Homework 1: State Space Realization - Kckong.info

Homework 1: State Space Realization
Digital Control Systems [MEE4003]
Department of Mechanical Engineering, Sogang University
Due on September 19, 2012
Problems 1–4 are for your review on rank and nullity.
1. Describe the column space and the null space of the matrices
[ ] 1 −1
(a)
0 0
[ ] 0 0 3
(b)
1 2 3
⎡ ⎤
1 0
(c) ⎣ 0 1 ⎦
1 1
2. Show that if A is an invertible matrix in R 8×8 , then the column space ofAisR 8 .
3. Suppose v 1 ,v 2 ,v 3 ,v 4 are vectors inR 3 .
(a) Explain why these four vectors are linearly dependent.
(b) If v 1 , v 2 , and v 3 are linearly independent, find the range space of[v 1 v 2 v 4 ] ∈ R 3×3 .
4. Suppose that v 1 ,v 2 , andv 3 are independent vectors inR 3 , andw 1 = v 2 −v 3 ,w 2 = v 1 −v 3 , andw 3 = v 1 −v 2 are
their differences.
(a) Find a matrix A such that VA = W where V = [v 1 v 2 v 3 ] and W = [w 1 w 2 w 3 ].
(b) Find rank(V) and nullity(V).
(c) Find rank(W) and nullity(W).
(d) Find rank(A) and nullity(A).
(e) Discuss about the relationship between the ranks of V , W , and A.
5. For a matrix
F =
[ 1 −1
2 4
]
∈ R 2×2
(a) Find the eigenvalues and eigenvectors of F .
(b) Solve 1 a state space model, ẋ = Fx, wherex(0) = [ 0 6 ].
(c) If B = F −7I, what are the eigenvalues and eigenvectors ofB and how are they related to those of F
1 That is, findx(t) by the convolution theorem.
1

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

0.09
0.08
The ¼ mass of a car M 1
The mass of
a tire M 2
u
c , k 1 1
c1
u
M 1
k 1
x 1
Position of M 1
0.07
0.06
0.05
0.04
0.03
0.02
M 2
x 2
0.01
c 2
, k 2
c 2
k 2
0
0 5 10 15 20
Time(sec)
(a) A quarter vehicle model.
(b) A simulation result.
Figure 1: Settings for Problem 5.
(d) Find V and Λ such that F = VΛV −1 and Λ is a diagonal matrix with the eigenvalues of F .
(e) Calculate the position of M 1 (i.e., x 1 (t)), right after a driver of 70kg (approximately 700N) gets on the
vehicle. That is,
{
700 for t ≥ 0
u(t) =
0 for t < 0
You may use the Matlab function, lsim.m.
(f) The driver left the vehicle at t = 10, i.e.
⎧
⎪⎨ 0 for t < 0
u(t) = 700 for 0 ≤ t < 10
⎪⎩
0 for 10 ≤ t
Calculate the position ofM 1 using lsim.m and plot y(t) versus t. 4 The graph should look like Fig. 1b.
10. Given the system
ẋ =
[ −4 1
−2 −1
] [ 0
x+
1
with zero initial conditions, find the steady-state value 5 of x for a unit step input u.
]
u
4 Hint: you may define
t = [0:0.001:20]; and
u = zeros(1,20001); u(1:10000)=700*ones(1,10000);
5 Hint: notice thatẋis zero at the steady state.
3