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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 π₯Μ = ππ₯ + ππ’.
a) What is the system state? What is the system state space?
b) What is the control input?
c) Given the initial condition π₯(0) = π₯0 and a linear controller π(π₯) = βππ₯, find π₯(π‘).
d) When does the feedback-controlled system in c) converge?
Problem 3
Consider an π-dimensional linear time-invariant system π₯Μ = π΄π₯ + π΅π’.
a) What is the state space?
b) Suppose that π’ β β2. What are the dimensions of matrices π΄ and π΅?
c) Discretize the system into discrete time with step size πΏ. You need to express π₯((π +1)πΏ) in terms of π΄,π΅, πΏ,π₯(ππΏ),π’(ππΏ). To make the notation easier to read, you can use π₯[π + 1] instead of π₯((π + 1)πΏ) to denote the discrete-time state and express π₯[π + 1] in terms of π₯[π]; note that you still need to consider the impact of πΏ.
d) Suppose that we use a linear controller π(π₯) = βπΎπ₯. Write the difference equation for the discretized system; i.e., write how to obtain π₯[π + 1] from π₯[π] . Find π₯[π] in terms of π΄, π΅,πΎ and the initial condition π₯[0] = π₯0 β βπ.
e) [Bonus] Suppose that π’ β βπ. What are the domain and the range for π? Hint: the range will depend on the rank of πΎ.