Consider the following system whose state space representation is as follows:
x'1 -1 1 α x1 0
x'2 = 0 -2 1 x2 + 0 u
x'3 0 0 -3 x3 1
y = [1 0 0]x
a) Draw the block diagram representation of the system.
b) Find the transfer function of the system.
c) Are there any values of α for which the system would be unstable?
d) Using the transfer function, determine for which value(s) of α the system is unstable.
e) Is the system controllable when α tends to infinity? Explain your answer.
f) Using the controllability matrix, check the answers found in (d) and (e).
Solution includes block diagram drawing and explanations using transfer function.
Find the transfer function from state space representation
Consider the following system (see attachment) whose state space representation is as follows:
a) Find the transfer function of the system.
b) Compute eAt using the eigenvalues and eigenvectors method.
c) Compute eAt using the partial fraction method.
d) If u (t)=0 for t ≥ 0,compute x (t), y(t), and x(1) =[ -1 2]T
e) Assume that the initial conditions are zero. Using MATLAB ,Plot x(t), y(t), given that u(t)= -5 for 0 ≤ t ≤ 3 and u(t)=5 for 3 ≤ t ≤ 6