Boundary Feedback Control Research Articles

Well-posedness and exponential stability of piezoelectric beam with thermal effect and boundary or internal distributed delay

In this work, we focus on a one-dimensional initial-boundary value problem of a thermoelastic piezoelectric beam in the presence of magnetic effects with distributed delay acting in the mechanical equation. The well-posedness of the system is initially demonstrated by applying semigroup theory (Hille-Yosida theorem). Also, based on the construction of a Lyapunov functional, which is equivalent to the energy functional of the problem through multiplier techniques, we show that a unique dissipation through frictional damping is strong enough to ensure exponential stabilization of the model. Next, the results are compared to those of the electrostatic or quasi-static approaches. Moreover, we consider the fully dynamic and electrostatic or quasi-static piezoelectric beams with thermal effects, boundary feedback controllers and boundary distributed delays and using the multipliers technique, we establish an exponential stability result of the solution. Our results are related to the distributed delay weights. Finally, we give some numerical tests to illustrate the theoretical result.

Sliding mode control to stabilization of coupled time fractional parabolic PDEs subject to disturbances

AbstractIn this article, the stabilization of coupled time fractional parabolic partial differential equations subject to external disturbances is investigated. By using sliding mode control method and backstepping approach, a boundary state feedback controller is designed to reject the matched disturbance and achieve the Mittag‐Leffler input‐to‐state stability of closed‐loop system. The existence of the generalized solution to the closed‐loop system is proven by Galerkin approximation scheme. Simulations are presented to illustrate the validity of our theoretical results.

Adaptive boundary feedback control of a rigid–flexible system under unknown time-varying disturbance

This paper studies the boundary feedback control of a rotating disk-cable-mass system (DCMS). The system consists of a flexible cable with a rigid disk connected at its upper end and a tip mass at the lower boundary. We assume that the disk and the tip mass are affected by unknown time-varying disturbances, while the cable is influenced by distributed disturbance. We design two boundary controls to suppress the vibration of the flexible cable and simultaneously adjust the system’s rotating speed to a desired value. The first controller is designed assuming the system’s parameters are accurately known. The second controller is conceived under circumstances where some system parameters are unknown, for which an adaptive technique is employed to regulate the DCMS. For these two controls, we choose two Lyapunov functions to prove the stability of the closed-loop system. Theoretical results demonstrate that the states of the closed-loop system are ultimately uniformly bounded. At the end of the paper, we validate two boundary controls through numerical tests.

Neural Operator Approximations for Boundary Stabilization of Cascaded Parabolic PDEs

ABSTRACTThis article proposes a novel method to accelerate the boundary feedback control design of cascaded parabolic difference equations (PDEs) through DeepONet. The backstepping method has been widely used in boundary control problems of PDE systems, but solving the backstepping kernel function can be time‐consuming. To address this, a neural operator (NO) learning scheme is leveraged for accelerating the control design of cascaded parabolic PDEs. DeepONet, a class of deep neural networks designed for approximating nonlinear operators, has shown potential for approximating PDE backstepping designs in recent studies. Specifically, we focus on approximating gain kernel PDEs for two cascaded parabolic PDEs. We utilize neural operators to map only two kernel functions, while the other two are computed using the analytical solution, thus simplifying the training process. We establish the continuity and boundedness of the kernels, and demonstrate the existence of arbitrarily close DeepONet approximations to the kernel PDEs. Furthermore, we demonstrate that the DeepONet approximation gain kernels ensure stability when replacing the exact backstepping gain kernels. Notably, DeepONet operator exhibits computation speeds two orders of magnitude faster than PDE solvers for such gain functions, and their theoretically proven stabilizing capability is validated through simulations.

Exponential Boundary Control for 2-D Spatial Distributed Parameter Systems Under Boundary Collocated and Planar Linear Measurements.

For a class of 2-D spatial distributed parameter systems (DPSs) with space-dependent diffusivity, this article aims to achieve exponential realization of their desired profiles. To reduce the number of required sensors and actuators, a planar output feedback boundary control strategy is proposed with combining two nonfull-domain measurement methods, boundary collocated measurement and planar linear measurement, in which only two boundaries of the considered 2-D spatial DPSs are controlled and a little output information is measured. Moreover, by employing the Poincaré-Wirtinger inequality and variable substitution dexterously, the final exponential convergence criteria of the error system can be obtained with method of "Diverse treatment for same term." Finally, we provide a general numerical example and an application example in 2-D heat conduction systems to illustrate the effectiveness and practicability of the proposed measurement and control schemes.

Mean-square exponential stabilization of mixed-autonomy traffic PDE system

Control of mixed-autonomy traffic where Human-driven Vehicles (HVs) and Autonomous Vehicles (AVs) coexist on the road has gained increasing attention over the recent decades. This paper addresses the boundary stabilization problem for mixed traffic on freeways. The traffic dynamics are described by uncertain coupled hyperbolic partial differential equations (PDEs) with Markov jumping parameters, which aim to address the distinctive driving strategies between AVs and HVs. Considering that the spacing policies of AVs vary in mixed traffic, the stochastic impact area of AVs is governed by a continuous Markov chain. The interactions between HVs and AVs such as overtaking or lane changing are mainly induced by impact areas. Using backstepping design, we develop a full-state feedback boundary control law to stabilize the deterministic system (nominal system). Applying Lyapunov analysis, we demonstrate that the nominal backstepping control law is able to stabilize the traffic system with Markov jumping parameters, provided the nominal parameters are sufficiently close to the stochastic ones on average. The mean-square exponential stability conditions are derived, and the results are validated by numerical simulations.

Feedback boundary control of multi-dimensional hyperbolic systems with relaxation

This paper is concerned with boundary stabilization of multi-dimensional hyperbolic systems of partial differential equations. By adapting the Lyapunov function previously proposed by the second author for linearized hyperbolic systems with relaxation structure, we derive certain control laws so that the corresponding solutions decay exponentially in time. The result is illustrated with an application to water flows in open channels. The effectiveness of the derived control laws is confirmed by numerical simulations.

Boundary feedback control of Navier–Stokes equations around arbitrary bluff body with prescribed drag and lift coefficients

This paper presents a boundary feedback control method for the two- and three-dimensional Navier–Stokes equations around an arbitrary bluff body, with prescribed drag and lift coefficients. In order to determine the feedback control law, we consider an extended system coupling the equations governing the Navier–Stokes problem with an equation satisfied by the control on the bluff body, which is a part of the domain boundary. We utilize the artificial compressibility method, which approximates the Navier–Stokes equations in the limit, along with a priori estimation techniques, to establish a boundary control. Such a control law ensures that the prescriptions of the drag and lift coefficients are attained. Subsequently, a compactness result allows to let the compressibility parameter tend to zero. The application of the Faedo-Galerkin method is essential in reaching this result.

Sufficient and necessary conditions of exponential stabilisation for a class of distributed second-order semilinear systems with time delay

In this paper, we investigate the problem of exponential stabilisation for a class of retarded distributed second-order semilinear systems by means of a bounded feedback control. The existence and uniqueness of mild solution for the considered systems are proved in Hilbert space framework. Sufficient and necessary conditions for exponential stabilisation are given. Some illustrating applications are provided.

PDE-based control synthesis for a planar cable-driven continuum arm

Classical wave and beam models cannot describe the arc-like planar configuration of a continuum arm. Based on the Cosserat rod theory, a strain vector and the rotation matrix are introduced into this work. Two groups of nonlinear partial differential equations (PDEs) are obtained by analyzing kinetic and potential energy densities and virtual power done by the cable tension, which are explicit both in the spatial coordinates and in the physical parameters. Although involving algebraic–trigonometric descriptions, the obtained dynamical model is the most concise formulation but difficult to use for control design. We develop a feedforward plus proportional–derivative feedback boundary control scheme for the nonlinear PDE system to regulate the planar shape of the continuum arm and then linearize the dynamical equations about the equilibrium to assess the control performance. Some interesting results are the closed-loop stability and the well-posedness analysis of the linearized model, which have not previously been noted. Finally, numerical results validate the dynamical equations and the performance of the controller.

Stabilization of the Kawahara equation with saturated internal or boundary feedback controls

Stabilization of the Kawahara equation with saturated internal or boundary feedback controls

Boundary feedback control for hyperbolic systems

We are interested in the feedback stabilization of general linear multi-dimensional first order hyperbolic systems in ℝd. Using a Lyapunov function with a suited weight function depending on the system under consideration we show stabilization in L2 for the studied system using a suitable feedback control. Therefore the controllability of the studied system is related to the feasibility of an associated linear matrix inequality.We show the applicability discussing the barotropic Euler equations.

Local stabilization of the Burgers–Fisher equation via boundary control

AbstractLocally exponential stabilization for the Burgers–Fisher system is addressed by boundary control in this paper. For the nonlinear partial differential equation, a linear boundary feedback control law is applied to control the Burgers–Fisher system. Locally exponential stabilization of the closed loop system is established based on the relationship between operator theories and relations of different norms. Finally, the theory is validated through numerical simulations.

Uniform exponential stabilization of distributed bilinear parabolic time delay systems with bounded feedback control

Uniform exponential stabilization of distributed bilinear parabolic time delay systems with bounded feedback control

Backstepping Boundary Control for a Class of Gantry Crane Systems.

In this article, two boundary feedback controllers are designed via the backstepping approach for a class of gantry crane systems. To provide an accurate and concise representation of the dynamic behavior, the gantry crane with a flexible cable is described by a hybrid system. The hybrid system is formed by an ordinary differential equation coupled with a partial differential equation. In the first control strategy, a backstepping-based boundary state-feedback controller is proposed for the gantry crane to transport a payload to an expected position with less shaking. In the second control strategy, a boundary output-feedback controller is explored with an observer estimating the inaccessible states. By using the backstepping technique and kernel functions, the original systems with different control strategies are transformed into target systems. By using the operator semigroup and Lyapunov stability theories, the target system is proven to be well-posed and exponentially stable, respectively. Finally, numerical simulations and comparisons are provided to illustrate the efficiency and the advantages of the proposed methods.

Robust boundary tracking control of wave PDE: Insight on forcing slow-aseismic response

The output boundary tracking problem is addressed for a wave partial differential equation (PDE) with uncertain multiplicative parameters and additive boundary and in-domain terms. To solve this problem, a homogeneous boundary feedback controller is designed using collocated boundary measurements only. Global exponential Input-to-State Stability (eISS) of the closed-loop system and its insensitivity to matched disturbances are additionally achieved. The ultimate bound estimate is diminished by tuning of the controller gains and/or by an appropriate pre-selection of a reference signal to follow. The proposed homogeneity-based strategy is further evaluated in a simplified earthquake model for controlled dissipation of its stored energy and producing slow-aseismic response. Such a case study possesses important applications in earthquake prevention, renewable energy production and storage. Numerical simulations are additionally conducted to support the robustness and the smoothness of the resulting closed-loop system depending on the homogeneous control parameter selection.

State feedback controller design of fractional ordinary differential equations coupled with a fractional reaction–advection–diffusion equation with delay

The boundary control problem of fractional ordinary differential equations coupled with a time fractional reaction–advection–diffusion equation with delay is studied in this paper. To ensure the asymptotic stability of the system we studied, a state feedback boundary controller is proposed. By backstepping method, we transform the fractional coupled system into a chosen target system under a controller. Furthermore, we obtain the existence and uniqueness of the state solution of the considered system. A Lyapunov functional is constructed to show the asymptotic stability of the fractional coupled system by the special fractional Halanay inequality. The asymptotic stability criterion of the fractional coupled system is described by Linear Matrix Inequalities (LIMs). Which can be easily solved and verified. Finally, the applicability of our theoretical results is showed by a numerical simulation.

Switching clusters’ synchronization for discrete space-time complex dynamical networks via boundary feedback controls

Unlike the existing literatures that consider only discrete-time networks, this paper explores the double effects of both discrete time and discrete spatial diffusions in a switching complex dynamical networks. By means of the knowledge of clusters controls, a clusters synchronous frame of space-time discrete switching complex networks with boundary feedback controller is newly proposed and established. With the helps of some indispensable vector-valued sequence inequalities and Lyapunov function with switching signals and clusters’ information, the boundary feedback controllers are designed to synchronize space-time discrete switching complex networks coupled with nodes’ states or spatial diffusions in the form of clusters. Additionally, a realizable computer algorithm is given to make the derived results of this paper easier to enforce. The current work is pioneering in consideration of discrete spatial diffusions and provides a theoretical and practical basis for future research in this regard.

Solid Boundary Output Feedback Control of the Stefan Problem: The Enthalpy Approach.

By taking enthalpy-an internal energy of a diffusion-type system-as the system state and expressing it in terms of the temperature profile and the phase-change interface position, the output feedback boundary control laws for a fundamentally nonlinear single-phase one-dimensional (1-D) PDE process model with moving boundaries, referred to as the Stefan problem, are developed. The control objective is tracking of the spatiotemporal temperature and temporal interface (solidification front) trajectory generated by the reference model. The external boundaries through which temperature sensing and heat flux actuation are performed are assumed to be solid. First, a full-state single-sided tracking feedback controller is presented. Then, an observer is proposed and proven to provide a stable full-state reconstruction. Finally, by combining a full-state controller with an observer, the output feedback trajectory tracking control laws are presented and the closed-loop convergence of the temperature and the interface errors proven for the single-sided and the two-sided Stefan problems. Simulation shows the exponential-like trajectory convergence attained by the implementable smooth bounded control signals.

Adaptive stabilization for a wave equation subject to boundary control matched harmonic disturbance

Abstract In this paper, we are concerned with adaptive stabilization for a wave equation subject to boundary control matched harmonic disturbance. We use the adaptive and Lyapunov approach to estimate unknown disturbance and construct an adaptive boundary feedback controller. By the semigroup theory and Lasalle‘s invariance theorem, the well-posedness and asymptotic stability of the closed-loop system is proved, respectively. At the same time, it is shown that the parameter estimates involved in the constructed controller converge to their own real values as time goes to infinity. Some numerical simulations are offered at the end of the paper to illustrate the effectiveness of theoretical results.