Bratislav Dankovic

A CLASS OF ALMOST ORTHOGONAL FILTERS

In this paper we define a new class of the almost orthogonal filters. These filters are a generalization of the classical orthogonal filters commonly used in the circuit theory, control system theory, signal processing and process identification. Almost orthogonal filters generate the series of almost orthogonal Legendre functions on the interval (0, ∞). They can be successfully used for the analysis, synthesis and designing of imperfect technical systems. They can also be used for approximation of arbitrary functions on (0, ∞). A new method for obtaining the models of imperfect systems is presented as well. Simulations were performed in order to validate the theoretical results and the comparison of new filters with classical filters is given.

Modelling of dynamical systems based on almost orthogonal polynomials

A new class of the almost orthogonal filters is described in this article. These filters are a generalization of the classical orthogonal filters commonly used in the circuit theory, control system theory, signal processing, and process identification. Almost orthogonal filters generate the series of almost orthogonal Legendre functions over the interval (0, ∞). It is well known that all real systems suffer from some imperfections, so the models of these systems should reflect this fact. A new method for obtaining an imperfect system model is proposed. This method uses an almost orthogonal filter, which is based on almost orthogonal functions. Experiments with modular servo drive were performed to validate theoretical results and demonstrate that the method described in the article is suitable for modelling of imperfect systems.

Some Mu¨ntz orthogonal systems

Abstract Let Λ = ∗λ 0 , λ1,…∗ be a given sequence of complex numbers such that ¦λ i ¦ > 1 and Re (λ i λ j − 1) > 0 for each i and j. For Muntz polynomials P(x) = ∑i=0npixλi and Q(x) = ∑j=0mqjxλj we define an inner product [P, Q] by [P,Q]= ∫ 0 1 (P Q )(x) d x x 2 where (P⊙ Q )(x) = ∑ i=0 n ∑ j=0 m p i q j x λ i λ j . We obtain the Muntz polynomials (Qn(x)) orthogonal with respect to this inner product and connect them with the Malmquist rational functions W ν (z)= ∏ v=0 n−1 ( z−1 λ v ) ∏ v=0 n (z−λ v ) orthogonal on the unit circle with respect to the inner product (u,v)= 1 2τ i ∮ || =1 u(z) v(z) d z z Several interesting properties of polynomials Qn(x) are given.

Approximation based on orthogonal and almost orthogonal functions

Abstract In this paper, we define a class of almost orthogonal rational functions of Legendre type in a new manner. Relations of these functions with classical exponentional functions orthogonal over interval (0, ∞), as well as classical polynomials orthogonal over (0, 1) are explained. Defining relations of these functions can be used for designing almost orthogonal filters. These filters are generators of orthogonal signals and can be successfully applied in finding the best signal approximation in the sense of the mean square error. The filters orthogonal property enables building of physical (in this case electrical) models of dynamical systems (the sources of signals to be approximated) either with less components for the same model accuracy or higher accuracy for the same number of components than the other known models. New filters represent further improvement of previously designed filters, by the same authors, in the sense of simplicity, higher accuracy, lesser approximation time and even a possibility to approximate signals generated by systems with built-in imperfections. Series of experiments were performed to analyze the dependence of approximation accuracy and the number of filters sections.

Systems modeling based on Legendre polynomials

A new method for obtaining models of continuous systems, based on Legendre orthogonal filters, is proposed in this paper. Recent results in the field of orthogonal functions were used to improve accuracy of modeling. First, Legendre orthogonal filters were designed on the basis of orthogonal functions. Then, adjustable models were formed. Models parameters were optimized using genetic algorithm. As a case study, an experimental simple DC drive was considered and described in details. Simulations were performed out to approve theoretical results and demonstrate that the method described in the paper is very suitable for modeling continuous systems. It achieves excellent results in the sense of model accuracy and modeling algorithm speed.

The Almost Orthogonal Polynomial Sequences

Since the orthogonal polynomials have a lot of applications in mathematics (theory of special functions, spectral operators, probability, analytic numbers and approximations) and other sciences (physics, mechanics, electrotechnics), it appeared numerous extensions of the concept of standard orthogonality. We will discuss the concept of almost orthogonality which is less known, but very useful in modeling of electronic systems which generate orthonormal basis. They are very suitable for analysis and synthesis of imperfect technical systems which are projected to generate orthogonal polynomials, but in the reality generate almost orthogonal polynomials. Like the most important application we will consider approximations of the functions.

On a correlation between sensitivity and identificability of dynamical systems

A correlation between sensitivity and identificability of the continuous and discrete systems is considered in this paper. First, a method for determining parametric sensitivity is presented and then identification is performed and system identificability is derived directly from the identification error. Based on these results, we have calculated the correlation coefficient between these two systems features. Experiments have confirmed very strong correlation between sensitivity and identificability for continuous as well as for discrete time systems.

Systems modelling based on orthogonal filters

A new method for obtaining the models of continuous systems based on the Legendre orthogonal filters is proposed in this paper. New results in the field of orthogonal functions are used to improve accuracy of modelling. First, Legendre orthogonal filters are designed on the basis of orthogonal functions. Then, based on them, adjustable models are formed. Models parameters are optimized using the genetic algorithm. As a case study, an experimental simple DC drive was considered and described in details. Simulations are performed to approve theoretical results and demonstrate that the method described in the paper is suitable for systems modelling.

Modeling of imperfect system based on almost orthogonal polynomials

A new class of the almost orthogonal filters is defined in this paper. These filters are a generalization of the classical orthogonal filters commonly used in the circuit theory, control system theory, signal processing, and process identification. Almost orthogonal filters generate the series of almost orthogonal Legendre functions on the interval (0, ¿). Its a well known fact that every real system is imperfect, so the models of these systems should be imperfect, also. A new method for obtaining an imperfect system model by using almost orthogonal filters based on almost orthogonal functions is proposed. Simulations are performed to approve theoretical results and demonstrate that the method described in the paper is suitable for modelling of imperfect systems.

Correlation between System Sensitivity and Identification Error for Continuous Systems

The correlation between sensitivity and identification error of continuous linear dynamic systems is investigated in this paper. First, the parametric sensitivity of dynamic system is determined. Using the integration method, the error of parametric identification is calculated and, finally, the correlation is determined