95 research outputs found
On the Discretization of a Class of Homogeneous Differentiators
This paper proposes explicit and implicit discrete-time realizations for a class of homogeneous sliding-mode-based differentiators. The proposed approach relies on the exact discretization of the continuous differentiator. Also, it is demonstrated that the proposed implicit discretization always exists, is non-anticipative and unique. A numerical simulation shows the better performance of the implicit scheme over the proposed and the referenced explicit implementations.ITESO, A.C
Optimal Robust Exact Differentiation via Linear Adaptive Techniques
The problem of differentiating a function with bounded second derivative in
the presence of bounded measurement noise is considered in both continuous-time
and sampled-data settings. Fundamental performance limitations of causal
differentiators, in terms of the smallest achievable worst-case differentiation
error, are shown. A robust exact differentiator is then constructed via the
adaptation of a single parameter of a linear differentiator. It is demonstrated
that the resulting differentiator exhibits a combination of properties that
outperforms existing continuous-time differentiators: it is robust with respect
to noise, it instantaneously converges to the exact derivative in the absence
of noise, and it attains the smallest possible -- hence optimal -- upper bound
on its differentiation error under noisy measurements. For sample-based
differentiators, the concept of quasi-exactness is introduced to classify
differentiators that achieve the lowest possible worst-case error based on
sampled measurements in the absence of noise. A straightforward sample-based
implementation of the proposed linear adaptive continuous-time differentiator
is shown to achieve quasi-exactness after a single sampling step as well as a
theoretically optimal differentiation error bound that, in addition, converges
to the continuous-time optimal one as the sampling period becomes arbitrarily
small. A numerical simulation illustrates the presented formal results
Discretization of the Robust Exact Filtering Differentiator Based on the Matching Approach
This paper presents a time discretization of the robust exact filtering differentiator, a sliding mode differentiator coupled to filter, which provides a suitable approximation of the derivatives for some noisy signals. This realization rely on the stabilization of a pseudo linear discrete-time system, it is attained through the matching approach. As in the original case, the convergence of the robust exact filtering differentiator depends on the bound of a higher-order derivative. Nevertheless, this new realization can be implemented with or without the knowledge of such constant. It is demonstrated that the system trajectories converge to a neighborhood of the origin for a free-noise input. Finally, comparisons between the behavior of the differentiator with different design parameters are presented.ITESO, A.C
Designing predefined-time differentiators with bounded time-varying gains
There is an increasing interest in designing differentiators, which converge
exactly before a prespecified time regardless of the initial conditions, i.e.,
which are fixed-time convergent with a predefined Upper Bound of their Settling
Time (UBST), due to their ability to solve estimation and control problems with
time constraints. However, for the class of signals with a known bound of their
-th time derivative, the existing design methodologies are either only
available for first-order differentiators, yielding a very conservative UBST,
or result in gains that tend to infinity at the convergence time. Here, we
introduce a new methodology based on time-varying gains to design
arbitrary-order exact differentiators with a predefined UBST. This UBST is a
priori set as one parameter of the algorithm. Our approach guarantees that the
UBST can be set arbitrarily tight, and we also provide sufficient conditions to
obtain exact convergence while maintaining bounded time-varying gains.
Additionally, we provide necessary and sufficient conditions such that our
approach yields error dynamics with a uniformly Lyapunov stable equilibrium.
Our results show how time-varying gains offer a general and flexible
methodology to design algorithms with a predefined UBST
Robust exact differentiators with predefined convergence time
The problem of exactly differentiating a signal with bounded second
derivative is considered. A class of differentiators is proposed, which
converge to the derivative of such a signal within a fixed, i.e., a finite and
uniformly bounded convergence time. A tuning procedure is derived that allows
to assign an arbitrary, predefined upper bound for this convergence time. It is
furthermore shown that this bound can be made arbitrarily tight by appropriate
tuning. The usefulness of the procedure is demonstrated by applying it to the
well-known uniform robust exact differentiator, which the considered class of
differentiators includes as a special case
Smooth non linear high gain observers for a class of dynamical systems
High-gain observers are powerful tools for estimating the state of nonlinear systems. However, their design poses several challenges due to the need of dealing with phenomena such as peaking and chattering. To address these issues, we propose a differentiator operator design based on a non linear second order high-gain observer, which is suited to a class of dynamical systems. Our method includes a procedure to determine high gains in order to avoid chattering in the case of noise-free models, and cut-off frequency based gain design in the case of noisy measurements. Complementary, we suggest performing observability analyses to ensure a priori the feasibility of the estimation. The main strengths of our approach are its simplicity and robustness. We demonstrate the performance of the proposed method by applying it to two processes (chemical and biological).Xunta de Galicia | Ref. ED431F 2021/003MCIN/AEI/10.13039/501100011033 | Ref. RYC-2019-027537-
Discrete-time differentiators: design and comparative analysis
This work deals with the problem of online differentiation of noisy signals. In this context, several types of differentiators including linear, sliding-mode based, adaptive, Kalman, and ALIEN differentiators are studied through mathematical analysis and numerical experiments. To resolve the drawbacks of the exact differentiators, new implicit and semi-implicit discretization schemes are proposed in this work to suppress the digital chattering caused by the wrong time-discretization of set-valued functions as well as providing some useful properties, e.g., finite-time convergence, invariant sliding-surface, exactness. A complete comparative analysis is presented in the manuscript to investigate the behavior of the discrete-time differentiators in the presence of several types of noises, including white noise, sinusoidal noise, and bell-shaped noise. Many details such as quantization effect and realistic sampling times are taken into account to provide useful information based on practical conditions. Many comments are provided to help the engineers to tune the parameters of the differentiators
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