An Equipartition of Energy Principle for Oscillatory Systems

Copyright © IFAC Large Scale Systems: Theory and Applications. Osaka, Japan. 2004

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AN EQUIPARTITION OF ENERGY PRINCIPLE FOR OSCILLATORY SYSTEMS P. Rapisarda· J .C. Willems··

• Department of Mathemafic8. University of Maastricht. P . O. Box 616. 6200 MD Maa8tricht. The Nethe1'lands , e-mail P.Rapisarda~math.unimaas.nL

•• ESA T -SISTA. K. U. Leuven. K asteelpark A 1'enberv /0. B-300 1 Leuven-Hetlerlee. Belgium. e-mail Jan.WiZZems~esat . kuZeuven . ac . be

Abstract: \Ve consider oscillatory systems consisting of identical subsystems symmetrically coupled,. We show that the time average of allY quadratic functional of the variables of a subsystem and their derivatives equals the time average of the same functional Oil any other subsystem. Copyright © 2004 IFAC Keywords: Behavioral approadl; oscillatory systems; quadratic differential form; bilinear differential form; equipartition of energy.

1. INTRODUCTION

(kinetic+potential) energy of symmetrically coupled undamped oscillators is asymptotically the same for every oscillator. The latter is a result of (I3ernstein and I3hat , 2002) which we obtain in a different context and as a special case of a more general result.

In this paper we consider oscillatory systems, whose traject.ories are linear combinations of sinusoidal functions 11'(t) = Lk=l, ... ,.. AA- sin(wl;t +
We obtain our result using concepts and techniques developed in the behavioral framework (see (Polderman and Willems, 1998)). A special role in our investigation is played by the concept of quadra.tic differential form (QDF) , introduced in (\Villems and Trent.elman, 1998). The paper is structured as follows : after reviewing the basic notions regarding oscillatory systems in the behavioral framework (section 2) , we proceed 1.0 give an introduction to quadrat.ic differential forms and their calculus (section 3). We then define t.he notion of conserved- and of zero-mean quantit,v. Equipped with such notions , we state Olll' equipartition of energy principle. Finally. we show some applications of such principle to simple systems consisting of few oscillators.

In this paper we state and give a sketch of the proof of a deterministic equipartition of enervy principle for oscillatory systems. We prove that if an oscillatory system consists of "symmetrically coupled" (such notion will be formally introduced further in the communication) identical subsystems. I hen the difference between the value of any quadratic fUllctional of the variables of the olle subsystem and their deri vat iVffi , and its value on the variables of the other and their derivatives is zero-mean. In particular, the time-averaged total

The notation used in this paper is standard : the space of n dimensional real, respect.ively complex,

179

vectors is denoted by )RD. respectively en. and the space of m X n real matrices by JRDl "' n. \Vhenever one of t he two dimensions is not specifip
are railed the characteristic frequ.encies of 'B . ""hen considering nonminimal kernel representations. the nonzero invariant pol~' noll1ials in the Smith form of any matrix H.' E ]R."'V[(l such that '13 = ker R' ( also equal the invariant polynomials of '13 (see Corollary 3.6.3 in (Polderman and Willems, 1998)).

1t ).

\-VI:' now define linear oscillatOIT behaviors. Definition 1. '13 E Cv is an oscilla.i.or.lJ behal'ior if

wE'B ==>

The ring of polynomials with real coefficients in the indeterminate ~ is denoted by R[~l: the set of two-variable polYllomials with real coefficient.s in the indeterminates ( and.,., is denoted by R[(.'71. The space of all n x m polynomial matrices in the indeterminate { is denoted by IRD X III[~], and that consisting of all n x m polynomial matrices in the indeterminates ( and 17 by IRD""[(.111. We denote with Q:X()R, IRq) the set of infinitely often differentiable functions from JR to IRq.

U'

is bounded on (-00, +oc)

Prom the definition it follows immediately that an oscillatory system is necessarily autonomous. The following is a characterization of oscillatory systems in terms of properties of its kernel representation . Proposition 2. Let '13

=

ker R(

ft),

with R E

]R."'V[(l . Then '13 is oscillatory if and only if every nonzero invariant polynomial of '13 has distinct and purely imaginary roots.

2. LINEAR OSCILLATORY I3EHAVIORS Sketch of proof: We use a classic technique in behavioral system theory, namely reducing the problem to the scalar case by resorting to the Smith form. Without loss of generality assume that the kernel representation induced by R is minimal. Compute the Smith form H = U ~ V of H, with U, V ullimodular and ~ the diagonal matrix of the invariant polynomials V'i of R. With a change of variable, we reduce to the scalar case, namely proving that 'Bj := ker V'j (ft) is oscillatory if and only if 'If.'j E IR[~l has distinct and purely imaginary roots.

A linear differentiol beho.vior is a linear subspace '13 of (!:OC(IR,IRV) consisting of all solutions w of a system of linear constant-coefficient differential equations: d H(-Z )w Lt

= 0,

(1)

where R E JR."'v[~], is called a kernel representation of the behavior

'13:= {w E (!:OO(IR,IRV) I w satisfies (I)}, and w is called the external variable of 'B. The class of all such behaviors is denoted with .cv.

(If) Observe that if the characteristic frequencies k = 1. .. . ,deg( l/lj) of'Bj lie on the imaginary axis and are distinct , then w) E 'Bj if and only if

In the following, a special role is played by linear different.ial autonomous systems. Informally, a system is autonomous if the fut ure of every trajectory in '13 is uniquely determined by its past, equivalently by its present "state" (see (Polderman and Willems, 1998) for a formal definition): in other words, if the system has no inputs. It can be shown that the behavior of an autonomous system admits kernel representations (1) in which t.he matrix H. is square and nonsingular: moreover (see Theorem 3.6.4 in (Polderman and Willems , 1998)). such a representation has the minimal number of equations (w. the number of variables of the s.vstem) needed in order to describe an autonomous behavior '13 , and is consequently called a minimal representation.

Wjk,

(2) for 0jk E C, k = 1, ... ,deg(l/'j)' Observe that the corresponding to conjugate characteristic frequencies ±iWjk are also conjugate. since each entry of 1/'j(O has real coefficients. Conclude that (2) describes a linear combination of sinusoidal functions : thus, 'Bj ~s oscillatory. Qjk'S

(Only if) The proof is by contradiction. Assume that there is a characteristic frequency of'Bj not lying on the imaginary axis ; it is easy to verify that this is in contradiction with the boundedness of the trajectories in 'Bj on the whole real axis. Now assume by contradiction that there is a characteristic frequency iWjk which is not simple. Then there would exist one trajectory wj in 'Bj of the form wj(t) = tsin(wjkt + 9jd. Since such

It can be shown that all minimal kernel representations have the same Smith form: for this reason, the diagonal elements in such Smith form are called the invariant polynomials of '13: their product is denoted by X'B, and is called the characteristic polynomial of 'B. The roots of \:'13

180

1l'j

is unbounded, this is in contradiction with the oscillatory nature of 'Bj .

Definition :7. Let 'B E CV be an oscillatory system . and let 4> E JR';{"V[(,1]] . Then Q1' IS a zeromean quantity for 'B if

3. BILINEAR- AND QUADRATIC DIFFERENTIAL FOR~IS

wE'B

( w,

N

lh

~

= L

h,k=ll

( dt h

'1' )

io(T Q.dw)(t)dt = 0

A parametrization of zero-mean QDFs in terms of algebraic properties of the corresponding twovariable polynomial matrices is given next.

dk W2

h ,k dt k

1

_'1,

lim T- x

A b·ilinem· differential form (BDF) is a functional from (!::X: (]R, JR v,) x e: (lR, JRv,) to (!:X (JR, JR), defined as: L.dWI ' U'2)

==}

Proposition 4. Let 'B E Cv be oscillatory, and let R E JRVXV[~] be such that 'B = ker R({h). Then

.

cl> E JR';{" V[(, TJl is a zero-mean quantity if and only if there exist lit, X E ]Rvxv[(, '//1 such that

where cI>'.,k E ]Rv, xv, and N is a nonnegative integer. Let N

<1>(( , 11)

L

=

<1>( ( , '1]) = (( + 1]) lit ((, '1])

4>h,k(hll,

h,k= O

+H(()'l'X(( , I])

This two-variable 101 X 102 polynomial matrix 4>((, TJ) induces the bilinear differential form Lt, defined above.

A proof of this result can be found in Proposition 15 of (Rapisarda and Willems , 2004) .

A I3D F L is a symmetric two-variable polynomial matrix, i.e. if 101 = 102 and 4>((.11) = 4>(TJ,()T . The set of symmetric two-variable polynomial matrices of dimension TO x TO in the indeterminates ( and '1 is denoted with RSXV [(,711.

In (Rapisarda and Willems, 2004) it is shown that certain zero-mean quantities are such for every oscillatory system , i.e. their zero-mean nature has nothing to do with the dynamics of the particular system at hand, but follows instead from the fact that such quadratic differential forms are derivatives of some other QDF. We call them "trivially zero-mean QD Fs": a parametrization is given in Proposition 22 of (Rapisarda and Willems , 2004).

If is symmetric then it. also induces a quadratic functional acting on (!:oc (JR , ]RV) as

QoJ. : C:(JR. JR V)

--+

+ x (I]. ()'1'H(1/)

(!:"X(JR.JR) Example 5. Consider the single oscillator described by the different.ial equation m. ~ + ,,~W = O. The result of Proposition 4 allows us to conclude that the following are zero-mean quantities for

Q4(W) := L 4 (w, w) .

We call Q1' the qlwdratic differential form (QDF) a'Ssociated with <1>.

ker(m~ + I.,):

The association of two-variable polynomial matrices with I3DF's and QDF's allows to develop a calculus that has applications in many areas of systems and control (see (Willems and Trentelman. 1998) for a thorough exposition). An important role in the following is played by the notion of derivative of a QDF. Given a QDF Q .;., we define its derivative as the QDF Q. defined by

'111(1/ - k

1

= (( + 1/)2(( + TJ)m ~

>VI

-(m(

2

+ ,,~ ) . -1 - -1 . (1Jl.1]2 + I.,) 2

2

(( + TJ)'" = -(( + 1])m(1]

1>

'-v-"

d Q. (w) := dt (Q1'(W)) 1·

for all w E (!:X(lR,JRV) . In terms of the twovariable polynomial matrices associated with the QDF 's, the relationship between a QDF Q1' and its derivative Q. is expressed as

Observe that the first. of these zero-mean quantities is none other than the Lagrangian of the system, while the second one is evidently a trivially 2 zero-mean quantity, being .

1t",w

1·

4. A DETERMINISTIC EQUIPARTITIO~ OF ENERGY PRINCIPLE

•

{=:}

<1>( (, TJ) = (( +1] )<1>( (,1])

We end this section with the definition of zeromean quantity.

We begin this section by fonnalizing the notion of symmetry in a behavioral framework. As usual ,

181

we do it in an int.rinsic way, i.e. at. t.he level of the trajectories of the behavior (see (Fagnani and Willems, 1993) for a thorough riiscllssion of symmetries and representational issues in a behavioral framework).

U'm+2, . . . , W2m from Wm+l imply that there exists an F E IR (.,-1)" J [~l such that U'2 )

:

=

d

(

F( elt )wJ and

n=

(0 IlU)

Consequently, Q,t>(w) = Q..,,(wl! - Q..,,(wm+l), where t.he symmetric two-variable polynomial I{!' (( , 11) is defined as

= (1

FT(Cl) I{!((,I]) (

h~'

or equivalently, we consider systems wit.h 2m external variables U'i, i = 1, ... , 2m for which

(

1[1'(,1])

o

0

-IjI'((,II)

) E ]R2X2[( 171 ,

is zero-mean. In order to do so, observe that the projection of '13 on the ll'J- andu'.,+ J \'ariable 'Bw1,W"", := {(tl'J,wm+tlI3wi,2::;

11.'

E

'13

=

{=}

F~II»)

We now prove that the QDF induced by (3)

I .. 0

d

F( rll )wm+ I

tl'2m

1jI'((,I])

In the following we use the s.vmmet.ry induced the permutation matrix

=

( 'U'm

Definition ti. Let '13 fw a linear Jiffereutial behavior with w external variables, and let n E IR""" he a linear involution, i.e. n2 = I". '13 is caliE'd Ilsymmetric if 1l'B = 'B.

U'm +2 ) :

i::;

2m , i

1= I , ll

such that ('Il'l, . . . , W2m) Uti' E

'13

(4)

is oscillatory. Such behavior is symmetric with re.spect to tOm

\Ve now introduce the notion of ohservability. Let. '13 E C", with its external variable w partitioned as W = (Wl' W2); then W2 is observable from Wl if for all (WI,11'2), (11'l'W~) E'B implies '11'2 = w~. Thus, the variable '11.'2 is observable from WJ if WJ and the dynamics of the s.\'stem uniquely determine W2: in other words, the w\fiable WJ contains all the information about the trajectory 11' = ('I1'l,W2). An algebraic characterization of observability in terms of properties of the matrix R of a kernel representation of '13, and further consequences of this properly are given in (Polderman and Willems, 1998) .

Using the results of (Fagnani and 'Villems, 1993) conclude that such a system admits a kernel representation like

with ri E IR[~J, i = 1, 2. Observe that det(J(') = r~2 - r'.] is an even polynomiaL since '13 is oscillatory (see Proposition 2). Condude from this that and r~ are even polynomials. Use the fact that the second external variable is observable from the first one to eondude that there exist a, b E ]R[~l such that ar; +br~ = 1. Observe that. since r; and r~ are even, a and b can also be taken to be even polynomials.

r;

The main result of this communication is the following. Theorem 7. Let '13 be an oscillat.ory behavior wit.h w = 2m external variables. Assllme that '13 is [1symmetric, with n given by (3), i.e. (-1) holds. Moreover, assllme t.hat

Now let

r

E IRs[C 171, and define \: (C) '= 8[(0 , .., .

(a(~ .) b(O

-b(t:) ) -o(t:)

(fl) 1112, . . . ,11'.. , U'm+ J observable from U'J: and (b) Wm+2 , .. . ,U'2m observable from W.,+I·

It is a matter of straightforward manipulations to see that

Let IjI E IR., ,, m[c 1/], and consider the QDF Q'l' induced by the 2m x 2m two-variable matrix

J{r(_~)X(O + Xl(_t:)R'(~) = (a[d~) -()~(O)

<1>((, '11) := (

-

1{!((,l/) 0 ) 0 -1{!((,1])

It can be shown that such equation is equivalent with equation (;l); from this we conclude that Qr( wJ) -Qr( W2) is zero-mean. This concludes the proof of the claim.

on 'B . Then Qif, is a zero-mean quantity for 'B. Sketch of proof: We Hrs! reduce ourselws to the case of w = 2 in the following way. Symmetry of '13 and observability of lI'2, . .. , Wm from WJ' and of

Example 8. Assume that two equal masses III connected to "walls" by springs of equal stitfnt>Ss ~',

182

are COli pled togetitn with a spriug of stiffness k' . \\'e consider this as the symmetric interconnection, t.hrough the spring with e1ast.ic constant J.-', of two identical oscillators. each consisting of a mass III and a spring with elastic constant k . Take as external variables t.he displacements -utI and U'2 of the masses frolll their equilibrium positions: in such case t.wo equations describing the syst.em are (PWI III

=~:

dt 2

d2 tl'2 '111-dt 2

, (U"2 -

= k'(w)

-

Willems ..LC. and H.L. Trentelman (1998) . On quaJrati(; difft'H-'lltial forms. SIAM 1. Control Opt. 36(5) , 1703-1749.

U'J) - ku') 'W2) -

~:U'2

It is easy to veri(y that these equations describe a synmwtric behavior in the sense of Definition 6. From the result. of Theorem 7, we can conclude that t he difference between the kinetic energies of the two oscillators, represented by the twovariable polynomial llIatrix (

m(17 0

o

.' )

-m(,l/

is zero mean. Also the difference between the potent.ial energies of the two oscillators, induced by

(~~k ) Of course, this implies that on average, also the total energy of the two oscillators is the same.

5. CONCLUSIONS

In this communicat.ion we have stated the equipartition principle stated in Theorem 7 and given a sketch of its proof. Instrumental in such investigation is the behavioml framework , and the concept of quadratic differential forms. ~Jrther results in the direction outlined in this communication can be found in (Rapisarda and Willems, 2004). where also the concept of conserved quantit." is discussed. and a decomposition t.heorem for QDFs is stated.

6. REFERENCES

I3ernstein, D.S. and S.P. I3hat (2002) . Energy equipartition and the emergence of damping in lossless systems. Proc. 41st IEEE CDC pp. 2913-2918. Fagnani, F. and J.C . Willems (1993). Representations of symmet.ric linear dynamical systems. SIAM 1. Conf1·. Opt. 31(5), 1267-1293. Polderman, J.W . and J .C. Willems (1998). Introduction to Mathematic.al System theory: A Behal1ioral Approach. Springer-Verlag. Rapisarda, P. and J.C. Willems (2004). Conserved and zero-mean quadratic quantities in oscillatory systems. Submitted for publication.

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