State feedback gain for observer/estimator

[PainterGuy]

Homework Statement

I cannot figure out how to solve Part 4. It requires to apply the gain to observer/estimator.

Relevant Equations

Please check the posting.

For the state equations given above calculate state feedback gain k so that the state feedback system has -1 and -2 as its poles. Then, find TF from r to y for the given state feedback system. Then create a 2 dimensional estimator where the poles of estimator are -2±2j. Finally calculate TF from r to y for the estimator if the found feedback gain is applied to estimated state of estimator. Do you find that two transfer functions are same?

I don't know how to solve for Part 4. All the other answers are correct. Could you please help me with Part 4?

 

[Valkarie]

Thank you very much.For Part 4, we can solve for the transfer function of the state feedback system by first rearranging the state equations to get an expression for x1 and x2 in terms of the inputs r, u, and y. Then, we can substitute this into the equation for y to get the transfer function from r to y.The transfer function of the estimator is found by rearranging the estimator equations to get an expression for x1_hat and x2_hat in terms of the inputs r, u, and y_hat. We then substitute this expression into the equation for y to get the transfer function from r to y.The two transfer functions should be the same since they are both derived from the same state equations.