Dynamical estimation of a noisy input in a system with a Caputo fractional derivative. The case of continuous measurements of a part of phase coordinates

Platon Surkov

doi:10.3934/mcrf.2022020

Article Contents

2023, Volume 13, Issue 3: 895-917. Doi: 10.3934/mcrf.2022020

This issue Previous Article Second-order problems involving time-dependent subdifferential operators and application to control Next Article A geometric approach of gradient descent algorithms in linear neural networks

Dynamical estimation of a noisy input in a system with a Caputo fractional derivative. The case of continuous measurements of a part of phase coordinates

Platon Surkov

Ural Federal University, 12, Mira street, Krasovskii Institute of Mathematics and Mechanics UB RAS, 16, S. Kovalevskaya street, Ekaterinburg, 620108, Russia

Received: December 2021

Revised: March 2022

Early access: April 2022

Published: September 2023

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Abstract

The problem of estimating (reconstructing) an unknown input for a system of nonlinear differential equations with the Caputo fractional derivative is considered. Information on the position of the system is available for observations and only a part of system's parameters can be measured. The case of measuring all phase coordinates is also presented. The measurements are continuous and the data obtained in them are noisy. The considered problem is ill-posed and, to solve it, we use the method of dynamic inversion. It is based on regularization methods and constructions of positional control theory. In particular, we use the Tikhonov regularization method also known as the smoothing functional method and the Krasovskii extremal aiming method. The approach to estimating an unknown input implies introducing an auxiliary system (a model) with an appropriate rule of forming a control. The proposed estimation algorithm gives approximations of an unknown input and is stable under informational noises and computational errors. As an example illustrating the elaborated technique, a biological model of human immunodeficiency virus disease is used for simulation. The simulation results demonstrate the importance of the approach to on-line estimating unobservable parameters in real processes.

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Mathematics Subject Classification: Primary: 34A08, 93C41; Secondary: 26A33, 49N45.

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Figure 1. The component $ z $ of system (76) and the component $ w_1 $ of model (79)

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Figure 2. The unovservable component $ x $ of system (76) and the component $ w_2 $ of model (79)

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Figure 3. The input action $ u $ and the reconstructed action $ v^h $

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Figure 4. The components u₁, u₂ and the components v₁^h, v₂^h

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Figure 5. The input action Bu and its estimation Bv^h

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