A new performance index is taken as
00
J= f [x'(t)Q (t) + '(t)RU(t)]dt (4-58)
0
The standard algebraic Riccati equation may be solved yielding an optimal feedback gain K for the system (4-56). It just so happens that
if this same feedback gain is used for the original system then the integrand of (4-55) is minimized with respect to the original system. In addition, the eigenvalues of the closed-loop system will all have real parts less than -a. This is the desired result. A similar
technique can be used in discrete-time control [21].