Homework 1: State Space Realization - Kckong.info

6. Calculate e Ft and F k , where k is an integer, for the matrices:
[ ] 0 0
(a) F =
1 0
[ ] 4 3
(b) F =
1 2
[ ]
0 1
(c) F =
−3 −2
7. Consider a matrix
⎡
F =
⎢
⎣
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
−45 −111 −104 −48 −11
⎤
⎥
⎦
(a) Using the Matlab function eig.m, show that the eigenvalues of F are−1, −2±j, −3, and −3.
(b) Find V ∈ R 5×5 such that 2
where
⎡
A =
⎢
⎣
F = VAV −1
−1 0 0 0 0
0 −2 1 0 0
0 −1 −2 0 0
0 0 0 3 1
0 0 0 0 3
(c) Noting that the matrix A consists of three block matrices, solve e Ft .
(d) For a state space equation, ẋ = Fx, findx(1) for x(0) = [ 1 1 1 1 1 ] T .
⎤
⎥
⎦
8. Suppose a state space model is given byẋ = Fx, where F ∈ R 2×2 . Suppose it is known that
[ ] [ ] [ ] [
4 1 5 1
x(1) = for x(0) = and x(1) = for x(0) =
−2 1
−2 2
]
(a) Find e Ft .
(b) Find x(10) for x(0) = [ 1 0 ].
9. Consider a quarter vehicle model shown in Fig. 1a, where u is the force (input) applied to the vehicle.
(a) Find the equation of motion of the simplified model shown in the figure.
(b) Defining x 1 as the output of the system, find a state space model corresponding to the equation of motion. 3
In other words, find the matrices F ,G, and H with appropriate dimensions:
ẋ = Fx+Gu
y = Hx
(c) Assuming M 1 = 1000, M 2 = 100, k 1 = 20000, k 2 = 15000, c 1 = 10000, and c 2 = 40000, find the
eigenvalues and eigenvectors of the state matrix F by the Matlab function, eig.m.
2 Since V ∈ R 5×5 , all the elements of V should be real.
3 Hint: you may define the statexas [ x 1 x 2 ẋ 1 ẋ 2
] T
∈ R 4 .
2