Lecture 2: Controllability of nonlinear systems

More documents

Recommendations

Info

DISC Systems and Control Theory of Nonlinear Systems 18Example 8 Consider the following set of partial differentialequations0 = x 1∂φ∂x 1+ x 2∂φ∂x 2+ x 3∂φ∂x 30 = ∂φ∂x 3Define the vector fieldsf 1 (x) =⎡⎢⎣⎤ ⎡x 1x 2⎥⎦ , f 2(x) = ⎢⎣x 3001⎤⎥⎦ .It is checked that D := span(f 1 , f 2 ) has constant dimension = 2 onthe set X = {x ∈ R 3 | x 2 1 + x 2 2 ≠ 0} (that is, R 3 excluding the x 3 -axis),and is involutive. Thus, by Frobenius’ theorem, D is integrable.
DISC Systems and Control Theory of Nonlinear Systems 19Consequently, for each x 0 ∈ X, there exists a neighborhood N of x 0and a real-valued function φ(x) with dφ(x) ≠ 0 that satisfies thegiven set of partial differential equations. In fact, φ(x) = lnx 1 − lnx 2is a (global) solution.Note that the solution is not unique. In particular, φ(x) = tan −1 x 2x 1isalso a global solution.
Page 1 and 2: DISC Systems and Control Theory of
Page 17: DISC Systems and Control Theory of

Lecture 2: Controllability of nonlinear systems

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?