1.2 KiB
1.2 KiB
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
- Compute Controllability Matrix C = [B AB … A^(n-1)B]
- If rank( C) = n <==> controllable
- Singular value decomposition (SVD) tells us about:it orders singular vectors to show most controllable to least controllable states
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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
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Reachable set R_t: all vectors in Rn that can be reached zome where sys is controllable
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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.
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Controllability Gramian:W_t = 
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Stabilizibility: all unstable directions (eigenvectors) are controllable.
- Unstable and lightly damped directions should be controllable!
