Consider the continuous time state space model governed by:
A = [-1 0;1 0]
B = [1; 0]
C = [1 -1]
D = 0
a) suppose we are using the following state feedback and observer gains
K = [k1 k2]
L = [l1;l2]
find the loop gain symbolically using MATLAB (aka find process & controller gains)
b) combine the observer and the state feedback and simulate the system for a step input and non-zero initial conditions on x_tilda
Other parts of this question that I omitted called for designing the observer & feedback matrix for given poles. The result is:
K = [7.0000 12.0000]
L = [-44.5000; -55.5000]
For part a) i used the following block diagram from my textbook and course in trying to figure out the controller and process gains:
The Attempt at a Solution
for part a) I turned the state space model into a second-order transfer function:
P(s) = (s - 1)/(s^2 + s)
I believe I blanked on turning the controller portion into a transfer function because I could not derive C(s).
With P(s) and C(s) the closed-loop transfer function would be defined by:
G(s) = P(s)C(s) / (1 + P(s)C(s))
From G(s), I am unsure how to determine the gain. Would I examine the maximum of the bode plot?
for b) the x_tilda value is what goes into the A block in the controller so I just need to look at that value. Implementing this simulation in MATLAB is where I am having issues.
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