## 1 Introduction
In the work of [11; 12], the filtering result (ratio of final and initial covariance matrix) can indeed reflect the measure of observability in different directions, but this method does not clarify the reason for the filtering result. Meanwhile, the methods of [2; 7; 9; 10] have not taken into account the influence of the cost function or weighting matrix \(P\), \(Q\), \(R\). [PERSON], [PERSON], and [PERSON] demonstrated the difference between the standard controllability and controllability with \(P\), \(Q\), and \(R\) in [15; 16; 17]. However, they briefly discussed these differences in measures of controllability. Therefore, in this work, we propose a new method for determining the controllability and observability, which takes into account both the influences of the initial values of \(P_{0}\), \(Q\), and \(R\) and the controllability (observability) Gramian on the \(P\) convergence process for each state variable. This method is based on local controllability (observability) [18], explaining the relationship between the standard controllability (observability) matrix and the convergence results of \(P\). Furthermore, the existing criteria always use determinant, trace, and maximal eigenvalue of the inverse characteristic controllability or observability matrix as controllability measures. The proposed measures of controllability (observability) was developed for specific state variables, which allows us to get a priori information about the control performance of each state variable in advance. It is very convenient for calculation and application in practice. Since the determination of the degree of controllability is complex and requires laborious preliminary calculations, for the purpose of effectively studying the degree of controllability, we first compose a dual system and determine its degree of observability (a simple criterion for degree of observability is proposed in this paper). After that, according to the duality theorem [19; 20; 21], this value is equivalent to the degree of controllability of the original system. The prospects for future research include using the wavelet transform and fractal principle [22; 23; 24; 25; 26; 27; 28; 29; 30] as the qualitative analysis tools for the system models. The degree of observability of state variables depends on the features of the models' description. Investigating the influence of these features on the properties of system models is a promising area of research. The study of the dynamic object was carried out by modeling the object, represented by time series, using wavelet and fractal analysis. The determination of the chaos indices and the analysis of special states were carried out using the wavelet transform. The analysis of the wavelet coefficients allows us to reveal the nonstationarity of the models and obtain the characteristics of the nonstationarity. The study of the degree of observability and controllability using these methods makes it possible to obtain a more complete connection between the qualitative characteristics of the systems under study. The paper has the following structure: In Section 2, we present our study of the duality of recursive process \(\mathbf{P}\) in linear quadratic regulation (LQR) and estimation (LQE). In Section 3, we present a method for determining the degree of observability of state variables. In Section 4, using the duality theorem, the degree of controllability is calculated in a similar way by using the duality theorem. In Section 5, we determined the dependences of the degree of controllability on the parameters of model matrix and control matrix. The numerical example of inverted pendulum on a carriage shows that the degree of controllability analysis can help us predict the control performance of different state variables. As the degree of the controllability increases, the settling time of the response process becomes shorter and the amplitude of the response process decreases.