Second Mid-Term
CS 599: Autonomous Cyber-Physical Systems
Instructor: Jyotirmoy V. Deshmukh
Assigned: April 29, 2019. Due (by email): May 8, 2019
Instructions:
1. In this exam we will use a unique single digit number that we
will generate from your USC student ID. Add all digits of your
USC ID. Then add all the digits of the result. Keep doing this
till you have a single positive integer in [1, 9] – we will call this
your key. For example, if my USC ID is 7895, then my key is 2.
2. Please feel free to refer to any material that you deem fit.
3. Discussions among fellow students are generally not encouraged. If you
do wish to ask questions, please do it on Slack or through email where
Nicole or I will answer. Please adhere to the academic integrity policy.
4. We prefer that you turn in a pdf by email before class to both Nicole and
me. You can also hand in a printed/handwritten copy in class.
Problem 1. [20 points] Consider a 2D linear dynamical system as given below.
Here, k is your key.
x˙1 −10 k x1
= (1)
x˙2 −1 −1 x2
Given a set of initial states I and a set of unsafe states F , recall that a strict
barrier certificate for a system of the form ẋ = f (x) is defined using a function
B(x) which has the following properties:
1. ∀x ∈ I: B(x) ≤ 0,
2. ∀x ∈ F : B(x) > 0,
∂B
3. ∀x s.t. B(x) = 0, ∂x f (x) <0
Let I be defined as: −1 ≤ x1 ≤ 1 and −1 ≤ x2 ≤ 1. Let F be defined as
x2 > 10 or x2 < −10. The value of x1 is unconstrained for the set F . Your task
is to find a barrier certificate that proves safety for the above system.
1
CS 599: Autonomous Cyber-Physical Systems
Instructor: Jyotirmoy V. Deshmukh
Assigned: April 29, 2019. Due (by email): May 8, 2019
Instructions:
1. In this exam we will use a unique single digit number that we
will generate from your USC student ID. Add all digits of your
USC ID. Then add all the digits of the result. Keep doing this
till you have a single positive integer in [1, 9] – we will call this
your key. For example, if my USC ID is 7895, then my key is 2.
2. Please feel free to refer to any material that you deem fit.
3. Discussions among fellow students are generally not encouraged. If you
do wish to ask questions, please do it on Slack or through email where
Nicole or I will answer. Please adhere to the academic integrity policy.
4. We prefer that you turn in a pdf by email before class to both Nicole and
me. You can also hand in a printed/handwritten copy in class.
Problem 1. [20 points] Consider a 2D linear dynamical system as given below.
Here, k is your key.
x˙1 −10 k x1
= (1)
x˙2 −1 −1 x2
Given a set of initial states I and a set of unsafe states F , recall that a strict
barrier certificate for a system of the form ẋ = f (x) is defined using a function
B(x) which has the following properties:
1. ∀x ∈ I: B(x) ≤ 0,
2. ∀x ∈ F : B(x) > 0,
∂B
3. ∀x s.t. B(x) = 0, ∂x f (x) <0
Let I be defined as: −1 ≤ x1 ≤ 1 and −1 ≤ x2 ≤ 1. Let F be defined as
x2 > 10 or x2 < −10. The value of x1 is unconstrained for the set F . Your task
is to find a barrier certificate that proves safety for the above system.
1
