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# Event-driven boundary control for time fractional diffusion systems under time-varying input disturbance

**Conference Paper**· June 2018

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DOI: 10.23919/ACC.2018.8431000

Cite this publicationConference: 2018 Annual American Control Conference (ACC)

- ... In our previous work [17] , the event-driven boundary state feedback control problem for time fractional reaction-diffusion systems with Dirichlet boundary conditions has been studied. The present paper is an extension of [17] , where we propose both the state and the output feedback event-triggered boundary control strategies to stabilize the subdiffusion system at hand. ...... In our previous work [17] , the event-driven boundary state feedback control problem for time fractional reaction-diffusion systems with Dirichlet boundary conditions has been studied. The present paper is an extension of [17] , where we propose both the state and the output feedback event-triggered boundary control strategies to stabilize the subdiffusion system at hand. It is shown that both two kinds of event-triggered controllers could significantly asymptotically stabilize the estimation. ...Article
- Oct 2018
- INFORM SCIENCES

This paper is concerned with the event-triggered boundary feedback control problems for networked reaction-subdiffusion processes governed by time fractional reaction-diffusion systems with unknown time-varying input uncertainties over sensor/actuator networks. The event-triggered boundary state feedback controller is first designed and implemented via backstepping technique. Moreover, we realize that the availability of full-state measurements in many practical applications may be impossible due to the difficulties in measuring. To solve this limitation, we design an extended Luenberger observer that embeds within the networked sensor to estimate the whole states of the systems under consideration. Based on this, the boundary output feedback event-triggered implementation of the studied system in the context of sensor/actuator networks is then proposed. It is shown that both two kinds of event-triggered strategies could significantly asymptotically stabilize the estimation with the Zeno phenomenon being excluded. Two numerical illustrations are finally included.

- Article
- Nov 2001

We consider the coagulation dynamics A+A -> A and the annihilation dynamics A+A -> 0 for particles moving subdiffusively in one dimension, both on a lattice and in a continuum. The analysis combines the "anomalous kinetics" and "anomalous diffusion" problems, each of which leads to interesting dynamics separately and to even more interesting dynamics in combination. We calculate both short-time and long-time concentrations, and compare and contrast the continuous and discrete cases. Our analysis is based on the fractional diffusion equation and its discrete analog. - Book
- Jan 2018

This monograph provides an accessible introduction to the regional analysis of fractional diffusion processes. It begins with background coverage of fractional calculus, functional analysis, distributed parameter systems and relevant basic control theory. New research problems are then defined in terms of their actuation and sensing policies within the regional analysis framework. The results presented provide insight into the control-theoretic analysis of fractional-order systems for use in real-life applications such as hard-disk drives, sleep stage identification and classification, and unmanned aerial vehicle control. The results can also be extended to complex fractional-order distributed-parameter systems and various open questions with potential for further investigation are discussed. For instance, the problem of fractional order distributed-parameter systems with mobile actuators/sensors, optimal parameter identification, optimal locations/trajectory of actuators/sensors and regional actuation/sensing configurations are of great interest. The book’s use of illustrations and consistent examples throughout helps readers to understand the significance of the proposed fractional models and methodologies and to enhance their comprehension. The applications treated in the book run the gamut from environmental science to national security. Academics and graduate students working with cyber-physical and distributed systems or interested in the the applications of fractional calculus will find this book to be an instructive source of state-of-the-art results and inspiration for further research. - ArticleFull-text available
- Jan 2017

We investigate the dynamics of water confined in soft ionic nano-structures, an issue critical for a general understanding of the multi-scale structure-function interplay in advanced materials. We focus on hydrated perfluoro-sulfonic acid polymers employed as electrolytes in fuel cells. These materials form phase-separated morphologies that show outstanding proton-conducting properties, directly related to the state and dynamics of the absorbed water. We quantify water motion and ion transport by combining Quasi Elastic Neutron Scattering, Pulsed Field Gradient Nuclear Magnetic Resonance, and Molecular Dynamics computer simulation. Effective water and ion diffusion coefficients are determined at the relevant atomic, nanoscopic and macroscopic scales, together with their variation upon hydration. We demonstrate that confinement at the nano-scale and direct interaction with the charged interfaces produce anomalous sub-diffusion within the ionic nano-channels, irrespective to the details of the chemistry of the hydrophobic matrix. - ArticleFull-text available
- Oct 2015

Cyber-physical systems (CPSs) are man-made complex systems coupled with natural processes that, as a whole, should be described by distributed parameter systems (DPSs) in general forms. This paper presents three such general models for generalized DPSs that can be used to characterize complex CPSs. These three different types of fractional operators based DPS models are: fractional Laplacian operator, fractional power of operator or fractional derivative. This research investigation is motivated by many fractional order models describing natural, physical, and anomalous phenomena, such as sub-diffusion process or super-diffusion process. The relationships among these three different operators are explored and explained. Several potential future research opportunities are then articulated followed by some conclusions and remarks. - Article
- Jan 2016
- IEEE T IND INFORM

This paper provides an overview and makes a deep investigation on sampled-data-based event-triggered control and filtering for networked systems. Compared with some existing event-triggered and self-triggered schemes, a sampled-data-based event-triggered scheme can ensure a positive minimum interevent time and make it possible to jointly design suitable feedback controllers and event-triggered threshold parameters. Thus, more attention has been paid to the sampled-data-based event-triggered scheme. A deep investigation is first made on the sampled-data-based event-triggered scheme. Then recent results on sampled-data-based event-triggered state feedback control, dynamic output feedback control, H∞ filtering for networked systems are surveyed and analyzed. An overview on sampleddata- based event-triggered consensus for distributed multi-agent systems is given. Finally, some challenging issues are addressed to direct the future research. - Article
- Jan 2016
- IEEE T AUTOMAT CONTR

In this paper, the problem of event-trigger based adaptive control for a class of uncertain nonlinear systems is considered. The nonlinearities of the system are not required to be globally Lipschitz. Since the system contains unknown parameters , it is a difficult task to check the assumption of the input-to-state stability (ISS) with respect to the measurement errors, which is required in most existing literature. To solve this problem, we design both the adaptive controller and the triggering event at the same time such that the ISS assumption is no longer needed. In addition to presenting new design methodologies based on the fixed threshold strategy and relative threshold strategy, we also propose a new strategy named the switching threshold strategy. It is shown that the proposed control schemes guarantee that all the closed-loop signals are globally bounded and the tracking/stabilization error exponentially converges towards a compact set which is adjustable. - Article
- Mar 2016
- IET CONTROL THEORY A

In this study, the authors attempt to explore the boundary feedback stabilisation for an unstable heat process described by fractional-order partial differential equation (PDE), where the first-order time derivative of normal reaction-diffusion equation is replaced by a Caputo time fractional derivative of order α e (0, 1]. By designing an invertible coordinate transformation, the system under consideration is converted into a Mittag-Leffler stability linear system and the boundary stabilisation problem is transformed into a problem of solving a linear hyperbolic PDE. It is worth mentioning that with the help of this invertible coordinate transformation, they can explicitly obtain the closed-loop solutions of the original problem. The output feedback problem with both anti-collocated and collocated actuator/sensor pairs in one-dimensional domain is also presented. A numerical example is given to test the effectiveness of the authors' results. - Article
- Mar 2016
- INT J CONTROL

This paper is devoted to analyzing the actuator characterizations for the fractional sub-diffusion equation under consideration to become approximately controllable. Two different cases are considered, where the control inputs emerge in the differential equation as distributed inputs and as boundary inputs in the boundary conditions. The dual system for fractional sub-diffusion equation is solved and the necessary and sufficient conditions for the approximate controllability of the system are established. Several examples are worked out in the end to illustrate our results. - Book
- Jan 2015

This book reports on the latest advances in the study of Networked Control Systems (NCSs). It highlights novel research concepts on NCSs; the analysis and synthesis of NCSs with special attention to their networked character; self- and event-triggered communication schemes for conserving limited network resources; and communication and control co-design for improving the efficiency of NCSs. The book will be of interest to university researchers, control and network engineers, and graduate students in the control engineering, communication and network sciences interested in learning the core principles, methods, algorithms and applications of NCSs. - Book
- Nov 2014

As a result of researchers and scientists increasing interest in pure as well as applied mathematics in non-conventional models, particularly those using fractional calculus, Mittag-Leffler functions have recently caught the interest of the scientific community. Focusing on the theory of the Mittag-Leffler functions, the present volume offers a self-contained, comprehensive treatment, ranging from rather elementary matters to the latest research results. In addition to the theory the authors devote some sections of the work to the applications, treating various situations and processes in viscoelasticity, physics, hydrodynamics, diffusion and wave phenomena, as well as stochastics. In particular the Mittag-Leffler functions allow us to describe phenomena in processes that progress or decay too slowly to be represented by classical functions like the exponential function and its successors. The book is intended for a broad audience, comprising graduate students, university instructors and scientists in the field of pure and applied mathematics, as well as researchers in applied sciences like mathematical physics, theoretical chemistry, bio-mathematics, theory of control and several other related areas. - In this paper, the uncertainty and disturbance estimator (UDE)-based robust control is applied to the control of a class of nonaffine nonlinear systems. This class of systems is very general and covers a large range of nonlinear systems. However, the control of such systems is very challenging because the input variables are not expressed in an affine form, which leads to the failure of using feedback linearization. The proposed UDE-based control method avoids the inverse operator construction, which might result in the control singularity problem. Moreover, the general assumption on the uncertainty and disturbance term is relaxed, and only its bandwidth information is required for the control design. The asymptotic stability of the closed-loop system is established. The proposed approach is easy to be implemented and tuned while bringing very good robust performance. The important features and performance of the proposed approach are demonstrated through both simulation studies and experimental validation on a servo system with nonaffine uncertainties.
- In this paper, we are concerned with the boundary stabilization of a one-dimensional unstable heat equation with the external disturbance flowing into the control end. The active disturbance rejection control (ADRC) and the sliding mode control (SMC) are adopted in investigation. By the ADRC approach, the disturbance is estimated through an external observer and cancelled online by the approximated one in the closed-loop. It is shown that the external disturbance can be attenuated in the sense that the resulting closed-loop system under the extended state feedback tends to any arbitrary given vicinity of zero as the time goes to infinity. In the second part, we use the SMC to reject the disturbance with the assumption in which the disturbance is supposed to be bounded. The reaching condition, and the existence and uniqueness of the solution for all states in the state space via SMC are established. Simulation examples are presented for both control strategies.
- ArticleFull-text available
- Apr 2014

Water molecules play an important role in providing unique environments for biological reactions on cell membranes. It is widely believed that water molecules form bridges that connect lipid molecules and stabilize cell membranes. Using all-atom molecular dynamics simulations, we show that translational and rotational diffusion of water molecules on lipid membrane surfaces exhibit subdiffusion and aging. Moreover, we provide evidence that both divergent mean trapping time (continuous-time random walk) and long-correlated noise (fractional Brownian motion) contribute to this subdiffusion. These results suggest that subdiffusion on cell membranes causes the water retardation, an enhancement of cell membrane stability, and a higher reaction efficiency. - Article
- Jul 2017

In this paper, a novel robust control strategy named uncertainty and disturbance estimator (UDE) is applied to the stabilization of an unstable heat equation with Dirichlet type boundary actuator and unknown time-varying input disturbance. The system is stabilized by the backstepping approach and the unknown input disturbance is compensated by the UDE-based method which constructs an estimation of the disturbance through filtering the system input U(t) and boundary state u(1, t). Compared to other existing disturbance compensation methods, the UDE-based method only requires the spectrum information of the disturbance signal. Furthermore, the output feedback version of the proposed control is also derived for the practical implementation purpose. Stability analysis of the closed-loop system for both state feedback and output feedback cases are carried out and simulation examples are also provided to verify the proposed method. - Article
- Nov 2011
- NONLINEAR ANAL-THEOR

We consider a scalar fractional differential equation, write it as an integral equation, and construct several Lyapunov functionals yielding qualitative results about the solution. It turns out that the kernel is convex with a singularity and it is also completely monotone, as is the resolvent kernel. While the kernel is not integrable, the resolvent kernel is positive and integrable with an integral value of one. These kernels give rise to essentially different types of Lyapunov functionals. It is to be stressed that the Lyapunov functionals are explicitly given in terms of known functions and they are differentiated using Leibniz’s rule. The results are readily accessible to anyone with a background of elementary calculus. - This paper proposes a robust control strategy for uncertain LTI systems. The strategy is based on an uncertainty and disturbance estimator (UDE). It brings similar performance as the time-delay control (TDC). The advantages over TDC are: (i) no delay is introduced into the system; (ii) there are no oscillations in the control signal; and (iii) there is no need of measuring the derivatives of the state vector. The robust stability of LTI-SISO systems is analyzed, and simulations are given to show the effectiveness of the UDE-based control with a comparison made with TDC.
- Article
- Dec 2012
- IET CONTROL THEORY A

In this study, the event-triggered average consensus control for discrete-time multi-agent systems (MASs) is investigated. Based on a Lyapunov function, a sufficient condition is derived to give an event condition, which is designed based on the measurement error and the disagreement vector. The sufficient condition, described in terms of a linear matrix inequality (LMI), is easily solved by available LMI toolbox. Under this event condition, the event-triggered MAS reaches average consensus. Furthermore, the results are extended to the self-triggered consensus control, where the next task release time can be decided depending on the current sampled data. In addition, a certain restriction on the event condition is proposed in order to avoid Zeno-behaviour. Finally, two simulation examples illustrate the effectiveness of the theoretical results. - Conference Paper
- Jul 2009

This article focuses on dynamic output feedback and robust control of quasi linear parabolic partial differential equations (PDE) systems with time-varying uncertain variables. Especially processes that are described by dissipative PDEs are considered. The states of the process required for designing controllers are dynamically estimated from limited number of noisy process measurements employing an Extended Kalman filter. The issue of utilizing these estimated states in a robust controller to achieve the desired process objective is investigated. The controller design needs to address both model uncertainty and sensor noise. The methodology is employed on an representative example wherein the desired objective is to stabilize an unstable operating point in a catalytic rod, where an exothermic reaction occurs. A finite dimensional robust controller, utilizing dynamically estimated states, is used to successfully stabilize the process to an open-loop unstable steady-state. - Article
- Feb 2011
- AUTOMATICA

This paper considers the stabilization problem of a one-dimensional unstable heat conduction system (rod) modeled by a parabolic partial differential equation (PDE), powered with a Dirichlet type actuator from one of the boundaries. By applying the Volterra integral transformation, a stabilizing boundary control law is obtained to achieve exponential stability in the ideal situation when there are no system uncertainties. The associated Lyapunov function is used for designing an infinite-dimensional sliding manifold, on which the system exhibits the same type of stability and robustness against certain types of parameter variations and boundary disturbances. It is observed that the relative degree of the chosen sliding function with respect to the boundary control input is zero. A continuous control law satisfying the reaching condition is obtained by passing a discontinuous (signum) signal through an integrator. - Article
- Dec 2000
- PHYS REP

Fractional kinetic equations of the diffusion, diffusion–advection, and Fokker–Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns. These fractional equations are derived asymptotically from basic random walk models, and from a generalised master equation. Several physical consequences are discussed which are relevant to dynamical processes in complex systems. Methods of solution are introduced and for some special cases exact solutions are calculated. This report demonstrates that fractional equations have come of age as a complementary tool in the description of anomalous transport processes. - Nowadays control systems are mostly implemented on digital platforms and, increasingly, over shared communication networks. Reducing resources (processor utilization, network bandwidth, etc.) in such implementations increases the potential to run more applications on the same hardware. We present a self-triggered implementation of linear controllers that reduces the amount of controller updates necessary to retain stability of the closed-loop system. Furthermore, we show that the proposed self-triggered implementation is robust against additive disturbances and provide explicit guarantees of performance. The proposed technique exhibits an inherent trade-off between computation and potential savings on actuation.
- Article
- Feb 2000
- PHYSICA A

We derive a fractional reaction–diffusion equation from a continuous-time random walk model with temporal memory and sources. The equation provides a general model for reaction–diffusion phenomena with anomalous diffusion such as occurs in spatially inhomogeneous environments. As a first investigation of this equation we consider the special case of single species fractional reaction–diffusion in one dimension and show that the fractional diffusion does not by itself precipitate a Turing instability. - Article
- Jul 2008
- Comm Nonlinear Sci Numer Simulat

The article is devoted to theoretical description of charge carrier transport in disordered semiconductors. The main idea of the approach lies in the use of fractional calculus. The physical reasons of introducing fractional derivatives in semiconductor theory are discussed, the process of derivation of fractional differential equations is demonstrated, the tied link of their solutions with non-Gaussian stable processes is shown. The last section of the article contains solutions of some concrete problems: multiple trapping, transient photocurrent and drift mobility, dispersive transport percolation model of semiconductors, transport in bilayer semiconductor and so on. Some numerical results are obtained and their agreement with experimental data is demonstrated. - Book
- Jan 2006

*In the title of this book (monograph), please immediately correct the word "Fractinal" to read "Fractional". Thanks!* - We consider the coagulation dynamics A + A --> A and the annihilation dynamics A +A --> 40 for particles moving subdiffusively in one dimension, both on a lattice and in a continuum. The analysis combines the "anomalous kinetics" and "anomalous diffusion" problems, each of which leads to interesting dynamics separately and to even more interesting dynamics in combination. We calculate both short-time and long-time concentrations, and compare and contrast the continuous and discrete cases. Our analysis is based on the fractional diffusion equation and its discrete analog.
- Article
- Dec 2004
- BIOPHYS J

Macromolecular crowding dramatically affects cellular processes such as protein folding and assembly, regulation of metabolic pathways, and condensation of DNA. Despite increased attention, we still lack a definition for how crowded a heterogeneous environment is at the molecular scale and how this manifests in basic physical phenomena like diffusion. Here, we show by means of fluorescence correlation spectroscopy and computer simulations that crowding manifests itself through the emergence of anomalous subdiffusion of cytoplasmic macromolecules. In other words, the mean square displacement of a protein will grow less than linear in time and the degree of this anomality depends on the size and conformation of the traced particle and on the total protein concentration of the solution. We therefore propose that the anomality of the diffusion can be used as a quantifiable measure for the crowdedness of the cytoplasm at the molecular scale. - Energy-efficient wireless communication network design is an important and challenging problem. Its difficulty lies in the fact that the overall performance depends, in a coupled way, on the following subsystems: antenna, power amplifier, modulation, error control coding, and network protocols. In addition, given an energy constraint, improved operation of one of the aforementioned subsystems may not yield better overall performance. Thus, to optimize performance one must account for the coupling among the above subsystems and simultaneously optimize their operation under an energy constraint. In this article we present a generic integrated design methodology that is suitable for many kinds of mobile systems and achieves global optimization under an energy constraint. By pointing out some important connections among different layers in the design procedure, we explain why our integrated design methodology is better than traditional design methodologies. We present numerical results of the application of our design methodology to a situational awareness scenario in a mobile wireless network with different mobility models. These results illustrate the improvement in performance that our integrated design methodology achieves over traditional design methodologies, and the tradeoff between energy consumption and performance.
- ArticleFull-text available
- Mar 1997

In this paper, stability results of main concern for control theory are given for finite-dimensional linear fractional differential systems. For fractional differential systems in state-space form, both internal and external stabilities are investigated. For fractional differential systems in polynomial representation, external stability is thoroughly examined. Our main qualitative result is that stabilities are guaranteed iff the roots of some polynomial lie outside the closed angular sector jarg(oe)j ff=2, thus generalizing in a stupendous way the well-known results for the integer case ff = 1. 1. INTRODUCTION Fractional differential systems have proved to be useful in control processing for the last two decades (see [21, 22]). The notion of fractional derivative dates back two centuries; some references that have now become classical were written two decades ago (see [20, 25]). Several authors published reference books on the subject very recently: see [26] for a thorough mathem...