DESIGN FOR A BRAIN 20/7 



Example: The equation A 3 + 15 A 2 + 2 A + 8 = has roots 

 — 14-902 and — 0-049 ± 0-729 V — 1, so the system is stable. 



20/7. A test which avoids finding the latent roots is Hurwitz' : 

 a necessary and sufficient condition that the linear system is 

 stable is that the series of determinants 



etc. 



Example: The system with characteristic equation 

 A 3 + 152 2 + 2A + 8 = 

 yields the series 



+ 15, 



15 



8 





 15 



8 



15 



8 

 



These have the values —15, + 22, and +176. So the system 

 is stable, agreeing with the previous test. 



20/8. Another test, related to Nyquist's, states that a linear 

 system is stable if, and only if, the polynomial 



l n + m^"" 1 + m 2 A n " 2 + . . . + m n 

 changes in amplitude by nn when A, a complex variable 

 (X = a + hi where i = V — 1), goes from — i oo to + i oo along 

 the 6-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 is of little use in the 

 theory of adapting systems, and will not be discussed here. 



20/9. Some examples will illustrate various facts relating to 

 stability in linear systems. 



Example 1: The diagonal terms a u represent the intrinsic 

 stabilities of the variables; for if all variables other than x t are 

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



dx { /dt = a H Xi + c, 

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