10,253 research outputs found
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
Fractional Systems’ Identification Based on Implicit Modulating Functions
This paper presents a new method for parameter identification based on the modulating function method for commensurable fractional-order models. The novelty of the method lies in the automatic determination of a specific modulating function by controlling a model-based auxiliary system, instead of applying and parameterizing a generic modulating function. The input signal of the model-based auxiliary system used to determine the modulating function is designed such that a separate identification of each individual parameter of the fractional-order model is enabled. This eliminates the shortcomings of the common modulating function method in which a modulating function must be adapted to the investigated system heuristically
Fractional order differentiation by integration: an application to fractional linear systems
International audienceIn this article, we propose a robust method to compute the output of a fractional linear system defined through a linear fractional differential equation (FDE) with time-varying coefficients, where the input can be noisy. We firstly introduce an estimator of the fractional derivative of an unknown signal, which is defined by an integral formula obtained by calculating the fractional derivative of a truncated Jacobi polynomial series expansion. We then approximate the FDE by applying to each fractional derivative this formal algebraic integral estimator. Consequently, the fractional derivatives of the solution are applied on the used Jacobi polynomials and then we need to identify the unknown coefficients of the truncated series expansion of the solution. Modulating functions method is used to estimate these coefficients by solving a linear system issued from the approximated FDE and some initial conditions. A numerical result is given to confirm the reliability of the proposed method
Bias-free Parameter Identification of a Fractional Order Model
This paper deals with the parameter identification of a fractional system considering a noisy observation of the output signal. The novelty is that the instrumental variable method is applied to the modulating function method applied to a fractional system. A simulated output signal which is not correlated to noise is required as the instrumental variable. Because all known simulation algorithms only consider zero initial conditions, the simulated output signal converges against the true output signal in an undefined time if the zero initial conditions are penalized. Therefore, an algorithm is extended with the short-memory principle. The benefit is that after a fixed time the error between the simulated and true output signal is small and can be used as the instrumental variable. Considering this extension of the simulation algorithms, it is shown that a consistent estimation is yield with the instrumental variable method
Non-Asymptotic State and Disturbance Estimation for a Class of Triangular Nonlinear Systems using Modulating Functions
Dynamical models are often corrupted by model uncertainties, external
disturbances, and measurement noise. These factors affect the performance of
model-based observers and as a result, affect the closed-loop performance.
Therefore, it is critical to develop robust model-based estimators that
reconstruct both the states and the model disturbances while mitigating the
effect of measurement noise in order to ensure good system monitoring and
closed-loop performance when designing controllers. In this article, a robust
step by step non-asymptotic observer for triangular nonlinear systems for the
joint estimation of the state and the disturbance is developed. The proposed
approach provides a sequential estimation of the states and the disturbance in
finite time using smooth modulating functions. The robustness of the proposed
observer is both in the sense of model disturbances and measurement noise. In
fact, the structure of triangular systems combined with the modulating
function-based method allows the estimation of the states independently of
model disturbances and the integral operator involved in the modulating
function-based method mitigates the noise. Additionally, the modulating
function method shifts the derivative from the noisy output to the smooth
modulating function which strengthens its robustness properties. The
applicability of the proposed modulating function-based estimator is
illustrated in numerical simulations and compared to a second-order sliding
mode super twisting observer under different measurement noise levels.Comment: 24 page
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