EEE 582
HW # 6 SOLUTIONS
Problem 14.4 To verify the stated identity, multiply from left and right with the quantities in the inverses: P (I QP )1 = (I P Q)1 P , (I P Q)P (I QP )1 (I QP ) = (I P Q)(I P Q)1 P (I QP ) , (I P Q)P = P (I QP ) Now, the transfer
EEE 582
Problem 9.1 The controllability matrix is Wc = B
HW # 5 SOLUTIONS
AB
A2 B
3 1 + 2 2 + 2 5 1 1 =4 1 1 0 0
2
= = For controllability we need that det(Wc ) 60. But det(Wc ) = , hence the system is controllable for all 60. The observability matrix is
EEE 582
HW # 4 SOLUTIONS
Problem 7.3 Consider the Lyapunov function candidate V = x> F 1 x, for which _ V = x> (A> F )F 1 x + x> F 1 F Ax = x> (A> + A)x < 0 _ Hence, V is negative denite and the state equation is exponentially stable. Problem 7.4 Consider
EEE 582
Problem 5.2a-b
HW # 3 SOLUTIONS
1. The characteristic polynomial is given by det(I A) = 0, 2 + 2 + 1 = 0 hence 1;2 = 1. The exponential matrix is given by: eAt = 0 (t)I + 1 (t)A. The functions 0 and 1 are given by: et tet so 1 = tet and 0 = et + t
EEE 582
HW # 1 SOLUTIONS
Problem 1.2 Let i and pi be the i-th eigenvalue and eigenvector of A, so Api = i pi 1. A2 pi Ak pi = AApi = A (i pi ) = 2 pi i . . . = AAk1 pi = k pi i (1)
Hence the eigenvalues of Ak are k for i = 1; :; n. i 2. Multiply both side