SHORT PAPERS

large as 45°. The assumption that R = R2 which reduced (4) to theconvenient form of (6) is necessary for this result to hold. A morelengthy analysis shows that for arbitrary p, the minimum value ofIVm. occurs when

pRR.2 + I Z12(2p -1)R, + R I Z12(p - 1) = 0 (7)from which it is difficult to extract any useful information. Inciden-tally, the analysis here for p= a justifies the intuitive measurementtechnique of setting R= R. for a bridge of the type in Fig. 1, whenperfect balance cannot be obtained because of a small reactive com-ponent in Z. The simple analysis presented here is easily within thereach of sophomore students. The use of these concepts in a bridge ex-periment leads to a more lively and interesting experiment than isprovided by the usual dc wheatstone bridge.

Anke [4] and Biickner [5 ] have presented methods for calculating(1) for a system described by the differential equation

n dixE ai- =0

i-o dt' (3)

without solving explicitly and under the hypothesis that all solutionsof (3) approach zero as t-+ so. The present paper uses the results ob-tained by Anke and a transformation to calculate the more generalsynthesis objectives expressed by integral (2) above. More precisely,we consider here the following problem: Given a dynamical systemdescribed by the differential equation (3) and an associated set ofinitial conditions

diqi-odtxU_t-l = qi,i =i1) 2, * * * n

and that, further,

lim --=0 i=1,2, , ,

(4)

Time Domain Calculation of SystemPerformance Criteria

J. E. DIAMESSIS

Abstract-In the design of dynamical systems, described by thelinear differential equation !_0 ai(dix/dti) =0, integrals of the formIk=Jo°°tKx2dt, k= 0, 1, 2, * . are often used as criteria of perfor-mance. Itis then desirable to be able to calculate these integrals with-out having to solve explicitly the differential equation of the system.Anke [4] showed that this can be done for integrals of the formIo=Jo'x0dt. This paper uses the results obtained by Anke and achange of variable to calculate the more general performance cri-teria Ik=Jotx2dt, K=0, 1, 2, * .

I. INTRODUCTION AND PROBLEM FORMULATION

In the synthesis of general dynamical systems, and of control sys-tems in particular, a problem of central interest is the calculation ofcertain time integrals of the response of the system. These integralsexpress in mathematical language certain objectives of the process ofsynthesizing a given system. It is important to be able to calculatethese integrals in terms of the system parameters and the initial con-ditions of the variables without having to solve the differential equa-tion of the system explicitly.

One such integral, which is used very frequently mainly because itcan be computed rather easily, is of the form

1o = f x2dt (1)

where x may represent the difference, or error, between the desiredand the actual values of a system variable. To keep the error x smallby keeping Io small, is a commonly used synthesis objective. Yet inmany practical synthesis problems, a large error x is quite unavoid-able during the initial interval of the operation of the system, a factwhich can make Io large although the error x might become small astime progresses. This points to the desirability of a time-weighted in-tegral of the form

Ik =] tkx2dt, k = 1, 2, * * * (2)

which is less sensitive to the value of x during the initial interval ofoperation. While the usefulness of (2) has been recognized by manyinvestigators [1 ], it has heretofore been considered difficult to handleanalytically. Recently, methods have been developed for calculatingintegrals of the form (2) using Liapunov functions and state variabledescription of the dynamical system [2 ], [3 ].

Manuscript received August 1, 1966.The author is with the Department of Electrical Engineering, Villanova Uni-

versity, Villanova, Pa.

(5)

compute the integral (2) in terms of the coefficients ai and the initialconditions qi, without solving (3) explicitly.

II. ANKE'S METHOD

To illustrate the method developed by Anke [4], consider thesecond-order differential equation

a2 d+ a,- + aox = 02dt2 dt

with initial conditions

x(0) = qo,dx

ql.dt to

It is desired to calculate the integral

lo = f xsdt

in terms of ao, a,, a2 and qo, q, under the assumption that

lim x = 0 lim -=0.t--+00 t+t-os dt

(6)

(7)

(8)

(9)

Multiplying (6) successively by x and dx/dt and integrating the re-sulting equations from zero to infinity we obtain

a2 d xdt + a, - xdt + a) x2dt = 0~dFx -i-s dt+ f

c d2x dx o dx\2 C0 dxa2 J -dt J t d- dt+a = . (10)a2' 2 -dt+ao dt ox~jd1=

Evaluating the integrals appearing in (10) by parts and substitutingfrom (9), there result

d2x dx c¢ dx4A2 0(dx 20-xdt Jt J - dt qoql- dtdx 12

-Txdt - q-~JXd2x dx 12- dxdxdt = - Iqs.

dt2 dt 2 q2

Substituting in (10) yields

aol0 + a2J1 = aiqO2 + 2a2qoql

aJ =- aoqo2 - a2ql2, Jl = r dt- dt,

or, in matrix form,

(11)

(12)

[A]I] = [Q]a] (13)

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IEEE TRANSACTIONS ON EDUCATION, MARCH 1967

where

21] ']=1Q' a]=as1 (14)

L-0 al- Io L ° 0 -q12 a2

From (12) we can now find Io, the required quantity, in terms of ao,a,, a2 and qo, ql. In the case of (3), the form of the matrices [A], a]and [Q] is given in Anke [4].

III. THE CALCULATION OF IkWith the background of Sections I and II, we proceed now to de-

velop the main result of this paper, that is, an expression for the in-tegral (2). If we make the change of variable

x = 6-("/2)9ywhere v is a real positive number, (3) and (4) become

n diyE a,(a , v)- = 0i-O dti

anddiy q,(qi,sv), j=0,1. n- 1

diit-0e=

(15)

(16)

and

y(O) = qo = x(O) = 1dyId=i=-dt t-o d t -o 2 2

Using (12), we have for (23) and (25)1 fa, 1 v2 1 1v2 ao a/ 8 aoas ao 2

Then, from (21),

= f tx2dt = [dIO(W)]odv v,0and

12 = f t2X2dt Io(od2 v=o

(25)

(26)

(27)

(28)

where Io(v) is given by (26). Evaluating (27) and (28) we obtain

(17) and

where the new coefficients ai and the transformed initial conditions qiare functions of the original coefficients as and the original initial con-ditions q;.

Using now Anke's [4] results, that is (13), we can compute

lo(v) =fy2dt (18)

in terms of the a,'s and q,'s. Making use of (15), (18) becomes

io(v) etx2dt. (19)

Expanding e't in a power series in v yields, for (19),v2 pk co k

I0(v) = Io + VII + 2!I2 + ***+ k! 1k + * k(20)2! k k-0 V'!

where

Ik tkx2dt k =0, 1,*.

From (20) it is seen that

Ik f tkx2dt = -(p) 1 (21)o dvk J-0

Expression (21) shows that the integrals Jk, which express the syn-thesis objectives, can be obtained from knowledge of the system pa-rameters ai and the initial conditions qi. Equation (21) is particularlyuseful for higher-order systems and is believed to be a new result inthe synthesis of dynamical systems.

IV. AN APPLICATIONTo illustrate the method of Section III, let it be desired to calcu-

late the integrals I,=fo tx2dt and I2=fo t2x2dt for the system de-scribed by (6) and (7). Let

a2= 1, qo=1, q, = 0 (22)which is no essential restriction. Using (15), (6) and (7) become

d2y dy- +a,-+aoy= (23)di' dt

wherea, = a, -

p 2

aO = aO--a- +2 4

all + 2as2I11=- -

4ao2a,2

aj6 -aoal4 + 4ao3 + 4ao2a13 - 3ao2a,212 = _ _

4ao3a,3

(29)

(30)

The values of ao and a, which minimize (29) or (30) can be easily com-puted.

REFERENCES[1] D. Graham and R. C. Lathrop, "The synthesis of optimum transient response:

criteria and standard forms,' Trans. AIEE (Application and Industry), vol. 72,pp. 273-288, November 1953.

121 A. MacFarlane, "The calculation of functionals of a linear dynamical system,'Quart. J. Mech. and Appl. Math., vol. 16, p. 259, 1963.

131 J. E. Diamessis, 'A new method for calculating system performance measures,Proc. IEEE, vol. 52, p. 1240, October 1964.

[41 K. Anke, "Eine neue Berechnungsmethode der quatratischen Regelflache,"Z. Angew. Math. u. Phys., vol. 6, pp. 327-331, 1955.

[51 H. Buckner, 'A formula for an integral occurring in the theory of linear servo-mechanisms and control systems,' Quart. Appi. Math., vol. 10, 1952.

Demands on Engineering EducationP. A. LIGOMENIDES, MEMBER, IEEE

Abstract-Science will not set the goals for mankind. It cannottell us where we should go. We must decide that for ourselves. Therelation between a wandering humanity and science is like that be-tween Alice and the Cheshire Cat.

" Would you tell me, please, which way I ought to go fromhere?"

"That depends a good deal on where you want to get to,"said the Cat.

"I don't much care where... ," said Alice.

"Then it doesn't matter which way you go," said the Cat.

In our days, scientific views permeate modern life as a kind of sub-stitute for values and convictions which have been lost. People learnto "believe" in science and in engineering achievements, where theyused to trust in God. Everything is expected to be cured with the helpof science and engineering. Research is expected to abolish socialantagonism, economic depressions, broken marriages, frayed nerves,and maybe even war. Technological advancements will make ushealthy, free, harmonious and secure.

Manuscript received October 25, 1966.(24) The author is with the Department of Engineering, University of California,Los Angeles, Calif.

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