Description
1: 20 points
Consider two bases for space R
3
.
{a1, a2, a3} =
2
1
4
,
3
−2
−2
,
4
2
1
, {b1, b2, b3} =
−2
3
1
,
−4
−3
−2
,
5
−2
0
If a linear operator is given in the {a1, a2, a3} basis by
A =
8 −2 −1
4 −2 −3
2 −3 −3
(a) Find the representation for this operator in the basis {b1, b2, b3}.
(b) If we are given a vector x =
2 −1 −4
T
in the basis {a1, a2, a3}, determine the
representation for x in the basis {b1, b2, b3}.
2: 10 points
For the pair of matrices
A =
2 3 1 4 −9
1 1 1 1 −3
1 1 1 2 −5
2 2 2 3 −8
, y =
17
6
8
14
Determine all the possible solutions to the system Ax = y.
3: 10 points
Find the best solution, in the least-squared error sense, to the equations
−2 = x1 − 2×2
5 = x1 − 2×2
1 = −2×1 + x2
−3 = x1 − 3×2.
4: 15 points
Find the eigenvalues and corresponding eigenvectors of the following matrices.
(a)
1 −2 0
−1 2 −1
0 −1 1
(b)
1 3 3
3 1 3
−3 −3 −5
(c)
0 1
−ω
2
n 0
5: 10 points
Compute the singular values of the following matrices.
(a)
−1 0 1
2 −1 0
(b)
−1 2
2 4
6: 30 points
We again revisit the problem of controlling the sliding mass in discrete time. With a sampling
period of 0.01 s, the plant dynamics are given by
xk+1 =
1 0.01
0 1
xk +
0
0.01
uk
yk =
1 0
xk
(a) Using Matlab, find the optimal feedforward control sequence (in the sense of kuk2) that drives
the state from x0 =
1 0T
to x10 =
0 0T
. Plot the response of the system and the optimal
control input on separate graphs.
(b) Let’s now take a feedback approach to the same problem, using a control of the form
u =
k1 k2
| {z }
K
x. As we will see later, the closed loop system is stable ⇐⇒ the eigenvalues of
A + BK satisfy |λi
| < 1 ∀i. Find the values k1 and k2 that place both eigenvalues at λ1 = λ2 = 0
(this is known as deadbeat control). Plot the response of the closed loop system and the control
usage in this case.
