System Theory

Download Algebraic Methods for Nonlinear Control Systems by Giuseppe Conte, Claude H. Moog, Anna Maria Perdon PDF

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By Giuseppe Conte, Claude H. Moog, Anna Maria Perdon

This can be a self-contained advent to algebraic keep watch over for nonlinear structures appropriate for researchers and graduate scholars. it's the first booklet facing the linear-algebraic method of nonlinear keep an eye on structures in any such designated and vast model. It offers a complementary method of the extra conventional differential geometry and offers extra simply with numerous vital features of nonlinear structures.

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Extra resources for Algebraic Methods for Nonlinear Control Systems (Communications and Control Engineering)

Example text

3). Instead of {s1 , . . , sp }, it is possible to use the observability indices as defined in Chapter 4 to derive an analogous input-output equation. 2. For the system ⎧ x˙ 1 ⎪ ⎪ ⎪ ⎪ ⎨ x˙ 2 x˙ 3 ⎪ ⎪ y1 ⎪ ⎪ ⎩ y2 = = = = = x3 u1 u1 u2 x1 x2 we have y˙ 1 = x3 u1 , y¨1 = u2 u1 + x3 u˙ 1 , and finally y¨1 = u2 u1 + (y˙ 1 /u1 )u˙ 1 The last equation holds at every point in which u1 = 0. For the second output, y˙ 2 = u1 immediately. The following example shows that for a more general nonlinear system, where x˙ does not appear explicitly, such as F (x, x, ˙ u, .

For the second output, y˙ 2 = u1 immediately. The following example shows that for a more general nonlinear system, where x˙ does not appear explicitly, such as F (x, x, ˙ u, . . 7) the method described above cannot be applied. 3. Consider the system (x˙ − u)2 = 0 y=x The implicit function theorem cannot be invoked to obtain x, since for every x and every u, ∂(x˙ − u)2 /∂x = 0. By the way, an input-output relation for this example is given by (y˙ − u)2 = 0 or by (y˙ − u) = 0 26 2 Modeling Results similar to those described above may be found in [156].

1. 2). Proof. The first step in constructing an input-output representation consists of applying a suitable change of coordinates. To this aim, let us denote by s1 the minimum nonnegative integer such that (s −1) rank ∂(h1 , . . , h1 1 ∂x ) (s ) = rank ∂(h1 , . . , h1 1 ) ∂x If ∂h1 /∂x ≡ 0 we define s1 = 0. Analogously for 1 < j ≤ p, let us denote by sj the minimum integer such that (s −1) rank ∂(h1 , . . , h1 1 (sj −1) ; . . ; h j , . . , hj ∂x (s −1) = rank ∂(h1 , . . , h1 1 If (s −1) (sj ) ; .

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