Direct adaptive model-free control of a class of uncertain nonlinear systems using Legendre polynomials

Abstract

In this paper, a simple model-free controller for a class of uncertain nonlinear systems is presented using Legendre polynomials. According to the orthogonal functions theorem, Legendre polynomials are universal approximators. From this point of view, they are similar to fuzzy systems and can be used as a controller to approximate the ideal control law. Legendre coefficients are estimated online using the adaptation rule obtained from stability analysis. In comparison with fuzzy systems and neural networks, Legendre polynomials are simpler and less computational. Moreover, there are very few tuning parameters in Legendre polynomials. The case studies are an inverted pendulum and an articulated robot manipulator. Simulation results verify the effectiveness of the proposed controller and its robustness against large uncertainties. Comparisons with model-free controller using extended state observer and adaptive fuzzy systems show the superiority of proposed controller in disturbance rejection.

An observer-based output tracking controller for electrically driven cooperative multiple manipulators with adaptive Bernstein-type approximator

“…The function approximation technique (FAT) is a relatively new method for approximating diverse functions such as uncertainties. Various approaches based on the FAT have been formed, such as Legendre polynomials [35], differential equations [36,37], and Fourier series [38,39], and Bernstein-Stancu operator [40]. Simple structures of FATs in comparison with neural or fuzzy-based methods made them ideal candidates for function approximation [41][42][43].…”

Observer-based adaptive control of robot manipulators using reinforcement learning and the Fourier series expansion

“…Due to the excellent uncertainty approximation ability of function approximation techniques such as the Fourier series (FS) (Izadbakhsh et al, 2019), Szász–Mirakyan operator (Izadbakhsh et al, 2020), and Legendre polynomials (Zarei and Khorashadizadeh, 2019) much interest has been generated in this area (Chien and Huang, 2012; Kai and Huang, 2013). Similar to other estimators such as fuzzy systems and NNs, a function can be expanded by an FS.…”

Adaptive control of electrically‐driven nonholonomic wheeled mobile robots: Taylor series‐based approach with guaranteed asymptotic stability

“…For instance, Fourier series and an adaptive technique were integrated [41] to design a model-free control scheme for electrically-driven manipulators. In order to approximate the solution of the shortest path problems especially in industrial robotic applications with boundary and interior barriers [42] and to propose a direct adaptive controller to control an inverse pendulum [43], Legendre polynomials are utilized. Chebyshev polynomials alongside neural networks are developed in the nonlinear system identification [44] and to design an adaptive output feedback to control attitude of a spacecraft [45].…”