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.stversions/OneNote/OSD/Control System/Controllability & Observability~20250203-063556.md

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+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!