59
Using (4-33) and (4-34) it readily follows that
and
* o
sup IlF (t) F II = (2ou + 40,00 = s
t>0 lie.
sup IIG (t) G^ II = 0 = e,
t>0
(4-38)
(4-39)
In order to show stability by the Poincare-Liapunov theorem we must then
have
e^m a < 0 (4-40)
where -a corresponds to the real part of the right-most eigenvalue of
the matrix given in equation (4-35) when F*(t) is replaced by F. The
constant m depends on the eigenvectors of this matrix.
In order to meet the stability requirement, the feedback gains
and K2 are selected using a standard technique for eigenvalue assign
ment. (Note: it can be shown that the linearized system is control
lable and hence arbitrary eigenvalue assignment can be made). The
closed-loop eigenvalues are chosen to be:
-4.0
-5.0
-5.0 j3.0
-4.0 j2.0
This gives a = 4, however, we shall not concern ourselves with the
calculation of m. Just as an example, let us assume m = 1, = 1/2