By Solomon Lefschetz (Eds.)
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Extra resources for Stability of Nonlinear Control Systems
G. 14) p2 - qq > 0. ), Then if B = - 1( Po 2 we have P o = -P P 9 40 qo=- 4 p 40 Po + iv r , ro =-. P 37 3. 15) po + 40 + c = 0. 17) I-: = PPO = + qo)(L' + = 40) - d c + 40). - (p2 + VZ)q,qo > 0. 16) that c Hence we may write c =M + ig, (70 ~9,340> 0. + q,, must be real and negative. 18) a+y
If B = h'B, ' h thcn + h'B, ' ~ ) B , ( x ,+ hB; + paZ. Hence if one chooses as coordinates (x, + hB, la,a)our situation will x'Bx =(x~' '0) be unchanged save that with the new coordinates /J = 0. This is assumed henceforth. 5) becomes p > h'B, 'h. Since this must hold for p arbitrarily small positive and B , > 0, necessarily h = 0. Thus under our stringent conditions we have B = diag(B,,O). Upon taking into account the properties of the admissible class cp it is easily verified that V fulfills all its expected requirements regarding absolute stability.
It will be convenient to take the system as n + 2 dimensional. If A,,... denotes the usual quantities A,... 1) then by a suitable choice of coordinates A, = diag(02, A ) where A is an n x n stable matrix and O2 has one of two forms J 02=(0 J. 0 0 0 0 or 02=( If O2 is of the first form the system reduces to lil = -B ld4 t 2 = - B2Cp(a) f = Ax - bq(a) a = 7151 + Y 2 t 2 + c’x It is clear though that there exist nonzero constants that the above system has the solution t1 = gl, t2 = 52 r2,5, xand 0,E2a such0.