Choosing desired self-conjugate closed-loop pole locations

Question

If i am given:

A = [1 -2; 0.5 -1];
B = [2; 2];
C = [-1 1];
D = 0;
\zeta = 0.707;
\omega_0 = 2 rad/s;

I need to design a state controller for the system such that the closed-loop system becomes two complex poles. I want to use the place function in matlab but how do i find p, the desired self-conjugate closed loop pole locations?

Answer 1

We have with null initial conditions

$$ sX(s)=AX(s)+BU(s)\Rightarrow X(s) = (sI-A)^{-1}B U(s) $$

but $Y(s) = C X(s)$ and calling the controller $K(s)$ we have

$$ U(s) = K(s)(Y_r(s)-Y(s)) $$

then in closed loop

$$ Y(s) = C(sI-A)^{-1}B K(s)(Y_r(s)-Y(s)) $$

or

$$ \frac{Y(s)}{Y_r(s)} = \frac{C(sI-A)^{-1}B K(s)}{1+C(sI-A)^{-1}B K(s)} = \frac{K(s)}{K(s)+s^2}=\frac{\omega^2}{s^2+2\zeta\omega s + \omega^2} $$

now solving for $K(s)$ we obtain

$$ K(s) = \frac{\omega^2s}{s+2\zeta\omega} $$