2020/01/29 by Fredy Vides, Vides, Fredy
Computer Science · Decision Sciences · Engineering · Mathematics · Physics and Astronomy · #47N70 (primary) and 93C57 #93B28 #93B40 (secondary) #Control Systems and Identification #FOS: Electrical engineering #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Operator Algebras (math.OA) #Optimization and Control (math.OC) #Probabilistic and Robust Engineering Design #Systems and Control (eess.SY) #cs.NA #cs.SY #eess.SY #electronic engineering #information engineering #math.NA #math.OA #math.OC #msc:47N70 #msc:93B28 #msc:93B40 #msc:93C57
paper · pdf · doi:10.48550/arxiv.2001.11133
arxiv created 2020/01/29 · openalex publication_date 2020/01/29 · arxiv updated 2020/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this document, some structured operator approximation theoretical methods for system identification of nearly eventually periodic systems, are presented. Let ℂn× 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 ℂn, 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 Σ=\xt\1≤ t≤ m⊂ Σ according to the rule T(xt,1)=xt+1 for each 1≤ t≤ m-1, and given x∈ Σ. We study the solvability of the existence problems for two triples (p,A,φ) and (p,Aη,Φ) determined by a polynomial p∈ ℂ[z] with °(p)≤ m, a matrix root A∈ℂm× m and an approximate matrix root Aη∈ℂr× r of p(z)=0 with r≤ m, two completely positive linear multiplicative maps φ:ℂm× m→ ℂn× n and Φ:ℂr× r→ ℂn× n, such that ‖T(x,t)-φ(At)x‖≤ε and ‖Φ(Aηt)x-φ(At)x‖≤ε, for each integer t≥ 1 such that ‖T(x,t)-y‖≤ ε for some y∈ Σ. Some numerical implementations of these techniques for the reduced-order predictive simulation of dynamical systems in continuum and quantum mechanics, are outlined.