## 1 Introduction  
The derivative estimation of a measured signal has considerable importance in signal processing, numerical analysis, control engineering, and failure diagnostics, among others [1]. Derivatives and structures using derivatives of signals are often used in industrial applications, for example PD controllers. These kinds of controllers are often used practically in different fields of application. In applications, signals are corrupted by measurement and process noise, therefore a filtering procedure needs to be implemented. A number of different approaches have been proposed based on least-squares polynomial fitting or interpolation for off-line applications [1; 2]. Another common approach is based on high-gain observers [3; 4; 5]. These observers adjust the model by weighting the observer output deviations from the output of the system to be controlled. In [6], a sliding mode control (SMC) using an extended Kalman filter (EKF) as an observer for stimulus-responsive polymer fibers as soft actuator was proposed. Because of the slow velocity of the fiber, the EKF produces poor estimation results. Therefore, a derivative approximation structure is proposed to estimate the velocity through the measurement of the position. This approach realized the approximation of the derivative using also a high gain observer. The method presented in the different applications cited above is a method that approximates the derivative in an infinite horizon of time. In this sense, the proposed differentiator is an asymptotic estimator of the derivative. Several researchers studied this problem by applying the SMC approach. This paper emphasizes some mathematical aspects of an algorithm that was used in the past for practical applications such as, for instance, in [7; 8]. In particular, in [7] this algorithm is used in designing a velocity observer. In [8], a similar algorithm is used to estimate parameter identification in an application in which a synchronous motor is proposed. In recent years, real-time robust exact differentiation has become the main problem of output-feedback high order of sliding mode (HOSM) control design. Even the most modern differentiators [9] do not provide for exact differentiation with finite-time convergence and without considering noise. The derivatives may be calculated by successive implementation of a robust exact first-order differentiator [10] with finite-time convergence but without considering noise, as in [11; 12]. In [13], an arbitrary-order finite-time-convergent exact robust differentiator is constructed based on the HOSM technique. In [10], the proposed differentiator provides the proportionality of the maximal differentiation error to the square root of the maximal deviation of the measured input signal from the base signal. Such an order of the differentiation error is shown to be the best possible when the only information known of the base signal is an upper bound for Lipschitz's constant of the derivative. According to Theorem 2, the proposed algorithm to produce the optimal approximation needs to work the knowledge of the maximal Lipschitz's constant. More recently in [14], sliding mode (SM)-based differentiation is shown to be exact for a large class of functions and robust with respect to noise. In this case, Lipschitz's constant must be known to apply the algorithm. Those methods, using the maximal Lipschitz's constant, perform an approximation in a finite horizon of time. In practical applications, the presence of noise and faults does not allow a Lipschitz's constant to be set. In fact, the noise is not distinguishable from the input signal. In this sense, it appears impossible to apply these algorithms in real applications. This paper proposes an approximated derivative structure to be taken into account for such types of applications, so that spikes, noise, and any other kind of undesired signals that occur from the derivatives can be reduced.