Stability of Nonlinear Control Systems by Solomon Lefschetz (Eds.)

By Solomon Lefschetz (Eds.)

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Let us turn our attention then to the more interesting case of a double zero root. 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.

Here we have the fortunate circumstance that - contains a only through cp(a), and is a quadratic form in y and cp. (This is the great merit of the Lurie and Postnikov type of function V ) It suffices therefore to demand that - V be a positive definite quadratic form in y and cp. c. due to Sylvester, is that the principal minors of the matrix c 2) all be positive. In particular this must hold for C and so C > 0, hence ICI # 0. Beyond this we still require that the determinant Referring then to (IX, 52) this yields the fundamental inequality (Fi) p > d’C-’d.

Thus, our assumption that both h,, h, # 0 means that the control is completely effective. Under our assumption then, the change of coordinates y j + hjyj, j = 1 , 2 will yield the same system but with This scheme will somewhat simplify the calculations. We continue to assume, of course, that the matrix A is stable. There are then the following three possible normal forms for A : I. 11. 111. (: diag(A,,A,), diag(1, I), A, and I , real and A real and I complex.

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