Dr. Samuels [1] is to be congratulated on a most interesting paper. It is unfortunate that a number of errors appear in Sec. III which invalidate both that section and Sect. IV

Caughey, T. K.

English

Caltech Authors - Main

THE JOURNAL OF THE ACOUSTICAL $OCIETV OF AMERICA VOLUME 32, NUMBER l0 OCTOBER, 1960 
Letters to the Editor 
Erratum: Measurements of the Sound Absorption 
Coefficient and the Sound Transmission 
Loss at the Kobayasi Institute 
of Physical Research 
['J. Acoust. Soc. Am. 32, 376-379 (1960)] 
KOZI SATO AND MASARU KOYAS•/ 
Koba•asi Institute of Physical Researth, Kokubunji, Tokyo. Japan 
On page 378, second column, line 6, for "500 cps" read "125 cps.' 
Comments on ttOn the Stability of 
Random Systems" 
T. K. CAt, anEV 
Division of Engineering, California Institute of Technology, 
Pasadena, CaliJornia 
(Received June 13, 1960; revised manuscri0t received July 11, 1960} 
R. Samueh I is to be congratulated ona most interesting 
paper. It is unfortunate that a number of errors appear in 
Sec. III which invalidate both that section and Sec. IV. The 
errors are; 
(1) l•[• 2in Eq. (55) should read 
•+i6# •- 4 (wo 2+ 2•) •o-i80• •' 
(2) If Eq. (54,) is multiplied out, there results the equation 
1- w•S•Wa--4•&W•-O. (2) 
If the corrected expressions in Eq. (55) are substituted into (2), 
the basic frequency equation is 
where 
Sa+ A aSa+ A •S + A o = O, (3) 
A, =4[wn'+2•(l --0S,)] . (4) 
•e fr•uency •uation is thus •en to be a cubic equation •d 
not a sixth-order •uatlon as given by Samuds. Equation O) 
above was obtaln• by Caughey and Dienes a by •tting up the 
Fokker Planck equation for the system. 
0) If St is •t equal to •ro in •. (3) herein, the correct 
equation for (58) of Samuels' paper is 
•alysis of the Routh-Hu•itz stabi•ty chter• shmvs t•t for 
stability 
4fl>wdSo. (6) 
(4) If S0 is set equal to •ro in •. (3) herein, the corr•t 
•uation for (•) of Samuels' paper is 
S•+2fl O-- 2•S0•+4[•0•+2• (• --•S0 IS 
+•(t-•&) =0. (•) 
Analysis of the Routh-Hu•itz criteria shows that a n•essary 
condition for stability is that 
Hence the system is not unconditionally stable as Samuels finds. 
(5) If So is set equal to zero in Eq. (3) herein, and 0 is replaced 
by --0, Eq. (62) should read 
S a -- 20 (3 +20St) Sa+4Ewo •+2• (1 +ilS0 ]S 
Examination of the Routh-Hurwitz stability conditions shows 
that this system is unstable for all St. This means that it is not 
possible to stabilize an unstable system by means of random noise. 
Samuels' results are, therefore, shown to be incorrect. 
It should also be pointed out that mean squared stability is a 
necessary--but not a sufficient•ondition for the stability of a 
system. In order to ensure stability of a system, all the moments 
must be stable. 
For example, if we consider the first moment of Q in Eq. (40), 
it is easily shown that (Q) satisfies the differential equation 
d• (Q)+ 20d (Q)+wo•(Q)=O. (10) dF dt 
The requirements for stability are that 
w0 •> 0J' 
Hence, even if Samuels' analysis of the mean squared stability 
were correct, the stochastic mean of Q would be unstable if either 
0 or wo • were made negative; therefore, the system would be 
unstable. 
sj. C. Samuels. J. Acoust. Soc. Am. 32. 594--601 (1960). 
a T. K. Caughey and J. K. Dienes (unpublished}. 
Remarks on Dr. Caughey's Comments on My 
Paper entitled "On the Stability of 
Random Systems, Etc." 
J. CL1FION SAMUELg 
International Business Machines Corporation, Federal Systems Divisio•t. 
Oz•ego, Nrezo 
{R•ived July 6. 1960) 
HE author is grateful to Dr. Caughey for having pointed out 
some errors in S<. III of his paper• wMch inv•idate certain 
conclusions concerning the possibility of stabil•ing systems wi• 
random noir. •e corrections to the formulas which he gives are 
eorr<t. It should be noted, however, that the possibifity of 
stabil•ing systems with some form of random noise is not dead. 
The error in •e •alysis which led the author to believe that 
systems could be stabiliz• with random noi• prompt• him to 
t• to do it with a s<ond-order system on • analog computer. 
While the results have been far from conclusive, we have •d mine 
encouragement. It is possible, however, •at the situation is con- 
fus• by the •tering in of certain nonlinmr eff•ts in the computer 
multipliers. 
We should remember that •. (10) of Dr. Caughey's comments 
and his conclusions hold only for •hlte n•se parameter v•iations. 
It would be desirable to extend the stability •eory to systems 
with nonwMte noise parameter variations. The author feels t•t 
further theoretical and ex•rimental work is required to cl•r up 
the situatiou. 
• J. C. •muels, J. A•ust. S•. Am. 32, 59•1 (1•0). 
1356 


Comments on "On the Stability of Random Systems"

THE JOURNAL OF THE ACOUSTICAL $OCIETV OF AMERICA VOLUME 32, NUMBER l0 OCTOBER, 1960 
Letters to the Editor 
Erratum: Measurements of the Sound Absorption 
Coefficient and the Sound Transmission 
Loss at the Kobayasi Institute 
of Physical Research 
['J. Acoust. Soc. Am. 32, 376-379 (1960)] 
KOZI SATO AND MASARU KOYAS•/ 
Koba•asi Institute of Physical Researth, Kokubunji, Tokyo. Japan 
On page 378, second column, line 6, for "500 cps" read "125 cps.' 
Comments on ttOn the Stability of 
Random Systems" 
T. K. CAt, anEV 
Division of Engineering, California Institute of Technology, 
Pasadena, CaliJornia 
(Received June 13, 1960; revised manuscri0t received July 11, 1960} 
R. Samueh I is to be congratulated ona most interesting 
paper. It is unfortunate that a number of errors appear in 
Sec. III which invalidate both that section and Sec. IV. The 
errors are; 
(1) l•[• 2in Eq. (55) should read 
•+i6# •- 4 (wo 2+ 2•) •o-i80• •' 
(2) If Eq. (54,) is multiplied out, there results the equation 
1- w•S•Wa--4•&W•-O. (2) 
If the corrected expressions in Eq. (55) are substituted into (2), 
the basic frequency equation is 
where 
Sa+ A aSa+ A •S + A o = O, (3) 
A, =4[wn'+2•(l --0S,)] . (4) 
•e fr•uency •uation is thus •en to be a cubic equation •d 
not a sixth-order •uatlon as given by Samuds. Equation O) 
above was obtaln• by Caughey and Dienes a by •tting up the 
Fokker Planck equation for the system. 
0) If St is •t equal to •ro in •. (3) herein, the correct 
equation for (58) of Samuels' paper is 
•alysis of the Routh-Hu•itz stabi•ty chter• shmvs t•t for 
stability 
4fl>wdSo. (6) 
(4) If S0 is set equal to •ro in •. (3) herein, the corr•t 
•uation for (•) of Samuels' paper is 
S•+2fl O-- 2•S0•+4[•0•+2• (• --•S0 IS 
+•(t-•&) =0. (•) 
Analysis of the Routh-Hu•itz criteria shows that a n•essary 
condition for stability is that 
Hence the system is not unconditionally stable as Samuels finds. 
(5) If So is set equal to zero in Eq. (3) herein, and 0 is replaced 
by --0, Eq. (62) should read 
S a -- 20 (3 +20St) Sa+4Ewo •+2• (1 +ilS0 ]S 
Examination of the Routh-Hurwitz stability conditions shows 
that this system is unstable for all St. This means that it is not 
possible to stabilize an unstable system by means of random noise. 
Samuels' results are, therefore, shown to be incorrect. 
It should also be pointed out that mean squared stability is a 
necessary--but not a sufficient•ondition for the stability of a 
system. In order to ensure stability of a system, all the moments 
must be stable. 
For example, if we consider the first moment of Q in Eq. (40), 
it is easily shown that (Q) satisfies the differential equation 
d• (Q)+ 20d (Q)+wo•(Q)=O. (10) 
dF dt 
The requirements for stability are that 
w0 •> 0J' 
Hence, even if Samuels' analysis of the mean squared stability 
were correct, the stochastic mean of Q would be unstable if either 
0 or wo • were made negative; therefore, the system would be 
unstable. 
sj. C. Samuels. J. Acoust. Soc. Am. 32. 594--601 (1960). 
a T. K. Caughey and J. K. Dienes (unpublished}. 
Remarks on Dr. Caughey's Comments on My 
Paper entitled "On the Stability of 
Random Systems, Etc." 
J. CL1FION SAMUELg 
International Business Machines Corporation, Federal Systems Divisio•t. 
Oz•ego, Nrezo 
{R•ived July 6. 1960) 
HE author is grateful to Dr. Caughey for having pointed out 
some errors in S<. III of his paper• wMch inv•idate certain 
conclusions concerning the possibility of stabil•ing systems wi• 
random noir. •e corrections to the formulas which he gives are 
eorr<t. It should be noted, however, that the possibifity of 
stabil•ing systems with some form of random noise is not dead. 
The error in •e •alysis which led the author to believe that 
systems could be stabiliz• with random noi• prompt• him to 
t• to do it with a s<ond-order system on • analog computer. 
While the results have been far from conclusive, we have •d mine 
encouragement. It is possible, however, •at the situation is con- 
fus• by the •tering in of certain nonlinmr eff•ts in the computer 
multipliers. 
We should remember that •. (10) of Dr. Caughey's comments 
and his conclusions hold only for •hlte n•se parameter v•iations. 
It would be desirable to extend the stability •eory to systems 
with nonwMte noise parameter variations. The author feels t•t 
further theoretical and ex•rimental work is required to cl•r up 
the situatiou. 
• J. C. •muels, J. A•ust. S•. Am. 32, 59•1 (1•0). 
1356 


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