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Fractal Fract
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Advances in Fractional Order Systems and Robust Control
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Advances in Fractional Order Systems and Robust Control

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: closed (10 December 2023) | Viewed by 9518

Special Issue Editor

Prof. Dr. Feng Liu

E-Mail Website
Guest Editor
School of Automation, China University of Geosciences, Wuhan 430074, China
Interests: complex network system analysis and control; nonlinear system dynamics analysis and control; intelligent control
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Recently, fractional differential calculus has been applied to the study of dynamic systems. Fractional calculus is a generalization of the classical integration and differentiation operators to non-integer order. Because of the unique characteristics of historical memory, the dynamic process of a real problem can be described more accurately by fractional order systems. Meanwhile, most systems in reality are highly coupled linear or nonlinear systems and there are always dynamic uncertainties in the modeling. These uncertainties reduce the performance of controllers designed based on the accurate mathematical models. Therefore, it is necessary to study the robust control of uncertain linear or nonlinear fractional order systems.

The focus of this Special Issue is on continuing to advance research on topics relating to the theory, design, implementation, and application of fractional order systems and robust control. Topics that are invited for submission include (but are not limited to):

  • Stability analysis and robust control of linear fractional order systems;
  • Stability analysis and robust control of linear time-varying periodic fractional order systems;
  • Stability and robust control of fractional order T-S fuzzy systems；
  • Robust control methods for interval uncertain fractional order systems;
  • Robust control for uncertain linear fractional order systems with external uncertain perturbations;
  • Robust controller design method for nonlinear uncertain fractional order systems;
  • Robust control for nonlinear fractional order systems with parametric uncertainties;
  • Robust control for uncertain nonlinear fractional order systems with time-varying uncertainties;
  • Robust control method for uncertain nonlinear fractional order systems with stochastic perturbations.

Prof. Dr. Feng Liu
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional order systems
  • fractional differential/difference equations
  • robust control
  • fractional controllers
  • stability analysis
  • modeling

Published Papers (7 papers)

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17 pages, 6241 KiB  
Article
Dynamic Analysis and Control of a Financial System with Chaotic Behavior Including Fractional Order
by Angelo M. Tusset, Maria E. K. Fuziki, Jose M. Balthazar, Dana I. Andrade and Giane G. Lenzi
Fractal Fract. 2023, 7(7), 535; https://doi.org/10.3390/fractalfract7070535 - 11 Jul 2023
Cited by 4 | Viewed by 1003
Abstract
This paper presents the results of investigating the dynamics of an economic system with chaotic behavior and a suboptimal control proposal to suppress the chaotic behavior. Numerical results using phase portraits, bifurcation diagrams, Lyapunov exponents, and 0-1 testing confirmed chaotic and hyperchaotic behavior. [...] Read more.
This paper presents the results of investigating the dynamics of an economic system with chaotic behavior and a suboptimal control proposal to suppress the chaotic behavior. Numerical results using phase portraits, bifurcation diagrams, Lyapunov exponents, and 0-1 testing confirmed chaotic and hyperchaotic behavior. The results also proved the effectiveness of the control, showing errors below 1%, even in cases where the control design is subject to parametric errors. Additionally, an investigation of the system in fractional order is included, demonstrating that the system has periodic, constant, or chaotic behavior for specific values of the order of the derivative. Full article
(This article belongs to the Special Issue Advances in Fractional Order Systems and Robust Control)
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17 pages, 6080 KiB  
Article
Bifurcation Analysis and Fractional PD Control of Gene Regulatory Networks with sRNA
by Feng Liu, Juan Zhao, Shujiang Sun, Hua Wang and Xiuqin Yang
Fractal Fract. 2023, 7(7), 497; https://doi.org/10.3390/fractalfract7070497 - 22 Jun 2023
Cited by 1 | Viewed by 980
Abstract
This paper investigates the problem of bifurcation analysis and bifurcation control of a fractional-order gene regulatory network with sRNA. Firstly, the process of stability change of system equilibrium under the influence of the sum of time delay is discussed, the critical condition of [...] Read more.
This paper investigates the problem of bifurcation analysis and bifurcation control of a fractional-order gene regulatory network with sRNA. Firstly, the process of stability change of system equilibrium under the influence of the sum of time delay is discussed, the critical condition of Hopf bifurcation is explored, and the effect of fractional order on the system stability domain. Secondly, aiming at the system’s instability caused by a large time delay, we design a controller to improve the system’s stability and derive the parameter conditions that satisfy the system’s stability. It is found that changing the parameter values of the controller within a certain range can control the system’s nonlinear behaviours and effectively expand the stability range. Then, a numerical example is given to illustrate the results of this paper. Full article
(This article belongs to the Special Issue Advances in Fractional Order Systems and Robust Control)
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27 pages, 9584 KiB  
Article
Improved Performance in the Control of DC-DC Three-Phase Power Electronic Converter Using Fractional-Order SMC and Synergetic Controllers and RL-TD3 Agent
by Marcel Nicola and Claudiu-Ionel Nicola
Fractal Fract. 2022, 6(12), 729; https://doi.org/10.3390/fractalfract6120729 - 9 Dec 2022
Cited by 4 | Viewed by 1238
Abstract
In this article, starting from a benchmark represented by a Direct Current-to-Direct Current (DC-DC) three-phase power electronic converter used as an interface and interconnection between the grid and a DC microgrid, we compare the performances of a series of control structures—starting with the [...] Read more.
In this article, starting from a benchmark represented by a Direct Current-to-Direct Current (DC-DC) three-phase power electronic converter used as an interface and interconnection between the grid and a DC microgrid, we compare the performances of a series of control structures—starting with the classical proportional integrator (PI) type and continuing with more advanced ones, such as sliding mode control (SMC), integer-order synergetic, and fractional-order (FO) controllers—in terms of maintaining the constant DC voltage of the DC microgrid. We present the topology and the mathematical modeling using differential equations and transfer functions of the DC-DC three-phase power electronic converter that provides the interface between the grid and a DC microgrid. The main task of the presented control systems is to maintain the DC voltage supplied to the microgrid at an imposed constant value, regardless of the total value of the current absorbed by the consumers connected to the DC microgrid. We present the elements of fractional calculus that were used to synthesize a first set of FO PI, FO tilt-integral-derivative (TID), and FO lead-lag controllers with Matlab R2021b and the Fractional-order Modeling and Control (FOMCON) toolbox, and these controllers significantly improved the control system performance of the DC-DC three-phase power electronic converter compared to classical PI controllers. The next set of proposed and synthesized controllers were based on SMC, together with its more general and flexible synergetic control variant, and both integer-order and FO controllers were developed. The proposed control structures are cascade control structures combining the SMC properties of robustness and control over nonlinear systems for the outer voltage control loop with the use of properly tuned synergetic controllers to obtain faster response time for the inner current control loop. To achieve superior performance, this type of cascade control also used a properly trained reinforcement learning-twin delayed deep deterministic policy gradient (RL-TD3) agent, which provides correction signals overlapping with the command signals of the current and voltage controllers. We present the Matlab/Simulink R2021b implementations of the synthesized controllers and the RL-TD3 agent, along with the results of numerical simulations performed for the comparison of the performance of the control structures. Full article
(This article belongs to the Special Issue Advances in Fractional Order Systems and Robust Control)
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24 pages, 63909 KiB  
Article
Sensitivity of Fractional-Order Recurrent Neural Network with Encoded Physics-Informed Battery Knowledge
by Yanan Wang, Xuebing Han, Languang Lu, Yangquan Chen and Minggao Ouyang
Fractal Fract. 2022, 6(11), 640; https://doi.org/10.3390/fractalfract6110640 - 2 Nov 2022
Cited by 2 | Viewed by 1375
Abstract
In the field of state estimation for the lithium-ion battery (LIB), model-based methods (white box) have been developed to explain battery mechanism and data-driven methods (black box) have been designed to learn battery statistics. Both white box methods and black box methods have [...] Read more.
In the field of state estimation for the lithium-ion battery (LIB), model-based methods (white box) have been developed to explain battery mechanism and data-driven methods (black box) have been designed to learn battery statistics. Both white box methods and black box methods have drawn much attention recently. As the combination of white box and black box, physics-informed machine learning has been investigated by embedding physic laws. For LIB state estimation, this work proposes a fractional-order recurrent neural network (FORNN) encoded with physics-informed battery knowledge. Three aspects of FORNN can be improved by learning certain physics-informed knowledge. Firstly, the fractional-order state feedback is achieved by introducing a fractional-order derivative in a forward propagation process. Secondly, the fractional-order constraint is constructed by a voltage partial derivative equation (PDE) deduced from the battery fractional-order model (FOM). Thirdly, both the fractional-order gradient descent (FOGD) and fractional-order gradient descent with momentum (FOGDm) methods are proposed by introducing a fractional-order gradient in the backpropagation process. For the proposed FORNN, the sensitivity of the added fractional-order parameters are analyzed by experiments under the federal urban driving schedule (FUDS) operation conditions. The experiment results demonstrate that a certain range of every fractional-order parameter can achieve better convergence speed and higher estimation accuracy. On the basis of the sensitivity analysis, the fractional-order parameter tuning rules have been concluded and listed in the discussion part to provide useful references to the parameter tuning of the proposed algorithm. Full article
(This article belongs to the Special Issue Advances in Fractional Order Systems and Robust Control)
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23 pages, 5819 KiB  
Article
Fractional-Order Control Strategy for Anesthesia–Hemodynamic Stabilization in Patients Undergoing Surgical Procedures
by Erwin T. Hegedus, Isabela R. Birs, Mihaela Ghita and Cristina I. Muresan
Fractal Fract. 2022, 6(10), 614; https://doi.org/10.3390/fractalfract6100614 - 20 Oct 2022
Cited by 9 | Viewed by 1479
Abstract
Fractional calculus has been opening new doors in terms of better modeling and control of several phenomena and processes. Biomedical engineering has seen a lot of combined attention from clinicians, control engineers and researchers in their attempt to offer individualized treatment. A large [...] Read more.
Fractional calculus has been opening new doors in terms of better modeling and control of several phenomena and processes. Biomedical engineering has seen a lot of combined attention from clinicians, control engineers and researchers in their attempt to offer individualized treatment. A large number of medical procedures require anesthesia, which in turn requires a closely monitored and controlled level of hypnosis, analgesia and neuromuscular blockade, as well maintenance of hemodynamic variables in a safe range. Computer-controlled anesthesia has been given a tremendous amount of attention lately. Hemodynamic stabilization via computer-based control is also a hot topic. However, very few studies on automatic control of combined anesthesia–hemodynamic systems exist despite the fact that hemodynamics is strongly influenced by hypnotic drugs, while the depth of hypnosis is affected by drugs used in hemodynamic control. The very first multivariable fractional-order controller is developed in this paper for the combined anesthesia–hemodynamic system. Simulation studies on 24 patients show the effectiveness of the proposed approach. Full article
(This article belongs to the Special Issue Advances in Fractional Order Systems and Robust Control)
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18 pages, 1050 KiB  
Article
Fuzzy Fractional-Order PD Vibration Control of Uncertain Building Structures
by Kang Xu, Tingli Cheng, António M. Lopes, Liping Chen, Xiaoxuan Zhu and Minwu Wang
Fractal Fract. 2022, 6(9), 473; https://doi.org/10.3390/fractalfract6090473 - 28 Aug 2022
Cited by 5 | Viewed by 1171
Abstract
A new control strategy is proposed to suppress earthquake-induced vibrations on uncertain building structures. The control strategy embeds fuzzy logic in a fractional-order (FO) proportional derivative (FOPD) controller. A new improved FO particle swarm optimization (IFOPSO) algorithm is derived to adjust the initial [...] Read more.
A new control strategy is proposed to suppress earthquake-induced vibrations on uncertain building structures. The control strategy embeds fuzzy logic in a fractional-order (FO) proportional derivative (FOPD) controller. A new improved FO particle swarm optimization (IFOPSO) algorithm is derived to adjust the initial parameters of the FOPD controller. An original fuzzy logic-FOPD (FFOPD) controller is then designed by combining the advantages of the fuzzy logic and FOPD control, to deal with large displacements on structures under earthquake excitation. Simulation experiments are carried out on uncertain building structures subjected to the effects of different kinds of seismic signals, illustrating the validity and feasibility of the proposed method. Full article
(This article belongs to the Special Issue Advances in Fractional Order Systems and Robust Control)
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17 pages, 6414 KiB  
Article
Asymptotic and Robust Stabilization Control for the Whole Class of Fractional-Order Gene Regulation Networks with Time Delays
by Zitong Li, Zhe Zhang, Qiong Liao and Mingqiang Rong
Fractal Fract. 2022, 6(8), 406; https://doi.org/10.3390/fractalfract6080406 - 24 Jul 2022
Cited by 2 | Viewed by 1122
Abstract
Throughout this article, a novel control strategy for fractional-order gene regulation networks (FOGRN) of all categories is designed by using the vector Lyapunov function in combination with the M-matrix measure. Firstly, a series of puzzles surrounding the asymptotic stability of two-dimensional FOGRN are [...] Read more.
Throughout this article, a novel control strategy for fractional-order gene regulation networks (FOGRN) of all categories is designed by using the vector Lyapunov function in combination with the M-matrix measure. Firstly, a series of puzzles surrounding the asymptotic stability of two-dimensional FOGRN are studied, and a new asymptotic stability control strategy is formulated based on the vector Lyapunov function in combination with the M-matrix measure, ensuring that the controlled FOGRN has a strong robust stability. In addition, the corresponding asymptotic stability criterion is deduced. On this basis, the problem of asymptotic stability of a three-dimensional FOGRN is studied. Based on the new method, a stabilization control strategy is also formulated with the corresponding asymptotic stability criterion deduced, ensuring that the controlled FOGRN has a strong robust stability as well. Finally, this novel method’s effectiveness and generality are authenticated via simulation experiments. Full article
(This article belongs to the Special Issue Advances in Fractional Order Systems and Robust Control)
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