41 lines
1.2 KiB
Markdown
41 lines
1.2 KiB
Markdown
Xdd = Ax + Bu
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Y = Cx + Du
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A and B decide if it is controllable
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C and D decide if it is observable
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# Controlability
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- System can be uncontrollable linearly, but controllable non-linearly
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## Linear Systems
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1. Compute Controllability Matrix C = [B AB … A^(n-1)B]
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2. If rank( C) = n <==> controllable
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3. 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:
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- Arbitrary eigenvalue (pole) placementu = -Kx <==> xd = (A-BK)x
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- Reachibility (get to any state in R^n)R_t = R_n
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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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- 
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- Stabilizibility: all unstable directions (eigenvectors) are controllable.
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- Unstable and lightly damped directions should be controllable! |