Section: Scientific Foundations

Finite time estimation of derivatives

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  [70] , [71] , [91] , [90] , [97] , [98] , and the references therein).

Model-free techniques for numerical differentiation

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  [101] , [99] , 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 $x_N\left(t\right)={\sum }_{\nu =0}^N{x}^{\left(\nu \right)}\left(0\right)\frac{t^{\nu }}{\nu !}\in ℝ\left[t\right]$, $t\ge 0$, of degree $N$. Rewrite it in the well known notations of operational calculus:

  $X_N\left(s\right)=\sum _{\nu =0}^N\frac{x^{\left(\nu \right)}\left(0\right)}{s^{\nu +1}}$

  Here, we use $\frac{d}{ds}$, which corresponds in the time domain to the multiplication by $-t$. Multiply both sides by $\frac{d^{\alpha }}{d{s}^{\alpha }}{s}^{N+1}$, $\alpha =0,1,\cdots ,N$. The quantities $x^{\left(\nu \right)}\left(0\right)$, $\nu =0,1,\cdots ,N$ are given by the triangular system of linear equations:

  $\frac{d^{\alpha }{s}^{N+1}{X}_N}{d{s}^{\alpha }}=\frac{d^{\alpha }}{d{s}^{\alpha }}\left(\sum _{\nu =0}^N{x}^{\left(\nu \right)}\left(0\right)s^{N-\nu }\right)$

  (17)

  The time derivatives, i.e., $s^{\mu }\frac{d^{\iota }{X}_N}{d{s}^{\iota }}$, $\mu =1,\cdots ,N$, $0\le \iota \le N$, are removed by multiplying both sides of Equation (17 ) by $s^{-\overline{N}}$, $\overline{N}>N$.

• 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 $x\left(t\right)={\sum }_{\nu =0}^{\infty }{x}^{\left(\nu \right)}\left(0\right)\frac{t^{\nu }}{\nu !}$, where $0\le t<\rho$. Approximate $x\left(t\right)$ in the interval $(0,\epsilon )$, $0<\epsilon \le \rho$, by its truncated Taylor expansion $x_N\left(t\right)={\sum }_{\nu =0}^N{x}^{\left(\nu \right)}\left(0\right)\frac{t^{\nu }}{\nu !}$ of order $N$. Introduce the operational analogue of $x\left(t\right)$, i.e., $X\left(s\right)={\sum }_{\nu \ge 0}\frac{x^{\left(\nu \right)}\left(0\right)}{s^{\nu +1}}$. Denote by ${\left[x^{\left(\nu \right)}\left(0\right)\right]}_{e_N}\left(t\right)$, $0\le \nu \le N$, the numerical estimate of $x^{\left(\nu \right)}\left(0\right)$, which is obtained by replacing $X_N\left(s\right)$ by $X\left(s\right)$ in Eq. (17 ). It can be shown [85] 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 [112] are inherently connected with the developed algebraic numerical differentiators. A least-squares interpretation then naturally follows [100] , [101] 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  [110] ). 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 $n\ge 2$, without using $n$ cascaded estimators) is expressed as the output of the linear time-invariant filter, with finite support impulse response $h_{\kappa ,\mu ,n,r}(·)$. 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.

Model-based estimation of derivatives

If we consider that the derivatives to be estimated are unmeasured states of the process that generates the signal, differentiation techniques can be regarded as left invertibility algorithms. In this sense, the previous algebraic estimation achieves a “model-free” left inversion. Now, when such a model is available, the finite-time observers, relying on higher order sliding modes [105] and homogeneity properties [106] , [102] , also represent possible non-asymptotic algorithms for differentiation(Usually, observer design yields asymptotic convergence of the estimation error dynamics. The main advantages of such a technique in the case of linear systems are simplicity of design, estimation with a filtering action and global stability property. Nevertheless, the filtering property is not ensured for nonlinear systems and the stability property is generally obtained only locally. For these reasons, in the case of nonlinear systems, finite-time observers and estimators have been proposed in the literature [98] , [106] , [107] , [86] ...). Using such model-based techniques appears to be complementary(The choice between the two approaches will be done after comparison with respect to the indicators 1, 2, 3, and taking into account the application (for instance, the system bandwidth, system dimension), the kind of discontinuity, the observer in the control loop or not...) and we already obtained left-inversion results for several classes of models: linear systems [87] , nonlinear systems [68] , delay systems [2] and hybrid systems [96] .