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The integer order differentiation by integration method based on the Jacobi orthogonal polynomials for noisy signals was originally introduced by Mboup, Join and Fliess. We propose to extend this method from the integer order to the fractional order to estimate the fractional order derivatives of noisy signals. Firstly, two fractional order differentiators are deduced from the Jacobi orthogonal polynomial filter, using the Riemann–Liouville and the Caputo fractional order derivative definitions respectively. Exact and simple formulae for these differentiators are given by integral expressions. Hence, they can be used for both continuous-time and discrete-time models in on-line or off-line applications. Secondly, some error bounds are provided for the corresponding estimation errors. These bounds allow to study the design parameters' influence. The noise error contribution due to a large class of stochastic processes is studied in discrete case. The latter shows that the differentiator based on the Caputo fractional order derivative can cope with a class of noises, whose mean value and variance functions are polynomial time-varying. Thanks to the design parameters analysis, the proposed fractional order differentiators are significantly improved by admitting a time-delay. Thirdly, in order to reduce the calculation time for on-line applications, a recursive algorithm is proposed. Finally, the proposed differentiator based on the Riemann–Liouville fractional order derivative is used to estimate the state of a fractional order system and numerical simulations illustrate the accuracy and the robustness with respect to corrupting noises. [ABSTRACT FROM AUTHOR]

In this paper, we present the globally optimal distributed Kalman filtering fusion with singular covariances of filtering errors and measurement noises. The following facts motivate us to consider the problem. First, the invertibility of estimation error covariance matrices is a necessary condition for most of the existing distributed fusion algorithms. However, it can not be guaranteed to exist in practice. For example, when state estimation for a given dynamic system is subject to state equality constraints, the estimation error covariance matrices must be singular. Second, the proposed fused state estimate is still exactly the same as the centralized Kalman filtering using all sensor raw measurements. [ABSTRACT FROM AUTHOR]

This paper is concerned with the moving horizon estimation (MHE) problem for linear discrete-time systems with limited communication, including quantized measurements and packet dropouts. The measured output is quantized by a logarithmic quantizer and the packet dropout phenomena is modeled by a binary switching random sequence. The main purpose of this paper is to design an estimator such that, for all possible quantized errors and packet dropouts, the state estimation error sequence is convergent. By choosing a stochastic cost function, the optimal estimator is obtained by solving a regularized least-squares problem with uncertain parameters. The proposed method can be used to deal with the estimation and prediction problems for systems with quantized errors and packet dropouts in a unified framework. The stability properties of the optimal estimator are also studied. The obtained stability condition implicitly establishes a relation between the upper bound of the estimation error and two parameters, namely, the quantization density and the packet dropout probability. Moreover, the maximum quantization density and the maximum packet dropout probability are given to ensure the convergence of the upper bound of the estimation error sequence. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM PUBLISHER]

We address the problem of state estimation for a class of nonlinear systems with measurement noise in the context of feedback control. It is well-known that high-gain observers are robust against model uncertainty and disturbances, but sensitive to measurement noise when implemented in a feedback loop. This work presents the benefits of a nonlinear-gain structure in the innovation process of the high-gain observer, in order to overcome the tradeoff between fast state reconstruction and measurement noise attenuation. The goal is to generate a larger observer gain during the transient response than in the steady-state response. Thus, by reducing the observer gain after achieving satisfactory state estimates, the effect of noise on the steady-state performance is reduced. Moreover, the nonlinear-gain observer presented in this paper is shown to surpass the system performance achieved when using comparable linear-gain observers. The proof argues boundedness and ultimate boundedness of the closed-loop system under the proposed output feedback. [ABSTRACT FROM AUTHOR]

This note presents a new algorithm that is designed to identify the frequency, magnitude, phase and offset of a biased sinusoidal signal. The structure of the algorithm includes an orthogonal system generator based on a second-order generalized integrator. The proposed strategy has the advantages of a fast and accurate signal reconstruction capability and a good rejection to noise. [ABSTRACT FROM PUBLISHER]