Lecture 2: Controllability of nonlinear systems

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DISC Systems and Control Theory of Nonlinear Systems 28Hence we conclude that C is spanned by all constant vector fieldsb i , Ab i , A 2 b i , . . ., i ∈ m, together with the linear drift vector field Ax,i.e.,C = {Ax, b i , Ab i , A 2 b i . . .,A n−1 b i , i = 1, . . .,m}.whileC 0 = columns of (B, AB, A 2 B . . .,A n−1 B)We see that for linear systems the rank condition for strongaccessibility coincides with the Kalman rank condition forcontrollability. Hence, if we would not have known anything specialabout linear systems, then at least a linear system which satisfiesthe Kalman rank condition is locally strongly accessible.
DISC Systems and Control Theory of Nonlinear Systems 29Example 12 (Actuated rotating rigid body)Consider⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤˙ω 1 A 1 ω 2 ω 3 α 1 0⎢⎣ ˙ω 2⎥⎦ = ⎢⎣A 2 ω 3 ω 1⎥⎦ + ⎢⎣ 0 ⎥⎦ u 1 + ⎢⎣α 2⎥⎦ u 2˙ω 3 A 3 ω 1 ω 2 0 0with α 1 ≠ 0, α 2 ≠ 0. Here the constants A 1 , A 2 , A 3 are determined bythe moments of inertia a 1 , a 2 , a 3 . Compute⎡ ⎤0[g 1 , f](ω) = ⎢⎣α 1 A 2 ω 3⎥⎦α 1 A 3 ω 2[g 2 , f](ω) =⎡ ⎤α 2 A 1 ω 3⎢⎣ 0 ⎥⎦α 2 A 3 ω 1
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Lecture 2: Controllability of nonlinear systems

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