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 illposed 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:

The coefficients of a polynomial time function are linearly identifiable. Their estimation can therefore be achieved as above. Indeed, consider the realvalued polynomial function , t0 , of degree N . Rewrite it in the well known notations of operational calculus:
Here, we use , which corresponds in the time domain to the multiplication by  t . Multiply both sides by , . The quantities , are given by the triangular system of linear equations:
The time derivatives, i.e., , , 0 N , are removed by multiplying both sides of Equation ( 17 ) by , .

For an arbitrary analytic time function, apply the preceding calculations to a suitable truncated Taylor expansion. Consider a realvalued analytic time function defined by the convergent power series , where 0 t< . Approximate x( t) in the interval (0, ) , 0< , by its truncated Taylor expansion of order N . Introduce the operational analogue of x( t) , i.e., . Denote by , 0 N , the numerical estimate of , which is obtained by replacing X_{N}( s) by X( s) in Eq. ( 17 ). It can be shown [16] that a good estimate is obtained in this way.
Thus, using elementary differential algebraic operations, we derive explicit formulae yielding pointwise 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 leastsquares 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 realtime 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 realtime implementation. Indeed, the n^{th} order derivative estimate (that can be directly managed for n2 , without using n cascaded estimators) is expressed as the output of the linear timeinvariant filter, with finite support impulse response . Implementing such a stable and causal filter is easy and simple. This is achieved either in continuoustime or in discretetime when only discretetime 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 equallyspaced abscissas.