Fractional-order control systems - theory and applications

• Main Author: Li, Xinyao
• Other Authors: Wen Changyun
• Format: Thesis-Doctor of Philosophy
• Language: English
• Published: Nanyang Technological University 2023
• Subjects: Engineering::Electrical and electronic engineering
• Online Access: https://hdl.handle.net/10356/165564
• Institution: Nanyang Technological University

As a generalization of integer-order calculus, fractional-order one is a branch of the mathematical field with over three centuries of history. Owing to its infinite memory and hereditary properties, fractional-order calculus possesses significant advantages in accurately modeling and characterizing dynamic properties of many real-world phenomena or systems arising from various fields such as physics, biology, mechanics, chemistry, finance, and engineering. In recent years, fractional-order calculus has attracted increasing attention in different areas, particularly in control engineering, yet compared with the widely-studied traditional integer-order case, the research on fractional-order control systems is still at its beginning stage. In this thesis, control schemes based on the adaptive backstepping technique and sampled-data are developed respectively for nonlinear and linear fractional-order systems. In addition, the fractional-order adaptive backstepping control problem for high-order nonlinear integer-order systems will also be investigated as employing fractional-order controllers in integer-order systems can help increase additional control design degrees of freedom and enable superior control performance. First of all, the adaptive backstepping smooth control problem for a class of commensurate fractional-order nonlinear systems in strict feedback form with system uncertainties and unknown time-varying external disturbance is studied. As one of the most prevalent techniques applied in controlling nonlinear systems, backstepping control demonstrates a recursive systematic design procedure. For fractional-order systems, the time derivative of composite functions, which is required for the backstepping controller design process, can not be easily obtained by the chain rule because of the specific mathematical characteristics of fractional-order calculus. To overcome this obstacle, a novel approach for approximating the fractional-order time derivatives of virtual control signals, which are composite functions about system states, is proposed and adaptive laws are designed to estimate the bounds of approximation errors. Besides, instead of using noncontinuous control signal, an auxiliary function is employed to obtain a smooth control input that is still able to achieve asymptotic tracking in the presence of arbitrarily bounded external disturbances. It is proved that the obtained closed-loop system is globally asymptotically stable under the assumption that the approximation errors between the actual values of fractional derivatives of virtual control signals and their approximated values remain bounded. Subsequently, in order to relax such an assumption, we propose an alternative adaptive backstepping smooth control strategy, under which the resulting closed-loop fractional-order system is guaranteed to be stable in the sense that all the closed-loop signals are uniformly ultimately bounded and the output tracking error can be driven towards a small region around the origin. Furthermore, it is also of great importance yet challenge to investigate the incommensurate fractional-order system, which is a universal form and could be found in various practical scenarios such as fractional-order phase-locked loop (PLL) and chaotic systems. Nevertheless, few results about the backstepping control of such systems have been reported. Consequently, we construct a smooth adaptive backstepping controller for a class of incommensurate fractional-order nonlinear systems with uncertainties and time-varying disturbance to achieve that the system output tracks a predefined signal asymptotically while ensuring the global boundedness of all closed-loop signals. In practice, control schemes are commonly implemented digitally so that digital control signals are generated, resulting in sampled-data control. Therefore, the consensus control problem for linear fractional-order multi-agent systems (MASs) in single-integrator form over directed topology is investigated using sampled-data control. Unlike the classical integer-order derivatives or integrals, fractional-order ones are non-local operators and own infinite memory and hereditary properties. As a result, a novel distributed control scheme with the consideration of these unique properties of fractional-order calculus is proposed to achieve asymptotic consensus and ensure global stability in the sense that all the signals in the closed-loop systems are globally bounded. Hereafter, for linear single-integrator fractional-order systems, a finite-dimensional sampled-data based control scheme, which is computationally tractable and more applicable in practical usage, is developed to guarantee the global stability of the corresponding closed-loop system in the sense that the closed-loop states can converge to an ultimate bound around the origin with control signals remaining bounded. A guideline on selecting the control signal storage space for the finite-dimensional controller to satisfy preset control accuracy is also provided. Then a new sampled-data controller is designed for a more general class of linear fractional-order systems whose solution contains the Mittag-Leffler function, one of the fundamental functions in fractional-order calculus which does not obey the basic exponentiation identity. It is ensured that the resulted closed-loop system is asymptotically stable with the hereditary and infinite memory characteristics of fractional-order calculus being taken into account. In the last part of this thesis, the fractional-order adaptive backstepping control problem for a class of high-order nonlinear integer-order systems with uncertainties and unknown external disturbance is addressed. Different from the existing works that are only for second-order integer systems, the considered class of uncertain nonlinear integer-order systems can be of arbitrary order. Based on the Lyapunov stability theorem, we provide theoretical proof to assure the global stability of the closed-loop system in the sense that all the closed-loop signals are uniformly ultimately bounded and the system output error converges to an arbitrarily small tunable bound by adjusting certain design parameters. In all the problems we considered above, rigorous theoretical analysis and demonstrative examples are provided to verify and illustrate the efficiency of the established results.