TECHNICAL NOTES AND CORRESPONDENCE 111

A simple computation shows that

( k ) zi = hiF”, k = 1,. . , , d i

( 4 + 1) zi = hiFd’+lx + hiFdiGu.

Therefore, Fo = TFT-’ contains ET==, di rows of the form [0 . . . 0 1 0 . . . 01 and C = TG contains , di zero row’s.

It is claimed that if the row-vector hlFj, j = 0. . . d i , i = 1 . . . J% span the n-dimensional state space q= , di = n - rn, and the integers di differ at most by unity

m a x [ ( i , . . - d ~ = m i n [ d , . . ~ d ~ + + . ~ = 0 , 1

the transformation T reduces the pair ( F , G) to the following canonical form:

r n l n - - m M

where D is nonsingular, as required, since G has full rank. Now the trans- formations z = Tx and L’ = Du reduce the system i = F x + Gu to the desired PVCF, t = Foz + GOc, where

To demonstrate this method, consider the example by Curran and Franklin :

The last n x n portion of the controllability matrix is singular, which prohibits the use of their method. Using the method presented here, one finds C, = G, h , = [l 0 01. d, = 1, C, = [IIT,(~,F)~], h, = [0 0 11, and d, = 0. Since d l + d, = n - in and dl and d 2 differ by unity, T trans- forms the pair ( F . G) to

A further transformation L: = Du yields the desired PVCF, i = F0z + G o r where

Consider a second example

PVCF does not exist. The transformation

0 0 0 1 ‘

1 0 0 0

0 1 0 0 T =

yields the canonical form j. 0

1

Finally, Curran and Franklin go on to say that “in light of the conditions derived here the results given in [3] are not as general as claimed.” This is a grossly incorrect statement and fails to point out what parts of the paper [3] are affected and to what extent because of the inability to find the de- sired transformation. The paper in question [3j deals with an entirely different problem. that of perfect model following. The PVCF is used there as a sideline to give some physical insight to the problem, and the inability to find a PVCF affects in no way the major results obtained.

S . J. Assm Cornell Aeronautical Lab. Buffalo, N.Y.

Comments on “On Necessary Conditions for Decoupling Multivariable Control Systems”

Abstract-In a recent correspondence two necessary conditions for decoupling mnltivariable systems by linear statevariable feedback are shown throngh unnecessarily long proofs. These necessary conditions are trivially proven here.

In the above correspondence’ Liu and Bergman show through an unnecessarily long proof that two necessary conditions for decoupling the n~-input m-output linear stationary system S described by

X = A x + Bu

y = c x (1)

using linear state-variable feedback (LSVF) described by the control law

II = FX + GW (2)

where the rank of the rn x m matrix G is J% i.e., r(G) = rn, are that r(B) = m z and r ( C ) = rn. In the preceding equations, A is n x n, B is n x m , C is ~n x n, and F is n1 x n. Falb and Wol~vich’-~ show the necessary and sufficient condition for the existence of F and G to decouple S. Falb and Wolovich’ show that a necessary condition for decoupling S by LSVF is that the rn x 111 transfer function of S, H(s). given by

H(s) = C(SZ - A ) - ’ B (3)

be nonsingular. Immediately, one can see that the rn x n matrix C and the n x rn matrix B must both be of rank 112 for LSVF to decouple S .

Howxver, these two necessary conditions are more fundamentally derived from the definition of decoupling. The system S is said to be de- coupled by LSVF if the transfer function of the overall system (1) au,mented

The last n columns of the controllability matrix are singular. hence Curran and Franklin’s method fails again. Using the method presented here, one obtains C , = [FG,G], h , = [0 0 0 13, d , = 2, C, = [ h r , ( h , F ) r . (h ,F2)T] , h, = [l 0 0 Oj, and d, = 0. Since d l and d, differ by more

hlanuscript received May 26, 1970. I C. K. Liu and N. J. Bergman, IEEE Trans. A l m m r . Conrr. (Corresp.), vol. AC-I5

P. L. Falb and W. A. Wolovich. “On the decoupllng multivartable sycrems.” in 1967

’ P. L. Falb and W. A. Wolovich. “Decouvlina in the desim of multivariable control

Feb. 1970, pp. 131-133.

Joinr Aurumaric Control Cunl.. Preprinrs, pp. 791-796.

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.