Problem 1
a) What is a mapping? What is a function?
b) What is asymptotic convergence?
Problem 2
Consider a one-dimensional linear time-invariant system 𝑥̇ = 𝑎𝑥 + 𝑏𝑢.
a) What is the system state? What is the system state space?
b) What is the control input?
c) Given the initial condition 𝑥(0) = 𝑥0 and a linear controller 𝜇(𝑥) = −𝑘𝑥, find 𝑥(𝑡).
d) When does the feedback-controlled system in c) converge?
Problem 3
Consider an 𝑛-dimensional linear time-invariant system 𝑥̇ = 𝐴𝑥 + 𝐵𝑢.
a) What is the state space?
b) Suppose that 𝑢 ∈ ℝ2. What are the dimensions of matrices 𝐴 and 𝐵?
c) Discretize the system into discrete time with step size 𝛿. You need to express 𝑥((𝑘 +1)𝛿) in terms of 𝐴,𝐵, 𝛿,𝑥(𝑘𝛿),𝑢(𝑘𝛿). To make the notation easier to read, you can use 𝑥[𝑘 + 1] instead of 𝑥((𝑘 + 1)𝛿) to denote the discrete-time state and express 𝑥[𝑘 + 1] in terms of 𝑥[𝑘]; note that you still need to consider the impact of 𝛿.
d) Suppose that we use a linear controller 𝜇(𝑥) = −𝐾𝑥. Write the difference equation for the discretized system; i.e., write how to obtain 𝑥[𝑘 + 1] from 𝑥[𝑘] . Find 𝑥[𝑘] in terms of 𝐴, 𝐵,𝐾 and the initial condition 𝑥[0] = 𝑥0 ∈ ℝ𝑛.
e) [Bonus] Suppose that 𝑢 ∈ ℝ𝑛. What are the domain and the range for 𝜇? Hint: the range will depend on the rank of 𝐾.
