by (2) J , ( K ) achieves its minimum at K* = 1.435, and the corresponding J: is H(F, G, S) = C ( S ~ - A - BI.?-’BG (4) 0.929 x i .

is nonsingular and can be made diagonal by interchanging rows and columns. (This dehition is equivalent to previous ones.)’.2 It is im- mediately obvious from (4) that C and B must be of rank n1 (G must be nonsingular) for H(F, C, s) to be nonsingular.

The requirements that r(B) = rn and r (C) = rn are generally fundamental assumptions for any S. If r(B) < rn, then some inputs are redundant and can be discarded. Similar comments apply to the case where r(C) < M.

CLAYTON R. PAUL Dep. Elec. Eng. Purdue University Lafayette, Ind. 47906

EXAMPLE 2

Find a linear feedback control law for

i = - x + ZI, x(0) = x.

such that the cost functional

J, = Joz (24r2x2 + U2)dl

is minimized. Applying Theorem 1’

u* = -x , J: = x;.

Again, the same problem is solved by parameter optimization. Let u = - K x , for any K such that the closed-loop system is asymptotically stable. Then

i = -(1 i K)x, x(0) = x0.

Comments On Of Linear Regulators Optimal for applying Parseval’s theorem, or by carrying out the integration directly. The corresponding J,(K) can be found either by applying the Lemma,’ by

Time-Multiplied Performance Indices” J,(K) is found to be

In the above paper’ and another [I], a procedure for design of linear feedback controls for linear time-invariant systems with time-multiplied quadratic performance indices has been presented. The equivalence between time-multiplied and constant weighting matrices shown in [2] is used to replace the timeweighting matrix by an equivalent matrix. The conventional procedure for deriving the optimal control law [3], [4] is then applied to obtain the linear feedback control law. However, it appears that the solution so obtained may not be optimal. Consider the following examples.

EXA~WLE 1

Find a linear feedback control law for

1 = - x + u, x(0) = x.

such that the cost functional

J , = (12rx2 + u2) dt

is minimized. Application of the result of Man and Smith’ leads to u* = - S,x, where,

from (14) and (15) in Man and Smith,’ S , satisfies the following equations:

-2s, - st + s, = 0

2(-1 - S,)S, + 12 = 0.

The positive real solution of this equation is S, = 1. The linear feedback law and the corresponding J , by the method of Man and Smith are

u* = -x, J* - 1 - xo.

Now the same problem is solved by parameter optimization. Let u = - K x . The system equation and the cost function then become

i = -(1 + K)x, x(0) = x0

J’(K) = Jrn (12t i K2)x2 dt 0

= ~om(12c + K2)X;exp[-2(1 i K ) r ] d t

- (1 + K ) K 2 + 6 2(1 + K ) ,

- X;.

Force under Grant AFOSR-68-1579B, the Joint Services Electronics Program under Manuscript received June 2, 1970. This work was supported in part by the U S . Air

Contract DAAB-07-67-C-0199, and the NSF under Grant GK-3893. F. T. Man and H. W. Smith, IEEE Tram. Aufomar. Confr. (Short Papers). vol. AC-14.

Oct. 1969, pp. 527-529.

J , ( K ) = 12 + KZ(l + K)’

2(1 + K)3 4

J , (K) achieves its minimum J ; = 0.833 xi. at K* = 1.554

CONCLUSIONS In both examples, the proposed design procedure in Man and Smith’

and Man [ 11 fails to minimize the given cost functionals. The reason for this failure appears to be the fact that there is an implicit relation between Sk and the feedback matrix K in (19),’ whereas the result of [3] which has been applied in Man and Smith’ and Man [I] is based on the assumption that the weighting matrix for x is independent of K .

CHESG-I CHEN Coordinated Sci. Lab. University of Illinois Urbana, Ill. 61801

REFERENCES [ I ! F. T. Man, “A computational scheme for optimal linear regulators relative to time-

welghted quadratic performance indices.” f. Mafh. AM/ . Appl.. vol. 29, Mar. 1970, pp. 58&588.

[2] A. G. J . MacFarlane. T h e calculatlon of functionals 01. tlme and frequency responses oca linear constant coefficient dynamical system,” Quarr. 1. Mech. Appl. Math.. “01. 16.

[3] R. E. Kalman. “Contributions to the theory ofoptimal control,” Bo/. SOC. Marh. ”e., pt. 2, May 1963. pp. 259-271.

[4] M. Athans and P. L. Falh, Oprima! Control. New York: McGraw-Hill. 1966. 1960. pp. 102-119.

Comments on “A Method to Determine Whether Two Polynomials Are Relatively Prime”

In the above correspondence’ the authors present what they claim to be an improved method for determining whether two given polynomials contain any common factors. The standard procedure for making this determination, which has been well known by algebraists and network theorists for years, is of course the use of Euclid’s algorithm.

The application of Euclid’s algorithm for this purpose is fully described in Bother,, where the condition involving the vanishing of the resultant

Manuscript received July 13, 1970. ’ W. G. Vogt and N. K. Box, IEEE Trans. Aufomaf. Conrr. (Corresp.), vol. AC-15,

M. Bocher, Imroducfiorr fo Higher Algebra. New York: Dover, 1964, pp. 191-196. June 1970. pp. 37!%380.

TECHNICAL NOTES AND CORRESPONDENCE 113

is also derived. The resultant is merely a compact theoretical way of stating the necessary and sufficient condition. Because of the tedium associated with the evaluation of a large determinant, in practice one conventionally reverts to the direct use of Euclid’s algorithm, which is the familiar con- tinued-fraction expansion, well known to network theorists in connection with Cauer synthesis.

The suggestion by Vogt and Bose’ to employ the Cayley-Hamilton theorem to simplify the calculations involved in their procedure leads to a sequence of operations which in fact partly parallel the operations involved in carrying out Euclid’s algorithm. The method of Vogt and Bose is more cumbersome and unwieldy, however, because of their introduction of matrix techniques.

In order to provide some comparison, Euclid’s algorithm was applied to Example 1 in Vogt and Bose. It almost immediately revealed a misprint in the text. The functionf(B) should be

f ( B ) = [34B2 + 59B + 251

not

f(B) = [34B2 + 59B i 2511.

Their succeeding results appear to be correct, however. Proceeding by Euclid’s algorithm, one easily finds that the factor common to the two given polynomialsf(l) and g(i) is d + 1.

R. E. MORTENSEN Dep. Syst. Sci. University of California Los Angeles, Calif. 90024

Authors’ Reply3 There are several shortcomings in Mortensen’s criticisms of the author‘s

recent paper.’ First, it is repeated that the method suggested by the authors is a direct test solely in terms of the coefticients of the two polynomials, which have to be examined for their relative .prime nature. The only other direct method solely in terms of the coefficients is the method of the resultant. Of course, the Euclid algorithm or the divide-invert-divide method, which is used in Cauer synthesis and which can be used to test for whether two given polynomials are relatively prime, is known even to most undergraduate students. However, the algorithm cannot be interpreted as a direct test in terms of the coefficients. Attention is directed to a very good paper by Fryer: where an approach has been suggested to perform the Euclid algorithm by Routh array. This comes very close to, but is not, strictly speaking, a direct test in terms of the coefficients, which the authors’ method indeed is.

Second, a brief discussion regarding the computational difficulties is given. Of course the method of resultants is very involved computationally. If one uses the Routh array, one can see that computation of the rows of the Routh array can often become very cumbersome, and special precautions must be taken, especially when the Routh array turns out to be singular. Taking all these into account, the Vogt-Bose approach seems to be con- siderably more straightforward and can be employed in a very routine manner.

Third, Mortensen’s remark that, “the suggestion by Vogt and Bose to employ the Cayley-Hamilton theorem to simplify the calculations involved in their procedure leads to a sequence of operations which in fact partly parallel the operations involved in carrying out Euclid’s algorithm,” is unsubstantiated, very possibly wrong, and irrelevant. However, as this is not a main issue, no attempt will be made to confirm the statement.

Finally, it is unfortunate that

f(B) = [34B2 + 59B + 251J

was (note that 1, represents an identity matrix of the third order) printed as

f(B) = [34B2 + 59B + 2511

as a result of a typographical error. More unfortunate, however, is Morten- sen’s attempted correction that the function f(B) should be

f(B) = [34B2 i 59B + 251.

This is a meaningless statement, as any elementary text in matrix algebra should reveal5

W. G. VOCT N. K. Bos~ Dep. Elec. Eng. University of Pittsburgh Pittsburgh, Pa. 15213

Cambridge Univ. Press, 1960. R. A. Frazer, W . J. Duncan, and A. R. Collar, E/emenrary Matrices. New York:

Comments on “A Method to Determine Whether Two Polynomials Are Relatively Prime”

The result given by Vogt and Bose in the above correspondence’ is well known and seems to have been first published by MacDuffee [l], although it has since been rediscovered [2], [3]. It should be noted that the desired matrix can be constructed very easily using a recurrence relation between successive rows. Direct extensions give the degree of the greatest common divisor of two polynomials [l] and the greatest common divisor itself [3].

S . B m m School Math. University of Bradford Bradford, Yorkshire, England

REFERENCE3

[ I ] C. C. MacDuffee, “Some applications of matrices in the theory of equations,” Amer.

[2] R. E. Kalman, “Mathematical description of linear dynamical systems,” S I A M J .

[3] S . Barnett. “Greatest common dlvisor of two polynomials," Linear Algebra, vol. 3,

Math. &ion., vol. 57, 1950, pp. 15G161.

Contr., ser. A, vol. 1. 1963. pp. 152-192.

1970, pp. 7-9.

Manuscript received August 24. 1970. ’ W. G. Vogt and N . K. Bo%, IEEE Trans. Aurumar. Cunrr. IC0rresp.J. bol. AC-15.

June 1970. pp. 379-380.

Correction to “Derivative Operations on Matrices”

In Section I V of the above paper,’ some transpose designations have inadvertently been omitted. The expressions for the Taylor formulas should have read as follows:

WILLIAM J. V m Dep. Elec. Eng. University of Waterloo Waterloo, Ont., Canada

W . D. Fryer, “Applications of Routh’s algorithm to network theory problems,” IRE ‘ W . J. Vetter, IEEE Trans. Aufomar. Conrr. (Short Papers), vel. AC-15, Apr. 1970, ’ Manuscript received July 27, 1970. Manuscript received August 3, 1970.

Trans. Circuit Theory, vol. CT-6, June 1959, pp. 144-149. pp. 241-244.