Control Systems: Uncertainties w/ State feedback & Observer controller

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Homework Statement


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

Homework Equations



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:

[PLAIN]http://img.photobucket.com/albums/v68/jumpboyb/Screenshot2010-12-08at21409PM.png [Broken]

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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