论文标题
通用代数控制器和系统标识
Universal Algebraic Controllers and System Identification
论文作者
论文摘要
在本文档中，提出了一些用于系统识别几乎最终周期系统的结构化操作员近似理论方法。令$ \ mathbb {c}^{n \ times m} $表示$ n \ times m $复杂矩阵的代数。给定$ \ varepsilon> 0 $，一种任意离散的时间动态系统$（σ，\ mathcal {t}）$，带有有限维度的Hilbert Space $ \ Mathbb {C}^n $中包含的状态空间$σ$，其状态转移映射$ \ \ m iffty {c} \ Mathbb {z}）\至σ$是未知或部分已知的，需要根据有限集中的一些样本数据来确定$ \hatς= \ {x_t \} _ {1 \ \ leq t \ leq t \ leq t \ leq t \ leq m} \ subset c $ \ subsetσ$ c $ \ nirter = $ \ nationcal $ \ natercal $ \ $ 1 $ 1 \ leq t \ leq m-1 $，并在\hatς$中给定$ x \。我们研究了两个三元$（p，a，φ）$和$（p，a_η，φ）$的存在问题的可解问题，由多项式$ p \ in \ mathbb {c} [c} [z] $带有$°（p）\ leq m $，matrix unimies $a_η\ in \ Mathbb {c}^{r \ times r} $ p（z）= 0 $ with $ r \ leq m $，两个完全积极的线性乘法映射$φ：\ m m缩$φ：\ MATHBB {C}^{r \ times r} \ to \ to \ Mathbb {c}^{n \ times n} $，因此$ \ | \ | \ m athcal {t}（x，x，x，t）-φ（a^t） $ \ |φ（a_η^t）x-φ（a^t）x \ | \ leq \ varepsilon $，对于每个整数$ t \ geq 1 $，以便$ \ | \ | \ m athcal {t}（x，x，x，x，x，x，x，t）-y \ | \ | \ leq \ leq \ leq \ varepsilon $ for Shore $ y y \ hat \ for for hat \ for。概述了这些技术的一些数值实现，以概述了连续和量子力学中动态系统的减少阶预测模拟。
In this document, some structured operator approximation theoretical methods for system identification of nearly eventually periodic systems, are presented. Let $\mathbb{C}^{n\times m}$ denote the algebra of $n\times m$ complex matrices. Given $\varepsilon>0$, an arbitrary discrete-time dynamical system $(Σ,\mathcal{T})$ with state-space $Σ$ contained in the finite dimensional Hilbert space $\mathbb{C}^n$, whose state-transition map $\mathcal{T}:Σ\times ([0,\infty)\cap \mathbb{Z})\to Σ$ is unknown or partially known, and needs to be determined based on some sampled data in a finite set $\hatΣ=\{x_t\}_{1\leq t\leq m}\subset Σ$ according to the rule $\mathcal{T}(x_t,1)=x_{t+1}$ for each $1\leq t\leq m-1$, and given $x\in \hatΣ$. We study the solvability of the existence problems for two triples $(p,A,φ)$ and $(p,A_η,Φ)$ determined by a polynomial $p\in \mathbb{C}[z]$ with $°(p)\leq m$, a matrix root $A\in\mathbb{C}^{m\times m}$ and an approximate matrix root $A_η\in\mathbb{C}^{r\times r}$ of $p(z)=0$ with $r\leq m$, two completely positive linear multiplicative maps $φ:\mathbb{C}^{m\times m}\to \mathbb{C}^{n\times n}$ and $Φ:\mathbb{C}^{r\times r}\to \mathbb{C}^{n\times n}$, such that $\|\mathcal{T}(x,t)-φ(A^t)x\|\leq\varepsilon$ and $\|Φ(A_η^t)x-φ(A^t)x\|\leq\varepsilon$, for each integer $t\geq 1$ such that $\|\mathcal{T}(x,t)-y\|\leq \varepsilon$ for some $y\in \hatΣ$. Some numerical implementations of these techniques for the reduced-order predictive simulation of dynamical systems in continuum and quantum mechanics, are outlined.
