Discrete linear input-state-output experiments

Let $d\ge 0$, $m,p\ge 1$, and $q\ge 1$ be integers, and let $h>0$ be a fixed sampling interval. A real discrete linear input-state-output experiment of state dimension $d$, input dimension $m$, output dimension $p$, and horizon $q$ consists of matrices $A\in M_{d\times d}(\mathbb{R})$, $B\in M_{d\times m}(\mathbb{R})$, and $C\in M_{p\times d}(\mathbb{R})$, an initial state $x_0\in {\mathbb{R}}^d$, and known inputs $(u_k{)}_{0\le k<q-1}$ in ${\mathbb{R}}^m$. The states are defined by $x_{k+1}=A{x}_k+B{u}_k(0\le k<q-1),$ and the exact sampled outputs at times $kh$ are $y_k=C{x}_k$ for $0\le k<q$. When observations are perturbed, the observed outputs are ${\tilde{y}}_k=y_k+{\eta }_k$ for deterministic error vectors ${\eta }_k\in {\mathbb{R}}^p$; the exact-data convention is ${\eta }_k=0$ for all $k$. No probability law is assigned to the errors. A numerical or discretization error that changes the forward map is distinct from this additive observation error.

For the fixed matrices $A,B,C$, the Markov parameters are $M_k:=C{A}^{k}B\in M_{p\times m}(\mathbb{R})(k\ge 0),$ interpreted using the infinite-horizon impulse experiment: extend the same state and output equations to all $k\ge 0$, with zero initial state, $u_0=v\in {\mathbb{R}}^m$, and $u_k=0$ for $k\ge 1$. Its output at sample $k+1$ is $y_{k+1}=M_{k}v$ for every $k\ge 0$. The horizon-$q$ restriction records this identity only for $0\le k<q-1$; when $q=1$, it records only $y_0=0$ and has no input samples. Matrix powers use $A^0=I_d$ and $A^{k+1}=A^{k}A$. If $d=0$, the state space consists of its single zero vector and the matrices have the corresponding empty shapes.

Finite real matrix spaces

Matrix products, powers, and identities

Standard Euclidean inner products

Dahleh, Dahleh, and Verghese, Lectures on Dynamic Systems and Control, Chapter 25

Lall, State-Space Realization, Engr207c lecture notes