Boundary Stabilization Research Articles

Exponential Decay Rate of port-Hamiltonian Systems with one side Boundary Damping

The multiplier approach is applied to a class of port-Hamiltonian systems with boundary dissipation to establish exponential decay . The exponential stability of port-Hamiltonian systems has been studied and sufficient conditions obtained. Here the decay rate Me-αt is established with M and α are in terms of system parameters. This approach is illustrated by several examples, in particular, boundary stabilization of a piezoelectric beam with magnetic effects.

Predictor-based output feedback stabilization of an input delayed parabolic PDE with boundary measurement

This paper is concerned with the output feedback boundary stabilization of general 1-D reaction diffusion PDEs in the presence of an arbitrarily large input delay. We consider the cases of Dirichlet/Neumann/Robin boundary conditions for the both boundary control and boundary condition. The boundary measurement takes the form of an either Dirichlet or Neumann trace. The adopted control strategy is composed of a finite-dimensional observer estimating the first modes of the PDE coupled with a predictor to compensate the input delay. In this context, we show for any arbitrary value of the input delay that the control strategy achieves the exponential stabilization of the closed-loop system, for system trajectories evaluated in H1 norm (also in L2 norm in the case of a Dirichlet boundary measurement), provided the dimension of the observer is selected large enough. The reported proof of this result requires to perform both control design and stability analysis using simultaneously the (non-homogeneous) original version of the PDE and one of its equivalent homogeneous representations.

Backstepping-Based Exponential Stabilization of Timoshenko Beam with Prescribed Decay Rate

In this paper, we present a rapid boundary stabilization of a Timoshenko beam with anti-damping and anti-stiffness at the uncontrolled boundary, by using PDE backstepping. We introduce a transformation to map the Timoshenko beam states into a (2+2) × (2+2) hyperbolic PIDE-ODE system. Then backstepping is applied to obtain a control law guaranteeing closed-loop stability of the origin in the H1 sense. Arbitrarily rapid stabilization can be achieved by adjusting control parameters. Finally, a numerical simulation shows that the proposed controller can rapidly stabilize the Timoshenko beam. This result extends a previous work which considered a slender Timoshenko beam with Kelvin-Voigt damping, allowing destabilizing boundary conditions at the uncontrolled boundary and attaining an arbitrarily rapid convergence rate.

Boundary stabilization and state estimation of ODE-transport PDE with in-domain coupling

This paper deals with the exponential stabilization of first order ODE-transport PDE coupled at the boundary point. A state feedback boundary control law has been formulated with the help of the backstepping method. The main novelty of this paper is that the stabilization of the coupled system is discussed by Lyapunov theory and the appropriate observer gain is designed by using the linear matrix inequalities (LMIs). An anti-collocated observer design for the corresponding dual system is also presented. The state feedback boundary controller, observer design and the stabilization of the closed-loop system are discussed in detail with illustrative numerical examples.

Mittag‐Leffler stabilization for coupled fractional reaction‐diffusion neural networks subject to boundary matched disturbance

In this paper, we study the boundary stabilization for a class of neural networks with non‐identical nodes described by the fractional Orr‐Sommerfeld (OS) equation cascaded with the fractional Squire equation and fractional ordinary differential equations (ODEs), in which the disturbances flow to the control end of both the OS equation and the Squire equation. By applying the Backstepping transform, the diffusion subsystems in the considered neural networks are decoupled, while only the ODEs are still cascaded with the diffusion terms through the Neumann interconnection. By the active disturbance rejection control (ADRC) approach, two auxiliary systems are constructed to compensate each disturbance by an estimator without high gain. Finally, a boundary feedback control law is designed to achieve the Mittag‐Leffler stability of the neural networks in the closed‐loop.

Finite-dimensional observer-based boundary stabilization of reaction–diffusion equations with either a Dirichlet or Neumann boundary measurement

This paper investigates the output feedback boundary control of reaction–diffusion equations with either distributed or boundary measurement by means of a finite-dimensional observer. A constructive method dealing with the design of finite-dimensional observers for the feedback stabilization of reaction–diffusion equations was reported in a recent paper in the case where either the control or the observation operator is bounded and also satisfies certain regularity assumptions. In this paper, we go beyond by demonstrating that a finite-dimensional state-feedback combined with a finite-dimensional observer can always be successfully designed in order to achieve the Dirichlet boundary stabilization of reaction–diffusion PDEs with either a Dirichlet or Neumann boundary measurement.

Stabilization and uncertainty analysis of a time‐fractional reaction diffusion equation cascaded with a time‐fractional hyperbolic partial differential equation

AbstractIn this paper, we consider a time‐fractional reaction diffusion equation cascaded with a time‐fractional hyperbolic partial differential equation (PDE), where the time‐fractional reaction diffusion equation possesses space‐dependent diffusivity and the time‐fractional hyperbolic equation acts as a boundary source for the time‐fractional reaction diffusion equation. Then, the control input is imposed at the boundary of the time‐fractional hyperbolic PDE. Two problems are investigated, namely, the stabilization by state feedback and the stability robustness of the closed‐loop system against small perturbations in the diffusion and reaction coefficients. By the backstepping transformation, we propose a boundary controller and show that this controller solves the boundary stabilization problem using the fractional Lyapunov method. Robustness to small perturbations in diffusion and reaction coefficients is proved. Finally, a numerical example is provided to test the effectiveness of the proposed synthesis for the stabilization problem when the kernel equations have not an explicit solution.

Finite-time boundary stabilization of the reaction-diffusion system with switching time-delay input

The paper is devoted to the study of boundary finite-time control for a reaction-diffusion (RD) system with switching time-delayed input. The RD system with switching time-delay input is converted to a switching system of RD equation cascaded with a transport equation with non-delay boundary input. Next, a novel switching controller is designed for the cascaded RD-transport system based on the backstepping technique, and this causes the closed-loop system to be convergence in a finite-time. Simulation results are provided to exhibit the effectiveness of the proposed method.

Social categories in the making: construction or recruitment?

Real kinds, both natural and social categories, are characterized by rich inductive potential. They have relatively stable sets of conceptually independent projectable properties. Somewhat surprisingly, even some purely social categories (e.g., ethnicity, gender, political orientation) show such multiple projectability. The article explores the origin of the inductive richness of social categories and concepts. I argue that existing philosophical accounts provide only a partial explanation, and mechanisms of boundary formation and stabilization must be brought into view for a more comprehensive account of inductively rich social categories.

Finite-time stabilization for a first-order hyperbolic system with integral kernel by boundary control

This paper consider the finite-time boundary stabilization for a first-order hyperbolic system with integral kernel and lower order term. Via choosing a suitable integral transformation converting the original system into a finite-time stable object system, then, we obtain the control law and certify the finite-time stabilization of the closed-loop system combined with the finite-time stability of object system and invertibility of forward transformation.

Boundary stabilization of an overhead crane with beam model

In this paper, we study the boundary feedback stabilization problem of a hybrid system consisting of a flexible beam attached to the platform moving a long a straight rail and carrying at the free end a load which is deplaced in a horizontal plan. The model proposed in this paper fits a large real applications such as an overhead crane with beam. Using the Riesz basis approach of general second-differential equation systems with non separated boundary conditions, it is shown that the Riesz basis property holds for the system and as consequence, the exponential stability is concluded. To verify the theoritical developments, numerical study of the spectrum is performed by Legendre approximation, also thenumerical simulations are presented to show the effectivnesse of the proposed control.

High-gain adaptive boundary stabilization for an axially moving string subject to unbounded boundary disturbance

In this paper, we are interested to stabilize an axially moving string subject to external disturbances. We assume that the disturbance may increases exponentially. We employ the active disturbance rejection control (ADRC) approach to estimate the disturbance. We design a disturbance observer that has time-varying gain so that the disturbance can be estimated with an exponential way. In order to stabilize the closed loop system, we use a control constructed through a high-gain adaptive velocity feedback. The existence and uniqueness of solution of the closed loop system is dealt with in the framework of the nonlinear semigroup theory by using a theorem due to Crandall-Liggett. It is shown that the formulated control is capable of stabilizing exponentially the closed loop system. The obtained results are also valid for the immobile case ($v=0$) and the present work improves certain previous results.

Adaptive boundary stabilization of a nonlinear axially moving string

AbstractIn this paper, we are interested in stabilizing a nonlinear axially moving string by an adaptive boundary control. The existence and uniqueness of a solution of the closed loop system is dealt with in the framework of the nonlinear semigroup theory by using a theorem due to Crandall‐Liggett. The control is constructed through a low‐gain adaptive velocity feedback. It is shown that the formulated control is capable of stabilizing exponentially the closed loop system.

Absolute boundary stabilization for an axially moving Kirchhoff beam

In this paper, we consider absolute stability for an axially moving Kirchhoff beam under nonlinear boundary feedback controls. The nonlinear boundary control which satisfies a slope-restricted condition is a negative feedback of the transverse velocity at the right eyelet of the moving beam. First, the well-posedness of the resulting closed-loop system is established by means of the Faedo–Galerkin approximation combined with some priori estimates. By invoking the integral-type multiplier method, the exponential stability of the closed-loop is developed. The numerical simulations by the finite element method are presented to demonstrate the effectiveness of the proposed controller.

Boundary Stabilization of Heat Equation with Multi-Point Heat Source

In this paper, we consider boundary stabilization problem of heat equation with multi-point heat source. Firstly, a state feedback controller is designed mainly by backstepping approach. Under the designed state controller, the exponential stability of closed-loop system is guaranteed. Then, an observer-based output feedback controller is proposed. We prove the exponential stability of resulting closed-loop system using operator semigroup theory. Finally, the designed state and output feedback controllers are effective via some numerical simulations.

Fractional boundary stabilization for a coupled system of wave equations

In this paper,we consider a system of a coupled wave equations in the presence of a boundary control of fractional derivative type. We prove well-posedness by using the semigroup theory. Also we establish an optimal decay result by frequency domain method and Borichev-Tomilov theorem.

Boundary stabilizing actuators for multi-phase fluids in a channel

This work is devoted to the problem of boundary stabilization of a mixture of two viscous and incompressible fluids in a three dimensional channel-like domain (x,y,z)∈R×(0,1)×R. The model consists of the Navier-Stokes equations, governing the fluid velocity, coupled with a convective Cahn-Hilliard equation for the relative density of the atoms of one of the fluids. The feedback stabilizing controller, we design here, is with actuation only on the tangential component of the velocity field on both walls y=0,1. Moreover, it is given in a simple and explicit form, easy to manipulate from the computational point of view.

Boundary Stabilization of Stochastic Delayed Cohen-Grossberg Neural Networks With Diffusion Terms.

This study considers the boundary stabilization for stochastic delayed Cohen-Grossberg neural networks (SDCGNNs) with diffusion terms by the Lyapunov functional method. In the realization of NNs, sometimes time delays and diffusion phenomenon cannot be ignored, so Cohen-Grossberg NNs with time delays and diffusion terms are studied in this article. Moreover, different from the previously distributed control, the boundary control is used to stabilize the system, which can reduce the spatial cost of the controller and is easy to implement. Boundary controllers are presented for system with Neumann boundary and mixed boundary conditions, and criteria are derived such that the controlled system achieves mean-square exponential stabilization. Based on the criterion, the effects of diffusion matrix, coupling strength, coupling matrix, and time delays on exponentially stability are analyzed. In the process of analysis, two difficulties need to be addressed: 1) how to introduce boundary control into system analysis? and 2) how to analyze the influence of system parameters on stability? We deal with these problems by using Poincaré's inequality and Schur's complement lemma. Moreover, mean-square exponential synchronization of stochastic delayed Hopfield NNs with diffusion terms, as an application of the theoretical result, is considered under the boundary control. Examples are given to illustrate the effectiveness of the theoretical results.

Event-Triggered Output-Feedback Backstepping Control of Sandwich Hyperbolic PDE Systems

Motivated by vibration control of a mining cable elevator avoiding frequent actions of a massive actuator, which is a hydraulic cylinder driving a head sheave, we present an event-triggered output-feedback backstepping boundary controller for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2\times 2$</tex-math></inline-formula> coupled hyperbolic partial differential equations (PDEs) sandwiched by two ordinary differential equations (ODEs), through a two-step design including an output-feedback low-pass-filter-based backstepping boundary stabilization law and the subsequent design of a dynamic event-triggering mechanism. The existence of a minimal dwell-time between two triggering times, and exponential convergence in the event-based closed-loop system are proved in this article. In numerical simulations, the proposed control design is validated in the application of axial vibration control of a mining cable elevator.

Event-Triggered Gain Scheduling of Reaction-Diffusion PDEs

This paper deals with the problem of boundary stabilization of 1D reaction-diffusion PDEs with a time- and space-varying reaction coefficient. The boundary control design relies on the backstepping...