Wave Partial Differential Equation Research Articles

Backstepping Boundary Control of a Wave PDE With Spatially Distributed Time Invariant Unknown Disturbances

An observer based boundary control scheme is suggested for an infinite dimensional system modeled as a wave partial differential equation (PDE) with in-domain and boundary disturbances. We consider a collocated case where the sensors and the actuators are located at the same boundary. The aim of the controller is to regulate the rate of change of the opposite boundary around a given reference. The unknown in-domain disturbance is spatially distributed but it is constant in time same as the unknown boundary disturbance. The problem is reformulated as a stabilization of an linear time invariant (LTI) system with simultaneous delays in input and output channels. The exponential stability of the equilibrium of the closed-loop system is proven. The performance of the controller is shown using numerical simulations.

State-Observer Design of a PDE-Modeled Mining Cable Elevator With Time-Varying Sensor Delays

A time-varying sensor delay is inevitable in a mining cable elevator operating at a maximum distance which is more than 2000-m underground, when the sensor signals are transmitted from a cage at the bottom of the cable to a control center on the ground through a set of wireless devices, including emitters and receivers. This sensor delay may degrade the estimation precision of the state observer or even destroy the stability of the closed-loop system. In this brief, we design an observer which can compensate the sensor delay with a time-varying and arbitrary length to estimate the vibration states of the mining cable elevator, including the time-varying-length mining cable and the cage. The vibration dynamics of the mining cable elevator is described by a wave partial differential equation (PDE)-ordinary differential equation (ODE) coupled system on a time-varying domain, and the sensor delay dynamics is represented by a transport PDE on a time-varying domain. A full-order infinite dimensional observer is constructed to estimate the states of such a wave PDE-ODE-transport PDE coupled system where PDEs are on time-varying domains, only using available boundary values without measurements at the connection points of the PDEs and ODE. The exponential stability of the observer error system is proved via Lyapunov analysis. The effectiveness of the proposed observer is verified via numerical simulation.

Output feedback vibration control of a string driven by a nonlinear actuator

• A wave partial differential equation sandwiched by nonlinear and linear ordinary differential equations is considered. • An observer is designed to estimate full states of the overall system only using available boundary values. • An observer-based output feedback boundary controller is proposed via the backstepping method. • The global exponential stability of the output feedback closed-loop system is proved by Lyapunov analysis. In this paper, we design an observer-based output feedback controller to exponentially stabilize a system of nonlinear ordinary differential equation-wave partial differential equation-ordinary differential equation. An observer is designed to estimate the full states of the system using available boundary values of the partial differential equation . The output feedback controller is built via the combination of the ordinary differential equation backstepping which is applied to deal with the nonlinear ordinary differential equation, and the partial differential equation backstepping which is used for the wave partial differential equation-ordinary differential equation. The controller can be applied into vibration suppression of a string-payload system driven by an actuator with nonlinear characteristics. The global exponential stability of all states in the closed-loop system is proved by Lyapunov analysis. The numerical simulation illustrates the states of the actuator, string, payload and the observer errors are fast convergent to zero under the proposed output feedback controller.

Robustness to In-Domain Viscous Damping of a Collocated Boundary Adaptive Feedback Law for an Antidamped Boundary Wave PDE

In this paper, the robustness to model mismatch of a preexisting collocated boundary adaptive feedback law is investigated. This control law was originally designed for an antidamped pure wave partial differential equation (PDE). Actuation and measurements are located at the same boundary. Adaptive terms account for uncertain parameters located at the antidamped boundary, opposite to the collocated actuation and measurement. By extending and transforming the system state using, in particular, backstepping, this paper establishes that this controller is robust to sufficiently small in-domain damping. In particular, both stability and attractivity (convergence) are established similarly as in the nominal case. Note moreover that, assuming that some parameters are known, the exponential stability to an attractor holds. Simulations are performed to illustrate the interest of this study to attenuate mechanical vibrations in an oil-drilling context.

Event-triggered damping of a linear wave equation

Abstract The paper addresses the design of an event-triggering mechanism for a partial differential wave equation posed in a bounded domain. The wave equation is supposed to be controlled through a classical damping term: the first order time derivative of the solution, distributed in the whole domain. In a continuous framework, such a feedback leads to the global exponential stability of the closed-loop system. Here, sufficient conditions based on the use of a suitable Lyapunov functional are proposed to guarantee that an event-triggered distributed damping ensures the exponential stability. Moreover, the designed event-triggering mechanism allows avoiding the Zeno behavior. The ‘existence and regularity’ prerequisite properties of solutions for the closed loop system are also proven.

Wave Equation-Based Time-Varying Formation Control of Multiagent Systems

This paper considers time-varying formation control for multiagent systems (MASs). The collective dynamics is modeled by a wave partial differential equation (PDE) in an angular symmetric annulus. The dynamical formation formed by a group of agents can be represented by wave PDEs due to the energy conservation property of wave PDEs. To achieve convergence of the MAS to desired time-varying formations, we design a boundary controller and a boundary observer by employing the backstepping method, which turns into a leader-enable actuation mode by using only local information. Closed-loop exponential stability with a desired decay rate in the $H^{1}$ norm is proven for both full state and output feedback designs, which means that the communication topology keeps unchanged and connected over time. Compared with the usual 1-D wave equation, the design of the controller, observer, and their stability proof are more involved due to the Laplace operator in polar coordinates. The effectiveness of the proposed control scheme of time-varying formation for MASs is illustrated via numerical simulations.

Backstepping Control of a Wave PDE With Unstable Source Terms and Dynamic Boundary

This letter presents the design of an exponentially stabilizing controller for a one-dimensional wave partial differential equation. The control is acting on a Robin’s boundary condition while the opposite boundary satisfies an unstable dynamic. The wave is also subject to unstable in-domain source terms. Closed-loop exponential stabilization is obtained via a full-state backstepping controller. The existence and uniqueness of this backstepping transformation is proven, using the method of successive approximations.

Axial Vibration Suppression in a Partial Differential Equation Model of Ascending Mining Cable Elevator

Lifting up a cage with miners via a mining cable causes axial vibrations of the cable. These vibration dynamics can be described by a coupled wave partial differential equation-ordinary differential equation (PDE-ODE) system with a Neumann interconnection on a time-varying spatial domain. Such a system is actuated not at the moving cage boundary, but at a separate fixed boundary where a hydraulic actuator acts on a floating sheave. In this paper, an observer-based output-feedback control law for the suppression of the axial vibration in the varying-length mining cable is designed by the backstepping method. The control law is obtained through the estimated distributed vibration displacements constructed via available boundary measurements. The exponential stability of the closed-loop system with the output-feedback control law is shown by Lyapunov analysis. The performance of the proposed controller is investigated via numerical simulation, which illustrates the effective vibration suppression with the fast convergence of the observer error.

Exponential regulation of the anti-collocatedly disturbed cage in a wave PDE-modeled ascending cable elevator

In cable elevators, large axial vibrations appear when a cage subject to disturbance is lifted up via a compliant cable. The axial vibration dynamics can be described by a wave partial differential equation (PDE) on a time-varying spatial interval with an unknown boundary disturbance. In this paper, we design an output feedback controller actuating at the boundary anti-collocated with the disturbance to regulate the state on the uncontrolled boundary of the wave PDE based on the backstepping idea and the active disturbance rejection control (ADRC) approach. The control law uses the state and disturbance information recovered from the state observer and the disturbance estimator, respectively, which are constructed via limited boundary measurements. The exponential convergence of the state on the uncontrolled boundary and uniform boundedness of all states in the closed-loop system are proved by Lyapunov analysis. Effective vibration suppression in the cable elevator with the designed controller is verified via numerical simulation.

Backstepping boundary observer based-control for hyperbolic PDE in rotary drilling system

It is well known that torsional vibrations in oil well system affect the drilling directions and may be inherent for drilling systems. The drill pipe model is described by second order hyperbolic Partial Differential Equation (PDE) with mixed boundary conditions in which a sliding velocity is considered at the top end. In this paper, we consider the problem of boundary observer design for one-dimensional PDE with the usually neglected damping term. The main purpose is the construction of a control law which stabilizes the damped wave PDE, using only boundary measurements. From the Lyapunov theory, we show an exponentially vibration stability of the partially equipped oil well drilling system. The observer-based control law is found using the backstepping approach for second-order hyperbolic PDE. The numerical simulations confirm the effectiveness of the proposed PDE observer based controller.

Formation tracking control for multi-agent systems: A wave-equation based approach

This paper considers the formation tracking control problem of large-scaled Multi-Agent Systems (MAS) for which the model is based on a system of mutually independent wave Partial Differential Equations (PDEs). The spatial derivatives in the equation correspond to the underlying communication topology of the agents. A leader-follower mode is employed in the control algorithm, with which the agents on the boundary of PDEs are chosen as leaders knowing the tracking trajectory and all the other agents are followers. Each follower has only the information of its own relative position and velocity to its topological neighbors. With a designed distributed controller, the formation tracking error is bounded by a constant proportional to the acceleration of the desired trajectory. Robustness of the control law to a perturbation in the velocity measurement is also discussed. Furthermore, some simulation studies are provided to show the effectiveness of the control algorithm.

Boundary Stabilization of Wave Equation With Velocity Recirculation

Nonlocal terms have been the mainstay of the applications of partial differential equation (PDE) backstepping methods to parabolic PDEs. The problem of similar nonlocal terms for wave equations is still open. For wave equations, similar nonlocal terms have not been studied. In this paper, we open the topic of exploration of control of wave PDEs with nonlocal terms. This paper is concerned with the wave equation with in-domain feedback/recirculation of a boundary velocity with a spatially constant coefficient. Due to this nonlocal term, the passivity of the wave equation is destroyed. We first design an explicit state feedback controller to achieve exponential stability for the closed-loop system. Then, we design the output feedback by using infinite-dimensional observer. The backstepping approach is adopted in investigation. It is shown that by using two measurements only, the output feedback makes the closed-loop system exponentially stable.

A new method to solve non-homogeneous wave equations of electromagnetic fields by fourier’s triple integral transform

This paper uses Fourier’s triple integral transform method to simplify the calculation of the non-homogeneous wave equations of the time-varying electromagnetic field. By adding several special definite conditions to the wave equation, it becomes a mathematical problem of definite condition. Then by using Fourier’s triple integral transform method, this three-dimension non-homogeneous partial differential wave equation is changed into an ordinary differential equation. Through the solution to this ordinary differential equation, the expression of the relationship between the time-varying scalar potential and electromagnetic wave excitation source is developed precisely. This method simplifies the solving process effectively.

A Conceptual Numerical Model of the Wave Equation Using the Complex Variable Boundary Element Method

In this work, a conceptual numerical solution of the two-dimensional wave partial differential equation (PDE) is developed by coupling the Complex Variable Boundary Element Method (CVBEM) and a generalized Fourier series. The technique described in this work is suitable for modeling initial-boundary value problems governed by the wave equation on a rectangular domain with Dirichlet boundary conditions and an initial condition that is equal on the boundary to the boundary conditions. The new numerical scheme is based on the standard approach of decomposing the global initial-boundary value problem into a steady-state component and a time-dependent component. The steady-state component is governed by the Laplace PDE and is modeled with the CVBEM. The time-dependent component is governed by the wave PDE and is modeled using a generalized Fourier series. The approximate global solution is the sum of the CVBEM and generalized Fourier series approximations. The boundary conditions of the steady-state component are specified as the boundary conditions from the global BVP. The boundary conditions of the time-dependent component are specified to be identically zero. The initial condition of the time-dependent component is calculated as the difference between the global initial condition and the CVBEM approximation of the steady-state solution. Additionally, the generalized Fourier series approximation of the time-dependent component is fitted so as to approximately satisfy the derivative of the initial condition. It is shown that the strong formulation of the wave PDE is satisfied by the superposed approximate solutions of the time-dependent and steady-state components.

Muscle Activity Map Reconstruction from High Density Surface EMG Signals With Missing Channels Using Image Inpainting and Surface Reconstruction Methods

The aim of this study was to reconstruct low-quality High-density surface EMG (HDsEMG) signals, recorded with 2-D electrode arrays, using image inpainting and surface reconstruction methods. It is common that some fraction of the electrodes may provide low-quality signals. We used variety of image inpainting methods, based on partial differential equations (PDEs), and surface reconstruction methods to reconstruct the time-averaged or instantaneous muscle activity maps of those outlier channels. Two novel reconstruction algorithms were also proposed. HDsEMG signals were recorded from the biceps femoris and brachial biceps muscles during low-to-moderate-level isometric contractions, and some of the channels (5-25%) were randomly marked as outliers. The root-mean-square error (RMSE) between the original and reconstructed maps was then calculated. Overall, the proposed Poisson and wave PDE outperformed the other methods (average RMSE 8.7 μVrms ± 6.1 μVrms and 7.5 μVrms ± 5.9 μVrms) for the time-averaged single-differential and monopolar map reconstruction, respectively. Biharmonic Spline, the discrete cosine transform, and the Poisson PDE outperformed the other methods for the instantaneous map reconstruction. The running time of the proposed Poisson and wave PDE methods, implemented using a Vectorization package, was 4.6 ± 5.7 ms and 0.6 ± 0.5 ms, respectively, for each signal epoch or time sample in each channel. The proposed reconstruction algorithms could be promising new tools for reconstructing muscle activity maps in real-time applications. Proper reconstruction methods could recover the information of low-quality recorded channels in HDsEMG signals.

Universal stabilisation design for general non‐linear systems under wave partial differential equation actuator dynamics with time‐varying moving boundary

A universal stabilisation formula is presented for globally (or locally) asymptotical stabilisation for general non-linear systems governed by wave actuator dynamics with time-varying moving boundary in this study. Stability analysis is developed with passivity-based synthesis approach and infinite-dimensional backstepping transformations. Under the backstepping transformations and the control law design, the original system is transferred to the target system. Then the authors prove that the target system is asymptotical stable. Moreover, it is shown that the stability of the original system is equivalent to that of the target system. Finally, three examples are given to illustrate the effectiveness of the proposed design technique.

A new framework for solving partial differential equations using semi-analytical explicit RK(N)-type integrators

This paper develops a new explicit semi-analytical approach to solving PDEs based on the variation-of-constants formula, or the equivalent integration equation. These new schemes avoid the discretization of spatial derivatives. Therefore, the accuracy of the semi-analytical explicit scheme depends entirely on time integration. We first establish a semi-analytical formula for wave partial differential equations and show the consistency of the semi-analytical formula with different boundary conditions. We then present semi-analytical explicit RKN-type integrators for solving wave PDEs. This methodology is also extended to dealing with first-order hyperbolic PDEs and parabolic PDEs, and for both of them the corresponding semi-analytical explicit RK-type integrators are derived as well. Numerical simulations are implemented and the results show the effectiveness of the new semi-analytical explicit schemes. Meanwhile, an operator-variation-of-constants formula with applications is introduced for high-dimensional nonlinear wave equations before presenting the idea of semi-analytical explicit RKN integrators.

Boundary Backstepping Control of Flow-Induced Vibrations of a Membrane at High Mach Numbers

We design a controller for flow-induced vibrations of an infinite-band membrane, with flow running across the band and only above it, and with actuation only on the trailing edge of the membrane. Due to the infinite length of the membrane, the dynamics of the membrane in the spanwise direction are neglected, namely, we employ a one-dimensional (1D) model that focuses on streamwise vibrations. This framework is inspired by a flow along an airplane wing with actuation on the trailing edge. The model of the flow-induced vibration is given by a wave partial differential equation (PDE) with an antidamping term throughout the 1D domain. Such a model is based on linear aeroelastic theory for Mach numbers above 0.8. To design a controller, we introduce a three-stage backstepping transformation. The first stage gets the system to a critically antidamped wave equation, changing the stiffness coefficient's value but not its sign. The second stage changes the system from a critically antidamped to a critically damped equation with an arbitrary damping coefficient. The third stage adjusts stiffness arbitrarily. The controller and backstepping transformation map the original system into a target system given by a wave equation with arbitrary positive damping and stiffness.

Nonlinear stabilization in infinite dimension

Significant advances have taken place in the last few years in the development of control designs for nonlinear infinite-dimensional systems. Such systems typically take the form of nonlinear ODEs (ordinary differential equations) with delays and nonlinear PDEs (partial differential equations). In this article we review several representative but general results on nonlinear control in the infinite-dimensional setting. First we present designs for nonlinear ODEs with constant, time-varying or state-dependent input delays, which arise in numerous applications of control over networks. Second, we present a design for nonlinear ODEs with a wave (string) PDE at its input, which is motivated by the drilling dynamics in petroleum engineering. Third, we present a design for systems of (two) coupled nonlinear first-order hyperbolic PDEs, which is motivated by slugging flow dynamics in petroleum production in off-shore facilities. Our design and analysis methodologies are based on the concepts of nonlinear predictor feedback and nonlinear infinite-dimensional backstepping. We present several simulation examples that illustrate the design methodology.

Circular polarization of obliquely propagating whistler wave magnetic field

The circular polarization of the magnetic field of obliquely propagating whistler waves is derived using a basis set associated with the wave partial differential equation. The wave energy is mainly magnetic and the wave propagation consists of this magnetic energy sloshing back and forth between two orthogonal components of magnetic field in quadrature. The wave electric field energy is small compared to the magnetic field energy.