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

ISBN-10: 1846285941

ISBN-13: 9781846285943

ISBN-10: 184628595X

ISBN-13: 9781846285950

It is a self-contained advent to algebraic keep watch over for nonlinear platforms compatible for researchers and graduate scholars. it's the first publication facing the linear-algebraic method of nonlinear keep an eye on structures in this type of distinct and vast type. It presents a complementary method of the extra conventional differential geometry and offers extra simply with numerous vital features of nonlinear platforms.

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**Example text**

Y (k−1) , u, u, ˙ . . , u(s) ). ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ y y˙ 0 ⎢ .. ⎥ ⎢ .. ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ . 15) ⎢ ⎢ ⎥ ⎥ ⎢ dt ⎢ u ⎥ ⎢ u˙ ⎥ ⎥ ⎢0⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ .. ⎥ ⎢ .. ⎥ ⎢ .. 14), deﬁne the ﬁeld K of meromorphic functions in a ﬁnite number of variables y, u, and their time derivatives. Let E be the formal vector space E = spanK {dϕ | ϕ ∈ K}. Deﬁne the following subspace of E ˙ . . , dy (k−1) , du, . . , du(s) } H1 = spanK {dy, dy, Obviously, any one-form in H1 has to be diﬀerentiated at least once to depend explicitly on du(s+1) .

Y (k) , u, . . 8) where u and y are, respectively, a scalar input and a scalar output, F is a meromorphic function of its arguments; and ∂y∂F(k) is generically nonzero. 8) can easily be constructed by introducing the new variable x = (x1 , . . , xk ), deﬁned by (x1 , . . , xk ) = (y, . . , y (k−1) ) This yields the following set of implicit state equations ⎧ x˙ 1 = x2 ⎪ ⎪ ⎪ ⎨ .. ⎪ = xk x ˙ ⎪ k−1 ⎪ ⎩ (s) F (x1 , . . , xk , x˙ k , u, . . 10) The assumption about ∂y∂F(k) and the implicit function theorem , now, allow us to write, at least locally, ⎧ x˙ 1 = x2 ⎪ ⎪ ⎪ ⎪ ..

A sequence of subspaces {Hk } of E such that each Hk , for k > 0, is the set of all one-forms with relative degree at least k. The sequence is deﬁned by induction as follows: H0 = spanK {dx, du}, Hj = {ω ∈ Hj−1 | ω˙ ∈ Hj−1 }. , E ⊃ H0 ⊃ H1 ⊃ H2 ⊃ · · · , and that, at the ﬁrst step, H1 = spanK {dx} An easy consequence of the construction is the following. 16. Hk is the space of one-forms whose relative degrees are greater than or equal to k. Furthermore, there exists an integer k ∗ > 0 such that: Hk ⊃ Hk+1 for k ≤ k ∗ , Hk∗ +1 = Hk∗ +2 = · · · = H∞ Hk∗ ⊇ / H∞ By deﬁnition, it follows that A = H∞ .

### Algebraic Methods for Nonlinear Control Systems (Communications and Control Engineering) by Giuseppe Conte, Claude H. Moog, Anna Maria Perdon

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