Differentiator

Abstract: The main problem in differentiator design is to combine differentiation exactness with robustness in respect to possible measurement errors and input noises. The proposed differentiator provides for 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 one when the only information known on the base signal is an upper bound for Lipschitz’s constant of the derivative.

Abstract: The state-of-the-art on generalized (or any order) derivatives in physics and engineering sciences, is outlined for justifying the interest of the noninteger differentiation. The problems subsequent to its use in real-time operations are then set out so as to motivate the idea of synthesizing it by a recursive distribution of zeros and poles. An analysis of the existing work is also proposed to support this idea. A comprehensive study is given of the synthesis of differentiators with integer, noninteger, real or complex orders, and whose action is limited to any given frequency bandwidth. First, a definition, in the operational and frequency domains, of a frequency-band complex noninteger order differentiator, is given in a mathematical space with four dimensions which is a Banach algebra. Then, the determination of its synthesized form, by a recursive distribution of complex zeros and poles characterized by complex recursive factors, is presented. The complex noninteger differentiation order is expressed as a function of these recursive factors. The number of zeros and poles is calculated to be as low as possible while still ensuring the stability of the synthesized differentiator to be synthesized. A time validation is presented. Finally, guidelines are proposed for the conception of the synthesized differentiator.

Abstract: It is shown that a general uncertain single-input-single-output regulation problem is solvable only by means of discontinuous control laws, giving rise to the so-called high-order sliding modes. The homogeneity properties of the corresponding controllers yield a number of practically important features. In particular, finite-time convergence is proved, and asymptotic accuracy is calculated in a very general way in the presence of input noises, discrete measurements and switching delays. A robust homogeneous differentiator is included in the control structure thus yielding robust output-feedback controllers with finite-time convergence. It is demonstrated that homogeneity features significantly simplify the design and investigation of a new family of high-order sliding-mode controllers. Finally, simulation results are presented.

Abstract: Second-order sliding modes are used to keep exactly a constraint of the second relative degree or just to avoid chattering, i.e. in the cases when the standard (first order) sliding mode implementation might be involved or impossible. Design of a number of new 2-sliding controllers is demonstrated by means of the proposed homogeneity-based approach. A recently developed robust exact differentiator being applied, robust output-feedback controllers with finite-time convergence are produced, capable to control any general uncertain single-input-single-output process with relative degree 2. An effective simple procedure is developed to attenuate the 1-sliding mode chattering. Simulation of new controllers is presented.

Abstract: The effective application of sliding mode control to mechanical systems is not straightforward because of the sensitivity of these systems to chattering. Higher-order sliding modes can counteract this phenomenon by confining the switching control to the higher derivatives of the mechanical control variable, so that the latter results are continuous. Generally, this approach requires the availability of a number of time derivatives of the sliding variable, and, in the presence of noise, this requirement could be a practical limitation. A class of second-order sliding mode controllers, guaranteeing finite-time convergence for systems with relative degree two between the sliding variable and the switching control, could be helpful both in reducing the number of differentiator stages in the controller and in dealing with unmodelled actuator dynamics. In this paper different second-order sliding mode controllers, previously presented in the literature, are shown to belong to the above cited class, and some cha...