State Feedback and Observer

Learning objectives

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

Can measurements distinguish states?

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.

import numpy as np
A=np.array([[0.,1.],[-2.,-3.]])
B=np.array([[0.],[1.]])
C=np.array([[1.,0.]])
for label,b,c in [('base',B,C),('no input',0*B,C),('no sensor',B,0*C)]:
    ctrb=np.column_stack([b,A@b]);obsv=np.vstack([c,c@A])
    print(label,np.linalg.matrix_rank(ctrb),np.linalg.matrix_rank(obsv))
# TODO: State what can and cannot be designed under each condition.

Record results and self-check

Condition:

Controllability rank:

Observability rank:

Design feasibility:

Points to check

• Are both ranks of the baseline model 2?
• Did you avoid applying pole placement directly to a rank-deficient model?

Connecting state feedback and observer

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.

from scipy.signal import place_poles,StateSpace,lsim
import matplotlib.pyplot as plt
K=place_poles(A,B,[-2.,-4.]).gain_matrix
L=place_poles(A.T,C.T,[-8.,-10.]).gain_matrix.T
a_aug=np.block([[A,-B@K],[L@C,A-B@K-L@C]])
b_aug=np.vstack([8*B,8*B])
t=np.linspace(0,4,801)
_,z,_=lsim(StateSpace(a_aug,b_aug,np.eye(4),np.zeros((4,1))),np.ones(len(t)),t,X0=[1,0,0,0])
fig,axes=plt.subplots(2,1,sharex=True)
for j,ax in enumerate(axes):
    ax.plot(t,z[:,j],label='state');ax.plot(t,z[:,j+2],'--',label='estimate');ax.legend();ax.grid()
axes[0].set_ylabel('Position [m]');axes[1].set_ylabel('Velocity [m/s]');axes[-1].set_xlabel('Time [s]')
print('final estimation error:',np.linalg.norm(z[-1,:2]-z[-1,2:]))
# TODO: Print and compare the control poles and observer poles.

Record results and self-check

K:

L:

Control poles:

Observer Pole:

Final error:

Points to check

• Do the eigenvalues of $A-BK$ and $A-LC$, after substituting the calculated $K$ and $L$, match the specified control and observer poles, respectively?
• At $t=4$ s, are the position and velocity estimation errors below 1e-6 m and 1e-6 m/s, respectively?

Is a faster observer always better?

Starting conditions

Add n(t)=0.02sin(40t) 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.

def noisy_observer(poles):
    L=place_poles(A.T,C.T,poles).gain_matrix.T
    aa=np.block([[A,-B@K],[L@C,A-B@K-L@C]])
    bb=np.block([[8*B,np.zeros((2,1))],[8*B,L]])
    inputs=np.column_stack([np.ones(len(t)),.02*np.sin(40*t)])
    _,z,_=lsim(StateSpace(aa,bb,np.eye(4),np.zeros((4,2))),inputs,t,X0=[1,0,0,0])
    u=8-(z[:,2:]@K.T).ravel();keep=t>=1
    return u,np.sqrt(np.mean((z[keep,:2]-z[keep,2:])**2,axis=0)),np.sqrt(np.mean(u[keep]**2))
for poles in [(-8,-10),(-20,-25)]:
    u,err,rms=noisy_observer(poles);plt.plot(t,u,label=str(poles));print(poles,err,rms)
plt.xlabel('Time [s]');plt.ylabel('u [N]');plt.legend();plt.grid()
# TODO: Explain which of the two settings you would choose, based on the metrics.

Record results and self-check

Observer Pole:

Position error RMS [m]:

Velocity error RMS [m/s]:

Input RMS [N]:

Reason for your choice:

Points to check

• Are noise, initial conditions, and time steps the same for both settings?
• Did you avoid conflating error and input in a single metric?

Official examples and data

• SciPy · Pole placement