Consider the LTI system
xË™ =
0 1
0 −1
x +
1
0
u x(0) = x0
and infinite horizon cost
R ∞
0 ∥x∥2 + ru2dt, where r > 0.
(a) Determine what Theorem 10.6 implies about the LQR for this problem.
(b) Compute by hand the LQR matrix ¯ P, the optimal feedback control law, the closed loop
state matrix Acl, and the poles of the closed loop system.
(c) Comment on how the feedback law and poles vary as r gets very large (expensive control)
or very small (cheap control).
(d) Find the eigenvalues of the Hamiltonian matrix H by hand. Is your answer consistent
with part (b)?