Identification of fractional order systems using modulating functions method

The modulating functions method has been used for the identification of linear and nonlinear systems. In this paper, we generalize this method to the on-line identification of fractional order systems based on the Riemann-Liouville fractional derivatives. First, a new fractional integration by parts formula involving the fractional derivative of a modulating function is given. Then, we apply this formula to a fractional order system, for which the fractional derivatives of the input and the output can be transferred into the ones of the modulating functions. By choosing a set of modulating functions, a linear system of algebraic equations is obtained. Hence, the unknown parameters of a fractional order system can be estimated by solving a linear system. Using this method, we do not need any initial values which are usually unknown and not equal to zero. Also we do not need to estimate the fractional derivatives of noisy output. Moreover, it is shown that the proposed estimators are robust against high frequency sinusoidal noises and the ones due to a class of stochastic processes. Finally, the efficiency and the stability of the proposed method is confirmed by some numerical simulations

Identification scheme for fractional Hammerstein Models with the delayed Haar Wavelet

The parameter identification of a nonlinear Hammerstein-type process is likely to be complex and challenging due to the existence of significant nonlinearity at the input side. In this paper, a new parameter identification strategy for a block-oriented Hammerstein process is proposed using the Haar wavelet operational matrix (HWOM). To determine all the parameters in the Hammerstein model, a special input excitation is utilized to separate the identification problem of the linear subsystem from the complete nonlinear process. During the first test period, a simple step response data is utilized to estimate the linear subsystem dynamics. Then, the overall system response to sinusoidal input is used to estimate nonlinearity in the process. A single-pole fractional order transfer function with time delay is used to model the linear subsystem. In order to reduce the mathematical complexity resulting from the fractional derivatives of signals, a HWOM based algebraic approach is developed. The proposed method is proven to be simple and robust in the presence of measurement noises. The numerical study illustrates the efficiency of the proposed modeling technique through four different nonlinear processes and results are compared with existing methods

Non-asymptotic fractional order differentiators via an algebraic parametric method

Recently, Mboup, Join and Fliess [27], [28] introduced non-asymptotic integer order differentiators by using an algebraic parametric estimation method [7], [8]. In this paper, in order to obtain non-asymptotic fractional order differentiators we apply this algebraic parametric method to truncated expansions of fractional Taylor series based on the Jumarie's modified Riemann-Liouville derivative [14]. Exact and simple formulae for these differentiators are given where a sliding integration window of a noisy signal involving Jacobi polynomials is used without complex mathematical deduction. The efficiency and the stability with respect to corrupting noises of the proposed fractional order differentiators are shown in numerical simulations

Fractional order differentiation by integration with Jacobi polynomials

The differentiation by integration method with Jacobi polynomials was originally introduced by Mboup, Join and Fliess. This paper generalizes this method from the integer order to the fractional order for estimating the fractional order derivatives of noisy signals. The proposed fractional order differentiator is deduced from the Jacobi orthogonal polynomial filter and the Riemann-Liouville fractional order derivative definition. Exact and simple formula for this differentiator is given where an integral formula involving Jacobi polynomials and the noisy signal is used without complex mathematical deduction. Hence, it can be used both for continuous-time and discrete-time models. The comparison between our differentiator and the recently introduced digital fractional order Savitzky-Golay differentiator is given in numerical simulations so as to show its accuracy and robustness with respect to corrupting noises

On the Selection of Tuning Methodology of FOPID Controllers for the Control of Higher Order Processes

In this paper, a comparative study is done on the time and frequency domain tuning strategies for fractional order (FO) PID controllers to handle higher order processes. A new fractional order template for reduced parameter modeling of stable minimum/non-minimum phase higher order processes is introduced and its advantage in frequency domain tuning of FOPID controllers is also presented. The time domain optimal tuning of FOPID controllers have also been carried out to handle these higher order processes by performing optimization with various integral performance indices. The paper highlights on the practical control system implementation issues like flexibility of online autotuning, reduced control signal and actuator size, capability of measurement noise filtration, load disturbance suppression, robustness against parameter uncertainties etc. in light of the above tuning methodologies.Comment: 27 pages, 10 figure

A rigorous and efficient asymptotic test for power-law cross-correlation

Podobnik and Stanley recently proposed a novel framework, Detrended Cross-Correlation Analysis, for the analysis of power-law cross-correlation between two time-series, a phenomenon which occurs widely in physical, geophysical, financial and numerous additional applications. While highly promising in these important application domains, to date no rigorous or efficient statistical test has been proposed which uses the information provided by DCCA across time-scales for the presence of this power-law cross-correlation. In this paper we fill this gap by proposing a method based on DCCA for testing the hypothesis of power-law cross-correlation; the method synthesizes the information generated by DCCA across time-scales and returns conservative but practically relevant p-values for the null hypothesis of zero correlation, which may be efficiently calculated in software. Thus our proposals generate confidence estimates for a DCCA analysis in a fully probabilistic fashion

Evidence of Intermittent Cascades from Discrete Hierarchical Dissipation in Turbulence

We present the results of a search of log-periodic corrections to scaling in the moments of the energy dissipation rate in experiments at high Reynolds number (2500) of three-dimensional fully developed turbulence. A simple dynamical representation of the Richardson-Kolmogorov cartoon of a cascade shows that standard averaging techniques erase by their very construction the possible existence of log-periodic corrections to scaling associated with a discrete hierarchy. To remedy this drawback, we introduce a novel ``canonical'' averaging that we test extensively on synthetic examples constructed to mimick the interplay between a weak log-periodic component and rather strong multiplicative and phase noises. Our extensive tests confirm the remarkable observation of statistically significant log-periodic corrections to scaling, with a prefered scaling ratio for length scales compatible with the value gamma = 2. A strong confirmation of this result is provided by the identification of up to 5 harmonics of the fundamental log-periodic undulations, associated with up to 5 levels of the underlying hierarchical dynamical structure. A natural interpretation of our results is that the Richardson-Kolmogorov mental picture of a cascade becomes a realistic description if one allows for intermittent births and deaths of discrete cascades at varying scales.Comment: Latex document of 40 pages, including 18 eps figure