## 2 OKID Formulation  
### Basic Observer Equation  
The discrete-time, state-space model of the multivariable linear system can be represented in the following  
\[x(k+1)=Gx(k)+Hu(k), \tag{1}\]  
\[y(k)=Cx(k)+Du(k), \tag{2}\]  
where \(x(k)\in\Re^{n}\), \(y(k)\in\Re^{p}\), and \(u(k)\in\Re^{m}\) are state, output, and control input vectors, respectively; \(G\in\Re^{n\times n}\), \(H\in\Re^{n\times m}\), and \(C\in\Re^{p\times n}\) are system, input, and output matrices, respectively, and \(D\) is transmission matrix of input function \(u(k)\). By adding and subtracting an observer term \(Fy(k)\)[5] to the right-hand side of the \(x(k+1)\) function in Equation (2), yield the observer equation  
\[ \begin{array}{ll}x(k+1)&=Gx(k)+Hu(k)+Fy(k)-Fy(k)\\ &=(G+F\mathbb{C})x(k)+(H+FD)u(k)-Fy(k)\\ &=\overline{G}x(k)+\overline{H}\overline{v}(k)\end{array}  \tag{3}\]  
where \(x(k)\) is considered as an observer state vector,  
\[\overline{G}=G+F\mathbb{C},\ \overline{H}=[H+FD,-F],\ \overline{v}(k)=\left[ \begin{array}{c}u(k)\\ y(k)\end{array} \right],\]  
and \(F\) is an \(n\times p\) arbitrary matrix that can be used to make the matrix \(\overline{G}\) as stable as desired. Therefore, the Markov parameters (\(\overline{G}\), \(\overline{H}\)) of the system in Equation (3) will be referred to as the observer Markov parameters. The mathematical development here can be interpreted as the point of view of [4] as attempting to place all the eigenvalues of \(\overline{G}\) at the origin. Consider the case where \(\overline{G}\) is asymptotically stable so that for some sufficiently large \(q\), \(\overline{G}^{q}\approx 0\) for all time steps \(k\geq q\). \[ \begin{array}{ll}x(k+1)&=\overline{G}x(k)+\overline{H}v(k),\\ x(k+2)&=\overline{G}x(k+1)+\overline{H}v(k+1),\\ &=\overline{G}^{2}x(k+1)+\overline{G}Hv(k)+\overline{H}v(k+1),\\ &\dots\dots\\ &\dots\dots\\ x(k+q)&=\overline{G}x(k+q-1)+\overline{H}v(k+q-1),\\ &=\overline{G}^{q}x(k)+\overline{G}^{q-1}\overline{H}v(k)+\overline{G}^{q-2} \overline{H}v(k+1)+\dots+\overline{H}v(k+p-1)\end{array}  \tag{4}\]Substituting \(x(k+q)\) in Equation (2) with Equation (4), results in  
\[ \begin{array}{ll}y(k+q)&=Cx(k+q)+Du(k+q)\\ &=C\overline{G}^{q}x(k)+C\overline{G}^{q-1}\overline{H}\overline{v}(k)+\dots+C \overline{H}\overline{v}(k+q-1)+Du(k+q).\end{array}  \tag{5}\]  
The set of Equation (5) for a sequence of \(k=0,\cdots,l-1\) can be written as  
\[\overline{y}=C\overline{G}^{l}x+\overline{V} \tag{6}\]  
where \(C\) is a unit matrix, \(\overline{G}^{q}\) is a constant,  
\[ \begin{array}{l}\overline{y}=\left[y(q)\ y(q+1)\dots y(l-1)\ \right]^{T}\\ \\ x=\left[x(0)\ x(1)\dots x(l-q-1)\ \right]^{T}\\ \\ \overline{Y}=\left[D\ C\overline{H}\ C\overline{G}\overline{H}\dots C \overline{G}^{(q-1)}\overline{H}\ \right]\end{array}  \tag{7}\]  
and  
\[\overline{V}=\left[ \begin{array}{cccc}u(q)&u(q+1)&\cdots&u(l-1)\\ \overline{v}(q-1)&\overline{v}(q)&\cdots&\overline{v}(l-2)\\ \overline{v}(q-2)&\overline{v}(q-1)&\cdots&\overline{v}(l-3)\\ \vdots&\vdots&\ddots&\vdots\\ \overline{v}(0)&\overline{v}(1)&\cdots&\overline{v}(l-q-1)\end{array} \right]. \tag{8}\]  
Since \(\overline{G}^{q}\) will be approximated to zero, thus Equation (6) will be simplified to  
\[\overline{y}=\overline{V}\overline{V} \tag{9}\]  
From Equation (9), the least-squares solution \(\overline{Y}\) is obtained  
\[\overline{Y}=\overline{y}\overline{V}^{+}, \tag{10}\]  
where \(\overline{V}^{+}\) is the pseudo-inverse of the matrix \(\overline{V}\). The matrices \(\overline{y}\) and \(\overline{V}\) are subsets of \(y\) and \(\overline{v}\), respectively, produced by deleting the first \(q\) columns. For nonzero unknown initial conditions, Equation (9) must be used in order to eliminate the effect of initial conditions because the initial conditions become negligible when they are multiplied by \(\overline{G}^{q}\). In other words, the initial conditions have negligible influence on the measured data after \(q\) time steps.