State Feedback and Observer

Estimate velocity to control a system that measures only position, and compare estimated velocity and noise sensitivity.

Starting conditions

With position [m] and velocity [m/s] as states, use $\dot x=Ax+Bu$, $y=Cx$. Use $A=\begin{bmatrix}0&1\\-2&-3\end{bmatrix}$, $B=\begin{bmatrix}0\\1\end{bmatrix}$, $C=\begin{bmatrix}1&0\end{bmatrix}$.

Procedure

1. Form the controllability matrix [B,AB] and observability matrix [C;CA], and find their ranks.
2. Compare the case where B is a zero vector with the case $C=\begin{bmatrix}0&0\end{bmatrix}$.
3. Explain the distinction between an ineffective input and an unavailable measurement.

Results to retain

• A comparison table of both ranks and feasible designs for the three conditions.

Starting conditions

$u=-K\hat x+8r$, $r=1$. Control poles: −2,−4; observer poles: −8,−10. Use $x(0)=\begin{bmatrix}1\\0\end{bmatrix}$, $\hat x(0)=\begin{bmatrix}0\\0\end{bmatrix}$.

Procedure

1. Calculate K and L, and compare the eigenvalues of $A-BK$ and $A-LC$.
2. Integrate actual and estimated states simultaneously, and plot position, velocity, and estimation errors over 0–4 s.
3. Distinguish the reduction in estimation error from the output approaching its reference.

Results to retain

• Two plots of actual and estimated position and actual and estimated velocity, and the final estimation errors.

Starting conditions

Add n(t)=0.02*sin(40*t) to the measurement. Compare observer poles (−8,−10) and (−20,−25), keeping controller K fixed.

Procedure

1. Use both observer settings in the practice notebook's function, and compare the control input u on the same time axis.
2. Calculate position-error RMS, velocity-error RMS, and control-input RMS separately after 1 s.
3. Interpret initial estimation-error convergence separately from the response to measurement noise.

Results to retain

• Control-input curves, a comparison table of the three RMS values, and justification for the selected setting.

• SciPy · Pole placement