By Giuseppe Conte, Claude H. Moog, Anna Maria Perdon
This is a self-contained creation to algebraic regulate for nonlinear structures compatible for researchers and graduate scholars. it's the first e-book facing the linear-algebraic method of nonlinear keep watch over structures in this kind of particular and wide style. It offers a complementary method of the extra conventional differential geometry and offers extra simply with numerous very important features of nonlinear systems.
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Extra resources for Algebraic methods for nonlinear control systems
Example text
Although the validity of such a representation is only local, it nevertheless turns out to be useful for understanding the system behavior and, more important, its construction helps in clarifying the inverse problem of defining state variables and state equations from an input/output relation. , representations of the 22 2 Modeling form x˙ = f (x, u, . . , u(s) ) y = h(x, u, . . 1) where, as usual, x ∈ IRn , u ∈ IRm , and y ∈ IRp , and the entries of f and h, which depend also on a finite number of time derivatives of the input, are analytic functions.
Dy (r−1) ; . . ; dyij , . . , dyij ij } be a basis for ( ) ∗ Xi+1,j := Xi+1,j−1 + Ds+2 ∩ spanK {dyij , ≥ 0} where rij = dimXi+1,j − dimXi+1,j−1 . Set Xi+1 = (r ) • If ∀ ≥ rij , dyij ij ∈ Xi+1 , set sij = −1. Xi+1,j ( ) If ∃ ≥ rij , dyij ∈ Xi+1 , then define sij as the smallest integer such that, abusing the notation, one has locally (r +sij ) yij ij (r +sij ) = yij ij (σ) (y (λ) , yij , u, . . , u(sij ) ) where 0 < λ < r, 0 < σ < rij + sij . 2 (r +s ) • If sij ≥ 0 and ∂ 2 yij ij ij /∂u(sij ) = 0 for some j = 1, .
Hj−1 ) ∂(h1 , . . , hj−1 = rank ∂x ∂x we define sj = 0. Write K = s1 + . . + sp . The vector rank −1) , hj ) S = (h1 , . . , h1s1 −1 , . . , hp , . . 1 State Elimination 23 It will be established in Chapter 4 that the case K < n corresponds to nonobservable systems. In this case, there exist analytic functions g1 (x), . . , gn−K (x) such that the matrix J = ∂(S, g1 , . . , gn−K ) ∂x has full rank n. Then the system of equations ⎧ x ˜1 = h1 (x, u, . . , u(α) ) ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ (s −1) ⎪ ⎪ = h1 1 (x, u, .