Optimal control based on neural observer with known final time for fractional order uncertain non-linear continuous-time
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Optimal control based on neural observer with known final time for fractional order uncertain non-linear continuous-time systems Gholamreza Nassajian1 · Saeed Balochian1 Received: 27 June 2020 / Revised: 21 October 2020 / Accepted: 1 November 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, an optimal control scheme with known final time is presented for continuous time fractional order nonlinear systems with an unknown term in the dynamics of the system. Fractional derivative is considered based on Caputo concept and fractional order is between 0 and 1. First, a neural network observer with fractional order dynamics is designed to estimate system states. Weights of the neural network are updated adaptively and the update laws are presented as equations of fractional order. By using the Lyapunov method, it is shown that state estimation error and weight estimation error are limited. Then, the optimal control problem with known final time for fractional order nonlinear systems is presented based on observed states. Finally, the simulation results show efficiency of the proposed method. Keywords Nonlinear system · Optimal control · Fractional derivative · Neural network · Nonlinear observer Mathematics Subject Classification 93C10 · 34H05 · 26A33 · 92B20
1 Introduction Scientists employ mathematical tools to describe phenomena of the surrounding world. One of these tools is the differential equation. Differential equations are used to describe dynamics phenomena and the relationships governing incremental variables. A generalization of differential equations which has attracted attentions in the recent decades is the fractional order differential equations. These equations have more flexibility to describe phenomena consid-
Communicated by José Tenreiro Machado.
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Saeed Balochian [email protected] Gholamreza Nassajian [email protected]
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Department of Electrical Engineering, Gonabad Branch, Islamic Azad University, Gonabad, Khorasan-e-Razavi 9691664791, Iran 0123456789().: V,-vol
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ering their degree of freedom in terms of the order of differentiation. These equations have attracted attentions as a proper tool for modelling. They are employed to describe diseases like Hepatitis (Ahmed and El-Saka 2010), cancer cells (Iomin 2006), viscoelasticity (Sun and Zhang 2007) and human emotions (Sprott 2005). Fractional order modelling of the systems results in employing various control methods and extending them to control such systems. Fractional order models describe behavior of the real system better; thus, any control and stabilization operation for these models has higher capability and performance compared to the integer order model of the same system. In addition, fractional order controllers have higher degree of freedom and more flexibility because they have more adjustment parameters (differentiation order parameter). Therefore, recently, researchers have been motivated to use these contro
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