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  [77] , [79] , [90] , [89] , [92] , [93] , 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  [95] , [25] , 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 real-valued 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:

  (17)

  The time derivatives, i.e., , , 0N , 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 real-valued analytic time function defined by the convergent power series , where 0t< . 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 , 0N , the numerical estimate of , which is obtained by replacing X_N(s) by X(s) in Eq. (17 ). It can be shown [87] that a good estimate is obtained in this way.

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 [101] are inherently connected with the developed algebraic numerical differentiators. A least-squares interpretation then naturally follows [94] , [95] and this leads to a key result: the algebraic numerical differentiation is as efficient as an appropriately chosen time delay. 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  [99] ). A second key feature of the algebraic numerical differentiators is its very low complexity which allows for a real-time 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 time-invariant filter, with finite support impulse response  . 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.