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Section: Scientific Foundations

Numerical differentiation

Numerical differentiation, i.e., determining the time derivatives of various orders of a noisy time signal, is an important but difficult ill-posed theoretical problem. This fundamental issue has attracted a lot of attention in many fields of engineering and applied mathematics (see, e.g. in the recent control literature  [65] , [67] , [80] , [79] , [82] , [83] , and the references therein). A common way of estimating the derivatives of a signal is to resort to a least squares fitting and then take the derivatives of the resulting function. In  [87] , [18] , this problem was revised through our algebraic approach. The approach can be briefly explained as follows:

Thus, using elementary differential algebraic operations, we derive explicit formulae yielding point-wise derivative estimation for each given order. Interesting enough, it turns out that the Jacobi orthogonal polynomials  [100] are inherently connected with the developed algebraic numerical differentiators. A least-squares interpretation then naturally follows  [86] , [87] and this leads to a key result: the algebraic numerical differentiation is as efficient as an appropriately chosen time delay is introduced. Though, such a delay may not be tolerable in some real-time applications. Moreover, instability generally occurs when introducing delayed signals in a control loop. Note however that since the delay is known a priori , it is always possible to derive a control law which compensates for its effects (see  [94] ). A second key feature of the algebraic numerical differentiators is its very low complexity which allows for a real-time implementation. Indeed, the  nth order derivative estimate (that can be directly managed for n$ \ge$2 , without using n cascaded estimators) is expressed as the output of the linear time-invariant filter, with finite support impulse response  Im42 ${h_{\#954 ,\#956 ,n,r}{(·)}}$ . Implementing such a stable and causal filter is easy and simple. This is achieved either in continuous-time or in discrete-time when only discrete-time samples of the observation are available. In the latter case, we obtain a tapped delay line digital filter by considering any numerical integration method with equally-spaced abscissas.


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