Using (4-33) and (4-34) it readily follows that
sup IF(t) - Foii = (2a 1 + 4a 1) = '1 (4-38)
t>O
and
sup IG (t) - G = 0 2 (4-39)
t )O
In order to show stability by the Poincare-Liapunov theorem we must then have
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 FO. The constant m depends on the eigenvectors of this matrix.
In order to meet the stability requirement, the feedback gains K1
and K2 are selected using a standard technique for eigenvalue assignment. (Note: it can be shown that the linearized system is controllable 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 =1/2