Download Hidden Order: How Adaptation Builds Complexity (Helix Books) by John Holland PDF

By John Holland

This paintings explains how scientists who research the sector of complexity are confident that yes consistent methods are at paintings in all types of unrelated complicated platforms. the writer additionally illustrates the relevance of clinical debate to the layman.

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Additional info for Hidden Order: How Adaptation Builds Complexity (Helix Books)

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, .

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