The paper deals with the problem of designing an unknown input observer for discrete-time non-linear systems. In particular, with the use of the Lyapunov method, it is shown that the proposed observer is convergent under certain, non-restrictive conditions. Based on the achieved results, a general solution for increasing the convergence rate is proposed and implemented with the use of stochastic robustness techniques. In particular, it is shown that the problem of increasing the convergence rate of the observer can be formulated as a stochastic robustness analysis task that can be transformed into a structure selection and parameter estimation problem of a non-linear function, which can be solved with the B-spline approximation and evolutionary algorithms. The final part of the paper shows an illustrative example based on a two phase induction motor. The presented results clearly exhibit the performance of the proposed observer.
Notationx k ,x k 2 R n state vector and its estimate y k ,ŷ k 2 R m output vector and its estimate e k 2 R n state estimation error " k 2 R m output error u k 2 R r input vector d k 2 R q unknown input vector, q m f k 2 R s fault vector g Á ð Þ, hðÁÞ non-linear functions E k 2 R nÂq unknown input distribution matrix L 1, k 2 R nÂs , L 2, k 2 R mÂs fault distribution matrices p 2 R n p design parameter vector of the observer
A subclass of Levy-stable distributions, i.e., symmetric a-stable distributions (SaS), is applied to mutation operators of evolutionary strategies (1 + 1)ES, and (1 + A)ESQ. The local convergence rate of algorithms is considered. Moreover, some conditions are established under which evolutionary algorithms with mutation based on distributions with heavy tails generally have better local as well as global convergence. In order to justify the theoretical deliberations, some illustrative numerical simulations are presented.
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