Xdd = Ax + Bu

Y = Cx + Du

  
  

A and B decide if it is controllable

C and D decide if it is observable

# Controlability

- System can be uncontrollable linearly, but controllable non-linearly
  

  

## Linear Systems

1. Compute Controllability Matrix C = [B AB … A^(n-1)B]
2. If rank( C) = n <==> controllable
3. Singular value decomposition (SVD) tells us about:it orders singular vectors to show most controllable to least controllable states

  

- If system is controllable then:
    
    - Arbitrary eigenvalue (pole) placementu = -Kx <==> xd = (A-BK)x
    - Reachibility (get to any state in R^n)R_t = R_n

Definitions

- Reachable set R_t: all vectors in Rn that can be reached zome where sys is controllable
- Controllability Matrix: C = [B AB … A^(n-1)B]This matrix is equivalent to an impulse response in dicrete time: basically matrix says wether the control input reaches all the states eventually.
- Controllability Gramian:W_t = 
    
    - ![冖 DDzYm ](Exported%20image%2020231126171851-0.png)
- Stabilizibility: all unstable directions (eigenvectors) are controllable.
    
    - Unstable and lightly damped directions should be controllable!