Linear algebra
1. For each of the following systems, determine if the system is: i) controllable, ii) observable, iii)
invertible, iv) strongly observable. (12 points)
(a) x[k + 1] = 1 1
0 !2
”
x[k] +
1
0
”
u[k], y[k] = ⇥
1 1⇤
x[k].
(b) x[k + 1] = 0.5 0.5
0 1 ”
x[k] +
1
1
”
u[k], y[k] = ⇥
1 !1
⇤
x[k].
2. For the systems in question 1, is it possible to find a matrix K so that the state-feedback input
u[k] = !Kx[k] stabilizes the system? If so, find the matrix K so that all of the controllable eigenvalues
are at 0. Is it possible to place all of the eigenvalues at zero in the above two systems? (8 points)
3. Suppose that the initial state of the systems in question 1 are unknown. Design a state-estimator for
each of the systems, if possible. Find the matrix L such that the eigenvalues of A ! LC are stable.
(10 points)
4. Consider a team of 3 mobile robots in a line, and denote the position of the i-th robot by the scalar
xi (we can easily extend this to motion in higher dimensions as well). Suppose that the robots move
according to their di↵erences in position from the other robots. Furthermore, robot 1 is a leader, and
is allowed to use an additional input u[k] in its motion. Consider the rules of motion
x1[k + 1] = x1[k] + ↵(x2[k] ! x1[k]) + ↵(x3[k] ! x1[k]) + u[k]
x2[k + 1] = x2[k] + ↵(x1[k] ! x2[k]) + ↵(x3[k] ! x2[k])
x3[k + 1] = x3[k] + ↵(x1[k] ! x3[k]) + ↵(x2[k] ! x3[k]),
where ↵ = 1
2 .
(a) Write the above rules of motion in the form x[k + 1] = Ax[k] + Bu[k], where x[k] 2 R3 is the
vector of positions of the three robots. (2 points)
(b) Suppose that the initial position of robot i is given by xi[0], and suppose that u[k] = 0 for all
k. What happens to the position of all the robots as k ! 1? Hint: Use question 8d) from HW
1. (5 points)
(c) Now suppose that robot 1 would like to choose u[k] in such a way that it puts the other robots
in some desired positions x1[L], x2[L], x3[L], for some time-step L. Can the robots be put into
any arbitrary final position starting from any initial position x[0] via updates by the leader?
Hint: what property of linear systems would this correspond to? (4 points)
(d) Suppose all robots start at the origin (i.e., x[0] = 0). Is it possible for the leader agent to
apply a sequence of inputs u[0], u[1],…,u[L] so that the positions of the robots are x1[L] = 2,
x2[L] = 1, x3[L] = 1, for some L? If so, find the corresponding sequence of inputs and the value
of L. If not, explain why not. (4 points)
5. Consider the F-8 airplane given by
x[k + 1] =
2
6
6
4
0.9987 !3.2178 !4.4793 !0.2220
0 10.1126 0.0057
0 00.8454 0.0897
0.0001 !0.0001 !0.8080 0.8942
3
7
7
5
x[k] +
2
6
6
4
!0.0329
0.0131
!0.0137
!0.0092
3
7
7
5
f[k]
y[k] =
0010
0001″
x[k],
where f[k] is an unknown actuator fault (we will assume that the known inputs to the system are zero, for simplicity). Construct an unknown input observer for this system. On the course website,there is a file named hw2_f8.mat that contains the output of the system over 100 time-steps. Use the command “load hw2_f8.mat” in MATLAB to import the data. This will create a variable y in your workspace, with 2 rows and 100 columns. The first column contains y[0], the second contains y[1]
and so forth. Use this to drive your unknown input observer and use the unknown input estimation scheme discussed in class to determine whether a fault has occurred (note that it will take a few time-steps for the observer to synchronize with the system, so the estimated inputs at the beginning should be disregarded). (20 points)
6. On the course website, there is a file named distillation_sf.mat, which contains a model of a chemical distillation column of the form
x[k + 1] = Ax[k] + Bu[k]
y[k] = Cx[k],
where x 2 R7 is the system state, u 2 R3 is the known system input, and y 2 R3 is the output. Use the command “load distillation_sf.mat” in MATLAB to import the data in the file (including the system matrices).We will be interested in diagnosing sensor faults in this system. Using the method described in class (Section 4.1 in the notes), design a state estimator for the system based on one of the outputs. In the file that you loaded, there is a variable labeled y, which is the output of the system over 200 time-steps. It has three rows and 200 columns; the first column contains y[0], the second contains y[1] and so forth. There is also a variable labeled u, which is a 3⇥200 vector that contains the known
inputs u[k] to the system; the k–th column contains u[k]. Use the provided inputs and outputs to drive your state estimator and use appropriate residuals to determine whether a sensor fault has occurred. Note that it will take a few time-steps for the observer to synchronize with the system, so you might notice that the residuals are very large for the first 10 or 20 time-steps – you can disregard this portion of the curves. Be sure to include a sketch of the residuals in your solution
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