Lecture 9: Controllability and Observability - Illinois Institute of ...
Lecture 9: Controllability and Observability - Illinois Institute of ...
Lecture 9: Controllability and Observability - Illinois Institute of ...
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<strong>Controllability</strong> Form<br />
Example<br />
Let<br />
⎡<br />
1 −1<br />
⎤<br />
0<br />
⎡ ⎤<br />
1<br />
A = ⎣−1 1 0 ⎦ <strong>and</strong> B = ⎣0⎦<br />
0 0 −1<br />
0<br />
Construct C(A, B) = [ B AB A 2 B ] .<br />
⎡<br />
1 −1<br />
⎤ ⎡ ⎤<br />
0 1<br />
⎡ ⎤<br />
1<br />
AB = ⎣−1 1 0 ⎦ ⎣0⎦ = ⎣−1⎦ ,<br />
0 0 −1 0 0<br />
⎡<br />
⎤ ⎡ ⎤ ⎡ ⎤<br />
1 −1 0 1 0<br />
A 2 B = A(AB) = ⎣−1 1 0 ⎦ ⎣−1⎦ = ⎣0⎦<br />
0 0 −1 0 0<br />
Thus<br />
⎡<br />
1 1<br />
⎤<br />
0<br />
C(A, B) = ⎣0 −1 0⎦<br />
0 0 0<br />
Thus rank C(A, B) = 2 < n = 3 which means not controllable.<br />
M. Peet <strong>Lecture</strong> 9: <strong>Controllability</strong> 14 / 15