## 3 A Method for Determining the Degree of Observability of State Variables  
A qualitative characteristic of the observed components of the state vector is the degree of observability. The degree of observability is an abstract characteristic, and the approach presented in [34; 35; 36] allows for determining which of the components of the state vector are observed better. The approach not only gives an assessment of the quality of observability of the system but also allows a comparison of the observability of the components of the state vectors of various systems. We assume that the directly measured components of the state vector have the maximum degree of observability. The degree of observability of the directly measured component of the state vector is consider to be 1. Assume that one component of the state vector is measured, the observation matrix \(\mathbf{H}_{k}= \begin{bmatrix}1&0&\cdots&0\end{bmatrix} \). Divide each measurement interval into \(n\) subintervals and denote the measurements \(z_{1},z_{2},\ldots,z_{n}\) by the state at the first subinterval \(\mathbf{x}_{1}\). \[ \begin{split} z_{1}=\mathbf{H}\mathbf{x}_{1}+v_{1},\\ z_{2}=\mathbf{H}\mathbf{r}\mathbf{x}_{1}+\mathbf{H}\mathbf{w}_{1}+v_{2},\\ \cdots&\cdots&\cdots\\ z_{n}=\mathbf{H}\mathbf{r}^{n-1}\mathbf{x}_{1}+\mathbf{H}\mathbf{r}^{n-2}\mathbf{w}_{1}+\cdots+\mathbf{H} \mathbf{w}_{n-1}+v_{n}.\end{split}  \tag{7}\]  
Or in the matrix form:  
\[\mathbf{z}^{*}=\mathbf{O}\mathbf{x}_{1}+\mathbf{v}^{*} \tag{8}\]  
where  
\[\mathbf{z}^{*}= \begin{bmatrix}\mathbf{z}_{1}\\ \mathbf{z}_{2}\\ \cdots\\ \mathbf{z}_{n}\end{bmatrix} ,\mathbf{O}= \begin{bmatrix}\mathbf{H}\\ \mathbf{H}\mathbf{r}\\ \cdots\\ \mathbf{H}\mathbf{r}^{n-1}\end{bmatrix} ,\mathbf{v}^{*}= \begin{bmatrix}\mathbf{v}_{1}^{+}\\ \mathbf{v}_{2}^{+}\\ \cdots\\ \mathbf{v}_{n}^{+}\end{bmatrix} = \begin{bmatrix}\mathbf{v}_{1}\\ \mathbf{H}\mathbf{w}_{1}+v_{2}\\ \cdots\\ \mathbf{H}\mathbf{r}^{n-2}\mathbf{w}_{1}+\cdots+\mathbf{H}\mathbf{w}_{n-1}+v_{n}\end{bmatrix} .\]  
From Equation (8), we can rewrite the state vector \(\mathbf{x}_{1}\) as follows:  
\[\mathbf{x}_{1}=\mathbf{O}^{-1}\mathbf{z}^{*}-\mathbf{O}^{-1}\mathbf{v}^{*} \tag{9}\]  
We introduce the concept of equivalent measurement noise. Then, according to the Equation (7), the equivalent measurement noise for an arbitrary component of the state vector has the following form:  
\[\xi^{*i}=\alpha_{i1}v_{1}+\alpha_{i2}v_{2}+\cdots+\alpha_{in}v_{n}. \tag{10}\]  
where \(\alpha_{i1},\alpha_{i2},\ldots,\alpha_{in}\) are elements in \(i\)th row of matrix \(\mathbf{O}^{-1}\). Determine the variance of the equivalent measurement noise for the \(i\)th component \(R^{*i}\) by the coefficients \(\alpha_{i1},\alpha_{i2},\ldots,\alpha_{in}\). \[R^{*i}=E\left[\left(\varsigma^{*i}\right)^{2}\right]=\left[\alpha_{i1}^{2}+ \alpha_{i2}^{2}+\ldots+\alpha_{in}^{2}\right]R \tag{11}\]  
where \(R=E\left[v_{j}v_{k}\right]\) is the variance of the measurement noise \(v\) of the directly measured component of the state vector. We can judge a measure of observability by two characteristics: accuracy of estimation and time of convergence. Thus, the criterion for the degree of observability has the following form [34; 35; 36]:  
\[DoO^{i}=\frac{P_{0}^{i}R^{1}}{P_{0}^{1}R^{*i}} \tag{12}\]  
where \(P_{0}^{1}\) is the initial variance of the directly measured component of the state vector. The criterion (12) may be used for any \(t_{0}\). The degree of observability for an arbitrary state component is a scalar in the criterion (12). Thus, the proposed criterion allows comparing the degrees of observability for a specific state component in various systems. This criterion directly uses the ratio of \(P\) and \(R\). If variance \(P\) of the directly measured state component converges in one step, then for the other state components \(P\) may converge in several steps.