Starting from:

$30

ECE4530J Homework 1 -Solved

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‡ = π‘Žπ‘₯ + 𝑏𝑒.

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œ‡(π‘₯) = βˆ’π‘˜π‘₯, find π‘₯(𝑑).

d)        When does the feedback-controlled system in c) converge?

Problem 3
Consider an 𝑛-dimensional linear time-invariant system π‘₯Μ‡ = 𝐴π‘₯ + 𝐡𝑒.

a)         What is the state space?

b)        Suppose that 𝑒 ∈ ℝ2. What are the dimensions of matrices 𝐴 and 𝐡?

c)         Discretize the system into discrete time with step size 𝛿. You need to express π‘₯((π‘˜ +1)𝛿) in terms of 𝐴,𝐡, 𝛿,π‘₯(π‘˜π›Ώ),𝑒(π‘˜π›Ώ). To make the notation easier to read, you can use π‘₯[π‘˜ + 1] instead of π‘₯((π‘˜ + 1)𝛿) 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 𝛿.

d)        Suppose that we use a linear controller πœ‡(π‘₯) = βˆ’πΎπ‘₯. 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 𝐴, 𝐡,𝐾 and the initial condition π‘₯[0] = π‘₯0 ∈ ℝ𝑛.

e)         [Bonus] Suppose that 𝑒 ∈ ℝ𝑛. What are the domain and the range for πœ‡? Hint: the range will depend on the rank of 𝐾.

More products