Universal Algebraic Controllers and System Identification

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 $(\mathrm{\Sigma },\mathcal{T})$ with state-space $\mathrm{\Sigma }$ contained in the finite dimensional Hilbert space ${\mathbb{C}}^n$, whose state-transition map $\mathcal{T}:\mathrm{\Sigma }\times ([0,\mathrm{\infty })\cap \mathbb{Z})\to \mathrm{\Sigma }$ is unknown or partially known, and needs to be determined based on some sampled data in a finite set $\widehat{\mathrm{\Sigma }}=\{x_t{\}}_{1\le t\le m}\subset \mathrm{\Sigma }$ according to the rule $\mathcal{T}(x_t,1)=x_{t+1}$ for each $1\le t\le m-1$, and given $x\in \widehat{\mathrm{\Sigma }}$. We study the solvability of the existence problems for two triples $(p,A,\phi )$ and $(p,A_{\eta },\mathrm{\Phi })$ determined by a polynomial $p\in $\mathbb{C}$[z]$with$\deg(p)\leq m$,amatrixroot$A\in$\mathbb{C}$^{m\times m}$andanapproximatematrixroot$A_\eta\in$\mathbb{C}$^{r\times r}$of$p(z)=0$with$r\leq m$,twocompletelypositivelinearmultiplicativemaps$\varphi:$\mathbb{C}$^{m\times m}\to $\mathbb{C}$^{n\times n}$and$\Phi:$\mathbb{C}$^{r\times r}\to $\mathbb{C}$^{n\times n}$,suchthat$|$\mathcal{T}$(x,t)-\varphi(A^t)x|\leq\varepsilon$and$|\Phi(A_\eta^t)x-\varphi(A^t)x|\leq\varepsilon$,foreachinteger$t\geq 1$suchthat$|$\mathcal{T}$(x,t)-y|\leq \varepsilon$forsome$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.