On the interpretation of the time-varying eigenvalues

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On the interpretation of the time-varying eigenvalues

M.A. Gutierrez de Anda, L.A. Sarmiento Reyes

Abstract In order to analyse the behaviourin small-signal regime of certain classes of dy-namic nonlinear circuits, dynamic linear time-varying (LTV) models must be considered. In anattempt to simplify the analysis of these models,time-varying eigenvalues may be used. In this ar-ticle, a simple interpretation of these quantities willbe given. Time-varying eigenvalues give informationon the growth rate of each of the normal solutionsof a LTV system in such a way that their dynamicsare effectively decoupled from each other. Some ar-guments are given to support this idea.

1 INTRODUCTION

In electronic circuit design, designers are confrontedwith the design of electronic systems which are in-herently nonlinear. The devices used in the con-struction of these systems inevitably display sometype of nonlinearity. Sometimes these nonlinear-ities are exploited in benefit of the design as it isdone in translinear circuits, for instance [1]. In themost common case, however, the designer has tominimise the undesirable effects introduced by thenonlinearities of each of the components of the sys-tem on the expected behaviour of the prototype un-

der design. The key to solve both problems lies onthe correct formulation of a suitable model whichblends those elements which play a fundamentalrole in the expected functionality of the circuit andthe unwanted elements which affect negatively itsbehaviour.

For most of the design paradigms that a designerhas to solve, linear time-invariant (LTI) modelsare usually enough to characterise the behaviourof nonlinear dynamic systems which will work insmall-signal regime. For other classes of systemswhich are intended to operate on relatively small

signals and show a linear behaviour (but not necess-arily time-invariant), or where the application ofLTI models to assess some aspects of their per-formance is formally not valid, other approaches tomodel their behaviour must be used. Mixers consti-tute the best example of circuits which may operatein small-signal regime and exhibit linear behaviourbut they cannot be represented with linear time-

Department of Electrical Engineering, Universi-dad Autonoma Metropolitana - Iztapalapa, email:[email protected], tel.: +52 (55) 58 04 46 33 ext.271 fax: +52 (55) 58 04 46 28

Department of Electronics, Instituto Nacional de As-

trofsica, Optica y Electronica, email: [email protected],tel.: +52 (222) 266 31 00 fax.: +52 (222) 2 47 05 17

invariant models to describe their frequency trans-lation properties [2]. Oscillators also constitute agood example of systems whose noise performancecannot be simply analysed using LTI models [3].

For the cases described above, LTV models maybe formulated to describe some or all of the aspectsof their small-signal behaviour. In general, it is noteasy to fully describe the behaviour of such modelsdue to the time-varying nature of its coefficients.However, a number of theories have been proposedfor determining analytically the general solution of

arbitrary dynamical LTV systems [4, 5]. The the-ories presented in references [4] and [5] are basedon the definition of quantities called either essen-tial D-eigenvalues or dynamic eigenvalues.

Although the terminology used in [4] and [5] dif-fers greatly, many of the concepts expressed thereare related [6]. Given the similarities betweenthese concepts, it is preferred to use the unifyingterm time-varying eigenvalues. Depending on themathematical framework considered for its defini-tion, the solutions proposed for an arbitrary dy-namical LTV system in terms of the time-varying

eigenvalues resemble those which can be formulatedfor a dynamical LTI system in terms of the solutionsof the characteristic equation for a scalar dynamicalLTI system [4] or in terms of eigenpairs for systemsof LTI differential equations [5].

The aforementioned theories have been used in anumber of design paradigms. In [7], for instance,the formulation presented in [4] was used to de-sign filters with time-varying bandwidth which canbe used in control applications. The concepts pre-sented in [5] were used in the assessment of the localbehaviour of a second-order oscillator implemented

using dynamic translinear synthesis techniques [8].Time-varying eigenvalues may be also used to val-idate the local behaviour of rather uncommon sys-tems such as the dynamic translinear circuit withchaotic behaviour presented in [9].

Although it is clear from [4] and [5] how the time-varying eigenvalues must be computed, it is notso evident how they should be interpreted unlikethe case of algebraic eigenvalues associated to LTIsystems. In this article, some arguments will bepresented in order to understand the meaning ofthe time-varying eigenvalues and which kind of in-formation they convey. The rest of this documentis organised as follows. In section 2, some funda-


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mentals of the theory proposed to formulate thesolutions in terms of time-varying eigenvalues forscalar LTV differential equations will be presented.Although this sort of equations can be used to de-

scribe input-output relations of single-input single-output systems only, the results are also extensiblefor LTV systems which are solved using the frame-work presented in [5]. Due to a lack of space, theseresults are to be presented in a separate publication.In section 3, a simple interpretation of these quan-tities will be given. In general, they give a measureof the growth in time of the normal solutions of aLTV system in the sense of Lyapunov [10] in sucha way that the rates of growth of each of the nor-mal solutions of the system are distinguished, evenin the case that linear combinations of normal sol-utions with different rates of growth are considered

for their formulation. Finally, in section 4 somefinal remarks will be made.

2 THE TIME VARYING-EIGENVALUES

Consider a homogeneous scalar LTV differentialequation of the form

y(n)(t) + an(t)y(n1)(t) + . . .

+ a1(t)y(t) + a0(t)y(t) = 0 (1)

wherey (t) is a function which depends on the timevariable t and a0(t), a1(t), . . . , an(t) represent time-

varying coefficients. The problem of finding a set oftime-varying eigenvalues 1(t), 2(t), . . . , n(t) as-sociated to the LTV differential equation (1) can bebetter understood if equation (1) is reformulated interms of a differential operatorDa of the form

Da = n + an(t)

n1 + . . . + a1(t) + a0(t) (2)

whereand stand, respectively, for the operatorsof differentiation with respect to t and scalar mul-tiplication, whereas a0(t), a1(t), . . . , an(t) are thetime-varying coefficients of equation (1). In ex-pression (2), n stands for the n-th derivative of

a scalar function and it can be seen as the compos-ition of n differential operators . Using operatorDa, it is possible to express equation (1) as follows

Da{y(t)}= 0 (3)

The differential operator Da of order n as givenin expression (2) can be expressed as a compositionofn differential operators of first order as follows

Da = [n(t)] . . . [2(t)] [1(t)] (4)

where 1(t), 2(t), . . . , n(t) are the time-varyingeigenvalues associated to equation (1) and repre-sents the composition operator. Intuitively speak-ing, equation (4) suggests that any n-th order LTV

system can be decomposed as a cascade of first-order LTV systems. This property is also foundin LTI scalar systems. However, there is a sub-stantial difference in the LTV case: given that in

general the composition of the operators present inexpression (4) is not commutative, the sequence inwhich the first-order LTV systems should be con-nected in order to build the original n-th order sys-tem cannot be changed. In the LTI case this ispossible since quantities 1(t), 2(t), . . . , n(t) canbe chosen as constants and therefore they can bepermuted freely in expression (4).

According to classical results presented in [4,11], if a set of n linearly independent solutionsy1(t), y2(t), . . . , yn(t) of equation (1) are known, itis possible to calculate a set of time-varying eigen-values 1(t), 2(t), . . . , n(t) as follows

i(t) = d

dtln

i(t)

i1(t) (5)

where i(t) and 0(t) are defined as

i(t) = det Wi(t) (6)

0(t) = 1 (7)

and Wi(t) is given by

Wi(t) =

y1(t) y2(t) . . . yi(t)y1(t) y

2(t) . . . y

i(t)

..

.

..

.

. ..

...

y(i1)1 (t) y

(i1)2 (t) . . . y

(i1)i

(t)

(8)

Notice that i(t) is nothing more than the Wron-skian of a set of i linearly independent solutions.Given a set ofi solutions of equation (1), it is poss-ible to formulate a new LTV differential equationof order i of the form

y(i)(t) + bi(t)y(i1)(t) + . . .

+ b1(t)y(t) + b0(t)y(t) = 0 (9)

whose solutions are given by y1(t), y2(t), . . . , yi(t)

provided that its time-varying coefficientsb0(t), b1(t), . . . , bn(t) satisfy the following setof linear algebraic equations

WTi (t)bi(t) = v(t) (10)

where bTi(t) = [b0(t), bl(t), . . . , bi(t)] and v

Ti(t) =

[y(i)1 (t), y

(i)2 (t), . . . , y

(i)i

(t)]. This statement will beused in the following discussion.

3 A SIMPLE INTERPRETATION FOR

THE TIME-VARYING EIGENVALUES

In this section, an interpretation for the time-varying eigenvalues will be given. Although the ar-


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guments which will be exposed here lack the necess-ary mathematical rigour to formulate a theoremwhich will cover the most general case, it shouldsuffice to get an intuitive idea of the meaning of

these quantities. To simplify the discussion, ex-pressions (6)-(8) will be applied in the computationof the time-varying eigenvalues for the second-orderdifferential equation with constant coefficients

y(t) +a1y(t) +a0y(t) = 0 (11)

If the following normal solutions of equation (11)y1(t) andy2(t) are used in the computation of a setof time-varying eigenvalues,

y1(t) = em1t (12)

y2(t) = em2t (13)

it turns out that1(t) = m1. This is not surprisingat all, because 1(t) is given by

1(t) = y1(t)

y1(t) (14)

and this expression can be readily identified as thelogarithmic derivative of y1(t). The computationof 2(t) is somewhat more involved. According toexpressions (6)-(8), 2(t) is given by

2(t) = d

dtln

det

y1(t) y2(t)y1(t) y

2(t)

y1

(t) (15)

In expression (15), independently of the choice ofsolutions y1(t) and y2(t), the Wronskian for equa-tion (11) is always the same and it is given by ex-pression [11]

det

y1(t) y2(t)y1(t) y

2(t)

= e

a1dt =e(m1+m2)t (16)

In consequence, 2(t) will be equal to m2.In the case that a set of linearly independent sol-

utions composed of arbitrary linear combinationsof functions em1t and em2t, the results still remain

valid. Assume that the solutions chosen for com-puting the time varying eigenvalues of equation (11)are now given by

y1(t) = Aem1t +Bem2t (17)

y2(t) = Cem1t +Dem2t (18)

where m1 > m2 and constants A , B, C and D aredependent on initial conditions given to equation(11) and also satisfy the relation AD C D = 0to guarantee that y1(t) and y2(t) are linearly in-dependent. For the given solutions, 1(t) is equalto

1(t) = Am1+Bm2e(m2m1)t

A+Be(m2m1)t (19)

It may occur that y1(t) may be equal to zero atsome time points and therefore 1(t) may be unde-fined at these points. However, for large t,1(t) willapproach m1. Given that the Wronskian of equa-

tion (11) is equal toe(m1+m2)t

,2(t) will tend tom2for large t. Therefore, the time-varying eigenvaluesallow the identification of the dynamics of each ofthe fundamental solutions which characterise equa-tion (11).

A closer examination to expressions (4) and (14)may help to further unveil the meaning of the time-varying eigenvalues. Equation (14) can be used topropose a homogeneous first-order LTV differentialequation of the form

y1(t) = 1(t)y1(t) (20)

whose general solution is given by

y1(t) = C e 1(t)dt (21)

whereCis a constant which depends on the initialconditions imposed on equation (20). Furthermore,equation (20) has the following associated differen-tial operator D1 :

D1 =1(t) (22)

Functiony1(t) may have any desired behaviour andtherefore any growth rate associated to it. If a sec-ond function with a different growth rate y2(t) is

added to y1(t) in order to form a set of linearlyindependent functions, it is possible to formulate ascalar LTV differential equation using the algebraicrelations given in expression (10) to determine itscoefficientsb0(t) and b1(t) of the scalar LTV equa-tion

y(t) +b1(t)y(t) +b0(t)y(t) = 0 (23)

If expression (4) is expanded for n= 2, the follow-ing differential operator is obtained

Da = 2 (1(t) +2(t))

+ (1(t) +1(t)2(t)) (24)

If the scalar function accompanying the differentialoperator in the differential operator associated toequation (23) and in expression (24), the followingrelation is obtained

b1(t) = 1(t) +2(t) (25)

As it is shown in [12,13], quantity b1(t) defines es-sential properties of the stability of the solutionsof equation (23). For any set of linearly indepen-dent solutions y1(t) and y2(t) of equation (23), theWronskian is given by

det

y1(t) y2(t)y1(t) y

2(t)

= e b

1(t)dt

(26)


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In the general case, if the scalar term accompany-ing the differential operator n1 in expression (2)and the expanded form of the differential operatorgiven in expression (4) are compared, the following

relation arisesan(t) = (1(t) +2(t) +. . .+n(t)) (27)

This result should be compared with the generalLTI case. In the LTI differential equation

y(n)(t) + any(n1)(t) + . . .

+ a1y(t) + a0y(t) = 0 (28)

the constant terman is equal to the negative of thesum of the solutions associated to the characteristicequation of equation (28). Similarly, when equation

(1) contains periodic coefficients with periodT

, thefollowing relation holds [12]:

T

0

an(t)dt= T(F1+F2+. . .+Fn) (29)

where F1, F2, . . . , F n are the Floquet exponents ofthe normal solutions of equation (1). Even for thecase of general LTV systems, a similar statement toequation (29) may be formulated in terms of Lya-punov exponents [13]. For regular LTV systems,the sumof the Lyapunov exponents associated toa set of normal solutions is given by

= limt

sup1

t

t

t0

an()d (30)

4 CONCLUSIONS

In this article it was shown that the time-varyingeigenvalues contain information on the growth ratesof the normal solutions of a scalar LTV system.These quantities can be formulated using sets oflinearly independent solutions whose growth ratesmay be influenced by the presence of a term witha dominant dynamic behaviour. The formulation

of the LTV eigenvalues guarantees that the dy-namics of each of the normal solutions of an arbi-trary scalar LTV system can be analysed separatelyby decoupling them from each other.

Acknowledgements

This work has been partially supported via aCONACyT grant under contract 42588-Y.

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On the interpretation of the time-varying eigenvalues

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