Universal Algebraic Controllers and System Identification

Abstract

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 $\epsilon >0$, an arbitrary discrete-time dynamical system $(\Sigma ,\mathcal{T})$ with state-space $\Sigma$ contained in the finite dimensional Hilbert space ${\mathbb{C}}^n$, whose state-transition map $\mathcal{T}:\Sigma \times ([0,\infty )\cap \mathbb{Z})\to \Sigma$ is unknown or partially known, and needs to be determined based on some sampled data in a finite set $\hat{\Sigma }=\{x_t{\}}_{1\le t\le m}\subset \Sigma$ according to the rule $\mathcal{T}(x_t,1)=x_{t+1}$ for each $1\le t\le m-1$, and given $x\in \hat{\Sigma }$. We study the solvability of the existence problems for two triples $(p,A,\phi )$ and $(p,A_{\eta },\Phi )$ determined by a polynomial $p\in \mathbb{C}[z]$ with $°(p)\le m$, a matrix root $A\in {\mathbb{C}}^{m\times m}$ and an approximate matrix root $A_{\eta }\in {\mathbb{C}}^{r\times r}$ of $p(z)=0$ with $r\le m$, two completely positive linear multiplicative maps $\phi :{\mathbb{C}}^{m\times m}\to {\mathbb{C}}^{n\times n}$ and $\Phi :{\mathbb{C}}^{r\times r}\to {\mathbb{C}}^{n\times n}$, such that $\Vert \mathcal{T}(x,t)-\phi (A^t)x\Vert \le \epsilon$ and $\Vert \Phi (A_{\eta }^t)x-\phi (A^t)x\Vert \le \epsilon$, for each integer $t\ge 1$ such that $\Vert \mathcal{T}(x,t)-y\Vert \le \epsilon$ for some $y\in \hat{\Sigma }$. Some numerical implementations of these techniques for the reduced-order predictive simulation of dynamical systems in continuum and quantum mechanics, are outlined.