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System state space representation

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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).

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Solution includes block diagram drawing and explanations using transfer function.

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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

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