STABILITY 



20/11 



20/9. If the coefficients in the characteristic equation are not 

 all positive the system is unstable. But the converse is not 

 true. Thus the linear system whose matrix is 



i V 6 ° 



— V 6 i ° 



—3^ 



has the characteristic equation A 3 +A 2 + A + 21=0; but the 

 latent roots are + 1 ± V— 6 and — 3 ; so the system is unstable. 



20/10. Another test, related to Nyquist's, states that a linear 

 system is stable if, and only if, the polynomial 



l n + mj"- 1 + m 2 X n ~ 2 +--•+«■ 

 changes in amplitude by nn when A, a complex variable 

 (A = a + hi where i = V— 1), goes from - t oo to + t co along 

 the fr-axis in the complex A-plane. 



Nyquist's criterion of stability is widely used in the theory 

 of electric circuits and of servo-mechanisms. It, however, uses 

 data obtained from the response of the system to persistent 

 harmonic disturbance. Such disturbance renders the system 

 non-absolute and is therefore based on an approach different from 

 ours. 



20/11. Some further examples will illustrate various facts 

 relating to stability. 



Example 1 : If a matrix [a] of order n x n has latent roots 

 A l9 . . . , A n , then the matrix, written in partitioned form, 



! / 



of order 2n x 2/?, where / is the unit matrix, has latent roots 



± VI7, . . . , ± a/A„. It follows that the system 



d 2 x- 



— ! = di 1 x 1 + a i2 x 2 + . . . + ainXn (« = 1, . . . , «) 



of common physical occurrence, must be unstable. 



Example 2 : The diagonal terms an represent the intrinsic 

 stabilities of the variables ; for if all variables other than xi are 

 held constant, the linear system's i-th. equation becomes 

 dxi/dt — auxi + c, 

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