Boundary Stabilization Research Articles

Output feedback stabilization of an unstable wave equation with observations subject to time delay

This paper focuses on boundary stabilization of a one-dimensional wave equation with an unstable boundary condition, in which observations are subject to arbitrary fixed time delay. The observability inequality indicates that the open-loop system is observable, based on which the observer and predictor are designed: The state of system is estimated with available observation and then predicted without observation. After that equivalently the authors transform the original system to the well-posed and exponentially stable system by backstepping method. The equivalent system together with the design of observer and predictor give the estimated output feedback. It is shown that the closed-loop system is exponentially stable. Numerical simulations are presented to illustrate the effect of the stabilizing controller.

Robust boundary stabilization of a vibrating rectangular plate using disturbance adaptation

SummaryThis paper considers robust boundary control with disturbance adaptation to stabilize the vibration of a rectangular plate under disturbances with unknown upper‐bounds. Disturbances are considered to be distributed both over the plate interior domain and along the boundary (in‐domain and boundary distributed disturbances). Applying Hamilton's principle, the dynamics of the plate is represented in the form of a fourth‐order partial differential equation subject to static and dynamic boundary conditions. The proposed model considers the membrane effect of axial force and the effect of actuator mass dynamics on the plate vibration. A robust boundary control is established that stabilizes the plate in presence of both in‐domain and boundary disturbances. A rigorous Lyapunov stability analysis shows that the vibration of the plate is uniformly ultimately bounded and converges to the vicinity of zero by proper selection of control gains. For the vanishing in‐domain disturbances, it is seen that exponential stability is achieved by the proposed control. Next, a disturbance adaptation law is introduced to stabilize the plate vibration in response to disturbances with unknown bounds, and the stability of the robust boundary control with disturbance adaptation is studied using Lyapunov. Simulation results verify the efficiency of the suggested control. Copyright © 2016 John Wiley & Sons, Ltd.

Boundary control of a singular reaction-diffusion equation on a disk

Recently, the problem of boundary stabilization for unstable linear constant-coefficient reaction-diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. As a first step, this work deals with radially-varying reaction coefficients under revolution symmetry conditions on a disk (the 2-D case). Under these conditions, the equations become singular in the radius. When applying the backstepping method, the same type of singularity appears in the backstepping kernel equations. Traditionally, well-posedness of the kernel equations is proved by transforming them into integral equations and then applying the method of successive approximations. In this case, the resulting integral equation is singular. A successive approximation series can still be formulated, however its convergence is challenging to show due to the singularities. The problem is solved by a rather non-standard proof that uses the properties of the Catalan numbers, a well-known sequence frequently appearing in combinatorial mathematics.

Boundary control of coupled reaction-diffusion systems with spatially-varying reaction

Recently, the problem of boundary stabilization for unstable linear constant-coefficient coupled reaction-diffusion systems was solved by means of the backstepping method. The extension of this result to systems with spatially-varying reaction (i.e., reaction coefficient depending on the spatial coordinate) is challenging due to complex boundary conditions that appear in the equations verified by the control kernels. In this paper we address this issue by showing that these equations are essentially equivalent to those verified by the control kernels for first-order hyperbolic coupled systems, which were recently found to be well-posed. The result therefore applies in this case, allowing us to prove H1 stability for the closed-loop system. It also shows an interesting connection between backstepping kernels for coupled parabolic and hyperbolic problems.

Boundary control of coupled reaction-advection-diffusion equations having the same diffusivity parameter

We consider the problem of boundary stabilization for a system of n coupled parabolic linear PDEs of the reaction-diffusion-advection type. Particularly, we design a state-feedback law with Dirichlet-type actuation on only one end of the domain and prove exponential stability of the closed-loop system with an arbitrarily fast convergence rate. The backstepping method is used for controller design, and the transformation kernel matrix is derived by using the method of successive approximations to solve the corresponding PDE. Simulation results support the effectiveness of the suggested design.

Boundary Feedback Control of Unstable thermoacoustic Oscillations in the Rijke Tube

The problem of boundary stabilization of thermoacoustic oscillations is investigated. The Rijke tube is used as prototype system to study such phenomena. We consider that this system is modelled as a 2 x 2 linear first-order hyperbolic system that behaves like a wave equation with the control variable at one boundary condition. Our control approach is based on employing characteristic coordinates to convert the system into a system of two delay elements. Analysing the periodicity of the solution of these equations a discrete transfer function is obtained. This enable us to employ a discrete time domain control design to guarantee the exponential stability of the closed-loop system. The performance of the controller is evaluated through simulation results.

Boundary stabilization of a Bresse‐type system

In this paper, we are concerned with asymptotic stability of a class of Bresse‐type system with three boundary dissipations. The beam has a rigid body attached to its free end. We show that exponential stabilization can be achieved by applying force and moment feedback boundary controls on the shear, longitudinal, and transverse displacement velocities at the point of contact between the mass and the beam. Our method is based on the operator semigroup technique, the multiplier technique, and the contradiction argument of the frequency domain method. Copyright © 2015 John Wiley & Sons, Ltd.

Nanoparticles at Grain Boundaries Inhibit the Phase Transformation of Perovskite Membrane.

The high-energy nature of grain boundaries makes them a common source of undesirable phase transformations in polycrystalline materials. In both metals and ceramics, such grain-boundary-induced phase transformation can be a frequent cause of performance degradation. Here, we identify a new stabilization mechanism that involves inhibiting phase transformations of perovskite materials by deliberately introducing nanoparticles at the grain boundaries. The nanoparticles act as "roadblocks" that limit the diffusion of metal ions along the grain boundaries and inhibit heterogeneous nucleation and new phase formation. Ba0.5Sr0.5Co0.8Fe0.2O3-δ, a high-performance oxygen permeation and fuel cell cathode material whose commercial application has so far been impeded by phase instability, is used as an example to illustrate the inhibition action of nanoparticles toward the phase transformation. We obtain stable oxygen permeation flux at 600 °C with an unprecedented 10-1000 times increase in performance compared to previous investigations. This grain boundary stabilization method could potentially be extended to other systems that suffer from performance degradation due to a grain-boundary-initiated heterogeneous nucleation phase transformations.

All Solid State Li-Ion Batteries Based on Cubic Garnets: From Material Optimization to Thin Film Battery Design

The next generation of energy storage devices relies on a broad and adaptable range of volumetric and gravimetric energies to compete with the challenges in stationary, mobility and portable electronic electricity supply. Here, all solid state batteries based on Li-garnet-based structures are interesting model systems as these allow for complete solid state battery types that: i.) can partially use industrial waste heat to increase charging times (increased Li+ diffusion) for stationary, ii.) with new high capacity electrode materials which show conventionally a low stability in standard liquid/polymer based Li-batteries, iii.) are easy transferrable to model thin film battery structures for powering of portable electronics on chip. Firstly, we report on Al:Li7-2xLa3Zr2O12 bulk pellet and thin film processing, their crystallization and ionic transport characteristics. Secondly, from a materials processing perspective we report on a new modified sol-gel synthesis-combustion method to lower the nano-particle production of the solids to ~ 600°C. Here, we exemplify stable electrolyte compounds based on Li7-3x(Gax)La3Zr2O12. Thirdly, for N-site doping Li7-xLa3Zr2-xTexO12 with x = 0.25 to 0.35 were prepared by synthesis in solid state, and Te is studied as alternative to Ga and Al stabilizers as battery electrolyte without grain boundary stabilization. Finally, we concluded on the suitability of these new materials for scaling down to thin film microstructures, as potential Li-garnet electrolytes in all solid state batteries for portable electronic applications. First strategies on how to design optimum thin film-microbatteries adaptable to chemo-mechanical variations during operation will be shown. Here, two different model designs are described: (i.) the fabrication of microbattery dot arrays and (ii.) wrapped anode/electrolyte/cathode microstructures with tunable strain.

Nonlinear Local Stabilization of a Viscous Hamilton-Jacobi PDE

We consider the boundary stabilization problem of the non-uniform equilibrium profiles of a viscous Hamilton-Jacobi (HJ) Partial Differential Equation (PDE) with parabolic concave Hamiltonian. We design a nonlinear full-state feedback control law, assuming Neumann actuation, which achieves an arbitrary rate of convergence to the equilibrium. Our design is based on a feedback linearizing transformation which is locally invertible. We prove local exponential stability of the closed-loop system in the $H^{1} $ norm, by constructing a Lyapunov functional, and provide an estimate of the region of attraction. We design an observer-based output-feedback control law, by constructing a nonlinear observer, using only boundary measurements. We illustrate the results on a benchmark example computed numerically.

Fresh Air Fraction Control in Engines Using Dynamic Boundary Stabilization of LPV Hyperbolic Systems

In this paper, we consider the boundary control of the fresh air mass fraction in a Diesel engine operated with low-pressure exhaust gas recirculation. The air mass fraction transport phenomenon is modeled using a cascade of first-order linear parameter-varying hyperbolic systems with dynamics associated with their boundary conditions. By means of Lyapunov-based techniques, sufficient conditions are derived to guarantee the exponential stability of this class of infinite dimensional systems. We develop a polytopic approach to synthesize a robust boundary control that guarantees the exponential stability for a given convex parameter set. Simulation results illustrate the effectiveness of the proposed boundary control to regulate the mass fraction of fresh air in a Diesel engine.

Stabilisation of Timoshenko beam system with a tip payload under the unknown boundary external disturbances

This paper is concerned with the boundary stabilisation of a Timoshenko beam with a tip payload under the boundary external disturbances. Nonlinear feedback control laws are designed to reduce the effects of the external disturbances. Since the controlled system is nonlinear, the well posedness of the nonlinear closed-loop system is investigated using the theory of nonlinear maximal monotone operators and the variational principle. In addition, we show the exponential stability of the nonlinear controlled system using the Lyapunov direct method. In the end, the numerical simulations are displayed to illustrate the effectiveness of the proposed boundary controls.

Adaptive boundary stabilization for first‐order hyperbolic PDEs with unknown spatially varying parameter

SummaryThe adaptive boundary stabilization is investigated for a class of systems described by first‐order hyperbolic PDEs with unknown spatially varying parameter. Towards the system unknowns, a dynamic compensation is first given by using infinite‐dimensional backstepping method, adaptive techniques, and projection operator. Then an adaptive controller is constructed by certainty equivalence principle, which can stabilize the original system in a certain sense. Moreover, the effectiveness of the proposed method is illustrated by a simulation example. Copyright © 2015 John Wiley & Sons, Ltd.

Boundary control of coupled reaction–diffusion processes with constant parameters

The problem of boundary stabilization is considered for some classes of coupled parabolic linear PDEs of the reaction–diffusion type. With reference to n coupled equations, each one equipped with a scalar boundary control input, a state feedback law is designed with actuation at only one end of the domain, and exponential stability of the closed-loop system is proven. The treatment is addressed separately for the case in which all processes have the same diffusivity and for the more challenging scenario where each process has its own diffusivity and a different solution approach has to be taken. The backstepping method is used for controller design, and, particularly, the kernel matrix of the transformation is derived in explicit form of series of Bessel-like matrix functions by using the method of successive approximations to solve the corresponding PDE. Thus, the proposed control laws become available in explicit form. Additionally, the stabilization of an underactuated system of two coupled reaction–diffusion processes is tackled under the restriction that only a scalar boundary input is available. Capabilities of the proposed synthesis and its effectiveness are supported by numerical studies made for three coupled systems with distinct diffusivity parameters and for underactuated linearized dimensionless temperature-concentration dynamics of a tubular chemical reactor, controlled through a boundary at low fluid superficial velocities when convection terms become negligible.

Nonlinear boundary stabilization for Timoshenko beam system

This paper is concerned with the existence and decay of solutions of the following Timoshenko system:‖u″−μ(t)Δu+α1∑i=1n∂v∂xi=0,inΩ×(0,∞),v″−Δv−α2∑i=1n∂u∂xi=0,inΩ×(0,∞), subject to the nonlinear boundary conditions:‖u=v=0inΓ0×(0,∞),∂u∂ν+h1(x,u′)=0inΓ1×(0,∞),∂v∂ν+h2(x,v′)+σ(x)u=0inΓ1×(0,∞), and the respective initial conditions at t=0. Here Ω is a bounded open set of Rn with boundary Γ constituted by two disjoint parts Γ0 and Γ1 and ν(x) denotes the exterior unit normal vector at x∈Γ1. The functions hi(x,s)(i=1,2) are continuous and strongly monotone in s∈R. The existence of solutions of the above problem is obtained by applying the Galerkin method with a special basis, the compactness method and a result of approximation of continuous functions by Lipschitz continuous functions due to Strauss. The exponential decay of energy follows by using the multiplier and the Zuazua method.

Polynomial stability of the Timoshenko system by one boundary damping

In this paper, we study the indirect boundary stabilization of the Timoshenko system with only one dissipation law. This system, which models the dynamics of a beam, is a hyperbolic system with two wave speeds. Assuming that the wave speeds are equal, we prove non-uniform stability and an optimal polynomial energy decay rate is obtained. Otherwise, if the ratio of the wave speeds is a rational number, we show that the decay rate is of polynomial type.

Control by Interconnection of Distributed Port-Hamiltonian Systems Beyond the Dissipation Obstacle

The main contribution of this paper is a general methodology for the definition of a new passive output that is instrumental for the stabilisation of a large class of distributed port-Hamiltonian systems defined on a one-dimensional spatial domain. This new output is in fact employed within the control by interconnection via Casimir generation paradigm for the synthesis of boundary stabilising control laws. It is well-known that this control technique is limited by the so-called \\dissipation obstacle" when the passive controller is interconnected to the natural input/output port of the plant. When it is the case, it is impossible to shape the energy of the system along the directions in which dissipation is present. In this paper, it is shown how these limitations can be removed by interconnecting the boundary controller to the new passive output of the system, and then how the control by interconnection can easily deal with the dissipation obstacle. The general theory is illustrated with the help of a concluding example, the boundary stabilization of the shallow water equation.

Boundary Stabilization of the Navier--Stokes Equations in the Case of Mixed Boundary Conditions

We study the boundary feedback stabilization, around an unstable stationary solution, of a two dimensional fluid flow described by the Navier--Stokes equations with mixed boundary conditions. The control is a localized Dirichlet boundary control. A feedback control law is determined by stabilizing the linearized Navier--Stokes equations around the unstable stationary solution. We prove that the linear feedback law locally stabilizes the Navier--Stokes system. Thus, we extend results previously known in the case when the boundary of the geometrical domain is regular and when only Dirichlet boundary conditions are present to the case with mixed boundary conditions.

Asymptotic Stabilisation of Distributed Port-Hamiltonian Systems by Boundary Energy-Shaping Control

This paper illustrates a general synthesis methodology of asymptotic stabilising, energy-based, boundary control laws, that is applicable to a large class of distributed port- Hamiltonian systems. Similarly to the finite dimensional case, the idea is to design a state feedback law able to perform the energy-shaping task, i.e. able to map the open-loop port- Hamiltonian system into a new one in the same form, but characterised by a new Hamiltonian with a unique and isolated minimum at the equilibrium. Asymptotic stability is then obtained via damping injection on the boundary, and is a consequence of the La Salle's Invariance Principle in infinite dimensions. The general theory is illustrated with the help of a simple concluding example, i.e. the boundary stabilisation of a transmission line with distributed dissipation.

The Active Disturbance Rejection Control to Stabilization for Multi-Dimensional Wave Equation With Boundary Control Matched Disturbance

In this paper, we consider boundary stabilization for a multi-dimensional wave equation with boundary control matched disturbance that depends on both time and spatial variables. The active disturbance rejection control (ADRC) approach is adopted in investigation. An extended state observer is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multi-dimensional system by infinitely many test functions. The disturbance is canceled in the feedback loop together with a collocated stabilizing controller. All subsystems in the closed-loop are shown to be asymptotically stable. In particular, the time varying high gain is first time applied to a system described by the partial differential equation for complete disturbance rejection purpose and the peaking value reduction caused by the constant high gain in literature. The overall picture of the ADRC in dealing with the disturbance for multi-dimensional partial differential equation is presented through this system. The numerical experiments are carried out to illustrate the convergence and effect of peaking value reduction.