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Additional info for Stability of Nonlinear Control Systems
Sample text
There is no need therefore to discuss this system any further. 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.
Dt Since by hypothesis o(t) + + 00 with t, and since Ily(t)II remains finite we may choose T so large that for t 2 T we have o(t) - C'A - ' y ( t ) > 0. Upon integrating then from T t o t we obtain [o(t) - C'A - 'Y(t)]'T < 0. As a consequence, since Ily(t)II is bounded, so is the proof of the theorem. o(t). This completes $3. Lurie's Method and a Variant By taking a more restricted type of matrix C and assuming that the characteristic roots of the matrix A are all distinct, Lurie obtained a narrower sufficiency condition for absolute stability than the inequality (Fi).
A more complete result is the following proposition due to LaSalle [l]. 2) Theorem of LaSalle. Let A be stable and C (hence B ) > 0. Then d C - ' d > c'A-'b. 3) 28 52. 1 1) is still valid without imposing on q(a) property I11 of I , $1 (divergence o f t h e integral @(a) as a -, f 00). 3a) Corollary. 1) in the strongerform p > c'A-'b. REMARK. 3) was obtained earlier by Yacubovich. 3). This is the crux of the matter and the more difficult part of the argument: we first show that if x , y are two n vectors then (Bx - y)'C- ' ( B x - y ) 2 2y'A- ' x .
