Description
1: 20 points
The set W of n × n matrices with real entries is known to be a linear vector space. Determine
which of the following sets are subspaces of W.
(a) The set of n × n skew-symmetric matrices.
(b) The set of n × n diagonal matrices.
(c) The set of n × n upper-diagonal matrices.
(d) The set of n × n singular matrices.
2: 20 points
Determine a basis for the space W ⊂ R
4
spanned by the four vectors
{y1, y2, y3, y4} =
−1
1
−5
7
,
2
1
7
−8
,
3
2
10
−11
,
5
5
15
−15
.
Test whether the following vectors are ∈ W.
• z =
3
−1
13
−17
• u =
4
9
12
−8
• v =
−1
−1
−3
3
3: 20 points
For the vectors below do the following.
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24-677 (LCS) Homework 2 Due 2/17/2021
• x1 =
2
−3
−1
• x2 =
1
1
−1
(a) Find the 1, 2, and ∞ norms.
(b) Find two orthonormal vectors that span the same space as {x1, x2}. This type of problem can
be used to represent the reachable subspace of a linear system (more later!).
4: 40 points
A unit mass sliding on a frictionless plane has dynamics
x˙ =
0 1
0 0
x +
0
1
u.
(a) Using the approximation ˙x(t) ≈
x[(k+1)T]−x[kT]
T
, find a discrete time system model for
T = 0.01 s.
(b) Read about the pidTuner functionality in Matlab here:
https://www.mathworks.com/help/control/ref/pidtuner.html. Design a PID controller that
has “good” performance using the tuner (“good” is up to you) for the discrete time system. Plot
the step response of the closed loop system using the step command.
(c) Create your PID controller in a Matlab script. Write a Matlab function that takes as
arguments the current value of the error, the last value of the error, and the reference set point
and sends as an output the control signal generated by your PID controller. Using your function,
simulate the step response of the closed loop system by directly solving the discrete time state
equations.
(d) Using the method of your choice, design an open loop controller (sequence of controls) that
takes the mass from state
0 0T
at t = 0 to state
1 0T
at t = 0.05 s. Plot the response and
the control sequence.
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