1) The document analyzes the boundedness and domain of attraction of a fractional-order wireless power transfer (WPT) system.
2) It establishes a fractional-order piecewise affine model of the WPT system and derives sufficient conditions for boundedness using Lyapunov functions and inequality techniques.
3) The results provide a way to estimate the domain of attraction of the fractional-order WPT system and systems with periodically intermittent control.
A COMPREHENSIVE ANALYSIS OF QUANTUM CLUSTERING : FINDING ALL THE POTENTIAL MI...IJDKP
Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is
accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a
σ value, a hyper-parameter which can be manually defined and manipulated to suit the application.
Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster
centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the
exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an
outstanding task because normally such expressions are impossible to solve analytically. However, we
prove that if the points are all included in a square region of size σ, there is only one minimum. This bound
is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new
numerical approach “per block”. This technique decreases the number of particles by approximating some
groups of particles to weighted particles. These findings are not only useful to the quantum clustering
problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics
and other applications.
Special Plenary Lecture at the International Conference on VIBRATION ENGINEERING AND TECHNOLOGY OF MACHINERY (VETOMAC), Lisbon, Portugal, September 10 - 13, 2018
http://www.conf.pt/index.php/v-speakers
Propagation of uncertainties in complex engineering dynamical systems is receiving increasing attention. When uncertainties are taken into account, the equations of motion of discretised dynamical systems can be expressed by coupled ordinary differential equations with stochastic coefficients. The computational cost for the solution of such a system mainly depends on the number of degrees of freedom and number of random variables. Among various numerical methods developed for such systems, the polynomial chaos based Galerkin projection approach shows significant promise because it is more accurate compared to the classical perturbation based methods and computationally more efficient compared to the Monte Carlo simulation based methods. However, the computational cost increases significantly with the number of random variables and the results tend to become less accurate for a longer length of time. In this talk novel approaches will be discussed to address these issues. Reduced-order Galerkin projection schemes in the frequency domain will be discussed to address the problem of a large number of random variables. Practical examples will be given to illustrate the application of the proposed Galerkin projection techniques.
A High Order Continuation Based On Time Power Series Expansion And Time Ratio...IJRES Journal
In this paper, we propose a high order continuation based on time power series expansion and time rational representation called Pad´e approximants for solving nonlinear structural dynamic problems. The solution of the discretized nonlinear structural dynamic problems, by finite elements method, is sought in the form of a power series expansion with respect to time. The Pad´e approximants technique is introduced to improve the validity range of power series expansion. The whole solution is built branch by branch using the continuation method. To illustrate the performance of this proposed high order continuation, we give some numerical comparisons on an example of forced nonlinear vibration of an elastic beam.
COMPUTATIONAL PERFORMANCE OF QUANTUM PHASE ESTIMATION ALGORITHMcscpconf
A quantum computation problem is discussed in this paper. Many new features that make quantum computation superior to classical computation can be attributed to quantum coherence
effect, which depends on the phase of quantum coherent state. Quantum Fourier transform algorithm, the most commonly used algorithm, is introduced. And one of its most important
applications, phase estimation of quantum state based on quantum Fourier transform, is presented in details. The flow of phase estimation algorithm and the quantum circuit model are
shown. And the error of the output phase value, as well as the probability of measurement, is analysed. The probability distribution of the measuring result of phase value is presented and the computational efficiency is discussed.
COMPUTATIONAL PERFORMANCE OF QUANTUM PHASE ESTIMATION ALGORITHMcsitconf
A quantum computation problem is discussed in this paper. Many new features that make
quantum computation superior to classical computation can be attributed to quantum coherence
effect, which depends on the phase of quantum coherent state. Quantum Fourier transform
algorithm, the most commonly used algorithm, is introduced. And one of its most important
applications, phase estimation of quantum state based on quantum Fourier transform, is
presented in details. The flow of phase estimation algorithm and the quantum circuit model are
shown. And the error of the output phase value, as well as the probability of measurement, is
analysed. The probability distribution of the measuring result of phase value is presented and
the computational efficiency is discussed.
A COMPREHENSIVE ANALYSIS OF QUANTUM CLUSTERING : FINDING ALL THE POTENTIAL MI...IJDKP
Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is
accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a
σ value, a hyper-parameter which can be manually defined and manipulated to suit the application.
Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster
centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the
exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an
outstanding task because normally such expressions are impossible to solve analytically. However, we
prove that if the points are all included in a square region of size σ, there is only one minimum. This bound
is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new
numerical approach “per block”. This technique decreases the number of particles by approximating some
groups of particles to weighted particles. These findings are not only useful to the quantum clustering
problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics
and other applications.
Special Plenary Lecture at the International Conference on VIBRATION ENGINEERING AND TECHNOLOGY OF MACHINERY (VETOMAC), Lisbon, Portugal, September 10 - 13, 2018
http://www.conf.pt/index.php/v-speakers
Propagation of uncertainties in complex engineering dynamical systems is receiving increasing attention. When uncertainties are taken into account, the equations of motion of discretised dynamical systems can be expressed by coupled ordinary differential equations with stochastic coefficients. The computational cost for the solution of such a system mainly depends on the number of degrees of freedom and number of random variables. Among various numerical methods developed for such systems, the polynomial chaos based Galerkin projection approach shows significant promise because it is more accurate compared to the classical perturbation based methods and computationally more efficient compared to the Monte Carlo simulation based methods. However, the computational cost increases significantly with the number of random variables and the results tend to become less accurate for a longer length of time. In this talk novel approaches will be discussed to address these issues. Reduced-order Galerkin projection schemes in the frequency domain will be discussed to address the problem of a large number of random variables. Practical examples will be given to illustrate the application of the proposed Galerkin projection techniques.
A High Order Continuation Based On Time Power Series Expansion And Time Ratio...IJRES Journal
In this paper, we propose a high order continuation based on time power series expansion and time rational representation called Pad´e approximants for solving nonlinear structural dynamic problems. The solution of the discretized nonlinear structural dynamic problems, by finite elements method, is sought in the form of a power series expansion with respect to time. The Pad´e approximants technique is introduced to improve the validity range of power series expansion. The whole solution is built branch by branch using the continuation method. To illustrate the performance of this proposed high order continuation, we give some numerical comparisons on an example of forced nonlinear vibration of an elastic beam.
COMPUTATIONAL PERFORMANCE OF QUANTUM PHASE ESTIMATION ALGORITHMcscpconf
A quantum computation problem is discussed in this paper. Many new features that make quantum computation superior to classical computation can be attributed to quantum coherence
effect, which depends on the phase of quantum coherent state. Quantum Fourier transform algorithm, the most commonly used algorithm, is introduced. And one of its most important
applications, phase estimation of quantum state based on quantum Fourier transform, is presented in details. The flow of phase estimation algorithm and the quantum circuit model are
shown. And the error of the output phase value, as well as the probability of measurement, is analysed. The probability distribution of the measuring result of phase value is presented and the computational efficiency is discussed.
COMPUTATIONAL PERFORMANCE OF QUANTUM PHASE ESTIMATION ALGORITHMcsitconf
A quantum computation problem is discussed in this paper. Many new features that make
quantum computation superior to classical computation can be attributed to quantum coherence
effect, which depends on the phase of quantum coherent state. Quantum Fourier transform
algorithm, the most commonly used algorithm, is introduced. And one of its most important
applications, phase estimation of quantum state based on quantum Fourier transform, is
presented in details. The flow of phase estimation algorithm and the quantum circuit model are
shown. And the error of the output phase value, as well as the probability of measurement, is
analysed. The probability distribution of the measuring result of phase value is presented and
the computational efficiency is discussed.
Finite element modelling of nonlocal dynamic systems, Modal analysis of nonlocal dynamical systems, Dynamics of damped nonlocal systems, Numerical illustrations
constant strain triangular which is used in analysis of triangular in finite element method with the help of shape function and natural coordinate system.
A Method for the Reduction 0f Linear High Order MIMO Systems Using Interlacin...IJMTST Journal
This paper presents a new mixed method for the reduction of linear high order MIMO system. This method
is based upon the interlacing property by which the denominator polynomial of the reduced order model is
obtained and the numerator is obtained by using factor division method. In general, the stability of the high
order system is retained in their models. Better approximation of the time response characteristics is
attained by using this suggested method. The number of computations has been reduced when compared to
several of the existing methods are in international literature. Another advantage of this method is that it is a
direct method. The suggested procedure is digital computer oriented.
Optimising Data Using K-Means Clustering AlgorithmIJERA Editor
K-means is one of the simplest unsupervised learning algorithms that solve the well known clustering problem. The procedure follows a simple and easy way to classify a given data set through a certain number of clusters (assume k clusters) fixed a priori. The main idea is to define k centroids, one for each cluster. These centroids should be placed in a cunning way because of different location causes different result. So, the better choice is to place them as much as possible far away from each other.
A novel nonlinear missile guidance law against maneuvering targets is designed based on the principles of partial stability. It is demonstrated that in a real approach which is adopted with actual situations, each state of the guidance system must have a special behavior and asymptotic stability or exponential stability of all states is not realistic. Thus, a new guidance law is developed based on the partial stability theorem in such a way that the behaviors of states in the closed-loop system are in conformity with a real guidance scenario that leads to collision. The performance of the proposed guidance law in terms of interception time and control effort is compared with the sliding mode guidance law by means of numerical simulations.
Finite Element Analysis is a widely used computational method in most of the engineering domains. But still, its considered as a difficult topic by most students. This presentation is an effort to introduce the very basics of FEA so as to build an intuitive feel for the method. Enjoy !
Event triggered control design of linear networked systems with quantizationsISA Interchange
This paper is concerned with the control design problem of event-triggered networked systems with both state and control input quantizations. Firstly, an innovative delay system model is proposed that describes the network conditions, state and control input quantizations, and event-triggering mechanism in a unified framework. Secondly, based on this model, the criteria for the asymptotical stability analysis and control synthesis of event-triggered networked control systems are established in terms of linear matrix inequalities (LMIs). Simulation results are given to illustrate the effectiveness of the proposed method.
Foundation and Synchronization of the Dynamic Output Dual Systemsijtsrd
In this paper, the synchronization problem of the dynamic output dual systems is firstly introduced and investigated. Based on the time domain approach, the state variables synchronization of such dual systems can be verified. Meanwhile, the guaranteed exponential convergence rate can be accurately estimated. Finally, some numerical simulations are provided to illustrate the feasibility and effectiveness of the obtained result. Yeong-Jeu Sun "Foundation and Synchronization of the Dynamic Output Dual Systems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29256.pdf Paper URL: https://www.ijtsrd.com/engineering/electrical-engineering/29256/foundation-and-synchronization-of-the-dynamic-output-dual-systems/yeong-jeu-sun
Finite element modelling of nonlocal dynamic systems, Modal analysis of nonlocal dynamical systems, Dynamics of damped nonlocal systems, Numerical illustrations
constant strain triangular which is used in analysis of triangular in finite element method with the help of shape function and natural coordinate system.
A Method for the Reduction 0f Linear High Order MIMO Systems Using Interlacin...IJMTST Journal
This paper presents a new mixed method for the reduction of linear high order MIMO system. This method
is based upon the interlacing property by which the denominator polynomial of the reduced order model is
obtained and the numerator is obtained by using factor division method. In general, the stability of the high
order system is retained in their models. Better approximation of the time response characteristics is
attained by using this suggested method. The number of computations has been reduced when compared to
several of the existing methods are in international literature. Another advantage of this method is that it is a
direct method. The suggested procedure is digital computer oriented.
Optimising Data Using K-Means Clustering AlgorithmIJERA Editor
K-means is one of the simplest unsupervised learning algorithms that solve the well known clustering problem. The procedure follows a simple and easy way to classify a given data set through a certain number of clusters (assume k clusters) fixed a priori. The main idea is to define k centroids, one for each cluster. These centroids should be placed in a cunning way because of different location causes different result. So, the better choice is to place them as much as possible far away from each other.
A novel nonlinear missile guidance law against maneuvering targets is designed based on the principles of partial stability. It is demonstrated that in a real approach which is adopted with actual situations, each state of the guidance system must have a special behavior and asymptotic stability or exponential stability of all states is not realistic. Thus, a new guidance law is developed based on the partial stability theorem in such a way that the behaviors of states in the closed-loop system are in conformity with a real guidance scenario that leads to collision. The performance of the proposed guidance law in terms of interception time and control effort is compared with the sliding mode guidance law by means of numerical simulations.
Finite Element Analysis is a widely used computational method in most of the engineering domains. But still, its considered as a difficult topic by most students. This presentation is an effort to introduce the very basics of FEA so as to build an intuitive feel for the method. Enjoy !
Event triggered control design of linear networked systems with quantizationsISA Interchange
This paper is concerned with the control design problem of event-triggered networked systems with both state and control input quantizations. Firstly, an innovative delay system model is proposed that describes the network conditions, state and control input quantizations, and event-triggering mechanism in a unified framework. Secondly, based on this model, the criteria for the asymptotical stability analysis and control synthesis of event-triggered networked control systems are established in terms of linear matrix inequalities (LMIs). Simulation results are given to illustrate the effectiveness of the proposed method.
Foundation and Synchronization of the Dynamic Output Dual Systemsijtsrd
In this paper, the synchronization problem of the dynamic output dual systems is firstly introduced and investigated. Based on the time domain approach, the state variables synchronization of such dual systems can be verified. Meanwhile, the guaranteed exponential convergence rate can be accurately estimated. Finally, some numerical simulations are provided to illustrate the feasibility and effectiveness of the obtained result. Yeong-Jeu Sun "Foundation and Synchronization of the Dynamic Output Dual Systems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29256.pdf Paper URL: https://www.ijtsrd.com/engineering/electrical-engineering/29256/foundation-and-synchronization-of-the-dynamic-output-dual-systems/yeong-jeu-sun
Design of State Estimator for a Class of Generalized Chaotic Systemsijtsrd
In this paper, a class of generalized chaotic systems is considered and the state observation problem of such a system is investigated. Based on the time domain approach with differential inequality, a simple state estimator for such generalized chaotic systems is developed to guarantee the global exponential stability of the resulting error system. Besides, the guaranteed exponential decay rate can be correctly estimated. Finally, several numerical simulations are given to show the effectiveness of the obtained result. Yeong-Jeu Sun "Design of State Estimator for a Class of Generalized Chaotic Systems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29270.pdf Paper URL: https://www.ijtsrd.com/engineering/electrical-engineering/29270/design-of-state-estimator-for-a-class-of-generalized-chaotic-systems/yeong-jeu-sun
Stability and stabilization of discrete-time systems with time-delay via Lyap...IJERA Editor
The stability and stabilization problems for discrete systems with time-delay are discussed .The stability and
stabilization criterion are expressed in the form of linear matrix inequalities (LMI). An effective method
allowing us transforming a bilinear matrix Inequality (BMI) to a linear matrix Inequality (LMI) is developed.
Based on these conditions, a state feedback controller with gain is designed. An illustrative numerical example
is provided to show the effectiveness of the proposed method and the reliability of the results.
THE LEFT AND RIGHT BLOCK POLE PLACEMENT COMPARISON STUDY: APPLICATION TO FLIG...ieijjournal1
It is known that if a linear-time-invariant MIMO system described by a state space equation has a number
of states divisible by the number of inputs and it can be transformed to block controller form, we can
design a state feedback controller using block pole placement technique by assigning a set of desired Block
poles. These may be left or right block poles. The idea is to compare both in terms of system’s response.
THE LEFT AND RIGHT BLOCK POLE PLACEMENT COMPARISON STUDY: APPLICATION TO FLIG...ieijjournal
It is known that if a linear-time-invariant MIMO system described by a state space equation has a number of states divisible by the number of inputs and it can be transformed to block controller form, we can design a state feedback controller using block pole placement technique by assigning a set of desired Block poles. These may be left or right block poles. The idea is to compare both in terms of system’s response.
Design of Resonators for Coupled Magnetic Resonance-based Wireless Power Tran...Quang-Trung Luu
Quang-Trung Luu, Duc-Hung Tran, Bao-Huy Nguyen, Yem Vu-Van, and Cao-Minh Ta, "Design of Resonators for Coupled Magnetic Resonance-based Wireless Power Transmission System," In Proc. Vietnam Conference on Control and Automation, Da Nang, Nov. 2013.
Centralized Optimal Control for a Multimachine Power System Stability Improve...IDES Editor
this paper introduces the application of wave variable
method in reducing the effect of delay during the transmission of
control signals via large distances in a multimachine power
system. Fast response and pure control signals applied to a power
system have great importance in control process. At first we
design a centralized optimal control considering no transport
delay and then we compare them to the one considering this effect
in order to demonstrate the deteriorating effect of transmission
delay. Then we apply the wave variable method and it is shown
that performance of centralized optimal controller is improved.
Development of deep reinforcement learning for inverted pendulumIJECEIAES
This paper presents a modification of the deep Q-network (DQN) in deep reinforcement learning to control the angle of the inverted pendulum (IP). The original DQN method often uses two actions related to two force states like constant negative and positive force values which apply to the cart of IP to maintain the angle between the pendulum and the Y-axis. Due to the changing of too much value of force, the IP may make some oscillation which makes the performance system could be declined. Thus, a modified DQN algorithm is developed based on neural network structure to make a range of force selections for IP to improve the performance of IP. To prove our algorithm, the OpenAI/Gym and Keras libraries are used to develop DQN. All results showed that our proposed controller has higher performance than the original DQN and could be applied to a nonlinear system.
Advanced Stability Analysis of Control Systems with Variable Parametersjournal ijrtem
The purpose of the current research is to advance further the D-Partitioning method and
emphasize on its practical application. It has the objective to clarify it in a user friendly manner in order to
simplify its implementation. By applying the basic initial ideas of the method, the main line of the research is the
development of a generalized stability analysis tool and demonstrating its application. With the aid of this tool,
proper parameter values can be chosen for a desirable performance and stability of a system. The analysis tool
can be practically used when one, two or more system’s parameters are varied independently or simultaneously.
Basically this tool defines regions of stability in the space of the system’s parameters.
Advanced Stability Analysis of Control Systems with Variable ParametersIJRTEMJOURNAL
The purpose of the current research is to advance further the D-Partitioning method and
emphasize on its practical application. It has the objective to clarify it in a user friendly manner in order to
simplify its implementation. By applying the basic initial ideas of the method, the main line of the research is the
development of a generalized stability analysis tool and demonstrating its application. With the aid of this tool,
proper parameter values can be chosen for a desirable performance and stability of a system. The analysis tool
can be practically used when one, two or more system’s parameters are varied independently or simultaneously.
Basically this tool defines regions of stability in the space of the system’s parameters.
In this paper, the tracking control scheme is presented using the framework of finite-time sliding mode control (SMC) law and high-gain observer for disturbed/uncertain multi-motor driving systems under the consideration multi-output systems. The convergence time of sliding mode control is estimated in connection with linear matrix inequalities (LMIs). The input state stability (ISS) of proposed controller was analyzed by Lyapunov stability theory. Finally, the extensive simulation results are given to validate the advantages of proposed control design.
2-DOF Block Pole Placement Control Application To: Have-DASH-IIBITT MissileZac Darcy
In a multivariable servomechanism design, it is required that the output vector tracks a certain reference
vector while satisfying some desired transient specifications, for this purpose a 2DOF control law
consisting of state feedback gain and feedforward scaling gain is proposed. The control law is designed
using block pole placement technique by assigning a set of desired Block poles in different canonical forms.
The resulting control is simulated for linearized model of the HAVE DASH II BTT missile; numerical
results are analyzed and compared in terms of transient response, gain magnitude, performance
robustness, stability robustness and tracking. The suitable structure for this case study is then selected.
2-DOF Block Pole Placement Control Application To: Have-DASH-IIBITT MissileZac Darcy
In a multivariable servomechanism design, it is required that the output vector tracks a certain reference
vector while satisfying some desired transient specifications, for this purpose a 2DOF control law
consisting of state feedback gain and feedforward scaling gain is proposed. The control law is designed
using block pole placement technique by assigning a set of desired Block poles in different canonical forms.
The resulting control is simulated for linearized model of the HAVE DASH II BTT missile; numerical
results are analyzed and compared in terms of transient response, gain magnitude, performance
robustness, stability robustness and tracking. The suitable structure for this case study is then selected.
Chaos Suppression and Stabilization of Generalized Liu Chaotic Control Systemijtsrd
In this paper, the concept of generalized stabilization for nonlinear systems is introduced and the stabilization of the generalized Liu chaotic control system is explored. Based on the time-domain approach with differential inequalities, a suitable control is presented such that the generalized stabilization for a class of Liu chaotic system can be achieved. Meanwhile, not only the guaranteed exponential convergence rate can be arbitrarily pre-specified but also the critical time can be correctly estimated. Finally, some numerical simulations are given to demonstrate the feasibility and effectiveness of the obtained results. Yeong-Jeu Sun | Jer-Guang Hsieh "Chaos Suppression and Stabilization of Generalized Liu Chaotic Control System" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-1 , December 2018, URL: http://www.ijtsrd.com/papers/ijtsrd20195.pdf
http://www.ijtsrd.com/engineering/electrical-engineering/20195/chaos-suppression-and-stabilization-of-generalized-liu-chaotic-control-system/yeong-jeu-sun
DESIGN OF OBSERVER BASED QUASI DECENTRALIZED FUZZY LOAD FREQUENCY CONTROLLER ...ijfls
This paper proposes Fuzzy Quasi Decentralized Functional Observers (FQDFO) for Load Frequency Control of inter-connected power systems. From the literature, it is well noticed about the need of
Functional Observers (FO’s) for power system applications. In past, conventional Functional Observers are used. Later, these conventional Functional Observers are replaced with Quasi Decentralized
Functional Observers (QDFO) to improve the system performance. In order to increase the efficacy of the
system, intelligent controllers gained importance. Due to their expertise knowledge, which is adaptive in
nature is applied successfully for FQDFO. For supporting the validity of the proposed observer FQDFO,
it is compared with full order Luenberger observer and QDFO for a two-area inter connected power system by taking parametric uncertainties into consideration. Computational results proved the robustness of the proposed observer.
DESIGN OF OBSERVER BASED QUASI DECENTRALIZED FUZZY LOAD FREQUENCY CONTROLLER ...Wireilla
ABSTRACT
This paper proposes Fuzzy Quasi Decentralized Functional Observers (FQDFO) for Load Frequency Control of inter-connected power systems. From the literature, it is well noticed about the need of Functional Observers (FO’s) for power system applications. In past, conventional Functional Observers are used. Later, these conventional Functional Observers are replaced with Quasi Decentralized Functional Observers (QDFO) to improve the system performance. In order to increase the efficacy of the system, intelligent controllers gained importance. Due to their expertise knowledge, which is adaptive in nature is applied successfully for FQDFO. For supporting the validity of the proposed observer FQDFO, it is compared with full order Luenberger observer and QDFO for a two-area inter connected power system by taking parametric uncertainties into consideration. Computational results proved the robustness of the proposed observer.
A sensorless approach for tracking control problem of tubular linear synchron...IJECEIAES
As well-known, linear motors are widely applied to various industrial applications due to their abilities in providing directly straight movement without auxiliary mechanical transmissions. This paper addresses the sensorless control problem of tubular linear synchronous motors, which belong to a family of permanent magnet linear motor. To be specific, a novel velocity observer is proposed to deal with an unmeasurable velocity problem, and asymptotic convergence of the observer error is ensured. Unlike other studies on sensorless control methods for linear motors, our proposed observer is designed by regrading unknown disturbance load in the tracking control problem whereas considering theoretical demonstrations. By adjusting controller parameters properly, the position and velocity tracking error converge in arbitrary small values. Finally, the effectiveness of the proposed method is verified in two illustrative examples.
Adaptive Projective Lag Synchronization of T and Lu Chaotic Systems IJECEIAES
In this paper, the synchronization problem of T chaotic system and Lu chaotic system is studied. The parameter of the drive T chaotic system is considered unknown. An adaptive projective lag control method and also parameter estimation law are designed to achieve chaos synchronization problem between two chaotic systems. Then Lyapunov stability theorem is utilized to prove the validity of the proposed control method. After that, some numerical simulations are performed to assess the performance of the proposed method. The results show high accuracy of the proposed method in control and synchronization of chaotic systems.
Similar to 1 Aminullah Assagaf_Estimation-of-domain-of-attraction-for-the-fract_2021_Nonlinear-Analysis--Hy.pdf (20)
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
2. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
components such as capacitor and inductor, the corresponding fractional-order model is more accurate than the integer-
order model in terms of modeling. Thus the fractional-order model of the system can more accurately describe parameters
of the system such as resonance frequency, inductance and capacitance, which is helpful to better analyze the frequency
drift phenomenon and estimate the domain of attraction. And recently, the effect of fractional orders on the transmission
power and efficiency of the fractional-order WPT system has been investigated [22]. Additionally, the WPT system also
belongs to a class of piecewise affine systems according to its working principle [23]. In the meanwhile, the problem
of the domain of attraction of the piecewise affine systems as one of the most challenging topics in the field of hybrid
systems has attracted great attention in the past few decades [24,25]. Stability regions for linear systems with saturating
control via circle and Popov criteria are estimated [24]. Results of global asymptotic stability of piecewise linear systems
based on impact maps and surface Lyapunov functions are given [25]. And compared with the paper [24], the results
in paper [25] show less conservatism, which leads to a larger stability region. Nonetheless, these works only focus on
the integer-order piecewise affine systems, so the obtained results cannot be applied to the fractional-order piecewise
affine systems. Besides, with regard to the fractional-order systems, the domain of attraction of the fractional-order linear
system is investigated [26]. However, up to now, to the best of our knowledge, the problem of the domain of attraction
for the fractional-order piecewise affine systems is still a gap.
The above mentioned works and observation inspire us to deal with the problem of the domain of attraction of the
fractional-order WPT system. First of all, the fractional-order model of the WPT system is built. Then, relevant results of
the fractional-order WPT system are derived by using Lyapunov function approach and inductive method, and further
extended to the fractional-order system with periodically intermittent control. The major contributions of this work are
as follows:
(a) To estimate the domain of attraction of the systems, sufficient conditions of the boundedness for the fractional-order
WPT system and the fractional-order system with periodically intermittent control are derived, respectively.
(b) Consider the conservatism of the results, the relevant inequality technique is used. And compared with the results
without using the inequality technique, the domains of attraction of the obtained results are larger. Besides, the
conservatism is also discussed from the practical point of view.
(c) The related results are also suitable for estimating the domain of attraction under the different τ.
(d) Estimation of the domain of attraction provides an important reference for the robustness evaluation of the WPT
system.
This paper is organized as follows. Section 2 introduces some relevant lemmas and definitions. Simultaneously, the
model of the fractional-order WPT system is described. In Section 3, the results of the boundedness of the fractional-
order WPT system and the fractional-order system with periodically intermittent control are demonstrated, respectively.
In Section 4, simulation is provided to verify the obtained results. Conclusions are drawn in Section 5.
2. Preliminaries and model of description
At first, some relevant definitions and lemmas are given as follows.
Definition 1 ([27]). For a continuous function f : [0, ∞) → R, the Caputo derivative of fractional order α is defined as
Dα
f (t) =
1
Γ (n − α)
∫ t
0
(t − s)n−α−1
f (n)
(s)ds (n − 1 < α < n, n = [α] + 1), (1)
where [α] denotes the integer part of the real number α. Γ (·) is the Gamma function.
Definition 2 ([28]). The Mittag-Leffler function with one parameter is defined as
Eα(z) = Eα,1(z) =
∞
∑
k=0
zk
Γ (kα + 1)
, (2)
where α > 0 and z ∈ C. The Mittag-Leffler function with two parameters appears most frequently and has the following
form:
Eα,β(z) =
∞
∑
k=0
zk
Γ (kα + β)
, (3)
where α > 0, β > 0, and z ∈ C. For β = 1 we obtain Eα,1(z) = Eα(z).
Lemma 1 ([29]). Let x(t) ∈ Rn
be a vector of differentiable functions. Then, for any time instant t ≥ t0, the following relationship
holds
1
2
C
t0
Dα
(xT
(t)Px(t)) ≤ xT
(t)P C
t0
Dα
x(t), ∀α ∈ (0, 1], ∀t ≥ t0, (4)
where P ∈ Rn×n
is a constant, square, symmetric and positive definite matrix.
2
3. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
Fig. 1. Typical WPT system.
Lemma 2 ([30]). For any matrices X and Y with appropriate dimensions, the following relationship holds
XT
Y + YT
X ≤ εXT
X +
1
ε
YT
Y, for any ε > 0. (5)
Lemma 3 ([31]). For any vectors a ∈ Rna , b ∈ Rnb , and any matrices N ∈ Rna×nb , X ∈ Rna×na , Y ∈ Rna×nb , Z ∈ Rnb×nb , if
[
X Y
YT
Z
]
≥ 0, then the following inequality holds
−2aT
Nb ≤
[
a
b
]T [
X Y − N
YT
− NT
Z
] [
a
b
]
(6)
Lemma 4 ([32]). Let H(t) be a continuous function on [0, ∞), if there exist k1 ∈ R and k2 ≥ 0 such that C
0 Dα
t H(t) ≤ k1H(t)+k2,
t ≥ 0, then
H(t) ≤ H(0)Eα(k1tα
) + k2tα
Eα,α+1(k1tα
), t ≥ 0. (7)
where 0 < α < 1, Eα(·) and Eα,α+1(·) are one-parameter Mittag-Leffler function and two-parameter Mittag-Leffler function,
respectively.
It is worth noting that k1 > 0 is required in work [32]. In fact, Lemma 3 still holds when k1 ∈ R. The proof is similar
to Lemma 3 of the work [32].
Lemma 5 ([33]). If α < 2, β is an arbitrary real number, η is such that πα
2
< η < min{π, πα} and m > 0 is a real constant,
then
| Eα,β(z) |≤
m
1+ | z |
, η ≤| arg(z) |≤ π, | z |≥ 0. (8)
The model of the typical WPT system is depicted in Fig. 1. E denotes DC power supply. The left LP CP resonant circuit is
called primary side and the right LS CS resonant circuit is called secondary side. LP , CP , RP and LS , CS , RS denote inductance,
capacitance and parasitic resistance of the primary side and the secondary side, respectively. RL denotes resistance of load.
MPS denotes mutual inductance. Besides, ip, up and is, us denote current and capacitance voltage of the primary side and
the secondary side, respectively.
Based on circuit theory, differential equations of the WPT system are listed as
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
s(t)E = up(t) + ip(t)Rp + Lp
dip
dt
+ Mps
dis
dt
0 = Ls
dis
dt
+ Mps
dip
dt
+ is(t)Rs + us + isRL
Cp
dup
dt
= ip(t)
Cs
dus
dt
= is(t)
(9)
Let vector x(t) = [ip, up, is, us]T
, then corresponding state space model of the WPT system is described as
ẋ(t) = Ax(t) + bi, i = 1, 2. (10)
3
4. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
where
A =
⎡
⎢
⎢
⎢
⎢
⎣
RpLs
M2
ps−LpLs
Ls
M2
ps−LpLs
(Rs+RL)Mps
LpLs−M2
ps
Mps
LpLs−M2
ps
1
Cp
0 0 0
RpMps
LpLs−M2
ps
Mps
LpLs−M2
ps
(Rs+RL)Lp
M2
ps−LpLs
Lp
M2
ps−LpLs
0 0 1
Cs
0
⎤
⎥
⎥
⎥
⎥
⎦
,
B =
[
ELs
LpLs−M2
ps
0
EMps
M2
ps−LpLs
0
]T
, s(t)=
{
1, nT ≤ t < nT + T
2
−1, nT + T
2
≤ t < (n + 1)T
,
bi = s(t)B =
{
b1 = B, nT ≤ t < nT + T
2
b2 = −B, nT + T
2
≤ t < (n + 1)T
.
T denotes the duty circle of the inverter. s(t) = 1 denotes that switch S1 and switch S4 are turned on while switch S2
and switch S3 are turned off. s(t) = −1 denotes that switch S1 and switch S4 are turned off while switch S2 and switch
S3 are turned on. n = 0, 1, 2, . . ..
On the basis of the analysis of the introduction part, one can see that the WPT system is the fractional-order piecewise
affine system. Hence the Caputo fractional-order model of the WPT system is established as
C
0 Dα
t x(t) = Ax(t) + bi, i = 1, 2. (11)
3. Main results
In this section, the boundedness and the domain of attraction with respect to the fractional-order WPT system and
the fractional-order system with periodically intermittent control are analyzed, respectively.
3.1. Boundedness analysis of the fractional-order WPT system
Theorem 1. The fractional-order WPT system (11) is bounded, and the trivial solution x(t) converges to the following compact
set
G = {x(t) ∈ R4
|∥ x(t) ∥≤
√
bT
1 b1m2
λmin(P)εγ1
} (12)
if there exist positive constants γ1, γ2, ε, a symmetric positive definite matrix P and matrices X1, X2, Y1, Y2, Z1 and Z2 with
appropriate dimensions satisfy the following conditions:
(i) (X1 + YT
1 + Y1 − NT
1 − N1 + Z1)/2 < 0, (13)
(ii) (X2 + YT
2 + Y2 − NT
2 − N2 + Z2)/2 < 0, (14)
(iii)
[
X1 Y1
YT
1 Z1
]
≥ 0,
[
X2 Y2
YT
2 Z2
]
≥ 0, (15)
where
N1 = −AT
P − PA − εP2
− γ1P, N2 = −AT
P − PA − εP2
− (γ1 + γ2)P.
Proof. Consider the following Lyapunov function
V = xT
(t)Px(t). (16)
Then, when t ∈ [nT, nT + T
2
), based on Lemma 1 we know that
C
0 Dα
t V ≤ (Ax + b1)T
Px + xT
P(Ax + b1)
= xT
(AT
P + PA)x + bT
1 Px + xT
Pb1.
(17)
Next, by Lemma 2 we can get that
C
0 Dα
t V ≤ xT
(AT
P + PA)x + εxT
P2
x +
1
ε
bT
1 b1
= −γ1V + xT
(AT
P + PA + εP2
+ γ1P)x +
1
ε
bT
1 b1
≤ −γ1V +
1
ε
bT
1 b1.
(18)
4
5. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
Then applying Lemma 3, and let N1 = −AT
P − PA − εP2
− γ1P, it yields
xT
(AT
P + PA + εP2
+ γ1P)x =
1
2
[−2xT
(−AT
P − PA − εP2
− γ1P)x]
=
1
2
(−2xT
N1x)
≤
1
2
[
x
x
]T [
X1 Y1 − N1
YT
1 − NT
1 Z1
] [
x
x
]
=
1
2
xT
[X1 + YT
1 + Y1 − NT
1 − N1 + Z1]x
(19)
Similarly, when t ∈ [nT + T
2
, (n + 1)T), we obtain that
C
0 Dα
t V ≤ xT
(AT
P + PA)x + bT
2 Px + xT
Pb2
= −(γ1 + γ2)V + xT
(AT
P + PA + εP2
+ (γ1 + γ2)P)x +
1
ε
bT
2 b2
≤ −(γ1 + γ2)V +
1
ε
bT
2 b2.
(20)
Let N2 = −AT
P − PA − εP2
− (γ1 + γ2)P, and by Lemma 3 we have
xT
(AT
P + PA + εP2
+ (γ1 + γ2)P)x =
1
2
[ − 2xT
( − AT
P − PA − εP2
− (γ1 + γ2)P)x]
=
1
2
(−2xT
N2x)
≤
1
2
[
x
x
]T [
X2 Y2 − N2
YT
2 − NT
2 Z2
] [
x
x
]
=
1
2
xT
[X2 + YT
2 + Y2 − NT
2 − N2 + Z2]x.
(21)
Then, when t ∈ [0, T
2
), by using Lemma 4 we have
V(t) ≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
), (22)
and
V(
T
2
) ≤ V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
). (23)
Considering the memory property of fractional calculus, when t ∈ [T
2
, T), using Lemma 4, and combining (22), (23)
and the condition b1 = −b2, we have
V(t) ≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) + V(
T
2
)Eα([−(γ1 + γ2) − (−γ1)]
· (t −
T
2
)α
) +
1
ε
(bT
2 b2 − bT
1 b1)(t −
T
2
)α
Eα,α+1([−(γ1 + γ2) − (−γ1)](t −
T
2
)α
)
≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1
· (
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(t −
T
2
)α
),
(24)
and
V(T) ≤ V(0)Eα(−γ1Tα
) +
1
ε
bT
1 b1Tα
Eα,α+1(−γ1Tα
) + [V(0)Eα(−γ1(
T
2
)α
)
+
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(
T
2
)α
).
(25)
Similarly, when t ∈ [T, 3T
2
), applying Lemma 4, and combining (24), (25) and the condition b1 = −b2, it yields
V(t) ≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1
5
6. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
· (
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(t −
T
2
)α
) + V(T)Eα([−γ1 − (−γ1 − γ2)]
· (t − T)α
) +
1
ε
(bT
1 b1 − bT
2 b2)(t − T)α
Eα,α+1([−γ1 − (−γ1 − γ2)](t − T)α
)
≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1
· (
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(t −
T
2
)α
) + {V(0)Eα(−γ1Tα
) +
1
ε
bT
1 b1
· Tα
Eα,α+1(−γ1Tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]
· Eα(−γ2(
T
2
)α
)}Eα(γ2(t − T)α
),
(26)
and
V(
3T
2
) ≤ V(0)Eα(−γ1(
3T
2
)α
) +
1
ε
bT
1 b1(
3T
2
)α
Eα,α+1(−γ1(
3T
2
)α
) + [V(0)
· Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2Tα
) + {V(0)
· Eα(−γ1Tα
) +
1
ε
bT
1 b1Tα
Eα,α+1(−γ1Tα
) + [V(0)Eα(−γ1(
T
2
)α
)
+
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(
T
2
)α
)}Eα(γ2(
T
2
)α
).
(27)
When t ∈ [3T
2
, 2T), using Lemma 4, and combining (26), (27) and the condition b1 = −b2, we obtain that
V(t) ≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1
· (
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(t −
T
2
)α
) + {V(0)Eα(−γ1Tα
) +
1
ε
bT
1 b1
· Tα
Eα,α+1(−γ1Tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]
· Eα(−γ2(
T
2
)α
)}Eα(γ2(t − T)α
) + V(
3T
2
)Eα((−(γ1 + γ2) − (−γ1))(t −
3T
2
)α
)
+
1
ε
(bT
2 b2 − bT
1 b1)(t −
3T
2
)α
Eα,α+1([−(γ1 + γ2) − (−γ1)](t −
3T
2
)α
)
≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1
· (
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(t −
T
2
)α
) + {V(0)Eα(−γ1Tα
) +
1
ε
bT
1 b1
· Tα
Eα,α+1(−γ1Tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]
· Eα(−γ2(
T
2
)α
)}Eα(γ2(t − T)α
) + {V(0)Eα(−γ1(
3T
2
)α
) +
1
ε
bT
1 b1(
3T
2
)α
· Eα,α+1(−γ1(
3T
2
)α
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]
· Eα(−γ2Tα
) + {V(0)Eα(−γ1Tα
) +
1
ε
bT
1 b1Tα
Eα,α+1(−γ1Tα
) + [V(0)
· Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(
T
2
)α
)}Eα(γ2(
T
2
)α
)}
· Eα(−γ2(t −
3T
2
)α
),
(28)
6
7. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
and
V(2T) ≤ V(0)Eα(−γ1(2T)α
) +
1
ε
bT
1 b1(2T)α
Eα,α+1(−γ1(2T)α
) + [V(0)Eα(−γ1(
T
2
)α
)
+
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(
3T
2
)α
) + {V(0)Eα(−γ1Tα
) +
1
ε
bT
1 b1
· Tα
Eα,α+1(−γ1Tα
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]
· Eα(−γ2(
T
2
)α
)}Eα(γ2Tα
) + {V(0)Eα(−γ1(
3T
2
)α
) +
1
ε
bT
1 b1(
3T
2
)α
· Eα,α+1(−γ1(
3T
2
)α
) + [V(0)Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]
· Eα(−γ2Tα
) + {V(0)Eα(−γ1Tα
) +
1
ε
bT
1 b1Tα
Eα,α+1(−γ1Tα
) + [V(0)
· Eα(−γ1(
T
2
)α
) +
1
ε
bT
1 b1(
T
2
)α
Eα,α+1(−γ1(
T
2
)α
)]Eα(−γ2(
T
2
)α
)}Eα(γ2(
T
2
)α
)}
· Eα(−γ2(
T
2
)α
).
(29)
In the following, inductive method is used to analyze.
When t ∈ [nT, nT + T
2
), we have
V(t) ≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) + V(
T
2
)Eα(−γ2(t −
T
2
)α
)
+ V(T)Eα(γ2(t − T)α
) + V(
3T
2
)Eα(−γ2(t −
3T
2
)α
) + V(2T)Eα(γ2(t − 2T)α
)
+ · · · + V(nT)Eα(γ2(t − nT)α
)
= V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) +
2n
∑
k=1
V(
kT
2
)Eα((−1)k
γ2(t −
kT
2
)α
).
(30)
When t ∈ [nT + T
2
, (n + 1)T), we can get that
V(t) ≤ V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) + V(
T
2
)Eα(−γ2(t −
T
2
)α
)
+ V(T)Eα(γ2(t − T)α
) + V(
3T
2
)Eα(−γ2(t −
3T
2
)α
) + V(2T)Eα(γ2(t − 2T)α
)
+ · · · + V(nT)Eα(γ2(t − nT)α
) + V(nT +
T
2
)Eα( − γ2(t − (nT +
T
2
))α
)
= V(0)Eα(−γ1tα
) +
1
ε
bT
1 b1tα
Eα,α+1(−γ1tα
) +
2n+1
∑
k=1
V(
kT
2
)Eα((−1)k
γ2(t −
kT
2
)α
).
(31)
Therefore, for any t ≥ 0, by Lemma 5, we can obtain that
∥ x(t) ∥ ≤
√
V(0) | Eα(−γ1tα) |
λmin(P)
+
√
1
ε
bT
1 b1tα | Eα,α+1(−γ1tα) |
λmin(P)
+
√∑2n+1
k=1 V(kT
2
) | Eα((−1)kγ2(t − kT
2
)α) |
λmin(P)
≤
√
V(0)m1
λmin(P)(1 + γ1tα)
+
√
1
ε
bT
1 b1tαm2
λmin(P)(1 + γ1tα)
+
√ 1
λmin(P)
·
2n+1
∑
k=1
V(kT
2
)m3
1 + γ2 | (t − kT
2
)α |
.
(32)
7
8. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
Besides, when t → ∞, we can know that
∥ x(t) ∥ ≤
√
bT
1 b1m2
λmin(P)εγ1
. (33)
Hence the system (11) is bounded. This completes the proof.
Remark 1. From (33), we can know that when time tends to infinity, V = xT
Px ≤
bT
1
b1m2
εγ1
. Thus the domain of attraction
of the system (11) can be estimated by this inequality.
Remark 2. During the derivation, some matrix variables by Lemma 3 are introduced. Conservatism of the obtained result
is improved in theory. This point can be seen in the analysis of later examples.
Especially, when γ1 = γ2 = γ , the following Corollary can be obtained.
Corollary 1. The fractional-order WPT system (11) is bounded, and the trivial solution x(t) converges to the following compact
set
G = {x(t) ∈ R4
|∥ x(t) ∥≤
√
bT
1 b1m2
λmin(P)εγ
} (34)
if there exist positive constants γ , ε, a symmetric positive definite matrix P and matrices X1, X2, Y1, Y2, Z1 and Z2 with
appropriate dimensions satisfy the following conditions:
(i) (X1 + YT
1 + Y1 − NT
1 − N1 + Z1)/2 < 0, (35)
(ii) (X2 + YT
2 + Y2 − NT
2 − N2 + Z2)/2 < 0, (36)
(iii)
[
X1 Y1
YT
1 Z1
]
≥ 0,
[
X2 Y2
YT
2 Z2
]
≥ 0, (37)
where
N1 = −AT
P − PA − εP2
− γ P, N2 = −AT
P − PA − εP2
− 2γ P.
3.2. Boundedness analysis of the fractional-order system with periodically intermittent control
To without loss of generality, consider the following fractional-order system with periodically intermittent control
C
0 Dα
t x(t) =
{
(A + K)x(t) + d, nT ≤ t < nT + τ,
Ax(t) + d, nT + τ ≤ t < (n + 1)T,
(38)
where d is an external bias vector. n = 0, 1, 2, . . ..
Theorem 2. The fractional-order system (38) is bounded, and the trivial solution x(t) converges to the following compact set
G = {x(t) ∈ Rl
|∥ x(t) ∥≤
√
dT dm2
λmin(P)εγ1
} (39)
if there exist positive constants γ1, γ2, ε, a symmetric positive definite matrix P, and matrices K, X1, X2, Y1, Y2, Z1 and Z2 with
appropriate dimensions satisfy the following conditions:
(i) (X1 + YT
1 + Y1 − NT
1 − N1 + Z1)/2 < 0, (40)
(ii) (X2 + YT
2 + Y2 − NT
2 − N2 + Z2)/2 < 0, (41)
(iii)
[
X1 Y1
YT
1 Z1
]
≥ 0,
[
X2 Y2
YT
2 Z2
]
≥ 0 (42)
where N1 = −PA − AT
P − Q − Q T
− εP2
+ γ1P, N2 = −AT
P − PA − εP2
+ (γ1 + γ2)P, Q = PK.
8
9. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
Proof. The process of the proof is similar to Theorem 1. Choosing the same Lyapunov function as Theorem 1. When
t ∈ [nT, nT + τ), by Lemmas 1 and 2 we get that
C
0 Dα
t V ≤ (xT
AT
+ xT
KT
+ dT
)Px + xT
P(Ax + Kx + d)
= xT
(PA + AT
P + PK + KT
P)x + dT
Px + xT
Pd
≤ xT
(PA + AT
P + PK + KT
P + εP2
)x +
1
ε
dT
d
= γ1V + xT
(PA + AT
P + PK + KT
P + εP2
− γ1P)x +
1
ε
dT
d
≤ γ1V +
1
ε
dT
d.
(43)
Then, let N1 = −PA − AT
P − PK − KT
P − εP2
+ γ1P, and by Lemma 3 we obtain that
xT
(PA + AT
P + PK + KT
P + εP2
− γ1P)x
=
1
2
[−2xT
(−PA − AT
P − PK − KT
P − εP2
+ γ1P)x]
=
1
2
(−2xT
N1x)
≤
1
2
[
x
x
]T [
X1 Y1 − N1
YT
1 − NT
1 Z1
] [
x
x
]
=
1
2
xT
[X1 + YT
1 + Y1 − NT
1 − N1 + Z1]x, (44)
when t ∈ [nT + τ, (n + 1)T), we have
C
0 Dα
t V ≤ (xT
AT
+ dT
)Px + xT
P(Ax + d)
= xT
(PA + AT
P)x + dT
Px + xT
Pd
≤ xT
(PA + AT
P + εP2
)x +
1
ε
dT
d
= (γ1 + γ2)V + xT
(PA + AT
P + εP2
− (γ1 + γ2)P)x +
1
ε
dT
d
≤ (γ1 + γ2)V +
1
ε
dT
d.
(45)
Similarly, let N2 = −AT
P − PA − εP2
+ (γ1 + γ2)P, and applying Lemma 3, it yields
xT
(AT
P + PA + εP2
− (γ1 + γ2)P)x =
1
2
[ − 2xT
( − AT
P − PA − εP2
+ (γ1 + γ2)P)x]
=
1
2
(−2xT
N2x)
≤
1
2
[
x
x
]T [
X2 Y2 − N2
YT
2 − NT
2 Z2
] [
x
x
]
=
1
2
xT
[X2 + YT
2 + Y2 − NT
2 − N2 + Z2]x.
(46)
Then, when n = 0, namely, t ∈ [0, T). For t ∈ [0, τ), using Lemma 4, we have
V(t) ≤ V(0)Eα(γ1tα
) +
1
ε
dT
dtα
Eα,α+1(γ1tα
), (47)
and
V(τ) ≤ V(0)Eα(γ1τα
) +
1
ε
dT
dτα
Eα,α+1(γ1τα
). (48)
When t ∈ [τ, T), utilizing Lemma 4 and combining and (47) and (48), we obtain that
V(t) ≤ V(0)Eα(γ1tα
) +
1
ε
dT
dtα
Eα,α+1(γ1tα
) + V(τ)Eα((γ1 + γ2 − γ1)(t − τ)α
)
+
1
ε
(dT
d − dT
d)(t − τ)α
Eα,α+1((γ1 + γ2 − γ1)(t − τ)α
)
≤ V(0)Eα(γ1tα
) +
1
ε
dT
dtα
Eα,α+1(γ1tα
) + (V(0)Eα(γ1τα
) +
1
ε
dT
dτα
· Eα,α+1(γ1τα
))Eα(γ2(t − τ)α
),
(49)
9
12. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
≤
√
V(0)m1
λmin(P)(1 + γ1tα)
+
√
1
ε
dT dtαm2
λmin(P)(1 + γ1tα)
+
√ 1
λmin(P)
·
n
∑
k=0
V(kT + τ)m3
1 + γ2 | (t − (kT + τ))α |
+
√ 1
λmin(P)
·
n
∑
k=1
V(kT)m4
1 + γ2 | (t − kT)α |
.
(57)
Consequently, when t → ∞, we have
∥ x(t) ∥ ≤
√
dT dm2
λmin(P)εγ1
. (58)
Thus the system (38) is bounded. It completes the proof.
Remark 3. Similarly, the domain of attraction of the fractional-order system with periodically intermittent control can be
estimated via the compact set S = {x(t) ∈ Rn
| V = xT
Px ≤ dT dm2
εγ1
}. Furthermore, the Theorem is appropriate not only for
the case τ = T
2
, but for other cases τ ∈ (0, T).
Additionally, if d = 0, then we can obtain the following fractional-order system with periodically intermittent control.
C
0 Dα
t x(t) =
{
(A + K)x(t), nT ≤ t < nT + τ,
Ax(t), nT + τ ≤ t < (n + 1)T,
(59)
where n = 0, 1, 2, . . ..
Thereupon, from the Theorem 2, relevant stability result can be derived as follow.
Corollary 2. The fractional-order system (59) is stable if there exist positive constants γ1, γ2, a symmetric positive definite
matrix P, and matrices K, X1, X2, Y1, Y2, Z1 and Z2 with appropriate dimensions satisfy the following conditions:
(i) (X1 + YT
1 + Y1 − NT
1 − N1 + Z1)/2 < 0, (60)
(ii) (X2 + YT
2 + Y2 − NT
2 − N2 + Z2)/2 < 0, (61)
(iii)
[
X1 Y1
YT
1 Z1
]
≥ 0,
[
X2 Y2
YT
2 Z2
]
≥ 0 (62)
where N1 = −PA − AT
P − Q − Q T
+ γ1P, N2 = −PA − AT
P + (γ1 + γ2)P, Q = PK.
Remark 4. From the Corollary, the stability conditions of the fractional-order system with periodically intermittent control
are obtained. Moreover, the cost of time and control can be also saved by the control method.
4. Examples and numerical simulations
Example 1. In order to demonstrate the effectiveness of the Theorem 1, relevant parameters of the WPT system are
adopted as follows:
E = 10 V, Cp = Cs = 4 µF, Rp = Rs = 0.15 , Lp = Ls = 90 µH, Mps = 30 µH, RL = 25 , T = 0.00012 s. Besides,
taking α = 1.01, ε = 100, m2 = 1, γ1 = 200, γ2 = 300.
And then, using linear matrix inequality (LMI) tool box to solve conditions (i), (ii) and (iii) of the Theorem 1, we get a
feasible solution of the matrix P as
P =
⎡
⎢
⎣
3.9701 −0.0401 1.6476 1.7081
−0.0401 0.2497 −5.6865 −0.2294
1.6476 −5.6865 432.8121 10.7114
1.7081 −0.2294 10.7114 6.6787
⎤
⎥
⎦.
In addition, Fig. 2 shows the phase plot of the fractional-order WPT system, from which one can see that the WPT
system is bounded. In the meantime, x(t) converges to the compact set
G = {x(t) ∈ R4
|∥ x(t) ∥≤
√
bT
1 b1m2
λmin(P)εγ1
= 2235.496}. (63)
12
13. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
Fig. 2. Phase plot of the fractional-order WPT system.
Fig. 3. Domain of attraction of the fractional-order WPT system.
Hence Theorem 1 is effective. Besides, from Remark 1, relevant two-dimensional projection of the domain of attraction
for the fractional-order WPT system is estimated and shown in Fig. 3. Consider the conservatism of the Theorem, Lemma 3
is used in this paper. From this picture, and by comparison, it can easily be seen that the domain of attraction obtained
by utilizing the inequality technique of Lemma 3 is larger. It indicates that the conservatism of the result is improved in
theory.
Remark 5. Via the above analysis, we know that the conservatism of the result is improved obviously. Nevertheless, from
the practical point of view, the conservatism of the system remains. Therefore, here we discuss it in two cases.
Case 1: The feasible solution exists in Theorem 1, which the domain of attraction can be estimated by this Theorem.
Thereby the robustness of the system can also be reflected though the domain of attraction.
Case 2: The feasible solution cannot be found by the Theorem, one can use state response to evaluate the robustness of the
system. First of all, performance indexes such as overshoot and regulation time can be obtained by the state response plot;
Secondly, construct a weight function about these performance indexes; Finally, evaluate the robustness of the system
by this function. Besides, one can utilize the robust control such as H∞ control and µ synthesis to increase the domain
of attraction and improve the robustness of the system.
13
14. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
Fig. 4. The linear fractional transformation system with uncertainties and controller.
Fig. 5. Phase plot of the system (38) without control.
Remark 6. In practice, it is not comprehensive to evaluate the robustness of the system only from the perspective of the
domain of attraction. The performance indexes such as overshoot and regulation time should also be considered so as to
fully evaluate the robustness of the system. Besides, uncertainties exist in reality. Thus, one can evaluate the robustness
of the system by means of the structured singular value µ analysis and the state response. Main procedure are as follows:
Step 1. Obtain the linear fractional transformation (Fig. 4) representation Q via the closed-loop system model. In Fig. 4,
△ denotes the set of the model uncertainties; G denotes the transfer function; H denotes the controller;
Step 2. Calculate the maximum structured singular value ρ of Q by using structured singular value theory, and obtain
stability boundary 1
ρ
. This boundary is similar to boundary of the domain of attraction in this paper;
Step 3. Obtain overshoot δ and regulation time ts via the state response plot;
Step 4. Construct a weight function H( 1
ρ
, δ, ts), and evaluate the robustness of the system according to given conditions.
Note that we only list the main steps here due to this topic will be investigated in the next paper.
Example 2. For the system (38), we take
A =
[
0.6 −0.1
0.1 0.3
]
, d =
[
2
2
]
, x(0) =
[
0.1 0.1
]T
. Besides, choosing α = 0.8, T = 2.5s, τ = 2s, γ1 = 1, γ2 = 1.5,
ε = 0.1, m2 = 1. Via solving the linear matrix inequalities (40), (41) and (42) of the Theorem 2. The feasible solution of
the matrices P and K is obtained as P =
[
0.0053 −0.0001
−0.0001 0.0064
]
, K =
[
−0.8757 −2.1755
1.8113 −0.7940
]
, respectively.
Phase plot of the system without intermittent control and with intermittent control is depicted in Fig. 5 and Fig. 6,
respectively. The Fig. 5 shows that the system (38) without control is of instability. But from Fig. 6, we can easily see that
the system (38) with periodically intermittent control is stable, which indicates that the designed controller is effective.
14
15. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
Fig. 6. Phase plot of the system (38) with periodically intermittent control.
Fig. 7. Domain of attraction of the system (38) with periodically intermittent control.
Additionally, substituting relevant parameters into (39), we can get that
G = {x(t) ∈ R2
|∥ x(t) ∥≤
√
dT dm2
λmin(P)εγ1
= 89.8933}. (64)
Meantime, by Remark 3, the corresponding domain of attraction is estimated and presented in Fig. 7, from which it is not
difficult to see that the conservatism of the result is reduced because of the domain of attraction becomes larger.
Remark 7. From this example, it is easy to find that the derived results can not only analyze the boundedness of the
fractional-order piecewise affine system, but also estimate the domain of attraction of the system.
5. Conclusion
The problem of the domain of attraction for the fractional-order WPT system by employing the Lyapunov function
approach and the inductive method was studied in this paper. The relevant results were derived, and further verified by
15
16. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
a few examples, respectively. Simulation shows that the obtained results are valid, and less conservatism. Furthermore,
the designed controller also presents the good control effect. However, uncertainties are not considered in this paper, and
the issue of the robustness evaluation has not been fully addressed, which would be worthwhile for a further research.
CRediT authorship contribution statement
Zhongming Yu: Writing - original draft, Visualization, Investigation, Software, Formal analysis, Resources. Yue Sun:
Conceptualization, Methodology, Data curation. Xin Dai: Funding acquisition, Supervision, Validation, Writing - review &
editing, Project administration.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to thank the associate editor and the anonymous reviewers for their suggestions and comments.
This work was supported by National Natural Science Foundation of China under Grant 62073047.
References
[1] P. Raval, D. Kacprzak, A. Hu, A ICPT system for low power electronics charging applications, in: 2011 6th IEEE Conference on Industrial
Electronics and Applications, 2011, pp. 520–525.
[2] S. Lee, J. Huh, C. Park, et al., On-line electric vehicle using inductive power transfer system, in: Energy Conversion Congress and Exposition,
2010, pp. 1598–1601.
[3] C. Wang, O. Stielau, G. Covic, Design considerations for a contactless electric vehicle battery charger, IEEE Trans. Ind. Electron. 52 (5) (2005)
1308–1314.
[4] H. Jiang, J. Zhang, D. Lan, K. Chao, H. Shahnasser, A low-frequency versatile wireless power transfer technology for biomedical implants, IEEE
Trans. Biomed. Circuits Syst. 7 (4) (2013) 526–535.
[5] W. Niu, J. Chu, W. Gu, et al., Exact analysis of frequency splitting phenomena of contactless power transfer systems, IEEE Trans. Circuits Syst.
I. Regul. Pap. 60 (6) (2013) 1670–1677.
[6] Y. Zhang, Z. Zhao, K. Chen, Frequency-splitting analysis of four-coil resonant wireless power transfer, IEEE Trans. Ind. Appl. 50 (4) (2014)
2436–2445.
[7] A. Radwan, A. Emira, A. Abdelaty, Modeling and analysis of fractional order DC-DC converter, Isa Trans. 82 (2018) 184–199.
[8] A. Anis, T. Freeborn, A. Elwakil, Review of fractional-order electrical characterization of supercapacitors, J. Power Sources 400 (2018) 457–467.
[9] D. Baleanu, A. Golmankhaneh, A. Golmankhaneh, On electromagnetic field in fractional space, Nonlinear Anal. RWA 11 (1) (2010) 288–292.
[10] M. Ortigueira, C. Matos, M. Piedade, Fractional discrete-time signal processing: Scale conversion and linear prediction, Nonlinear Dynam. 29
(1–4) (2002) 173–190.
[11] V. Martynyuk, M. Ortigueira, Fractional model of an electrochemical capacitor, Signal Process. 107 (2014) 355–360.
[12] V. Martynyuk, M. Ortigueira, M. Fedula, O. Savenko, Methodology of electrochemical capacitor quality control with fractional order model,
AEU-Int. J. Electron. Commun. 91 (2018) 118–124.
[13] S. Majidabad, H. Shandiz, A. Hajizadeh, Nonlinear fractional-order power system stabilizer for multi-machine power systems based on sliding
mode technique, Internat. J. Robust Nonlinear Control 25 (2015) 1548–1568.
[14] T. Freeborn, B. Maundy, A. Elwakil, Fractional-order models of supercapacitors, batteries and fuel cells: a survey, Mater. Renew. Sustain. Energy
4 (9) (2015) 1–7.
[15] S. Westerlund, L. Ekstam, Capacitor. theory, Capacitor theory, IEEE Trans. Dielectr. Electr. Insul. 1 (1994) 826–839.
[16] T. Hartley, J. Trigeassou, C. Lorenzo, N. Maamri, Energy storage and loss in fractional-order systems, Circuits Devices Syst. Iet 9 (3) (2015)
227–235.
[17] J. Trigeassou, N. Maamri, A. Oustaloup, Lyapunov stability of commensurate fractional order systems: a physical interpretation, J. Comput.
Nonlinear Dyn. 11 (2016) 051007-1-051007-8.
[18] J. Trigeassou, M. Nezha, O. Alain, Lyapunov stability of noncommensurate fractional order systems: an energy balance approach, J. Comput.
Nonlinear Dyn. 11 (2016) 041007-1-041007-9.
[19] S. Wang, Y. Yu, G. Wen, Hybrid projective synchronization of time-delayed fractional order chaotic systems, Nonlinear Anal. Hybrid Syst. 11
(2014) 129–138.
[20] S. Zhang, Y. Yu, H. Wang, Mittag-Leffler stability of fractional-order Hopfield neural networks, Nonlinear Anal. Hybrid Syst. 16 (2015) 104–121.
[21] Y. Tan, M. Xiong, D. Du, S. Fei, Observer-based robust control for fractional-order nonlinear uncertain systems with input saturation and
measurement quantization, Nonlinear Anal. Hybrid Syst. 34 (2019) 45–57.
[22] X. Shu, B. Zhang, The effect of fractional orders on the transmission power and efficiency of fractional-order wireless power transmission
system, Energies 11 (7) (2018) 1–9.
[23] Z. Yu, Y. Sun, X. Dai, Z. Ye, Stability and control of uncertain ICPT system considering time-varying delay and stochastic disturbance, Internat.
J. Robust Nonlinear Control 29 (18) (2019) 6582–6604.
[24] C. Pittet, Stability regions for linear systems with saturating controls via circle and popov criteria, in: IEEE Conference on Decision and Control,
2015, pp. 4518–4523.
[25] J. Goncalves, A. Megretski, M. Dahleh, Global analysis of piecewise linear systems using impact maps and surface Lyapunov functions, IEEE
Trans. Automat. Control 48 (12) (2003) 2089–2106.
[26] Y. Lim, K. Oh, H. Ahn, Stability and stabilization of fractional-order linear systems subject to input saturation, IEEE Trans. Automat. Control 58
(4) (2013) 1062–1067.
[27] D. Baleanu, S. Rezapour, H. Mohammadi, Some existence results on nonlinear fractional differential equations, Phil. Trans. R. Soc. A 371 (1990)
(2013) 1–7.
16
17. Z. Yu, Y. Sun and X. Dai Nonlinear Analysis: Hybrid Systems 42 (2021) 101062
[28] S.J. Sadati, D. Baleanu, A. Ranjbar, R. Ghaderi, T. Abdeljawad, Mittag-Leffler stability theorem for fractional nonlinear systems with delay, Abstr.
Appl. Anal. 2011 (12) (2014) 331–336.
[29] M. Duarte-Mermoud, N. Aguila-Camacho, J. Gallegos, R. Castro-Linares, Using general quadratic Lyapunov functions to prove Lyapunov uniform
stability for fractional order systems, Commun. Nonlinear Sci. Numer. Simul. 22 (1–3) (2015) 650–659.
[30] P. Khargonekar, I. Petersen, K. Zhou, Robust stabilization of uncertain linear systems: Quadratic stabilizability and H∞ control theory, IEEE
Trans. Automat. Control 35 (3) (1990) 356–361.
[31] Y. Moon, P. Park, W. Kwon, Y. Lee, Delay-dependent robust stabilization of uncertain state-delayed systems, Internat. J. Control 74 (14) (2001)
1447–1455.
[32] A. Wu, Z. Zeng, Boundedness, Mittag-Leffler stability and asymptotical ω-periodicity of fractional-order fuzzy neural networks, Neural Netw.
74 (2015) 73–84.
[33] X. Wen, Z. Wu, J. Lu, Stability analysis of a class of nonlinear fractional-order systems, IEEE Trans. Circuits Syst. II Express Briefs 55 (11) (2008)
1178–1182.
17