Miki Livne

Exact Differentiation of Signals With Unbounded Higher Derivatives

The proposed arbitrary-order differentiator based on high-order sliding modes is generalized to ensure exact robust nth-order differentiation of signals with a given functional bound of the (n + 1)th derivative. The asymptotic accuracies in the presence of noises and discrete sampling are estimated. The results are applicable for the global observation of system states when the dynamics is unbounded. Computer simulation confirms the applicability of the differentiator

Proper discretization of homogeneous differentiators

Homogeneous sliding-mode-based differentiators (HD) are known to provide for the high-accuracy robust estimation of derivatives in the presence of sampling noises and discrete measurements, provided that the differentiator dynamics evolve in continuous time. The popular one-step Euler discrete-time implementation is proved to cause differentiation accuracy deterioration, if the differentiation order exceeds 1. A novel discrete-time realization of the HD is proposed, which preserves the ultimate accuracy of the continuous-time HD also with discrete measurements.

Weighted homogeneity and robustness of sliding mode control

General features of finite-time-stable (FTS) homogeneous differential inclusions (DIs) are investigated in the context of sliding-mode control (SMC). The continuity features of the settling-time functions of FTS homogeneous DIs are considered, and the system asymptotic accuracy is calculated in the presence of disturbances, noises and delays. Performance of output-feedback multi-input multi-output homogeneous SMC systems is studied in the presence of relative degree fluctuations. The bifurcation of the kinematic-car-model relative degree is analyzed as an example.

Accuracy of disturbed homogeneous sliding modes

The asymptotic accuracy of disturbed homogeneous systems is studied. The results are applied to estimate the accuracy of homogeneous output-feedback high-order sliding-mode controllers in the presence of disturbances changing or eliminating the system relative degree. Two academic examples demonstrate the accuracy calculation.

Homogeneous discrete differentiation of functions with unbounded higher derivatives

Homogeneous sliding-mode-based differentiators (HD) are known for their high asymptotic accuracy. Their practical realization is computer-based and requires discretization. The corresponding combination of a discrete system with a continuous-time input signal produces hybrid dynamics. In the case of the most usual one-step Euler discretization that hybrid system lacks the homogeneity of its predecessor and loses its ultimate accuracy. Nevertheless, the discrete differentiator can be modified, restoring the homogeneity and the accuracy of HD. Similarly to HD, the proposed homogeneous discrete differentiator can also be used to differentiate signals with a variable upper bound of the highest derivative. Simulation results confirm the theoretical results.

Discrete-Time Sliding-Mode-Based Differentiation

Homogeneous sliding-mode-based differentiators provide for the high accuracy robust finite-time-exact estimation of derivatives. It is shown that their discrete-time implementation misses the homogeneity, and respectively features worse accuracy with respect to the sampling time interval. Detailed analysis of the asymptotic accuracy is provided in both cases of constant and variable sampling intervals.

Discrete sliding-mode-based differentiators

Sliding-mode-based differentiators of the input f(t) of the order k yield exact estimations of the derivatives f, ... f(k), provided an upper bound of |f(k+1)(t)| is available in realtime. Practical application involves discrete noisy sampling of f and numeric integration of the internal variables between the measurements. The corresponding asymptotic differentiation accuracies are calculated in the presence of Euler integration and discrete sampling, whereas both independently feature variable or constant time steps. Proposed discrete differentiators restore the optimal accuracy of their continuous-time counterparts. Simulation confirms the presented results.

Globally convergent differentiators with variable gains

ABSTRACT A new robust exact sliding mode (SM) based differentiator is proposed which provides for the fast global finite-time convergence of its outputs to the first n exact derivatives of its input f(t). The differentiator utilises the knowledge of a function L(t) providing the estimation |f (n + 1) 0| ⩽ L(t), and satisfying for a known bound M. The standard accuracy of the homogeneous SM-based differentiator is preserved in the presence of discrete sampling and noises in both f and L. The proposed discretisation scheme ensures the same accuracy in computer realisation.