Universal Algebraic Controllers and System Identification
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In this document, some structured operator approximation theoretical methods for system identification of nearly eventually periodic systems, are presented. Let C^n× m denote the algebra of n× m complex matrices. Given ε>0, an arbitrary discrete-time dynamical system (Σ,T) with state-space Σ contained in the finite dimensional Hilbert space C^n, whose state-transition map T:Σ× ([0,∞)∩Z)→Σ is unknown or partially known, and needs to be determined based on some sampled data in a finite set Σ̂={x_t}_1≤ t≤ m⊂Σ according to the rule T(x_t,1)=x_t+1 for each 1≤ t≤ m-1, and given x∈Σ̂. We study the solvability of the existence problems for two triples (p,A,φ) and (p,A_η,Φ) determined by a polynomial p∈C[z] with (p)≤ m, a matrix root A∈C^m× m and an approximate matrix root A_η∈C^r× r of p(z)=0 with r≤ m, two completely positive linear multiplicative maps φ:C^m× m→C^n× n and Φ:C^r× r→C^n× n, such that T(x,t)-φ(A^t)x≤ε and Φ(A_η^t)x-φ(A^t)x≤ε, for each integer t≥ 1 such that T(x,t)-y≤ε for some y∈Σ̂. Some numerical implementations of these techniques for the reduced-order predictive simulation of dynamical systems in continuum and quantum mechanics, are outlined.
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