Section: New Results

Algebraic technique for estimation, differentiation and its applications

Participants : Cédric Join, Mamadou Mboup, Wilfrid Perruquetti, Lotfi Belkoura, Olivier Gibaru, Zoran Tiganj, Dayan Liu.

Elementary techniques from operational calculus, differential algebra, and noncommutative algebra lead to a new algebraic approach for estimation and detection. It is investigated in various areas of applied sciences and engineering. The following lists only some applications:

• [28] presents a partial derivatives estimation method for multidimensional signals. On a small interval the signal is represented by a truncated Taylor expansion. Then the application of multivariate Laplace transform together with adequate algebraic manipulations enabled us to express the desired partial derivative of any order as a function of iterated integrals of the noisy signal. Several recurrence relations and structural properties were provided. An interpretation of the estimators as least square minimization is also done by expressing the estimators in an orthogonal basis constituted by Jacobi polynomials. This projection enabled us not only to show a spacial shifting inherent to a specific class of estimators but also to synthesize a new class of estimators minimizing the truncation remainder of the Taylor local model. We provided also another class of estimators minimizing the noise influence. Finally we provided a numerical implementation scheme in the form of a finite impulse digital filters.

• A fast identification algorithm is proposed in [20] for systems with delayed inputs. It is based on a non-asymptotic distributional estimation technique initiated in the framework of systems without delay. Such a technique leads to simple realization schemes, involving integrators, multipliers and piecewise polynomial or exponential time functions. Thus, it allows for a real time implementation.

• A new approach to estimate vehicle tire forces and road maximum adherence is presented in [30] . Contrarily to most of previous works on this subject, it is not an asymptotic observer based estimation, but a combination of elementary diagnosis tools and new algebraic techniques for filtering and estimating derivatives of noisy signals. In a first step, instantaneous friction and lateral forces will be computed within this framework. Then, extended braking stiffness concept is exploited to detect which braking efforts allow to distinguish a road type from another. A weighted Dugoff model is used during these “distinguishable" intervals to estimate the maximum friction coefficient. Very promising results have been obtained in noisy simulations and real experimentations for most of driving situations.

• [27] proposes a diagnosis approach of sensor and actuator modeled as structured signals acting on a particular class of uncertain linear dynamical systems. The main advantage of this approach is that it is possible under certain assumptions, to detect, isolate and identify faults using only input and output measurements without having to identify model parameters. The method is based on the generation and analysis of analytical redundancy relations and exploits the fact that a structured signal satisfies a differential equation. The decision rule is based entirely on the temporal behaviour of the estimates of some fault characteristics.

• Numerical causal derivative estimators from noisy data are essential for real time applications especially for control applications or fluid simulation so as to address the new paradigms in solid modeling and video compression. By using an analytical point of view, [23] revisited the $n$th order algebraic derivative estimators. Thanks to a given noise level and a well-suitable integration length window, we analyzed the derivative estimator error;

• Recent algebraic parametric estimation techniques provide an estimate of the derivatives by using iterated integrals of a noisy observation signal. These algebraic parametric differentiation techniques give derivative estimations which contain two sources of errors: the bias term error and the noise error contribution. In order to reduce these errors, [25] extends the parameter domains used in the estimators, and studies some error bounds which depend on these parameters. This allows us to minimize these errors. It is shown that a compromise choice of these parameters implies an “optimized" error among the noise error contribution, the bias term error and the time delay.

• The numerical differentiation by integration method based on Jacobi polynomials originally introduced by Mboup, Fliess and Join is revisited in [24] for the central case where the used integration window is centered.

• [59] proposed new algebraic techniques to estimate the amplitude, frequency and phase of a biased and noisy sinusoidal signal. The methods which are popular today seem unable to obtain a robust estimation of those parameters within a fraction of the signal's period. The efficiency of our approach is illustrated by several computer simulations;

• A “practical" comparison between high-order sliding modes and the recently introduced model-free control is made in [56] . The perfect knowledge of the relative degree of the output variable, which is a standard assumption for sliding modes, is assumed. The comparisons are based on two concrete case-studies and on numerous computer simulations. The smoothness of the input variables, the robustness with respect to noises and the straightforward extendibility of the model-free controllers to MIMO systems are highlighted.

• [43] and [66] present a parameter estimation algorithm for a magnetic bearing. Such process are inherently unstable systems with strongly nonlinear dynamics. A simplified model of the magnetic bearing is developed, which enables to obtain a linear expression with respect to the unknown parameters. These parameters are measurable with difficulties, and may slightly vary over time. The expression of the estimates is written as a function of integrals of the inputs and outputs of the system. The simulations and the experiments show a fast and robust on-line identification

• Estimators of the frequency, amplitude and phase of a noisy sinusoidal signal with time-varying amplitude by using the algebraic parametric techniques is studied in [52] , in which a similar strategy to estimate these parameters by using modulating functions method is applied. The convergence of the noise error part due to a large class of noises is studied to show the robustness and the stability of these methods. We also show that the estimators obtained by modulating functions method are robust to "large" sampling period and to non zero-mean noises.

• In the framework of the SYSIASS project, a single landmark based localization algorithm for non-holonomic mobile robots is studied in [58] . In the case of a unicycle robot model, the localization problem is equivalent to the system observability. Based on this observation, the proposed localization method consists in finding a vector function which depends on the measurement vector and its derivatives, for which a numerical differentiation method is used in [58] .