, it is not surpr~s~ng that a rather sophisticated geometrical interpretation has evolved for differential systems of the form

(x is an N vector)

f(x,t) ( lol)

.

x

Some of the most straightforward elements of this i~erpretation are reviewed at the beginning of Sec.

3.

As more and more details are specified on the form of f(x,t), these details i~evitably appear as geometrical properties of the

visual i~terpretation. In particular, if f(x,t) is linear in x , the general form must be

The subcase i~ which

f(x,t) = A(t) x + u(t)

A(t+T)= A(t) u(t+T)

= u(t)

is ~he object of discussion in the sequel. Some of the mathematical and physical origins of such systems are noted in Sec. 2, and the geometrical counterparts of the restrictions (1.3) are derived in Sec. 3.

(1.2)

11. OCCURRENCE OF LlNEAR VARIABLE-COEFFICIENT SYSTEMS IN AEROSPACE FLIGHT DYNAMICS

In order to provide motivation for the state space visualization herein proposed, some examples of linear variable-coefficient differential systems are given in this section. A quite gefteral viewpoint is taken in Sec.

~.l in which the two major sources of such systems are examined. Then specific

examples from recent flight dynamics research papers are discussed in Sec.

2.2.

This field is chosen since it provides a broad spectrum of such examples and be-cause it falls within the acquaintance of the author.

'2.1 The Two Ma~or Sources of Linear-Variable-Coefficient Systems

Source {A) The most natural occurrence of linear variable-coefficient

systems is in the linearized approximation to nonlinear systems with periodic

inputs. Such a nonlinear system may be written in the form (here x is an N vector)

x

=

f(x,t) f(x, t+T)== :f!{ x, t)

x(o)

=

b

The linearization process proceeds formally as follows,

f(x,t)

=

f(o,t) +

~I

x + •••

x=O

which leads to a system of the form

where x = A( t) x + u{ t) A{t+T) == A(t) u( t+T) == u( t)

x{ 0)

= b A( t) =

~;

I

x=O u{t)

=

f(o,t)

If the original system happen to be homogeneous, then u(t)

=

f(Q,t) == 0

(2.1)

(2.2)

Source (B) The second orlgln of 1i~ear variab1e-coefficient sys~ems arises from stabi1ity investigations uti1izing the first method of Liapunov. In

Liapunov's first method~ the stabi1ity of a particu1ar solution to a differentia1

system is examined via a neighbouring solution. If the solution whose stabi1ity

is in question happens to be periodic, then the stabi1ity prob1em reduces to the

properties of a 1inear periodic-coefficient system. For, suppose the origina1

system.is represented as (y is an N vector)

iJ = g(y,t) (2.6)

and that the solution y*(t} resu1ts from (2'9) and the initia1 condition y*(o)

=

c.

Fo11owing the first method of Liapunov, a neighbouring solutio~is cJnceived which

a1so satisfies (2.6), but which satisfies the initia1 condition y(o)

=

c + b.

Then the difference vector x,

must satisfy

x(t} y(t) - y*(t)

x y - y*

=

g(y,t) - g(y*,t)

g(Y*+ x,t) - g(y*,t)

Fo11owing Liapu~ov, we 1inearize and fi~d

x

=

A(t)x where

A(t)

=

~gl

y y

=

y*(-t;) Under certain conditions A(t) wi11 be periodic, i.e.

(i) , y*(t+T)

=

y*(t)

(2.8}-(2.9)

In this case, the origina1 system (2.6) is auto~omous a~d has the

periodic solution y*(t) . Therefore, from (2.9),

A(t+T)

=

A(t) (2.10)

(ii)

g(y,t + Tl )

=

g(y,t), y*(t + T2)

=

y*(t)

In this case, the origina1 system (2.6) has periodic inputs, of period Tl , and

possesses the periodic solution Y*(t), with period T2 . This does not guarantee

the periodicity of A(t) un1ess Tl and _T2 are mutua11y commensurab1e. If they are

then

where nl and n2 are integers.

where Tl nl T2 = n2 Then from (2.9) A(t + T)

=

A(t) (2.11) (2.12)

2.2 Examples of Linear Variable-Coefficient Systems

Some examples are now given which illustrate physical orlg~ns of

linear. variable-coefficient systems. Of the multitude of such sources, only some

of those from aerospace flight mechanics are discussed. Even in this relatively

narrow context, no attempt is made to be exhaustive; however, it is hoped the

ex-amples are representative.

2.2.1 Example 1: Dumbbell Librations in an Elliptic Orbit

If a dumbbell satellite is placed in an eccentric orbit

(eccentricity

=

e) then the equation of motion in pitch (i.e. in the orbital plane)

is given by(Refs.

6, 7,

and

8)

(1 + e cos,) 8" - 2 e sih,(8' + 1) + 3 sin 8 cos 8

o

(2.13)

where the pitch angle 8, is the angle between the dumbbell axis and the

earth-center vector, and where , is the ttue anomaly. Upon linearization (2.13) becomes

where 1 ) 2esin, , l+ecos~ (2.14) u(,) =

(2es~n)'

) l+ecos)'

This example of source

(A),

Sec. 2~1, will be studied further in Section

5.

2.~.2 Example 2: Flapping Behaviour of Helicop~er Rotors

When the flapping mot ion of a helicopter rotor is investigated

(e.g. Ref.9), the motion of the rotor (flapping angle

=

~) in terms of the azimuthal

angle of the rotor, 8, is governed by an equation of the form

~" + p (8) ~' + p {8) ~ = P (8) (2.15)

1 2 3

where p (8), ~ (8), and

P3(e)

are periodic with period 2~ , and involve trigonometrie

functio1s of

e:

The details depend on the type of rotor, the tip speed ratio, etc. The functions p and p may contain mechanical damping and spring constants,

respectively, bnt the

~eriodic

terms are aerodynamic in origine Evidently (2.15)

X'

=

A(8)x + u(8) where

This is another example of Source (A), Section e.l.

2.2.3

Example

3:

Motion of Aerodynamically Responsive Satellites

The motion of a satellite in an elliptic orbit has already been shown (Sec.

2.2.2)

to give rise to linear periodic-coefficient systems. A flight vehicle which travels in an elliptic orbit whose perigee lies within the sensible atmosphere also has system equations of the periodic-coefficient variety because the density is a periodic function of the orbital anomaly. The corrbined influence of these two effects is irrvestigated in Ref. 10, where the equation of motion for the pitch angle is shown to be of the form

8" + P (y)8' + p (),)8 p (y)

(2

.

16)

1 2 3

The coefficients Pl ' P2 ' and P3 are complicated but periodic functions of the orbital anomaly and all three functions contain both aerodynamic and gravity

gradient terms. Again the system equation

(2

.

16)

can be put into first-order form in the same manner as the pervious example.

2.2.4

Example

4:

Satellite Attitude Stability

One situation has already been cited (Example

1;

Secte

2.2

.

1)

in which satellite attitude motions were governed by an equation with period coe-fficients. The physical origin of these periodic inputs was the orbit itself. ~he orbital motion was seen to be a source of energy which drove the librational

motion of the dumbbell satellite. In this section, an example is studied in which the driving source is within the satellite attitude motion itself.

Consider a single rigid body which moves in a circular orbit and let 8₁ ' 8₂ ,and 8

3 denote body attitude angles measured around the radius

vector, the velocity vector and the orbit normal vector respectively. The

general equations -of attitude motion are, as always, highly nonlinear and cannot be solved generally. One solution of these equations is known, however, and it is a solution of practical importance, viz.,

8 == 0 8 == 0 ( ) I d

)

-

_dy

1 2-

(2.17)

8 " + k2 _{sin 2 8 0}

3 3

(2

.

18)

The term k2sin 2 8₃ represents the gravity-gradient torque. Equation

(2

.

18)

has the first integral

2

(~~~

+

2

k2 sin 83 constant

(2.19)

from which it is seen that 83 oscillates or is unbounded accordingly as

(2.20)

In the first case,

(2.18)

possesses periodic solutions (libration). In the second case, a tumbling motion about the orbit normal is indicated and it can be shown that, although 83(-1) is no longer periodic, 83'(')')' sin 83(')'), _{cos 83(')') are} periodic in ')'.

When the stability of

(2.17)

is investigated (source

(B),

Sec.

2.1)

the linearized homogeneous equations in 81 and. 82 (corresponding to

(2.8)

and (2.9)) contain _{coefficients which depend on 83}' , sin 83, cos 83(but not 83). This would be expected physically. Therefore whether the reference motion is libration about the or~it normal or tumbling about the orbit normal, the effect of this motion on the stability of the other two rotational degrees of freedom can be studied via linear equations wi th periodic coefficients. The libration case is reported in Ref.

11

and the tumbling case is treated in Ref.

12.

A related example, in which the periodic inputs for one attitude motion originate in other attitude motions, arises in connection with a gyroscope in a circular orbit (Ref.

13).

Or, again, if the moments of inertia of a spece vehicle vary with time, as they would for example with crew motion, time varying coefficients also appear. A situation of this kind is the subject of discussion in Ref.

14.

Each of these mentioned examples are of the Souree

(B)

variety (Sec.

2.1).

2.2.5

Example

5

:

Aircraft Attitude Stability

The type of behaviour just discussed (Sec.

2.2.4)

wherein the stability of an attitude motion or motions is studied in the presence of other (periodic)

attitude motions can also occur for flight vehicles within the atmosphere. The general equations of motion are found in Ref.

15.

It is shown that if the Euler angles are chosen to correspond to yaw, pitch, and roll attitude motion (~ , 8, ~ respectively), the equations of motion decouple into the so-called longitudinal and lateral submotions. For small attitude oscillations, a solution of the following form is possible

8(t)

F

0 (pitch)

(2.21 )

7f; == 0 (yaw) , ~ == 0 (roll)

(2.22)

The question arises as to the stability of the null lateral solution (2ö22) in view of the longitudinal motion

(2

.

21).

Unless continuously excited, the pitch motion will die out on account of aerodynamic damping and there is no threat of instability.

If the pitching motion were excited by a periodic input however*, the stability analysis of the solution

(2.22)

would foltow the development of Source

(B),

Sec.

2.1,

and the linear equations in ~ and ~ would have periodic coefficients. A partial examination of this behaviour is reported in Ref.

16

where the large effect of a periodicaltY varying lift coefficient is demonstrated.

III. LlNEAR DIFFERENTlAL EQUATIONS: GEOMETRICAL VISUALIZATION

The most general system of differential equations herein considered consists of the linear, ~ariable-coefficient, nonhomogeneous system which can be written in the form

x

=

A(t)x + ~(t)

x~o)

=

b

Inthis equation, x(t) is an unknown vector function of time and A(t) and u(t) are known functions of time, A,u, and x being N x N, N x 1, and N x 1 respectively. Since the motivation for this discussion has its origin in dynamics problems, N

ca~ normally be considered as aneven number.

The vector x is of ten referred to as the "st.ate vector" and it is said to lie in the "state space" associated with the system (3.1). The term

"state space" itself has strong geometrical connotations although the visualization can only be complete When N

<

3.

In particular, it is normal in such situations to picture the solution to (3'.1) as a "trajectory" in state space (Fig. 1). This "trajectory" is, of acourse, not the physical trajectory which may very well be the subject of the dynamical analysis. The notion of a trajectory in state space is particularly illuminating if the trajectory lies on a closed surface, or perhaps even on a closed curve.

Closely related to the above procedure is the one wherein an aug-mented state space is contemplated, and over which the vector (x t)T is defined. In this case, the visualization is not complete unless N

<

2. In this case, the axes are the Xl - , x2-, and t- axes (Fig.

2).

Despite the rather limited system order for which the solution vector can be visualized with ease, the concepts generated in the case N

=

2, A(t)

=

A, u(t)

=

0, have attained some considerable importance. For this case, a

second-order autonomous linear invariant system, the geometrical results have be-come popular text book material (see, for example, Refs. 17 and 18). The terms "stable node", unstable node", "saddle", "vortex" , "stable focus", and "unstable

focus" are familiar in this connection. These ideas, while incomplete for systems with N

>

2 give geometrical insight and provide a useful starting point for systems of higher order.

For the case N = 2, A(t)

=

A, u(t)

t=

0, the situation wherein u(t) is a periodic input vector can be simply interpreted provided the corresponding

* This system might be the result of a malfunctioning antopilot; an analysis showing the influence of a single spectral component of atmospheric turbulence would lead to a similar system.

homogeneous system is asymtotically stable. The well-known results. indicate that the effects of initial conditions (as represented by b) decay with time, and a

steady state periodic solution obtains in the limit as t ~oo. In substance then,

most of the situations of interest wherein A(t) is a constant matrix have been

dealt with by geometrical analogy. In contrast, geometrical models for systems

possessing a variable A(t) are, to the author's knowledge, in a relatively

hnder-developed state. Outside of the general state vector state space ideas outlined

briefly in the second and third paragraph above, there do not appear to be readily

available visualizations for the solution vector even under rather specific

re-strictions on A(t) and u(t).

It is not surprising that -the general linear case has not been

successfully modelled. It is known that the general solution to

(3.1)

canrtot be

obtained in terms of elementary functions. When one further considers the amount

of effort required successfully to visualize the case A(t)

=

A for the first

time, (a highly specialized case), it is little wonder that the general case re-veals only a state vector meandering somewhat vaguely through its state space.

There is at least one case, however, for which a rather precise

geometrical description can be evoked. Specifically, if A(t) and u(t) are periodic,

it is shown in Sect. 3.2 that a great deal can be said concerning the 'trajectory'

of the solution vector, x(t) in the augmented (x,t) space. Since this is

pre-cisely the sit~ation in many problems of interest (Sect. 2 for example), it is

worthwhile to study this case carefully.

3.1

Linear Invariant Systems

Prior to a study of the cases for which A(t) and u(t) are periodic,

corresponding results for linear invariant systems are derived. This is

conven-ient as an introduction to some of the notation in the sequel. However, the strongest motivation for glancing at the geometrical implications of linear

in-variant systems is th at A(t)

=

A is, af ter all, a particular case of a periodic

matrix. Viewed in this light, the considerations in Sec. 3.2 are generalizations of those in the present section. The results stated for the general periodic A(t) must necessarily reduce to those of this section as a particular case.

3.1.1

Liaear Invariant Homogeneous Systems

Under the conditions of invariance (A(t)

=

A) and homogeneity

(u(t) -

0)

the system

(3.1)

becomes

x

=

A x

x(o)

=

b

The solution to (3.2) is well known:

x(t)

=

exp (At)b

Let the eigenvalues of A be v~,--,

vN

and let

(the "i are the eigenvalues of exp A). Then (3.3) may be rewritten

where

and the columns of the N x N matrix U are the eigenvectors of A (and

In order for Ix(t) 1 to remain bounded it is necessary and sufficient that

1 --, N or, in terms of the v i' the more familiar criterion

dle(v, i)

<

0, i

=

1, --, N (3.5) (3.6) of exp A). to insist J:/' (3.7) (3.8) The conditions for boundedness (and, it happens, for stability) have been

stated in terms of the "i since this form is more convenient when comparison is made with the results of Sec. 3.~.

Again, looking forward to the developments of Sec. 3.2, let it be supposed that

trace A = _Nrr2

_<

₀

₍₃

_.

₉₎

Since the trace of a matrix have

is invariant under N

a similarity transformation, we

or, in terms of the "i

The case

L

v i

=

-N rr2 i=l N TI "i = exp (_Nrr2) i=l ~

=

0 (3.10 ) (3.11) (3.12)

is a logical starting point (no damping). For (3.11) combined with (3.7) now becomes

I". 1

= 1, i = 1, --, N

. l

and since A is real, the vi must be of the form

The above conclusions are now extended to the nonhomogeneous case.

3.1.2

Linear Invariant Nonhomogeneous Case

Under the conditions of invariance (A(t)

=

A), the system

(3

.1)

becomes

:ie = A x + u(t) x(o)

=

b

The solution to

(3.15)

is well known: t

x(t) = exp(At)b +

1

exp [A(t-t')] u(t' )dt'

o

(3.16)

Since the general case wherein u(t) is an arbitrary function of time does not lend itself to any specific geometric results, attention will be confined to the case where u(t) is periodic, with period ~

u(t + T)

=

u(t) together with only those stable situations where

/". / _~ <

_-

1, i = l,--,N

It is convenient to de fine a variable T as follows

T

=

t - nT, nT ~ t

<

(n+l)T

(3.18)

(3.19)

which implies that 0

<

T

<

T. The variable "t" is thus replaced by two variables

"T" and "n", the former referring to the time in the (n+l)st periode The

periodicity of u(t) may now be employed to advantage, and (3.16) is accordingly re.."cast thus:

T

x(t) = exp(AT) exp(nAT)b + exp(AT)

r

exp(-At')u(t')dt'

JO T

+ exp(AT) [exp(nAT}-I] [I - exp(-AT)fJ. JoexP(-At')U(t')d.t'

This solution may be more easily treated in the form

where

x*(T)

=

exp(AT)~(T) + exp(AT) [exp(-AT) - IJ -J. ~(T)

~(T)

=

lT

exp(-At') u(t' )dt' Po

=

b - [exp( -AT) - I fJ.~(T) (3.20 ) (3.21) (3.22)

(3.23)

(3.24 )

Since x*(T) does not contain n, it represents the component of the solution which is periodic with period T. In fact x*(T) may be written x*(t).

3.1.3

Geometrical Visualization for Second Order Systems

The results of the above analysis can be geometrically visualized for the case N = 2. Turning first to ~he homogeneous case, if there is no damping

(0-2 = ,0) then we may write """,

Àl

=

exp(ia), À₂

=

exp(-ia) and (3.5) may be more explicitly written as

where (see Appendix)

On expanding

(3.26),

x(t)

=

exp(zat)b Z =

lA

a

x(t)

=

b cos

a

t + Zb sin

a

t

(3.26)

(3.28)

the 'situation is easily interpreted: The vector x(t) continually follows a closed curve in the Xl - x2 plane, and this closed curve is further seen to be the ellipse

(Fig.

3):

(Note: since the left-hand-side of

(3.?9)

is evidently a positive definite quadratic form in x, it is represented geometrically as an ellipse.)

Anticipating further the more general problem where A(t) is periodic, the solution

(3.26)

is also displayed in Fig.

4

in terms of (x,t) space. The trajeetory is now a curve which winds around an elliptic cylinder whose cross-section is given by

(3.29).

Still with

a-2

=

0, consider the geometrical visualization of the non-homogeneous case as best represented for this purpose by

(3.21)

which is now wr i tten as

x(t)

=

x*(T) + p(t) where

p(t) exp(At) Po exp

(z

a

t) Po

from the developments in the Appendix. Equation (3.31) is of the same form as

(3.26)

and thus the p(t) vector lies on the ellipse

- The principal axes of this ellipse are found from the condition p .

p

=

0, or

[exp(zat} po] • [exp (zat) Z po]

=

°

(3.32)

(3.33)

(3.34)

whi~h leads to whence tan 2

ex

t p 2Po. Z Po lp o 12-lz Po 12

ex

t p X - tan 2 X

Note that a second solution to (3.35), viz

a t

p

-rr

X

+ -

₂

const.

gives the other perpendicular priRcipal axis. The principal axis vector corresponding to (3.36) is

P = p(t ) = exp(Zx) r5

P1 P 0

while the principal axis vector corresponding to (3.37) is

P

P2

=

exp (ZX) Z Po

Finally, the area of the ellipse is given by

s

-rrlp ~ P I

P1 P2

= _{-rr det { exp(zx)} Ipo}

~

Z Po I

But from (A.4) in the Appendix,

(3.38)

(3.40)

det { exp(zx)}

=

cos 2X + sin X cos X trace Z + sin2X det Z

=

1

whence

s

-rr lp ~ Z p I

o 0 (3.41)

The solution vector can now be visualized completely. In the

X1 - X2 plane, (3.30)shows that

x(t)

=

x*(t) + p(t)

where it has been possible to write x*(t) for x*(T) since the latter is a

periodic function of T with period T. Thus the solution vector consists of the

sum of two vectors x*(t) and p(t). The former follows a closed curve given by (3.22) with period T, while the latter travels continuouslyon an ellipse

(with period

2-rr/ex)

which is constant in shape and orientation but whose center

follows the locus of x*(t). This behaviour is shown in Fig. 5.

In (x,t) space (Fig. 6) the vector x*(t) is seen to be a curve, periodic with period T. With this curve as a locus of centers, a constant ellipse,

surface upon which the solutio~vector x must a1ways lieG The actual position of x on the surface is determined by the fact that the vector p rotates with a period

21rla.

The character of this surface in (x,t) space may be verified from a fundamental viewpoint. The solution to the nonhomogeneous system, x(t), must

be the sum of the general solution to the homogeneous system, XH(t) pl~s a particular integra1 which in this i~stance is x*(t),

subject of course to the initia1 condition x(o)

=

b. It is recognized then that p(t) corresponds to XH(t) , and in such a manner that the boundary conditions are satisfied.

If damping is present (rr2

>

0)

the above deve10pment must be accordingly modified. From (3.11) we have

and so (3.25) is replaced by

À₁

=

exp (_rr2 + ial, À₂

=

exp (_rr2 -i a) , For the homogeneous system (u(t)

=

0),

we must write

x(t)

=

exp(- rr2 t) exp

(z

a t)b where now {see Appendix)

z

_a 1

The solution vector must therefore satisfy

Ix

:§!l b

1

2 +

(3.40)

(3.42)

arelation which may be pictured as ·an exponentially decaying ellipse. In the X1 - X2 plane therefore the familiar spiral trajectory is indicated (Fig. 7) while, in (x,t) space, the solution vector lies on a surface generated by an ellipse which always has the t-axis as center but shrinks exponentially as it translates

in the t-direction (Fig.

8).

Turnins, now to the nonhomogeneous case, (3.21) leads to

x(t)

=

x*(t) + exp (_rr2 t) exp(zat) Po (3.44 )

In the X1 - X2 plane, the solution vector is the sum of the vector x*(t), which lies on a closed curve and has period T, together with a second vector (Fig. 9)

which lies on a moving ellipse which has its center at x*(t) and which decays exponentially. In (x,t) space (Fig. 10) the situation resembles Fig.

6

except that the surface decays into the single curve x*(t).

3.2

Linear Periodic Systems

In this section, the particular case of

(3.1)

is studied for which

x

=

A(t) x + u(t) x(o) = b

A(t + T) = A(t) tu(t + T) = u(t)

I~ is not necessary that T be the least period of A or u; it is only necessary that it be a common period (preferably the least common period). A number of examples of cases in which differential systems such as

(3.45)

occur in aerospace flight dynamics have been cited in Sect. 2.

The study made in Sec.

3.1

of the more familiar situation in which A(t)

=

A will now prove a solid foundation upon which to build the geometrical visualization of systems of the sort given ~y

(3.45).

There will be many results that are closely related; the results of the present section must of course re-duce to those of the last section as a particular case.

3.2.1

Linear Periodic Homogeneous Systems

Under the condition of homogeneity (u(t)

=

0) the system

(3.45)

becomes

x

=

A(t)x x( 0)

=

b

A(t + T) = A(t)

(3.46)

a system which, unfortunately,cannot be generally solved in closed form. It re-mains true, however, that the system is linear and is periodic, and every possible use will be made of this information. To utilize linearity, it is realized that if "N" linearly independent solutions are known ("N" is the system order), then any solution must be expressible as a linear combination of these solutions. To this end, suppose the following solutions are known:

T

cp~(o)

=

(1 0--0)

4J

_N = A (

t)

CPN

cP (o)T

=

(0 0 -- 1) N

The above N vector equations ~an be combined into one matrix equation if the matrix q, be defined as

Then

_•

q,

₌

_{A(t) q,}

(3

.

47)

q,(o) = I

t

I<p(t)

1=

exp

I

trace A(t')dt ' (3.48)

which te11s us, among other things, that the N solutions ~l(t),-- ~(t) which we

specifica11y chose to be 1inear1y independent at t

=

0, in fact remain 1inearly

independent since I<p (t) I cannot be zero. It is therefore true that the solution

to (3.46) can always be expressed as a linear combi nat ion of the ~i:

x(t)

=

<P(t)b

The development now utilizes the property of periodicity. It may

be shown that <P(t+T) is also a solution of

x

=

A(t)x (proof straightforward and

omitted}. Whence <P(t+T) must be expressible as a linear combinations of the

<p(t-tJr)

=

<P(t)R (3.50)

where, in fact,

R

=

<P(T)

This is a most usefu1 relationship; it says, in effect, that if the solution <P(t) is known over one period, T, then the solution for all subsequent times is easily calculable.

Suppose now that the eigenvalues of R are Àl' , ÀN and let

(3.52) where

A

=

diag ( Àl' _,ÀN)

Then, from (3.49) and (3.50), the solution to (3.46) may be written

x(t)

=

<P(T) U An U-1b

where t has been decomposed into Tand n (as in Sec. 3.1) according to

t

=

T + n T, n T ~ t «n+l)T

It is c1ear from (3.54) that in order for x(t) to remain bounded, it is necessary

and sufficient to insist that

IÀ. I

<

1, i

=

1, --, N

1.

-Now the determinant of a ~atrix is invariant under a simi1arity transformation.

The determinant of R, from (3.52) (3,51) and (3.48) yie1ds

N

[T

TI

Ài

=

exp trace A(T)dT

i=l

= exp{-N~2 T), say.

If there is no damping present, then ~2

=

0, and

The above conclusions (Floquet theory) are now extended to the nonhomogeneous case.

3.2

.

2

Linear Periodic Nonhomogeneous Case

Turning now to the more general case

(3.45)

we have the solution t

x(t) = <I>(t)b + <I>(t)

Jo

<I>-l.(t') u(t')dt'

(3.58)

where the periodicity of u(t), and the specific properties of <I> derived in the last paragraphs lead to

x(t) = <I>(T) RI), + <I>(T)

l~-l.(t,)

u(t' )dt'

+ <1>( T) (Rn_I)(I_R-1 ) y<l>-'(t, )u(t' )dt'

This solution may be more easily treated in the form

where

x*(T) = <I>(T) ~(T) + <I>(T) (R-l._I)-l.~(T)

~

( T)

=

foT

<I>-l. (t ' ) u( t ' ) dt ' P

=

b - (R-l._I)-l. ~(T) o

(3.60)

(3.61)

(3.62)

(3.63)

Since x*(T) does not contain n, it represents the component of the solution which is periodic with period T. In fact x*(T) may be written x*(t). It is also de-sirable to note that

<I>(t)

=

p(t) R(tjT)

_(3.64)

where

p(t + T) == p(t) and the corresponding form of

(3.60)

is

x(t)

=

x*(t) + p(t) R(tjT) Po

(3.66)

3.2.3

Geometrical Visualization for Second Order Systems

Again, the results of the above analysis can be geometrically visualized for the case N

=

2. Looking first at the homogeneous case, if there is no damping (rr2

₌

_{0) then we may write}

"l. = exp( i ex)

and

(3.49)

may be more explicitly written, using

(3.64),

as

where (see Appendix)

On expanding

(3.68),

x(t) x(t)

z

p(t) exp(~tjT)b 1

sinex (R-I cos ex)

P(t)b cos (atjT) + p(t)z b sin (atjT)

(3.67)

~he interpretation lies in the relation

I p( t ) b ~ p( t ) Z b 12

(3.71)

which represents, at fixed t, an ellipse parallel to the X1 - x2 plane. The

~eometrical _{visualization iIYthe x1 - X2plane itself is not very enlightening.}

However, the picture in (x,t) space is quite interesting. Consider anellipse

parallel to the _{x1 - x 2 plane whose center is always the t-axis} and which un-dulates according to

(3.71)

as it translates in the t-direction, thus generating

a surface (Fig.

11).

This surface will undulate with period T. The solution

vector always lies on this surface , rotating about the t-axis with a period

2TTT

j

a.

In general, the principal axes of this translating ellipse vary both in magnitude and direction as the surface is generated. To show this, let

at/T = ~, and consider the principal exes as obtained from the condition

èx

x ,

(ij)

=

0, that is

[pet) exp (Z~)bJ • [pet) exp (Z~)ZbJ

=

°

(3.73)

which leads to

== tan 2 X( t)

(3.73)

Equation

(3.73)

is regarded as defining the function ~p(t) - X(t), in which

case one principal axis of the ellipse is

x

P1 (t)

=

p( t ) exp [Z X ( t

lJ

b (

3 .74

)

while the orthogonal principal axis is

x (t)

=

pet) exp [ZX(t~ Z b P2

A calculation of ~è~(Xp.xp~~2J~=X shows that _{xp1 (t)} is the semimajor axis while

xp2 (t) is the sem1m1nor ax1S.

The cross-sectional area of the generating ellipse may be calculated from

set)

since

= ~ Ix (t) ~ xp (t) I

P1 2

= ~ 4et ( p( t) exp [Zx (t)] } Ib ~ Z b I = ~ det ( p( t) }det (exp [zx (t) ]} Ib ~ Z b I

= ~ det ( <p( t) } Ib ~ Z b I

det (exp [Zx(t)]}

=

1

despite the fact that X(t) is not constant; see the development of

(3.41).

From

(3.48),

it is seen that

provided

trace A(t) == 0

This condition is actually met in some of the examples of Sec. 2.2 (for example,

Refs. 11, 12). The geometrical result corresponding to this property of the

system equations is therefore that the generating ellipse has constant cross-sectional area given by

S = 7r Ib :Q!I Z b I (

3 .78 )

-Confining attention still to the undamped case (rr2 = 0) but

con-sidering inputs we use

(3.66),

viz.,

x(t) x*(t) + p(t)

where

(3.80)

The interpretation of p(t) has just been discussed; the only modification is that now the locus of the center of the generating ellipse is no longer the t-axis, but the curve described by the vector x*(t) which is periodie, with period T

(Fig. 12).

If rr2

>

0, the above results are accordingly modified. From

(3.56)

we have

and so

(3.67)

is replaced by

"1.

= exp( _rr2 T + ia}, "2= exp (- rr2T - ia) For the homogeneous system (u(t) == 0) we must write

where (see Appendix)

x(t)

=

exp(-rr2t) p(t) exp(ZatjT)b

1

Z = ,

-sin a [exp (rr2T) R - I cos a]

The solution vector must therefore satisfy

(3.82)

(3.84 )

Ix :Q!I p(t) bl 2 + IX:Q!l p(t) Z b 12 exp (_2rr2 t) Ip(t) b :Q!I p(t) Z bl 2

(3.85)

that is, the undulating surface generated as before but with an exponential factor causing decay to the t-axis.

Finally in the nonhomogeneous case we have

x(t) = x*(t) + p(t)

(3.86)

where

p ( t) = exp( _rr2t) p( t) exp(zatjT} p

. 0

IV • RELATIONSHIP TO LIAPUNOV' S SECOND METHOD

The first or direct method of Liapunov has been referred to in Sect. 2.1. Some of the features of the solution surfaces (disucssed in Sect. 3)

which relate to Liapunov's second, or indirect, method will now be pointed out.

This approach depends on successfully demonstrating that a function V(x,t) exists which has the following properties (Refs. 17, 19) .

a) V(x,t) is positive definite (4.1)

b)

~~(x,t)

is negative semidefinite (4.2)

In (4.2) the to~al derivative D/Dt signifies that the total rate of change of V is calculated along the solution trajectory. Thus, in the ge eral case, if

x

=

f(x,t),

DV _{oV + oV f( t)} Dt

dt dx

x, or, in particular i f

x

=

A(t)x,

DV _{oV +}

_~

A(t)x (4.3)

Dt

dt

The equations for the invariant surfaces demonstrated in Sect. 3 suggest Liapunov functions. Consider first the linear invariant case, and the solution surface given by (3.43) and sketched in Fig. 8. Then the funetion v(x)

. given by

V(x) ::: Ix

~

bl

2

+ Ix

~

Z bl2 b ~ Z b 12

can be shown to be a Liapunov funetion as follows. Certainly V(o) = 0 and (4.4)

V(x) ~ 0 when x

f

O. But in fact V(x)

>

0 (strictly) when x

f

0 since the con-trary would imply that band Zb were parallel; this ean never be true since the eigenvalues of Z are imaginary (± i). Finally, from (3.43) i t is seen that

~~ =

exp(-2O'2t) (4.5~

along the solution trajectory. This indicates stability and for 0'2

>

0 the

stability is asymptotic.

Similarily when periodic coefficients occur, the function V(x,t) Ix ~ p(t) bl

2

+ Ix ~ p(t) Z bl2

Ip(t)b ~ pet) Z bl2

(4.6)

merits consideration as a Liapunov funetion (c.f. (3.85)). Clearly,V(O,t)

=

0 as required. Also if x

f

0 the numerator cannot be zero since this would imply Pb

was parallel to PZb. Since p(t) is never singular this ~ain would lead to the wrong conclusion that b was parallel to Zb, To complete the demonstration of positive definiteness it must be shown that V(x,t) is bounded below by a positive definite function W(x) .• From the developments leading to (3.75)

since Xp cannot be O. Then the numerator in (4.6) satisfies the inequality

2

I x E p( t ) b 12 + I x E p( t ) Z b 12 ~ I X 12 E2 (4.8)

Similarly, the denominator can be given an upper bound

I p( t) b E p( t) Z b I = det ( p( t)) Ib E Z b I :s BIb l!!I Z b I

(4.9) where B is an upper bound for det (p(t)} ,O:S t:s T which can be inferred from

t

det (p(t)) = exp

{J

trace A( T)dT + 2CJ2t}

o

Therefore, from (4.6), (4.8) and (4.9)

V(x,t) :::: w(x)

(4.10)

(4.11) and this completes the proof of the positive definiteness of V(x,t). Finally, from (3.85),

~~

= exp (-2(J2t) ( 4 .12)

and therefore V(x,t) is a Liapunov function and stability is proved. The

solu-tion surfaces in Figs. 11 and 12 therefore represent Liapunov functions for the

stability of the solutions x(t)

=

0 and x(t)

=

x*(t) respectively.

It should be noted that xhis Liapunov function was in reality only

deduced af ter the stability of the solution had already been established, in fact,

two linearly independent solutions had to be known over 0

<

t

<

T in order to

de-fine p(t) and Z in (4.6). These two solutions may be thought of as two fibres

which lie in the invariant surf ace and which serve uniquely to determine that sur-face. This surface does serve, however, as a Liapunov surface whose equation is

explicitly time dependent.

V. EXAMPLE~ "DUMBBELL LIBRATIONS IN AN ELLIPTIC ORBIT

In this section, the visualization described for a linear second

order periodic-coefficient system will be emphasized by means of a numerical

example. The libration motion in pitch of a dumbbell satellite placed in an

eccentric orbit was mentioned in Sect. 2.2 together with the equation of motion,

(1 + e cos)')9" - 2 e sin)'

(8'

+1) + 3 sin 9 cos 9 = 0

This equation has been studied by Brereton and Modi (Ref. 7) with the object of

determining for what eccentricity, e, and initial conditions 91 , 9i the

satellite tumbles, i.e., Ie I

<

H/2,

Z

<

1

<

0 0 . Since a closed form solution

to (5.1) does not exist, they were obtiged to integrate (5.1) numerically over

a considerable period (0 < 1

<

100

H;

i.e., 50 orbits) and observe the occurrence

or not of tumbling. During the course of the investigation they discovered that the solutions to (5.1) tended to lie on tube-like surfaces in 9,9' ,1 space (Fig.

13) which were periodic in 1 with period 2H. In fact it was the striking

~.

Since it was not possible to show the existence of these periodic tubular surfaces in the general (nonlinear) case, (5.1), the linearized version of (5.1)was examined viz.,

(1 + e cos

y)

8" - 2 e sin

y8'

+ 38 = 2 e sin

y

which has been put in the canonical form (3.45) in (2.14). It is clear from (2.14· ) and (3.56) that

1

127T

~2

=

_

4; trace A(y)dy

=

0 7T 0

and herefore the product of the eigenval~es of ~(27T)

=

R obey the condition

À₁À₂

=

1. If À_{1 and} À₂ are real then one of them is greater than unity*and

in-stability results. When e

=

0, À_{1 and} À2 are given by

exp(±

2~ i). Since for

lel

<

1,8(y;e) depends continuouslyon e, therefore ~(27T) and hence À_{1 and} À₂

depend continuouslyon e. Since À_{1 and} À₂ are complex for e

=

0, then in some

neighbourhoed of e

=

0, À_{1 and} À2 are still complex and, in this neighbourhood,

the system is stable. (This is verified numerically in Ref.

7).

The geometrical visualization of Sect. 3.2.3 is therefore applicable.

In Figs. 14and 15, cross- sections of the solution surface of (5.2) are shown at fixed

y

=

Y2

,

namely

Y2

=

0 (+ 2n7T) , 7T/2 (+ 2n7T) ,

nft

2n7T) , and 37T/2(+2n7T). In Fig. 14, 81

=

8i

=

0 (b

=

0), while in Fig. 15 81

=

ei

=

0.1 [·i.e., bT = (0.1.0.1)] ; in both figures e

=

0.1. The solid lines are ellipses

plotted by the computer (for each of the above values of

Y2)

according to the equation [ xT

=

(8 8')]

where

X*(Y2)

=

~(Y2) ~(Yl) + ~(Y2) (R- 1_ I)-1~(27T)

~(Y2)

=

I

Y2

~-l(y)

u(y) dy

Po = b -

(R-

1 - I)-1~(27T)

(5.4)

and

f(Y)

is given in (2.14). The necessary information for the above expressions was obtained by numerically integrating (5.2) over 0

<

Y

<

27T (with u(y)

=

0) tWice, onee with bT

=

(1 0) and once with bT

=

(0 -1).- This yields ~(y),

o

~

y

~ 27T. Then R

=

~(27T) and the eigenvalues of R,

exp(±

i

a),

are calculated.

Z

is now available from

Z

=

.1 (R - I cos a)

Sln

a

(5.6)

The numerical integration of (5.2) over 0 ~ y ~

Y2

+ 1007T is also shown for comparison by the (+) symbols which represent

X(Y2

+ 2n7T ), n

=

0, 1, 2,--, 50. In some cases, the (+) symbols drift slightly from the solution surface. This is due to the errors in the numerical integration procedure (fourth order Runge-Kutta) over fifty orbits.

VI. CONCLUDING REMARKS

In the foregoing an effort has been made to carefully describe the periodic solution surfaces which exist for linear systems of differential

equat-ions with periodic coefficients. A great deal of detail was examined. in

connect-ion with the exact nature of the cross-sections of these surfaces and in dis-tinguishing between the contributions of the coefficient matrix,A(t) and the non-homogeneous input, u(t). However, the important underlying theme is that periodic solution surfaces of the general type described do exist for this frequently occurring class of differential equations, and that the mental conception of these surfaces is of considerable assistance in many instances .•.

A case in point, which was referred to ih Sect.

4,

is the con-sideration of stability in the sense of Liapunov. This is an essentially geometrical concept and is more readily applied when the geometrical behaviour of the solution in question is understood. Geometrical interpretation for sys-tem orders greater than two is of course not available in detail. Many of the second order results can be applied conceptually however, and the development and notation of Sect.

3

are suggestive of the extensions necessary.

10 Jourdain, PoE.P. 20 Descartes, Rene 30 Turnbull, H.Wo 40 Sawyer, W.W. 50 Lange, BoO. Smith, RoG.

6.

Beletsky,

V.V.

70 Brereton, R.C. Modi, V.J 0 80 SChechter, HoB. 90 Wilde, E. Bramwell, AoR.S. Summerscales, R. 10. Schrello, D.M. 11. Kane, ToR. 120 Kane, ToRo Shippy, DoJ. 13. Thomson,W.T. 14. Thoms on, W. T . Fung, Y oC. 15. Etkin, B. 16

.

Masak, M . REFERENCES

The Nature of Mathematics. Reprinted in The

World of Mathematics, Vol. I, Simon and Schuster, New York, 1956.

Discourse on Method. 1619.

The Great Mathematicians. Methuen, 1951.

Prelude to Mathematics, Penguin, 1955.

The Application of Floquet Theory to the Computation of Small Orbital Perturbations Over Long Time

Intervals Using the Tschauner-Hempel Equations.

Stanford University Report SUDAER No. 41, Aug. 1965.

_,

The Libration of a Satellite in an Elliptic Orbit.

published in Dynamics of Satellites. Maurice Roy,

Ed., Academie Press, 1963.

On The Stability of Planar Librations of a Dumbbell

Satellite in an Elliptic Orbit. Jour. of Royal Aero.

Soc., Dec. 1966.

Dumbbell Librations in Elliptic Orbits. AlAA J.

Vol. 2, No. 6, June 19640

The Flapping Behaviour of a Helicopter Rotor at High Tip Speed Ratios. Aero. Res. Coun. CP No. 877, 1966. Dynamic Stability of Aerodynamically Responsive Satelfites. JAS, Vol. 29, No. la, Oct. 1962. Attitude Stability of Earth-Pointing Satellites.

AlAA Jour., Vol. 3, No. 4, April 1965.

Attitude Stability of a .Spinning Unsymmetrical

Satellite in a Circular Orbit. J. Astro. Sci. Vol.

10, No. 4, Winter 1963.

Stability of Single Axis Gyros in a Circular Orbit.

AIAA Jour 0 Vol. 1, No. 7, July 1963.

Instability of Spinning Space Stations Due to Crew Motion, AlAA Jour. Vol. 3, No. 6, June 1965.

Dynamics of Flight. Wiley, 1959.

On the Lateral Instabilities of Aircraft Due to Parametrie Excitation. University of Toronto Report UTIAS TN 86, Jan. 1965.

17. stern, T.E.

18.

'Cunningham, W.J.

19.

Cesari, L.

Theory of Nonlinear Networks and Systems-an

Introduction. Addison-Wesley,

1965.

Intrcduction to Nonlinear Analysis. McGraw-Hil1,

1958

.

Asymptotic Behaviour and Stability Problems in

Ordinary Differential Equati ons. Academic Press,

1963.

APPENDIX

Raising a Real Second Order Matrix to a Power

In this appendix, a simple ex~ression will be derived for raising a 2 x 2 matrix M to the mth power. Snppose the eigenvalues of M are

Then

(A.l)

where D diag (À₁ ,À_{2 )} and the columns of E are the eigenvectors of M.

Then

Mm (E D Elm

=

E Dm E-1

=

E

~mex:(

i

0</> )

pmeXP~_imJ

E -1

We may write

m m(

M = P I cos

r«/J

+ Z sin m çp)

(A.2)

where

Z = E

(A.3)

Many of the properties of Z can be immediately tabulated from

(A.3),

viz. trace Z 0

Iz I

1 (A.4 )

and the most important one Z2 I

The matrix

Z

is most easily found from

(A.2)

with m

=

1,

M

=

p(I

cosçp

+ Z sin

çp)

that is

1

Z

=

psinP M - cot

çp

I

The above development in terms of Z is particularly useful when we write

(A.6)

exp(Zm )

+ Z

[mep

3~

+ --- ]

+ --- ]

exp(Zmi» == I cos

mi>

+ Z sin m

cp

Comparing (A.7) and

(A.2)

demonstrates that (A.6) is indeed correct. If it happens that there is interest in the case where

M

=

exp

P

(A.8)

and it is actually P that is known, rather than M, then a more useful expression for Z than (A.5) is available. In fact the (known) eigenvalues of

P

are just

i, + i

cp

, i, - i

cp

where

i, = i,n p (A.9)

Then

Mm

=

exp( mi,) exp(Zaxp) (A.1O)

where 1 _(A.ll)

Z

₌

- (p - Ii,)

cp

For, from (A.8), (A.6) and (A.9),

Mm

=

exp (mP)

=

exp (mri, +

mZcp)

and (A.ll) follows at once.

Only bounded situations are discussed in this report; in such cases p ~ 1. Accordingly, it is convenient to let

p exp

(_cr2~

(A

.

12)

or

_(A.13)

With this notation, the results of this appendix may be cast in the form

(A.14)

where cr2

Z == ~ M- cot

cp

I

sincp (A .15)

if M is the known matrix, and

Z _=~ 1 (P+cr2 I) (A.16)

FIG. 1 ' ...

...

x

= f(x,t)

x(o) = b

F tG. I

VISUALIZATION IN x SPACE OF THE SOLUrION TO THE DIFFERENrIAL SYSTEM SHOWN (N = 3)

x

= f(x,t)

x(o) = b

x=Ax trace A = 0 det A> 0 x(o) = b

FIG.3

t=O

FIG. 3 VISUALIZATION IN x SPACE OF THE SOLlJrION TO THE DIFFERENTIAL SYSTEM SHOWN (N = 2)

x = A x

trace A = 0

det A > 0

x = Ax + u(t) trace A = 0 det A > 0 u(t+T) = u(t) x(o) = b ---~~---Xl

FIG. 5 VISUALIZATION IN x SPACE OF TIIE SOLUrION TO TIIE DIP'P'ERENrIAL SYSTEM SHOWN (N = 2) T

FIG.6

ie = Ax + u(t) trace A = 0 det A > 0 u(t-+'r) = u(t) x(o) = b t. 2T

x;Ax trace A < 0 det A> 0 x(o) ; b FIG. 7

FIG.7

VISUALIZATION IN x SPACE OF THE SOLUTION TO THE DIFFERENTIAL SYSTEM SHOWN (N; 2) x;Ax trace A < 0 det A> 0 x(o) ~ b

FIG.8

x FIG. 9 je = AA + u(t) trace A < 0 det A > 0 u(t+T) = u(t) x(o) = b FIG.9

VISUALIZATION IN x SPACE Ol" THE SOL1lrION TO THE DIl"l"ERENrIAL SYSTEM SHOWN (N = 2) X,

FIG.IO

je = AA + u(t) trace A < 0 det A > 0 u(t+T) = u(t) x(o) = b 2T \

X2

x

=

A(t)

x

l

(t+T) = A(t)

o

trace

Ä.(

t)dt

=

0 trace <l>(T)

I

<

2

x(o)

= b FIG. 11

T

FIG. 11

VISUALIZATION IN

(x,

t) SPACE OF THE SOLillION TO THE DIFFERENrIAL SYSTEM SHOWN (N

=

2)

x

= A(t)

x

+ u(t)

A(t+T)= A(t)

u(t+T) = u(t)

~

trace A(t) dt =

0

>Ca,

Itrace q>(T) I

< 2 x(o)

= b

FIG. 12

~/

_~~

2T T XI

VISUALIZATION IN (x,t) SPACE OF THE SOLUTION TO THE DIFFERENTIAL SYSTEM SHOWN (N = 2)

~t

dl

o·o·

,.0 ,0-5 FIG. 13 ,0.

f~

,6

Cf

,,0·

,

,00.

"

,eO·

,,0·

'ZO· g(f

e

SOLUTION SURFACE FOR TEE PLANAR LIBRATIONS OF A DUMBBELL SATELLITE

7T/9

FIG. 14

'I=1T/2

dB

.40

di

.20

-.40

'lJ=37T/2

-7T/9

8

.40 7T19 -AO AO .20 7T/9 7T/18 dB dl" .40

~

-7T/18 -7T/9 B 7T/9 7T/18 1=0,27T' -1T1IS -7T/9 7T/9 7T/18 B r=7T' -.20 -.40 .40 -.20 -.40 dB

di

-7T/18 - i'9 B I=7T/2 -7T19 B '=37T/2

•