Boundary Feedback Control Research Articles

Exponential input-to-state stabilization of an ODE cascaded with a reaction–diffusion equation subject to disturbances

This paper addresses the exponential input-to-state stabilization of an ODE cascaded with a reaction–diffusion equation subject to disturbances appearing in all channels. By applying the backstepping method and the sliding mode control approach, the discontinuous boundary feedback controller is designed to reject the matched disturbance and achieve the exponential input-to-state stability of the closed-loop system. The existence of generalized solution to the closed-loop system is obtained using the Galerkin approximation scheme. The effectiveness of the proposed controller is illustrated by some numerical simulations.

A generalized exponential stabilization for a class of semilinear parabolic equations: Linear boundary feedback control approach

This paper investigates the generalized exponential stabilization (GES) of a class of semilinear parabolic equations, which is a generalization of the result of exponential stabilization and has the character of accelerating decay process locally. From the point of admissibility of unbounded input operators of view, the well‐posedness of the open‐loop system is analyzed. Then, under some sufficient conditions expressed by linear matrix inequality, a linear boundary output feedback controller only using boundary measurement is proposed such that the closed‐loop system satisfies GES in absence of external disturbance and satisfies an H∞ performance index when disturbance exists. The well‐posedness of the resulting closed‐loop system is also given via semi‐group theory. Finally, we give numerical examples to verify the performance of the designed boundary controller.

Uniform Stabilization of 3D Navier–Stokes Equations in Low Regularity Besov Spaces with Finite Dimensional, Tangential-Like Boundary, Localized Feedback Controllers

The present paper provides a solution in the affirmative to a recognized open problem in the theory of uniform stabilization of 3-dimensional Navier-Stokes equations in the vicinity of an unstable equilibrium solution, by means of a `minimal' and `least' invasive feedback strategy which consists of a control pair $\{ v,u \}$ \cite{LT2:2015}. Here $v$ is a tangential boundary feedback control, acting on an arbitrary small part $\widetilde{\Gamma}$ of the boundary $\Gamma$; while $u$ is a localized, interior feedback control, acting tangentially on an arbitrarily small subset $\omega$ of the interior supported by $\widetilde{\Gamma}$. The ideal strategy of taking $u = 0$ on $\omega$ is not sufficient. A question left open in the literature was: Can such feedback control $v$ of the pair $\{ v, u \}$ be asserted to be finite dimensional also in the dimension $d = 3$? We here give an affirmative answer to this question, thus establishing an optimal result. To achieve the desired finite dimensionality of the feedback tangential boundary control $v$, it is here then necessary to abandon the Hilbert setting of past literature and replace it with a Besov setting which are `close' to $L^3(\Omega)$ for $d=3$. It is in line with recent critical well-posedness in the full space of the non-controlled N-S equations. A key feature of such Besov spaces with tight indices is that they do not recognize compatibility conditions. The proof is constructive and is "optimal" also regarding the "minimal" number of tangential boundary controllers needed. The new setting requires establishing maximal regularity in the required critical Besov setting for the overall closed-loop linearized problem with tangential feedback control applied on the boundary. Finally, the minimal amount of tangential boundary action is linked to the issue of unique continuation of over-determined Oseen eigenproblems.

Output tracking for a radiative density optical communication system with unknown disturbance

This paper deals with boundary tracking problem for a radiative density parabolic equation (RDE) coming from an optical communication system. First, the radiative intensity is modeled by a radiative transfer equation (RTE). The idea of this work is to use the fact that the signal sent by the laser beam is collimated in consideration of the radiative density flux. Therefore, the control problem of the RTE model is transformed into a boundary control for an RDE based on the time-dependent SPN approximation. Then, we investigate a boundary output tracking problem of one dimensional parabolic density equation where the unknown disturbance enters the system from one part of the boundary. By utilizing one side boundary measurement, we construct an extended state observer to estimate the system state and the external disturbance. An output feedback boundary control is then designed using the states of the servo system and observer. It is shown that the closed-loop system and the control are uniformly bounded with output being tracking the reference and the closed-loop is internally exponentially stable. The numerical simulations are presented to illustrate the effectiveness of the proposed output feedback method.

Maximal L-regularity for an abstract evolution equation with applications to closed-loop boundary feedback control problems

In this paper we present an abstract maximal $L^p$-regularity result up to $T = \infty$, that is tuned to capture (linear) Partial Differential Equations of parabolic type, defined on a bounded domain and subject to finite dimensional, stabilizing, feedback controls acting on (a portion of) the boundary. Illustrations include, beside a more classical boundary parabolic example, two more recent settings: (i) the $3d$-Navier-Stokes equations with finite dimensional, localized, boundary tangential feedback stabilizing controls as well as Boussinesq systems with finite dimensional, localized, feedback, stabilizing, Dirichlet boundary control for the thermal equation.

Output feedback stabilization of a first‐order hyperbolic equation with multi‐point nonlocal term

AbstractThis paper presents the output feedback stabilization of a first‐order hyperbolic equation with multi‐point nonlocal term by boundary control. First of all, we utilize the backstepping method to design a boundary state feedback controller. On the basis of equivalence of original and target system, we guarantee the exponential stability of closed‐loop system. Hereafter, the output feedback control in the light of an infinite‐dimensional observer is constructed. We demonstrate that the resulting closed‐loop system is exponentially stable via operator semigroup theory. Finally, we display the validity of the proposed controller by some numerical simulations.

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 control of a fractional reaction-diffusion equation coupled with fractional ordinary differential equations with delay

This paper deals with boundary feedback control for a fractional reaction-diffusion equation with varying coefficient coupled with fractional ordinary differential equations with delay, which is a generalization of integer order coupled system. By designing a state feedback controller, we transform an unstable system into an asymptotic stable system via the backstepping method. The exact solution of the target system is given by the Prabhakar function. We also obtain the exact solution of the original system with the help of the invertible coordinate transformation. Furthermore, by the fractional Halanay’s inequality, we structure a Lyapunov functional to prove the asymptotic stability of the given system. Finally, a numerical simulation example is provided to illustrate the applications of our results.

Boundary Feedback Control of a Nonhomogeneous Wind Turbine Tower With Exogenous Disturbances

This article studies a vibration and disturbance rejection problem of a wind turbine tower under exogenous perturbations. The tower dynamics is captured by a nonhomogeneous Euler–Bernoulli beam model. The dissipativity of the system is realized by a boundary feedback control solution with a multivalued symbolic function. A Lyapunov-based stability analysis is established to assess the deflection of the tower is uniformly bounded even subject to exogenous disturbances. The extended Filippov framework and Galerkin approximation scheme are introduced to tackle the existence of the solution to the system with a discontinuous control input. Simulation results demonstrate the performance of the proposed control scheme.

Numerical stabilization of the Boussinesq system using boundary feedback control

We study the numerical stabilization of the two dimensional Boussinesq equations. The stabilization is achieved by boundary feedback controls of finite dimension applied on the velocity and temperature, and localized on parts of the domain boundary. We consider a stationary solution of the Boussinesq system which is linearly unstable. The linearized system around this unstable stationary solution is used to construct a feedback law. For that, we determine the projected linearized system onto an invariant subspace (including the unstable subspace), and we determine a feedback control law stabilizing this projected system, by solving a Riccati equation, of small dimension, associated with this system. The degrees of stabilizability of the different real generalized eigenspaces of the control system are used to determine the invariant subspace on which the linearized system is projected. Next, the linear feedback law, stabilizing the projected system, is applied to the nonlinear model to study its ability to locally stabilize the flow and its temperature.

Towards implementation of PDE control for Stefan system: Input-to-state stability and sampled-data design

The Stefan system is a representative model for a liquid–solid phase change which describes the dynamics of a material’s temperature profile and the liquid–solid interface position. Our previous work designed a boundary feedback control to stabilize the phase interface position modeled by the Stefan system. This paper resolves two issues our previous work did not study, that are, the robustness analysis under the unknown heat loss and the digital control action. First, we introduce the one-phase Stefan problem with a heat loss by modeling a 1-D diffusion Partial Differential Equation (PDE) dynamics of the liquid temperature and the interface position governed by an Ordinary Differential Equation (ODE) with a time-varying disturbance. We focus on the closed-loop system under the control law proposed in our previous work, and show an estimate of L2 norm in a sense of Input-to-State Stability (ISS) with respect to the unknown heat loss. Second, we consider the sampled-data control of the one-phase Stefan problem without the heat loss, by applying Zero-Order-Hold (ZOH) to the control law in our previous work. We prove that the closed-loop system under the sampled-data control law satisfies the global exponential stability in the spatial L2 norm. Analogous ISS result for the two-phase Stefan problem which incorporates the dynamics of the solid phase is also obtained. Numerical simulation verifies our theoretical results for showing the robust performance under the heat loss and the digital control implemented to vary at each sampling time.

Finite-time non-fragile boundary feedback control for a class of nonlinear parabolic systems

In this paper, the finite-time non-fragile boundary feedback control problem is investigated for a class of nonlinear parabolic systems, where both the multiplicative and additive controller gain variations are considered to describe the actuator parameter perturbation. Non-fragile boundary control strategies are designed with respect to two controller gain variations via collocated or non-collocated boundary measurement, respectively. In light of the finite-time stability and Lyapunov-based techniques, some sufficient conditions are presented in terms of linear matrix inequalities such that the resulting closed-loop system is well-posedness and practically finite-time stable. Finally, numerical examples are given to verify the effectiveness of the proposed design method.

Boundary feedforward and feedback control for the exponential tracking of the unstable high-dimensional wave equation

The problem of exponential tracking with a unstable high-dimensional wave equation in a cylindrical domain is considered. The signal to be tracked is assumed to be approximated by a Fourier polynomial or a polynomial of any degrees. The feedback controller for the tracking is applied on the top and lateral boundary of the cylindrical domain, while the feedforward controller is applied only on the top. To solve this problem, we introduce a straightforward method of translation of exogenous signal dependent equilibrium to the origin to convert it to a stabilization problem, decoupling the tracking problem into two separate problems: the wave regulator equations and the stabilization of the wave equation. The wave regulator equations are reduced to the classical Helmholtz's equations and then their solutions are explicitly expressed by using the solutions of the classical Helmholtz's equations. The feedforward controller is then synthesized from the solutions. On the other hand, the feedback controller is designed by the method of backstepping with the usual integral transformation used for the unstable heat equation. Since the convergence rate of the error is completely determined by the stabilized wave equation, which is completely decoupled from the wave regulator equations, it is independent of the structures of feedforward controllers. A numerical example is given to confirm our theoretical results.

On Wiener's violent oscillations, Popov's curves, and Hopf's supercritical bifurcation for a scalar heat equation

AbstractA parameter‐dependent perturbation of the spectrum of the scalar Laplacian is studied for a class of nonlocal and non‐self‐adjoint rank one perturbations. A detailed description of the perturbed spectrum is obtained both for Dirichlet boundary conditions on a bounded interval as well as for the problem on the full real line. The perturbation results are applied to the study of a related parameter‐dependent nonlinear and nonlocal parabolic equation. The equation models a feedback system that admits an interpretation as a thermostat device or in the context of an agent‐based price formation model for a market. The existence and the stability of periodic self‐oscillations of the related nonlinear and nonlocal heat equation that arise from a Hopf bifurcation are proved. The bifurcation and stability results are obtained both for the nonlinear parabolic equation with Dirichlet boundary conditions and for a related problem with nonlinear Neumann boundary conditions that model feedback boundary control. They follow from a Popov criterion for integral equations after reducing the stability analysis for the nonlinear parabolic equation to the study of a related nonlinear Volterra integral equation. While the problem is studied in the scalar case only, it can be extended naturally to arbitrary Euclidean dimension and to manifolds.

Boundary control design for conservation laws in the presence of measurement disturbances

Boundary feedback control design for systems of linear hyperbolic conservation laws in the presence of boundary measurements affected by disturbances is studied. The design of the controller is performed to achieve input-to-state stability (ISS) with respect to measurement disturbances with a minimal gain. The closed-loop system is analyzed as an abstract dynamical system with inputs. Sufficient conditions in the form of dissipation functional inequalities are given to establish an ISS bound for the closed-loop system. The control design problem is turned into an optimization problem over matrix inequality constraints. Semidefinite programming techniques are adopted to devise systematic control design algorithms reducing the effect of measurement disturbances. The effectiveness of the approach is extensively shown in several numerical examples.

A High-Efficient Finite Difference Method for Flexible Manipulator with Boundary Feedback Control

The paper presents a high-efficient finite difference method for solving the PDE model of the single-link flexible manipulator system with boundary feedback control. Firstly, an abstract state-space model of the manipulator is derived from the original PDE model and the associated boundary conditions of the manipulator by using the velocity and bending curvature of the flexible link as the state variables. Then, the second-order implicit Crank-Nicolson scheme is adopted to discretize the state-space equation, and the second-order one-sided approximation is used to discretize the boundary conditions with excitations and feedback control. At last, the state-space equation combined with the boundary conditions of the flexible manipulator is transformed to a system of linear algebraic equations, from which the response of the flexible manipulator can be easily solved. Numerical simulations are carried out to simulate the manipulator under various excitations and boundary feedback control. The results are compared with ANSYS to demonstrate the accuracy and high efficiency of the presented method.

Output feedback control of two-lane traffic congestion

This paper develops output feedback boundary control to mitigate traffic congestion of an unidirectional two-lane freeway segment. The macroscopic traffic dynamics are described by the Aw–Rascle–Zhang (ARZ) model respectively for both the fast and slow lanes. The traffic density and velocity of each of the two lanes are governed by coupled 2 × 2 nonlinear hyperbolic partial differential equations (PDEs). Lane-changing interactions between the two lanes lead to exchanging source terms between the two pairs second-order PDEs. Therefore, we are dealing with 4 × 4 nonlinear coupled hyperbolic PDEs. Based on driver’s preference for the slow and fast lanes, a reference system of lane-specific uniform steady states in congested traffic is chosen. To stabilize traffic densities and velocities of both lanes to the steady states, two distinct variable speed limits (VSLs) are applied at outlet boundary, controlling the traffic velocity of each lane. Using backstepping transformation, we map the coupled heterodirectional hyperbolic PDE system into a cascade target system, in which the traffic oscillations are damped out through actuation of the velocities at the downstream boundary. Two full-state feedback boundary control laws are developed. We also design a collocated boundary observer for state estimation with sensing of the densities at the outlet. Output feedback boundary controllers are obtained by combining the collocated observer and full-state feedback controllers. The finite time convergence to equilibrium is achieved and a performance improvement in fuel consumption, drivers’ comfort and total travel time is demonstrated with the proposed output feedback controllers for some parameter choice, compared with the open-loop system. The proposed output feedback design is also validated for different congested traffic scenarios and compared with one-lane backstepping and PI control designs.

Finite-Time Bounded Control for Coupled Parabolic PDE-ODE Systems Subject to Boundary Disturbances

In this paper, the finite-time bounded control problem for coupled parabolic PDE-ODE systems subject to time-varying boundary disturbances and to time-invariant boundary disturbances is considered. First, the concept of finite-time boundedness is extended to coupled parabolic PDE-ODE systems. A Neumann boundary feedback controller is then designed in terms of the state variables. By applying the Lyapunov-like functional method, sufficient conditions which ensure the finite-time boundedness of closed-loop systems in the presence of time-varying boundary disturbances and time-invariant boundary disturbances are provided, respectively. Finally, the issues regarding the finite-time boundedness of coupled parabolic PDE-ODE systems are converted into the feasibility of linear matrix inequalities (LMIs), and the effectiveness of the proposed results is validated with two numerical simulations.

State and output feedback boundary control of time fractional PDE–fractional ODE cascades with space‐dependent diffusivity

This study considers the boundary control problem of a time fractional partial differential equation (PDE) – fractional ordinary differential equation (ODE) cascaded system with spatially varying diffusivity and Dirichlet connection. Using the backstepping transformation, a full-state feedback law is obtained. Meanwhile, the well-posedness of kernel functions in this transformation is established theoretically. Subsequently, a Mittag-Leffler convergent boundary observer is derived, which composes with the proposed state feedback law to enable stabilisation by output feedback. With the designed state and output feedback controllers, the Mittag-Leffler stability of the closed-loop system is then proved by the fractional Lyapunov method. Finally, the results of numerical simulations are provided for fractional cascaded plants when neither kernel ODE nor kernel PDE has an explicit solution.

Input-to-State Stability of Non-uniform Linear Hyperbolic Systems of Balance Laws via Boundary Feedback Control

In this paper, a linear hyperbolic system of balance laws with boundary disturbances in one dimension is considered. An explicit candidate Input-to-State Stability (ISS)-Lyapunov function in $$L^2-$$ norm is considered and discretised to investigate conditions for ISS of the discrete system as well. Finally, experimental results on test examples including the Saint-Venant equations with boundary disturbances are presented. The numerical results demonstrate the expected theoretical decay of the Lyapunov function.