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

This can be a self-contained advent to algebraic regulate for nonlinear structures appropriate for researchers and graduate scholars. it's the first ebook facing the linear-algebraic method of nonlinear keep an eye on platforms in this type of exact and huge style. It offers a complementary method of the extra conventional differential geometry and bargains extra simply with numerous vital features of nonlinear structures.

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**Example text**

15). ∗ Consider H∞ = spanK {ω ∈ H1∗ | ω (k) ∈ H1∗ , ∀k ≥ 0} = 0. 14). 1 Irreducible Input-output Systems In this section, we will formalize a reduction algorithm to obtain the notion of input-output equivalence and a deﬁnition of realization. 6 (Irreducible input-output system). 7. The input-output system y¨ = yu2 + y u˙ is irreducible since ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ y 0 y˙ ⎟ ⎜ yu2 + y u˙ ⎟ ⎜ 0 ⎟ d ⎜ y ˙ ⎜ ⎟=⎜ ⎟ + ⎜ ⎟ u¨ ⎠ ⎝0⎠ u˙ dt ⎝ u ⎠ ⎝ 0 u˙ 1 is such that H∞ = 0. 13. 8. y¨ = u˙ + (y˙ − u)2 is not irreducible since ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ y 0 y˙ ⎟ ⎜ u˙ + (y˙ − u)2 ⎟ ⎜ 0 ⎟ d ⎜ y ˙ ⎜ ⎟=⎜ ⎟+⎜ ⎟u ⎠ ⎝0⎠ ¨ u˙ dt ⎝ u ⎠ ⎝ 0 u˙ 1 is not irreducible since d(y˙ − u) ∈ H∞ and we will claim that y˙ = u is an irreducible input-output system of y¨ = u˙ + (y˙ − u)2 .

15), one easily computes H2 = spanK {dy, dy, ˙ . . , dy (k−1) , du, . . 5 Input-output Equivalence and Realizations 29 More generally, deﬁne Hi as the subspace of E which consists of all oneforms that have to be diﬀerentiated at least i times to depend explicitly on du(s+1) . More precisely, the subspaces Hi are deﬁned by induction as follows for i ≥ 2. Hi+1 = spanK {ω ∈ Hi | ω˙ ∈ Hi } These subspaces will be used extensively later in this book and especially in Chapter 3. 15). ∗ Consider H∞ = spanK {ω ∈ H1∗ | ω (k) ∈ H1∗ , ∀k ≥ 0} = 0.

To investigate this structure, consider the dynamic system Σe whose input is u(s+1) and whose state is (y, y, ˙ . . , 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, . .