Download PDF by Cornelius T. Leondes: Advances in Theory and Applications : System Identification
By Cornelius T. Leondes
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Extra resources for Advances in Theory and Applications : System Identification and Adaptive Control, Part 2
Hence every system σ E S(n) belongs to one of the V therefore has a set of structure indices μ = tached to it. , n ) at- Those structure indices are usually called o b servability indices or left Kronecker indices; they determine the local coordinate system in which that system is described by the vector p . These structure indices and these coordinates will in turn define canonical S S , MFD, or ARMAX forms, as we shall see in the next section. They form a complete system of independent invariants for K(z) (see all cases d(\i) The canonical forms therefore £ d = n (p + s) .
2. Let μ and v be two partitions such that |μ| = The intersection U Π u is also dense in S(n). A point in that intersection can be represented by either τ or τ e Θ . G Θ Since S(n) is an analytic manifold, it follows that these two sets of coordinates are analytically related; that is the mapping % · φ ν 1 ; φ ν (ϋ μ Π V * V U y ° V (38) is analytic. Comment 3. S(n) can be covered by the Γη + p - ll [ P-1J open sets U , |μ| = n. Whether it can be covered by fewer sets is still an open question.
And to add e(t) to the right-hand side [see (33)]. An ARMA model would then be converted back into an ARMAX model. How ever, at this stage the only issue is one of parametrizing a SS, MFD, or ARMA model of a strictly causal rational transfer function. A. 1ια ι α 0 ... 0 ... 121 12η 1 2 0 0 0 •·· pll pin Ί 0 ·· 0 ^ pi ! '" ί 0 •· 0 ' | ... 0 0 ... ρ21 Ρ 2 η ρ2 0 ο 0 ' αΊ , · ! ΐρΐ 0 ·· 0 | _| j ... j • • 0 ο ! • ο [ • | j ο | . , (1, 1) ··· k (1, s) ' n, ' Β (nxs) (45c) k 1 (p / 1) ··· k1(p, kn P s) (pf 1) ··· k n (p, s) P Note that A and C have a very specific structure while B is fully parametrized.
Advances in Theory and Applications : System Identification and Adaptive Control, Part 2 by Cornelius T. Leondes