Boundary Control Research Articles

Controllability results of discrete boundary control system in Banach spaces

In this paper, we study the boundary controllability of the discrete-time control system in Banach spaces. The aim is to establish sufficient conditions for the existence and uniqueness of solutions, as well as for the approximate controllability of a discrete-time boundary control system. The form of the solution to the considered system and its existence and uniqueness are obtained using non-linear functional analysis, theory of difference equations, semigroup theory and contraction mapping principle. Furthermore, we achieve the approximate controllability results through a sequential approach. In the end, we present some examples to illustrate the application of these analytical results.

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.

Integral BLF-Based Adaptive Dynamic Event-Triggered Boundary Control for a Flexible Riser System.

For the flexible riser systems modeled with partial differential equations (PDEs), this article explores the boundary control problem in depth for the first time using a dynamic event-triggered mechanism (DETM). Given the intrinsic time-space coupling characteristic inherent in PDE computations, implementing a state-dependent DETM for PDE-based flexible risers presents a significant challenge. To overcome this difficulty, a novel dynamic event-triggered control method is introduced for flexible riser systems, focusing on optimizing available control inputs. In order to save computational costs from the controller to the actuator, a dynamic event-triggered adaptive boundary controller is designed to effectively reduce boundary position vibrations. Additionally, considering external disturbances, an adaptive bounded compensation term is incorporated to counteract the influence of external disturbances on the system. Addressing boundary position constraints, a new integral barrier Lyapunov function (iBLF) tailored specifically for flexible riser systems is introduced, thereby alleviating conservatism in the controller design of flexible risers modeled by PDEs. At last, the validity of the proposed method is demonstrated through a simulation example.

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.

PDE-Based Boundary Adaptive Consensus Control of Multiagent Systems With Input Constraints.

The leader-follower adaptive consensus control problem is addressed for partial differential equations (PDEs) multiagent systems (MASs), and these agents are composed of flexible manipulator systems with input nonlinearity, boundary uncertainties, and time-varying disturbances. Because of the spatial variables in the model, the design of adaptive protocols is more difficult than that of ordinary differential equation (ODE) MASs. By designing the Lyapunov function, a novel distributed boundary control (BC) protocol is constructed, which not only ensures the consensus of angular positions but also suppresses the boundary vibration of each agent. The hybrid effects of dead zones and input saturation on flexible manipulator systems are addressed using the approximation properties of neural networks (NNs). In addition, the disturbance adaptive laws are proposed to provide a control solution for bounded and time-varying disturbances. Furthermore, by applying the Lyapunov stability theory, the uniformly bounded stability of the multiflexible manipulator can be ensured. Finally, the feasibility of the presented control approach is verified using numerical examples.

Temperature field analysis in tunnel construction ventilation with emphasis on duct leakage and thermal conductivity

Mechanical ventilation is the primary measure for cooling during construction of high geo-temperature tunnel. Based on heat transfer theory and characteristics of press-in ventilation, a comprehensive calculation model considering the influence of construction process is proposed, and the reliability of the model is verified by on-site test. The effects of duct leakage rate and thermal conductivity of different excavation length on temperature field distribution during construction ventilation were analyzed based on the prediction model. The following conclusions are drawn: As the excavation length increases, the duct outlet temperature first increases and then decreases. The duct outlet temperature is mainly affected by geo-temperature and boundary control temperature of working face, the air leakage rate and excavation length are not the main factors. The highest temperature in the duct is about 40 °C. The distance between the position of highest temperature and working face increases with the increase of excavation length and decreases with the increase of air leakage rate. With the increase of excavation length, the impact of thermal conductivity on outlet temperature of duct is first enhanced and then stabilized under the same excavation length. The heat-insulated duct can effectively reduce the outlet air temperature and prolong the service life of the duct, while minimizing the air leakage of the duct.

An enhanced snow ablation optimizer for UAV swarm path planning and engineering design problems

The Snow Ablation Optimizer (SAO) is an advanced optimization algorithm. However, it suffers from slow convergence and a tendency to become trapped in local optima. To address these limitations, we propose an Enhanced Snow Ablation Optimization algorithm (ESAO). Initially, an adaptive T-distribution control strategy is employed to improve the algorithm's exploratory position adjustments, facilitating the identification of the global optimum. Furthermore, we introduce a Cauchy mutation strategy, endowing individuals with a robust capability to escape local extrema and steer the population towards more favorable directions. A leader-based boundary control strategy is also proposed to enhance the optimizer's search performance, significantly increasing the accuracy, speed, and stability of the algorithm in tackling complex problems. To validate the performance of ESAO, we utilize 29 CEC2017 benchmark functions for comparison against eight popular algorithms across various dimensions. Our algorithm ranked first in all comparisons, demonstrating ESAO's effectiveness. Additionally, to evaluate the practical applicability of the proposed method, we mathematically modeled the UAV swarm and solved the UAV swarm path planning problem using various competitor algorithms. Furthermore, we applied different competitor algorithms to two engineering design problems. The results demonstrate that ESAO performs the best. In general, ESAO outperforms its counterparts in terms of solution quality and stability, showcasing its superior application potential.

Boundary control problem for a parabolic equation with involution

In this paper, we consider a boundary control problem for a parabolic equation with involution in a bounded one-dimensional domain. On the part of the border of the considered domain, the value of the solution with control function is given. Restrictions on the control are given in such a way that the average value of the solution in the considered domain gets a given value. The problem given by the method of separation of variables is reduced to the Volterra integral equation of the first kind. The existence of the control function was proved by the Laplace transform method.

Stabilization for a class of fractional-order nonlinear reaction–diffusion systems with time-varying delay: Event-triggered boundary control approach

Based on the hybrid event-triggered mechanism (HETM), the boundary stabilization issue for fractional-order nonlinear reaction–diffusion systems (FNRDSs) with time-varying delay is studied by using two kinds of measurements. First, when the system state is measurable, a event-triggered feedback controller (ETFC) is designed directly based on the average measured output. Secondly, for the case that the state is unmeasurable, an event-triggered feedback controller based on observer framework is constructed through the boundary point measurement information. Utilizing the Lyapunov method and Wirtinger’s inequality, sufficient conditions for the asymptotic stability of the system are given in the form of linear matrix inequalities (LMIs), respectively, in which the Razumikhin theorem is used to deal with time-varying delay. Meanwhile, it is proved that Zeno behavior can be excluded by the designed HETM. Finally, numerical simulations demonstrate the validity and feasibility of the proposed control scheme.

Vibration control of 2-D variable-length flexible riser systems with unknown boundary disturbance

The riser system is subject to vibration due to marine environmental loads, which can affect the efficiency of the system or even destroy it. This paper proposes a solution for the vibration control problem of a 2-D variable-length flexible riser based on PDE model. Firstly, a disturbance observer is developed to estimate and offset the unknown disturbance in the feedback loop. Based on the above method, an effective boundary controller is designed to realize the elastic suppression goal of the 2-D variable-length flexible riser. Secondly, when considering the asymmetric output constraints, a time adjustment function is introduced to develop a new controller based on the barrier Lyapunov function, which realizes the control objective of suppressing the boundary vibration of the flexible riser within specified constraints in a finite time. Finally, the effectiveness of the controller is verified by simulation examples.

Soft‐switching boundary and piecewise control of LLC converter with hybrid modulation

AbstractThe voltage gain range of LLC converter can be effectively broadened by using the hybrid modulation of phase‐shift modulation (PSM) and pulse frequency modulation (PFM). However, the harmonic of resonant tank increases dramatically in PSM mode, and the first harmonic approximation is no longer suitable for solving the voltage gain. Moreover, the conditions for soft switching in PSM mode are not clear. Meanwhile, the small‐signal models of the converter in different modulation modes are quite different, and it is difficult for a fixed controller to ensure a good dynamic performance of the converter in both modulation modes. To address above issues, a voltage gain modeling method of PSM mode LLC converter is proposed. By establishing the function between input current and resonant current, the voltage gain model of the converter is obtained according to the power conservation. Moreover, considering the parasitic capacitance charge and resonant current commutation time, the condition for soft switching is derived. Furthermore, a control strategy of LLC converter in hybrid modulation is proposed. The piecewise proportional integral (PI) controller is adopted, and the parameters of the controller are dynamically adjusted with the modulation mode, so that the converter has excellent dynamic performance in both modes. And the hysteresis control is used to ensure the stability of mode switching. An experimental prototype with 200–340 V input and 28 V/1 kW output was built based on gallium nitride (GaN) devices. The peak efficiency was 97.82%, and the power density was 65.76 w/in3.

Observer-Based Asynchronous Boundary Stabilization for Stochastic Markovian Reaction-Diffusion Neural Networks.

This work investigates the observer-based asynchronous boundary stabilization for a kind of stochastic Markovian reaction-diffusion neural networks with exogenous disturbances. Specifically, parameter uncertainties are considered in the drift item. First, a hidden Markov model is introduced that guarantees the observer modes run asynchronously with the system modes. It should be noted that the asynchronous observer constructed in this work only uses the boundary measurement information. Then a nonfragile asynchronous observer-based boundary controller is designed. Taking advantage of inequality techniques and stochastic analysis method, sufficient criterion is provided to satisfy input-to-state exponentially mean-square stability, and the asynchronous boundary observer/controller gains are further derived. As a special case, the synchronous observer-based boundary stabilization is also obtained. Finally, a numerical example is exploited to manifest the validity of the established results.

Boundary stabilisation of stochastic delay impulsive Korteweg–de Vries–Burgers equations

Boundary stabilisation is addressed for stochastic delay impulsive Korteweg–de Vries–Burgers (SDIKdVB) equations. Initially, a novel boundary controller is devised. Utilizing the Lyapunov–Krasovskii functional approach and the impulsive theory, a sufficient condition is derived for achieving mean-square exponential stabilisation (MSES) of SDIKdVB equations. Secondly, the scenario is investigated in which uncertainties exist in the system parameters, and a sufficient condition is presented to attain the robust MSES. Additionally, the investigation is conducted on the impact of the diffusion term, the dispersion term, impulsive strengths and impulsive intervals on MSES. Lastly, the effectiveness of the theoretic findings is demonstrated via practical applications in both biomechanics and physics.

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.

Analysis of energy dissipation in hyperbolic problems influenced by internal and boundary control mechanisms

ABSTRACT This article primarily focuses on the rational stabilisation of the wave equation, supplied with a second-order dynamical boundary condition of hyperbolic type, while considering an additional internal damping mechanism within the specified ring. To achieve rational decay rates of the associated energy, it is imperative to exponentially stabilise a portion of the domain using a global Ingham's-type estimate. This paper will subsequently illustrate how this partially localised exponential stabilisation, combined with a Bessel analysis, leads to a rational decrease in the overall energy of the system considered.

A new wake superposition model and its application in collision prevention control of tri-riser array system

In the field of offshore oil exploitation, the compact multi-riser arrangement increases the risk of collision and has an adverse impact on production and the marine environment. The solution to this problem involves accurate motion prediction and collision prevention control under the effect of flow field superposition between risers, but there is little research on this related theory. This paper improves the traditional wake shielding effect model and further proposes a wake shielding superposition effect model for the mutual influence between risers, in order to establish a comprehensive accurate motion control model for tri-riser array system. A solution method is introduced for this model, both vibration and collision analysis are conducted for the system. The study revealed that solely augmenting the spacing inadequately mitigates system collisions, and active control is an inevitable choice. Based on this, this paper further designs an active boundary controller that is easy to implement in engineering, to achieve collaborative collision prevention control for three risers, and verifies its stability. Compared with traditional PD control, the proposed control method exhibits significant superiority, which can effectively reduce vibration amplitude, significantly increase dynamic spacing, and thus effectively prevent collisions between risers.

Prescribed-time and output feedback stabilization of heat equation with an intermediate-point heat source and boundary control

This paper addresses the problem of prescribed-time stabilization for the heat equation with an interior point heat source under output feedback. The study employs a two-step backstepping approach along with time-varying feedback techniques. A prescribed-time state observer and an observer-based boundary control law are developed. The proposed method extends the applicability of the prescribed time-stabilizing algorithm to heat equations with non-strict feedback terms. Numerical simulations validate its effectiveness.

Hierarchical exact controllability of a parabolic equation with boundary controls

This paper addresses the application of a multi-objective control problem to a parabolic equation. We present the Stackelberg-Nash strategy, which combines the concepts of controllability to trajectories with Nash equilibrium. Our assumption is that the system is influenced through a hierarchy of boundary controls, consisting of one main control (the leader) aiming to drive the solution to a specified target at a final time, and a pair of secondary controls (the followers) designed to minimize two prescribed cost functionals while adapting to the leader's objective. The main novelty of this work lies in the fact that all controls are located on the boundary. One of the primary challenges encountered is the derivation of suitable observability inequality of Carleman for an adjoint system with boundary coupling terms.

Secure consensus for PDEs-based multiagent systems under DoS attacks via boundary control approach

This paper focuses on secure consensus for leader-following multiagent systems (MASs) modeled by partial differential equations (PDEs) under denial of service (DoS) attacks. To mitigate the negative effects of DoS attacks, which can paralyze communication and cause agents to fail to receive valid control inputs, a buffer region is established in the communication channels among agents to temporarily store messages from neighbors. Additionally, since the states of the leader and followers are not always measurable, observers are used to estimate these states. To address these challenges, this paper proposes two boundary controllers to ensure leader-following consensus in both measurable and unmeasurable states. One controller is based on original boundary information, while the other utilizes observation information from both the leader and followers. To the best of our knowledge, this is the first attempt to use buffers to solve a class of PDEs-based MASs under DoS attacks. Furthermore, the boundary control approach has the potential to significantly reduce the number of actuators required, thereby lowering control costs. Finally, we present two numerical examples to validate the feasibility of the proposed methods.

Output feedback finite‐time boundary control for an unstable heat PDE with spatially varying coefficients

AbstractThis article studies the output feedback finite‐time boundary control for unstable heat systems with the spatially varying coefficient. First, a finite‐time observer with switched gains under a state‐dependent switching law is designed in order to estimate the states of the system in a finite‐time only exerting one displacement boundary measurement. Next, an observer‐based linear finite‐time control is planned. Namely, a linear switched control under a state‐dependent switching law is proposed to vanish every solution of an unstable heat partial differential equation with spatially varying coefficients in a finite time. We also present explicit forms for the proposed observer gains and output feedback finite‐time controller. Finally, some numerical simulations are provided to confirm the theoretical results.