24 – 677 Name:
Fall 2023
Mid-term Exam Andrew id:
10/24/23
Time: 24 Hours Print your initials on each
page that has your answers
This exam contains 8 pages (including this cover page) and 7 problems. Check to see if any pages
are missing. Enter all requested information on the top of this page, and put your initials on the
top of every page, in case the pages become separated.
You may use your equation sheet and calculator on this exam.
You are required to show your work on each problem on this exam. The following rules apply:
• Organize your work, in a reasonably neat and
coherent way, in the space provided. Work scattered all over the page without a clear ordering
will receive very little credit.
• Mysterious or unsupported answers will not
receive full credit. A correct answer, unsupported by calculations, explanation, or algebraic
work will receive no credit; an incorrect answer
supported by substantially correct calculations and
explanations will receive partial credit.
• You are allowed to use course slides, homework
solution sheets as references. You are allowed to
search for the knowledge needed on the internet.
• You must conduct the exam independently. Discussion or seeking help from others, online or inperson, is prohibited.
• All answers need to be derived by hand to get
points. You are allowed to use a calculator for basic calculation of scalars. You can use calculate/-
computer programs to verify your answers but the
effort does not account as credits.
• You can ask questions on campuswire but only to
the TAs and instructors.
• If you need more space, use the back of the pages;
clearly indicate when you have done this.
Do not write in the table to the right.
Problem Points Score
1 15
2 15
3 15
4 15
5 10
6 10
7 20
Total: 100
24 – 677 Mid-term Exam – Page 2 of 8 10/24/23
1. Please state whether each of the following statement is True or False. Explanation is not
required.
(a) (3 points) The system y(t) = sin(t)u(3t) is linear.
(b) (3 points) Assume that ˙x(t) = Ax(t) is an asymptotically stable continuous-time LTI
system. Assuming A−1
exists, the system ˙x(t) = A−1x(t) is asymptotically stable.
(c) (3 points) The following continuous time system is BIBO stable.
x˙ = u, y = x
(d) (3 points) The following DT system is controllable if a ̸= 0.
x[k + 1] =
1 a
0 1
x[k] +
1
1
u[k]
(e) (3 points) The system given in (d) is stabilizable when a = 0.
24 – 677 Mid-term Exam – Page 3 of 8 10/24/23
2. Consider a model of fisheries management. State x1 is the population level of a prey species,
x2 is the population level of a predator species, and x3 is the effort expended by humans in
fishing the predator species. The model is
x˙ 1 = (r1 − x2)x1
x˙ 2 = (r2 − x3)x2
x˙ 3 = u
y = x2
where u is the input, y is the measurement of the predator species, and r1 = 10 and r2 = 25
(a) (5 points) Find the equilibrium point if the prey species population is known to be ¯x1 = 20.
(b) (5 points) Linearize the model using the equilibrium point from (a)
(c) (5 points) Find the transfer function of the linearized state model from (b)
24 – 677 Mid-term Exam – Page 4 of 8 10/24/23
3. (15 points) For the following dynamical system
x˙(t) =
1 0
3 1
x(t) +
1
0
u(t)
compute x(0) when u(t) = 0 and x(2) =
1
0
24 – 677 Mid-term Exam – Page 5 of 8 10/24/23
4. Given an LTI system with state space representation
x˙(t) =
−1 −α
0 1 − α
x(t) +
1
α
u(t)
y(t) =
1 α
x(t) + u(t)
where α ∈ R.
(a) (5 points) Find the range of α for which the system is exponentially stable.
(b) (10 points) For the supremum (least upper bound) of the range of α determined in (a),
check whether the given system is BIBO stable.
24 – 677 Mid-term Exam – Page 6 of 8 10/24/23
5. Consider the following nonlinear system
x˙ 1 = −
x2
1 + x
2
1
− 2×1
x˙ 2 =
x1
1 + x
2
1
(a) (5 points) Using the candidate Lyapunov function V (x) = x
2
1 + x
2
2
and Lyapunov Direct
method, first find the equilibrium point and then find the stability of the system at the
equilibrium point
(b) (5 points) Linearize the system about the equilibrium point and find the stability of the
linearized system using Lyapunov indirect method
24 – 677 Mid-term Exam – Page 7 of 8 10/24/23
6. (10 points) Find the minimal realization for
G(s) = ”
s
s+1
1
s(s+1)#
.
24 – 677 Mid-term Exam – Page 8 of 8 10/24/23
7. Suppose you are invited as a control engineering consultant to investigate a critical safety issue
for an airplane company. You are provided with an approximate linear model of the lateral
dynamics of the aircraft which has the state and control vectors
x =
p r β ϕ T
and u =
δa δr
T
where p and r are incremental roll and yaw rates, β is an incremental sideslip angle, and ϕ is
an incremental roll angle. The control inputs are the incremental changes in the aileron angle
δa and in the rudder angle δr, respectively. In a consistent set of units, the linearized model is
given as ˙x = Ax + Bu with
A =
−10 0 −1 0
0 −1 1 0
0 −1 0 0
1 0 0 0
B =
10 0
0 −1
0 0
0 0
Answer the following questions with derivations. A simple yes or no without explanation will
not get 0 credit.
(a) (5 points) Is the linearized aircraft model asymptotically stable? Is the linearized aircraft
model stable i.s.L.?
(b) (5 points) Is the aircraft controllable with just δr? Is the aircraft controllable with both
δr and δa?
(c) (5 points) Suppose a malfunction prevents manipulation of the rudder angle δr, is it possible to control the aircraft using only the aileron angle δa?
(d) (5 points) If you only have budget to measure one state, which one to measure (choose
one from {p, r, β, ϕ}) so that the whole system is observable?
