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An inverse problem of nonlinear regulator design is posed and solved. A. I. IXTRODUCTIOX. STROSG motivation for designing optimal system is that such ...
460

IEEE TRWSACTIONS ON AUTOMATIC

CONTROL, VOL. .kc-18,

NO.

5,

OCTOBER

1973

Nonlinear Regulator Theory and an Inverse Optimal Control Problem PETER J. MOI’LAS

AND

Abstract-Nonlinear optimal regulators are discussed, and some useful properties are isolated. An inverse problem of nonlinear regulator design is posed and solved.

I. IXTRODUCTIOX

A

STROSG motivation for designing optimal s y s t e m is that such q-stemstendautomatically t o have properties thatare dosirable fromthe vic\vpoint of classical criteria: stability, reduced sensitivity t o parameter variations, and many other such properties. I n particular, it is \wll-known [ I ]that if the linear timeinvariant single-input system k = Fx

+ gu.

(u2

control problems can be found in [3] and [4]. In [3]?more general perfornmnce indices are considered than those of this paper (n-hich are quadratic in the control, but not necessarily the state)! but on the other hand, the main results arerestricted t o systems possessing anoptimal performance that is quadratic in the state. In [4], some structural properties are determined for the optinlum control law of a problem of the class considered in this paper. 11. SOLUTIOS OF THE DIRECT PROBLEII Consider the system k(t) = f [ x ( t ) ]

(1)

is opt.imized n-ith respect t o t h eperformance index

J

BRIAS D. 0. ANDERSOS

+ x’Q3:) dt

to

t,hen. under mild restrictions on Q ! the resulting feedback system is a ‘Lgood” system by most classical criteria. An equa.lly significant result isas follom [ 2 ] . Given a feedback control 1a.n-21 = -X-’.? for the system (1): \vhich is I;nou-n t o yield an asymptotically stableclosed-loop system, then a necessary and sufficient condition for this control law t o be optimal with respect t o a performance index of the type (2) is that.

+ Gu.(t)

(4)

where 3: and u are! respectively! the system state andcontrol and take as a performance index I’[z(fu), -LC(-), to,

2’1

=

loT

+

( u ’ ( t ) u ( f ) m [ z ( t ) ] }dt.

(5)

(The superscriptprimedenotesmatrixtransposition.’) It is assunwd that m(s) is nonnegative forall x , that ?n(O) = 0, and that f(0) = 0. In the sequel we shall a.lso require at times the following assumptions.? -4sszmptiun I : The system (4) is completely controllable; that is! for any initial state x(f,J and any other state rl, thew exist$ a square-integrable control u( .) and a time il such that x ( t J = .zl. k y j d - F ) -IBj2 1 (3) dssun~.ptiun 2: The functions f( .) and m(.) have sufficient propertiesincludingsmoothncsssuchthatan hold for all real w. This is the so-called “return difference optimal control exists. and that thc optimal performance condition,” \ d l kno\vn in classical controltheory as a index satisfies the Hamilton-Jacobi equation. desirable condition on a linear feedback control q-stem. A4sszmptiun 3: The free s?-strnl The purpose of the present paper is t o present corresponding resultsfor a class of nonlinear systems. We restrict ouri ( t ) = .t’[.?(t) ] y ( t j = IT1 [.r(f)] selves totinwinvariant systems,mostfrequentlywith the sense that .~n[x(t)]= 0, time-invariant control l a w . With greater notational com- is completely obserrable! in for all t E (fl,t2) with ti and t2 a r b i t r q - . implies r(t)= 0. plexith- and n-ith man>- more ad hoc assumptions. princi=Ilso ? I ) ( .) is such that the optimal performance index pally in respect t o existence of double limits. time-var?-ing approaclm infinity as iI.r(t,) approaches infinity. systems may be considered. and obvious gen(~ra1izations of Under ,&.sumption 2 . the optimal feedback control lawthe time-invariant results nil1 be obtained. is seen from the standard Hamilton-Jacobi theor?t o be We donot!however, restrictconsideration t o singleinput systems. Other results on nonlinear inverse optimal

11 +

/I

3Ianuscript received October 3: 1972. Paper recommended by E. Kreindler, Past Chairman of the IEEE 8-CS Optinlal Systems Committee. Thk xork was supported by the riustralian Itesenrch Grants Committee and the ShortlandCount?- Council. The authors are with the Department of Electricnl Engineering. University of Xewcastle, Sewcastle, X.S.lf-., 230s -4wxrali:I.

1 The matrix G, though assumed constant here, may, with wly minor smoothness assumptions, beassumed dependent on x. Further, the tern1 u ’ ( t i u ( f ) in (..?) can be weighted by npositive definite snlooth function of x with minor adjustment to later arguments. Althoughthese as~lmptionsare writtenin global terms, it wm1ld be reasonable to restrict them to local regions in which the state \\-as known to lie on n priori grounds.

46 1

REGULATOR THEORY

ARANDERSON: hIO€LAN AND

U* =

-$G’V+(x,t,T)

(6)

where +(x,t?T)is the solution of

!%! + v’+j(z) dt



$v‘+GG’v$

+ m(z) = 0

(7)

with boundary conditions +(O,t,T) = 0 for all t E [to, TI and +(x,T,T) = 0 for all x. Moreover,

+[z(b),fo,T] = inf V [ z ( t o ) , u ( . ) , t o , T ] . u(.)

Of special int.erest is the case T = m in ( 5 ) . Theorem I : WithAssumptions 1 and 2 holding, the optimal performa.nce index when T = a is given by

andthe optima,l cont,rol is givenby ( 6 ) , with +(x,t,T) repla,ced by $(x). Moreover, $(x) sat.isfies (7). Proof: Following arguments as for the linear-quadratic problem, due t o Kalman, see [I, p. 331 and [ 5 ] , monot,onicit; of +(z,t,T)as a function of T with fixed z and t can be shown. and the boundedness for all T follows from Assumpt.ion 1. Consequently the limit. on t,he right side of (8) ex&, and is easily checked t o be independent.of t. That, $(x> is in fact. the opt.inlal performance index is easily est>ablished, as in[ l , pp. 34, 3.51 and [ 5 ] . i. The corresponding resu1t.s for nonzero t.ern1ina.l wighting are considerably more difficult t,o obtain, although it is clear that one othersit.uat,ionfor n-hich Theorem 1 remains true is if a termina.1 n-eight.ing n [ r ( T ) added ] on to the right side of ( 5 ) is a steady-state solution of (7). [Theboundarycondition +(z,T,T) = 0 is replaced by +(z,T,T) = n [ z ( T ) ]= $ ( x ) . ] However, the presence of n ( x ) isirrelevant. if t.hc optimal solution in Theorem 1 leads t o a n as;mpt.oticallg stable system. Sufficient conditions for t.his tobet.ruearegiveninthe following theorem. Theorem 2: Under t.he conditions of Theorem 1: t.0gether with Assumption 3, t.he optimal closed-loop syst.em is globally asgnlptot.ically stable. Proof: The proof precisely parallels t,hat. used for the linear-quadratic problem [l: p. 411 a.nd [.I. For from (6) and (7), ‘the rate of change of 4(.z(t)) along an; opt,imal traject,ory is given by

-

~d4(z(t))

clt

-

V’$(x) [f(z) - $GG’V$(z)]

=

- ? n ( ~ )- $V“$(X)GG‘V$(X).

probably easier in most cases tlo check for controllability by simple inspection (perhapa, in a given example, from physical reasoning), while the observability requirement maybecircumvented, if necessary, bymaking 172(x) positive definite.

111. THERETURK DIFFEREKCE COXDITIOIV In the remainder of this paper, we shall be concerned withfeedbackcontrol1a.m for the syst.em (4) witch a property to bedescribed as the return difference condit,ion (RDC). The reason for this name is that, for linmr s ~ s tems, the condit,ion reduces t o t h e inequality (3). The precise st,atement of this condition is as follows. DeJinition.:rl function k(x) of t,he stateof the system (4) is said t o sat,isfy t,he RDC if, zcken.ezw x(&,) = 0,

Lom+ [u

Lrn

+ k(z)] rlf 2

k(z)]’[u

u’zc clt

(9)

for. all sqzme inteyrable u(.) such that x( a ) = 0, tchere x( .) i s the solution trajectory of (4). Sote that theinequality (9) is only required t o hold for zero init,ial states;nodirectcondition is placedon the response of (4) for nonzero a(fo),although of course the response in t.his caseis implicitly constra.ined by specifying the response for x(to) = 0 and arbitra.ry u(t). 4 s remarked above, in the linear s>-stem case, (9) reduces t o ( 3 ) . This is most. easily seen byconverting (9) t o a frequencydomaininequality.usingParseval’s t.heorem, and then making use of the essentially arbitrary nature of u(.).Note that. just as (3) is a condit.ion on the input.output behavior of a plant, and independent, therefore of initial conditions, so is t.his truc of (9). One can construct. physical interprct,a,tions for (9). For example, one could argue that (9) implies that a feedback law of -k(a) constitutesanegativefeedbacklau-in a sensesomewhatakin t o t,hat of classical control:with uextdenot,inga.n external control applied to thesyst.em (4), and n-ith -k(x) simultaneouslyafeedbackcontrol,one has u e x t = ZG k ( ~and ) ~ (9) implies that u(.) is smaller t.han ueXt(.)as measured bp a.n L2 norm. In other words, t.he feedback cancelsout. some of t.he a.pplied cont.rol. The importance of thc RDC is phon-n in thc follou-ing t,heorem. Theorem 3: For t,he system

+

+ Gu(t)

a(t) = f [ x ( t ) ]

(10)

the asymptotically st,able control law Bssumpt,ion 3 then implia that this dericat,ive can never u(t) = - k [ . x ( f ) ] (11) be identically zero, so t*hat.+(x) is a Lyapunov function for the optimal syst.em [6, p. 661. E optimal is t.he for problem of minimizing, subject t o Kote, hon-ever, that verification of -4ssumpt.ions 1-3 1-lmT.+rnz ( T ) = Ot a performance index of the form may be far from trivial; a considerable amount of resea.rch T is stillbeingcarriedout,on the obserra,bilit,yand con~ { zu ~ , j = Em [nz(z> u f u ]c ~ t (12) t,rollabilit,y of nonlinear systenx [i1-[9]. In practice, it. is T-i

S,

0

+

462

IEEE TRANSACTIONS O N AUTO~IATICCONTROL, OCTOBER

with m ( x ) nonnegat.ive forall 1: if. and only if, k(s) sat.isfies t.he RDC. Proof of Necessity: Suppose that, for somr nonnegative function m ( x ) , the control law (11) has been found t o be an optima.1 control for the system (10) that m i n i m i z e s (12). andthat B(x) isthe associatedminimum. Then 6 ( x ) satisfies ( i )which . may be rewitten as r n ( ~ )= $V’~(X)GG‘VB(.E) - V’~(X)~(X). Kon- for any u(.), not necessa.rily optimal, but for n-hich 1imt+- r(t) = 0, n-e have

[ ~ z ( x+ ) U ‘ U ] dt

=

Im [tv’BGG’V$

- V‘~(.T)

+ u‘21] dt = {(u + $G‘V4)’(u + +G’V$) to

1973

of the control Ian- u ( t ) = stabilityandstationarity - k [ s ( t ) ] ! use of any u E CTT (for any T ) vi11 lead t o lim r(t) = 0, so that C.TTis a subset.of the class of cont.rols

‘--

for which (9) holds. Consequently,

ICT

s,

T

2

u.’u dt.

Sote, incidentally, that the above lemma requires no direct restriction on s ( T ) . I t obviously is of independent interest, since it. ma.l-

(1;s)

d - [ B ( z ( t ) ) ] = V’$~(X) V‘BGU. dt

+

for all T 2 to. The reason for describing (1.5) as a passivity condition is Recalling from Theorem 1 that k(x) = +G’V$.we obtain a-s follom. Consider the system (14) 1xit.h input u 1 and zcchenerer x([,,) = 0 and x ( a ) = o? nith output yl = k ( x ) . Then, thinl&g of u 1 as, say, a volta.ge a.nd y1 as a Current., (15) is the condition that, the (21 k(x))’(u k(.E)) clf system be passive, i.e.. that xhen initiallyunexcited. it acts as an rnergy sink. Passivit>-has provrd an important for. all u(.),where x ( .) is the solution trajectory of (4). This completes the proof that the RDC is neccssa.rily concept in, for cxamplc. stability theoryof control systems. satisfied by an optimal system. Theconverse, namrly. that see, e.$.. [IO],and WP shall rcturn t o this in SectionT-. Inequalit>- (15) has a natural and important rxtension satisfaction of the conditionimplies the existence of a nonnegative function m ( - ) in (12) such that (11) is opti- t o cope with the case of a nonzero initial statcl, given in mal. is rather more difficult t o prove, a.nd it will first. be Lemma 2. -4version of this lcmma for the linear-quadratic [Ill. \\-here conncctionswithoptimal necessary t o establish some preliminary results conccrning caseappearsin properties of system? for which theRDC is satisfied. control a.nd stability arc noted. For our purposes though, To begin v-ith: it will be necessary t o hare a finite-time Lemma 2 is simply a tool for proving the sufficiency statement of Throrcm 8. version of the RDC. This inequality can be extended t o cope x-ith thecase of Lemma 1: If k ( s ) is such t,hat the RDC holds and such a nonzrro initial &atein the follox 0. (17)

k(x).

Nowconsider t,he following finiteperformanceindex associated mit,h (10) :

Assume there is sufEcient smoothnesst o apply the Hamilton-Jacobi theory. Then, using the bound (16), one ca,n V[x(td,u(.),t~,TI = $[z(T)l conclude t.he existence of a n optimal control law uI*(-) a.nd associated performa.nce index = -+,(zo,O,T) 2. 0 V[ZO,U~*,T,E]

=

+

T

[u’u

+ m(x)I c k (23)

Theoptimalperfornlanceindexisobtainedvia Hamilton-Jacobi theory as the solutionof

where the minus sign is a not.ationa1 covenience, and ,-,.) satisfies t.heHamilton-Jacobiequationwith v’+f(z) - $v’+GG’v4 - V’$f(x) appropriate bounda.ry conditions.Next., one can argue tha.t !! dt +c is monotoneincreasingwit,hT and bounded, a.nd accordingly, existenceis guaxant.eed of with the boundary conditions $€(x) = lim +€(x,t,T)5 0.

+E(.

+

+ $V‘$GG‘v&

the

=

0

T-m

Furt.her, following argumentslikethose of Theorem 1, $,( .) satisfies t.he limiting Hamilton-Jacobi equation

+

V ’ $ , [ ~ ( Z) + G ~ ( s ) ] E-’[~c(x)- $G’V+,]’

. [ k ( ~-) +G’V$,]

=

0. (18)

Define m , ( ~= ) -V‘+6[f(x)

- +Gk(x)].

(19)

From (lS), we see that m,(x) 2 0 for a.ll x. Next, one can show that +,(x) is monotonic increasing w-ith E , a.nd bounded. Accordingly, there exists

$(x)

=

lin14e(x) 2. 0

(20)

e-0

and, assuming3 lime_t0 V+,(x) = V [lim,,o

&(x) ]

=

It, is clear that 4(x,t,T)

= &(x) isonesolution of this equat,ion, a.nd t.he smoothness assumptions imposedso far imply that it, is the unique solut,ion. Letting T approach infinity in (23), and using the a.sympt.otic stabi1it.y of the contx-ol law u ( t ) = - k [ x ( t ) ] , it follon-s tha.t with m(x) = &(x) in (12), t,he nlinimum value of (12) is

+(x)

=

lim $(x) T-

=

$(x)

c?

and obviously &(x) istheoptimalperformance index. That t,he optimal control is u = - k ( x ) now follows immedia.tely from (22). 0

V$(x),

The full significance of this condition escapes us. It. would seem that &(x) is analytic in e, and t,hat the assumpt.ion of sufficient smoothness on f(. ) and .nz(. ) aould imply smoothnm of &(x) as a funct,ion of 2. But t.he joint smoothness in E and x necessary here seems harder toprovide for, via conditions on f( . ), etc.

TT.

PROPERTIES OF OPTIMAL SYSTENS

A keydevelopment in the preceding proof was the observationt,hat. (14)represent,sapassivesystem. It is

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I OCTOBER E E E TRANSACTIONS CONTROL, O X -4DTObIATIC

possible t o extend result this further, as shownthe in follon-ing lemma. Lemma. 3: Let #( .,t) be any time-varying gain or nonlinearity such that

1973

VI. COSCLUSIOW

In this paper an attempt has been made t o isolate those properties of a control systrm that are linked to a certain type of optimality, and it has been shown that a return difference condition, closely related t o loop gain concepts #(O,t) = 0, for all t. inlinearsystems,providesacriterionfor deciding the optimality or otherwise of a feedback law. It. should be noted that although the proof of optinlality of a control lawas gicen here is a constructive one, itdoes Then if x(!) is a solution of not normally lead t o computational1;v attractive construct.ion procedures. Also, only one of the man>- performance indices for which the control law is optimal has been presented. Bn example where the procedure can be reasonably where used t o const.ruct the performance index is inthe case of a 1inea.r system n i t h linear feedback. This case is! of course, included in the present results. It is vital to the results of this paper that the control then the RDC implies t,hat, appear quadraticall>- in the performance index; if a more general class of loss functions isconsidered, properties such as stability and t.olerance of input nonlinearities could no longer,in general, beguaranteed.(Inparticular,the inclusion of cross products between the control and the for all T 2 to? and any square-integrableuext(.). states would allon- the trivial result, apparentlyfirst noted Proof: The proof follows directly from the inequality in [ 2 ] , that any control lam- isoptimal.)Obviouslyour n results estrndtothc case \\-here inthe loss function, (13). Since the conditions (24) include the special case u’(t)u(t)is rrplaced byu’(t)Ru(f), with R a positivedefinit.e # ( ~ , t ) = U, Lemma 3 shows thatanoptimals>-stem is matrix; however, it is also interesting t o speculatr, on the always passive in thesense defined above.Howevrr. it also basis of the rrsults in[ E ] ! that many of the results of this shom that,t.he system remainspassive when a wide ra.nge paper could br extended t o t h ecase of a more grneral loss of nonlinearities is introduced into the feedback loop. function of the form r(u) m(.r)?n-ith sonw positivity If condition (24) is strengt.hened slight.ly t o constraint on ~ ( u ) . 3

+

REFERESCES [I] B. D. 0. Andelxon and J. B. Moore, Lirzrar Optimal Control Engles-ood Cliffs, N.J.: Prentice-Hall, 1971.

for somepositivcconstants el and E ~ ,t.hen the closed-loop R , E. Kalnlan, ‘ii+len is a linear control system optinlall,~l systemobtainrd by set,ting u = - # ( k [ x ( t ) ] ! t \ is also T r a m . ASJIE, J. Basic Eng., ser. D, pp. 1-10, Mar. 1964. [3] F. E. Thau, “On the inverse optimum cont.rol problem for a (Lyapunov) asymptotically stable. The proof, using class of nonlinear autorlon~ouas\-stems,” I E E E Tratrs. Automat. +[x(!)] asaLyapunorfunction,maybefoundin [E]. (-1 Confr., vol. AC-12, pp. 674-661, Dec. 1967. [4] K. Yokoyama and E. Kinnen, “The inverseproblem of the partialrcsult also appears in [3].) Since the closed-loop opt.inlal regulator,” ZEEE T r a m . d u f o ~ ~ aCo)itr.: t. vol. AC-15, pp. 497-.io4, Aug. 197”. system corrrsyonds t o setting uext(tj=0 in Lemma 3. the [5] R.E. Kalman, ‘Contributions to the theory of optimal control,” stability is simply Lyapunor stability of a passive system, Rot. SOC.X a t . X m . , pp. 102-119, 1960. whch is t o be expected a$ a consequence of thc passivity [6] J. La Salle and S. Lefschetz, Stability by Lyaptiizol,’s Direct X e t h o d . S e a Tork: A4caden1ic,1961.‘, proprrty. [7] E. W . Griffith and K. 6. P. Kumar,obsewability On the of In the linear quadratic problem, it is n-ell knon-n that the nonlinear systems: I,” J . X a h d , r a l . d p p l . , vol. 35, pp. 135147, July 1971. RDC implies thattheoptimal closrd-loop system ex[8] W. W. Schmaedeke, “The existence theolr of optimal control svstems,“ in A d m n c r s it{ Corltrol Systcnzs, vol. 3 , C. T. Leondes, hibits a l o w r sensitivity t o parameter variation than thr Ed. S e x Tork: Academic: 1066. equivalmt opcn-loop system. scr. e.g., [l]. The analogous [g] E. G . U b r e k h t and S . S . Krasovskii, ‘ T h e observability of a resulthcrc is morecamplicatcd.Thissensitivityimprovenonlinear controlledsystem in the neighborhood of a given motion,” d u f o m a t . Rcmofr Colrtr.? vol. 23, no. 5, pp. 934-9111, nlent follows in caw H z , 2 0, ~ h e r H c is thc Hamiltonian 1961. Of the s)-stcnl. ere [11];for nonlinear f(z),this condition is [lo] G. Zames, “On the input-output stability of time-varying nonlinear feedback syatems-Yarts I and 11,” I E E E T r a m . a w h - a r d , ini-olving thccostatevrctor. Ho\vrver, for Automat. Cottfr., vol. AC-11, pp. L’L‘X-%S, Apr. 1966, and vel. lincar f i r ) = F.r. it bwonws nz,, 2 0, and the sensitivity AC-11, pp. 1 6 . - A i 6 , JUIJ-1966. inlprovc,mcIlt call be thought of via a [11] J. C . Willems, ‘‘Least squares stational?. optimal control and the algebraic lliccati equation.’‘ I E E E Tram. Aufonlat. signal, or linearized. wrsion of the RDC. Contr., vol. XC-16, pp. 631-634, Dec. 1971.

IEEE TRANSACTIONS ON AUTOUTIC

CONTROL, VOL.

~018,NO. 5,

OCTOBER

B. D. 0. Anderson, “Stabilityresults for optimalsgstems,” EZeetron. Lett., vol. 5, p. 545, Oct. 1969. R.W. Bass and R. F. Webber, “Optimal nonlinear feedback controlderived from quart.ic and higher-order performance criteria,” IEEE Tram. Autond. Contr., vol. -4C-11, pp. _ _ 448454, July 1966. E. Kreindler, “On sensitiviB of closed-loop nonlinear optimal control syst,ems,” J . S I A X Contr., vol. 7, pp. 512-520,Aug. 1969.

Peter J. Moylan (M’73) was born in Hamilton, Victoria, Australia, on February 24,1948. He received the B.E. degree from t.he University of Melbourne, Melbourne, Australia, and the 31.E. degree and, in 1973, the Ph.D. degree both from the Vniversity of Newcastle, Pieacastle, Y.S.W., -4ustralia, all in electrical engineering. Since 19i2 he has been a Lect.urer in Electrical Engineering at, the University of Newcastle. His current. research interests are in

1973

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the fields of control of nonlinear systems, pattern recognition, and network synthesis.

Brian D. 0.Anderson (S’62->1’66) was born in Sydney, Aust,ralia, in 1941. He received bachelor degrees in pure mathemat.ics and electrical engineering from the University of Sydney,Sydney, Australia., andthePh.D. degree in electrical engineering from Stanford University, St.anford, Calif. Before joining the University of Newcastle, Nemastle, K.S.W., Australia, in 1965, where he is nox a Professor and Head of the Department of Electrical Engineering, he was a Facult,y Xember of the Depart,ment, of ElectricalEngineering a t %anford University and St,aff Consultant. at VidarCorporation, Xount,ain View, Calif. For the first six mont,hs of 1970 he was a Visit,ing Professor at. Southern 3Iethodist Universit.y, Ilallas, Tex. His research interests are in control, communication systems, and networks. He is coauthor of two books, Linear Optimal Control (w7it.h J. B. Moore) and :\‘efwork dnalysia a.nd Synthesis (wit.h 6. 1:ongpanitlerd), both published by Prentice-Hall.

Behavioral Consistency Tests of Econometric Models ALFRED L. NORMAN

AND

Absfracf-Most econometric models have many versions which differ from one another by 1) minor variations in the speciiication, or 2) choice of parameter estimator. This paper demonstrates how control theory can be employed to discriminate between alternative versions of an econometric model. For many policy problems involving a simple objective function and a single control variable, the investigator can hypothesize from the tenets of a given economic theory qualitative characteristics of the optimal economic behavior. By computing the optimal economic policy, the investigator can determine which versions are consistent with the hypothesis. This procedure was employed to determine which versions of a revised form of the Klein-Goldberger model are consistent with three hypothesesderived from economic theory. Forthefirst two hypotheses, the results indicate that consistency depends on the choice of parameter estimates. For the thudhypothesis, an inflation test at full employment, the results indicate that all versions are inconsistent with the hypothesis.

MORRIS R. NORMAS

I. INTRODUCTION

T

HE alternat,ive versionsof an econometric model ca.n bethought. of as a. set whosemembers differ from one another by variations in the model specification or choice of paramet.er estimator. One of the more difficult, problenls in econometric model const.ructionis determining which member of t.he set best rcprrsents the actual econonly [3]. As t,he t.ime interval between economic obserwtions is long, the possibility of quickly determining the best. member solely on the basis of predictive capacit5- is remote. As a consequence, t,ests d i c h can discern defrct.s in an econometric model in t,he absence o f forecast data a,re important. in economet.ric model e d u a t i o n . One category of such procedures is t.ests based on confronting themodel n-it.ha priori information. I n developing >lanuscript received February 18,1972; revised Kovember 6,1972. aneconometric model, the econometrician specifies the Paper recommended by I). Kendrick, Past Chairman of the IEEE S-CS Economic Syst.ema Committ.ee. This xork was supported in form of the model on the basis of economic theory. Estipart by the Economic Research Cnit, Univelyity of Pennsylvania, mates of the parameters are thrn determined by a statisPhiladelphia; t.he Comput,er Center, Univewit,y of Minnesota, But: as certain typcls of a Minneapolis; the Computing Cent.er, University of Texas, Austin; ticaI estimationprocedure. and PiSF Grant GS 32271. This xork combines various aspects of t,he pl*io~.i information are verJ-difficult t o incorporatr directly Ph.D. dissertations of the authors. A. L. Norman is wit.h the DeDartment of Economics. Cniversitv of intothe specification or estimation process! the cconoTexas, Austin, Tes. metricianusessuchinformation t.o evaluatc the model M . R. Xorman is with Econonletrica Int.ernationa1, Sant.a Barbara, after the estimates of the unknown parameters have been Calif. - I