Boundary Control Law Research Articles

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.

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.

Adaptive sliding mode boundary control of a perturbed diffusion process

SummaryThis paper proposes a sliding‐mode‐based adaptive boundary control law for stabilizing a class of uncertain diffusion processes affected by a matched disturbance. The matched disturbance is assumed to be uniformly bounded along with its time derivative, whereas the corresponding upper bounding constants are not known. This motivates the use of adaptive control strategies. In addition, the spatially‐varying diffusion coefficient is also uncertain. To achieve asymptotic stability of the plant origin in the ‐sense in the presence of the disturbance, a discontinuous boundary feedback law is proposed where the gain of the discontinuous control term is adjusted according to a gradient‐based adaptation law. A constructive Lyapunov analysis supports the stability properties of the considered closed‐loop system, yielding sufficient convergence conditions in terms of suitable inequalities involving the controllers' tuning parameters. Simulation results are presented to corroborate the theoretical findings.

Adaptive neural bounded formation tracking control of multiple autonomous surface vessels subject to actuator saturation and faults

This paper introduces an adaptive neural bounded formation tracking control strategy for a fleet of autonomous surface vessels with actuator saturation nonlinearity and faults. To begin with, by using the error transformation technique, the proposed algorithm derives the controller only based on the relative distance and angles measured by local sensor, which can avoid the requirement for supplementary leader-related information. Secondly, by employing neural networks (NNs) and auxiliary dynamic system, the system uncertainty and actuator saturation nonlinearity are handled, and a NNs-based bounded control law is derived. In addition, in order to solve actuator faults such as loss of validity, hard over faults and bias faults, the fault-tolerant adaptive law is introduced to compensate the weight matrix of the neural network and the upper bound of the fault parameters, instead of dealing with each individual matrix element. This can significantly decrease the computational workload of the algorithm, facilitating its practical implementation in engineering applications. Finally, it is demonstrated that all signals of the closed-loop system are semi-globally ultimately uniformly bounded. The performance and superiority of this control strategy are validated through numerical examples in various scenarios.

Continuum Modeling and Boundary Control of a Satellite with a Large Space Truss Structure

Due to its advantages of easy deployment and high stiffness-to-mass ratio, the utilization of truss structures for constructing large satellites presents an appealing solution for modern space missions, including Earth observation and astronomy. However, the dimensions of the traditional finite element model for a satellite with a large space truss structure become exceedingly large as the structure’s size increases. The control system design process based on the finite element model is complex and time-consuming. This paper employs the continuum modeling method to represent the truss structure as a continuous entity. The bending vibrations of the truss structure are encapsulated by a simplified partial differential equation (PDE), as opposed to the more intricate traditional finite element model. Simultaneously, the satellite’s attitude motion is characterized by an ordinary differential equation (ODE). Building upon this coupled PDE-ODE model, a boundary control law that only requires sensors/actuators at the boundary is formulated to effectively mitigate structural vibrations and regulate the satellite’s attitude. The exponential stability of this closed-loop system is scrutinized using Lyapunov’s direct method. The simulation results affirm that the continuum modeling method is indeed well-suited for satellites endowed with substantial truss structures, and the proposed boundary law proves to be highly effective in both attitude tracking and vibration suppression.

Finite-Time Control of Complex Networks With Nondifferential Delays by Bounded Control Laws

Finite-Time Control of Complex Networks With Nondifferential Delays by Bounded Control Laws

Input-to-state stability analysis of heat equation with boundary finite-time control

This paper addresses the problem of input-to-state stabilization for heat equation with external input via boundary control strategy. Following Polyakov et al., 2017, based on the backstepping approach, a switching boundary control law depending on the system state is designed. By estimating the upper bound of kernel functions, switching levels are determined and commutation law is further constructed. For the chosen switched control, the well-posedness property is verified. It is proved that the resulting system is input-to-state stable, and the solutions of the system will not exceed the highest admissible level dependent on the disturbance amplitude. Meanwhile, a stronger result is also obtained, that is the finite-time stability for the disturbance-free system. The numerical example is provided to support the derived result.

Analytic design of constrained control laws for nonlinear dynamic systems with symmetric and asymmetric limits

In this article, we formulate and investigate dynamic optimisation of systems with control limits. Aerial, electronic, electromechanical, mechatronic and robotic systems exhibit symmetric and asymmetric bounds. Applying a Hamiltonian optimisation, and, minimisation of specified-by-constraints nonquadratic performance functionals result in analytically designed bounded control laws. Closed form solution of an optimisation problem is found. Continuous control algorithms ensure consistency to physical systems and homogeneous performance. Findings in dynamic optimisation and constrained control are analytically, numerically and experimentally substantiated. Experimental studies and results are documented for a dc-dc switching regulator and permanent-magnet synchronous motor.

PDE-based prescribed performance adaptive attitude and vibration control of flexible spacecraft

This article presents a prescribed performance adaptive control scheme for the attitude tracking and vibration reduction of flexible spacecraft modeled by partial differential equations (PDEs). First, the original constrained problem is converted to an equivalent unconstrained one by adopting a prescribed error transformation. Then, a boundary control torque law and a boundary control force law are developed to track the desired attitude and reduce the vibration, respectively. Moreover, the parametric adaptation technique is integrated with the boundary control laws to estimate the disturbances. The closed-loop system is evaluated to be asymptotically stable through the Lyapunov stability theorem and LaSalle's invariance principle. The developed controller has two distinctive features as follows. One is the controller can guarantee the attitude tracking error and tip deformation always satisfy the prescribed performance constraints. Another is the controller can acquire the strong robustness owing to the feedforward disturbance compensation. Lastly, the derived results are demonstrated by simulated studies.

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.

Active disturbance rejection control to stabilization of coupled delayed time fractional-order reaction–advection–diffusion systems with boundary disturbances and spatially varying coefficients

In this paper we consider the boundary asymptotic stabilization of a coupled time fractional-order reaction–advection–diffusion (FRAD) system with boundary input disturbances, spatially varying coefficients and time varying delays. The complex system composition makes it difficult to analyze system property directly. For this issue, we here address it by introducing two transformations to convert the original system into the one where theories of fractional evolution equations and operator semigroup are easily applicable. The active disturbance rejection control (ADRC) and backstepping control are also utilized in investigation. Using ADRC, the disturbance is first estimated by constructing two auxiliary systems and then canceled through an approximation in the feedback-loop. For such auxiliary systems, one is used to take disturbances from the original system and to put them into a stable system. Another is used to estimate disturbances. In the second part, we use backstepping to develop a boundary state feedback control law with disturbance approximation to ’eliminate’ the disturbances and to achieve the closed-loop stability. The well-posedness and disturbance estimation are established by theories of fractional evolution equations. With fractional Halanay’s inequality, sufficient conditions are obtained to make the controlled system asymptotically stable and the auxiliary system bounded. Fractional simulation scheme is constructed to test the proposed synthesis generated by ADRC when the explicit solution of kernel equations does not exist.

Neural network–based active control of a rigid–flexible spacecraft with bounded input

This paper investigates the simultaneous control of a spacecraft with a rigid central hub and two flexible appendages. Taking into account the coupling between the vibration of flexible appendages and the motion of the hub, the rigid–flexible spacecraft subjected to external disturbance is modeled as a hybrid lumped–distributed parameter system. To attenuate the vibrations of the hub and flexible appendages and simultaneously track the given attitude angle, two boundary control laws are developed based on the original dynamic equation, where all system modes are controlled and spillover instability can be avoided. Using the active disturbance rejection control technique, two extended state observers are designed to obtain the estimation of external disturbances, which can be canceled in the negative feedback loop. The radial basis function neural network with adaptive weight law is employed to deal with the impact of input saturation. The well-posedness of the closed-loop system is proved by applying the semigroup theory. Under the rigorous Lyapunov analysis, the proposed control strategy ensures the practical stability of the rigid–flexible spacecraft system. Simulation verifies the performance of the designed control scheme.

Vibration Control for an Active Mass Damper of a High-Rise Building With Input and Output Constraints

Aiming at the windproof and seismic problems of a high-rise building with large flexible structures, this article introduces a boundary control technology for a high-rise building with active mass damper (AMD) under input and output constraints. Hamilton's principle is used to establish an infinite-dimensional distributed parameter system model of a high-rise building with AMD. Based on the partial differential equations model, a boundary control law with input and output constraints is proposed, where the bounded input constraints are handled by a smooth hyperbolic tangent function and the output constraints are dealt with by a barrier Lyapunov function. It is proved that the vibration is well suppressed by the controller, and the input and output constraints are not violated. Numerical simulations and experiments are used to verify the effectiveness of the control performance.

Neural Network-Based Adaptive Boundary Control of a Flexible Riser With Input Deadzone and Output Constraint.

In this article, vibration abatement problems of a riser system with system uncertainty, input deadzone, and output constraint are considered. For obtaining better control precision, a boundary control law is constructed by employing the backstepping method and Lyapunov's theory. The output constraint is guaranteed by utilizing a barrier Lyapunov function. Adaptive neural networks are designed to cope with the uncertainty of the riser and compensate for the effect caused by the asymmetric deadzone nonlinearity. With the designed controller, the output constraint is satisfied, and the system stability is guaranteed through Lyapunov synthesis. In the end, numerical simulation results are provided to display the performance of the developed adaptive neural network boundary control law.

Active vibration control of horizontal-axis wind turbine blades using disturbance observer-based boundary control approach

This paper addresses the issue of controlling vibration in horizontal-axis wind turbine (HAWT) blades under unknown boundary and distributed disturbances using an observer-based boundary control approach. In this regard, an actuator configuration, involving four active tendons mounted on the tip of the blade, is proposed to generate the out-of-plane and in-plane boundary control forces. The out-of-plane and in-plane governing equations are given as a set of PDEs and ODEs under parametric and direct excitations using Rayleigh beam theory including rotary inertia. A new disturbance observer is introduced to estimate boundary disturbances and subsequently combined with a new boundary control law for alleviating the HAWT blade vibration in the presence of unknown but bounded distributed disturbances. The uniform boundedness of the closed-loop system under unknown bounded distributed disturbances is confirmed via employing the Lyapunov’s direct method together with LaSalle–Yoshizawa’s theorem. Moreover, it is proved that the closed-loop system is uniformly exponentially stable in the absence of distributed and boundary disturbances. The reference NREL 5 MW HAWT blade is selected as the case study and the blade element momentum theory (BEMT) is employed to compute the aerodynamic load. Then, the open-loop and closed-loop responses of the blade under wind shear inflow and a predefined boundary disturbance are compared to demonstrate the performance of the closed-loop system in operational conditions.

Sliding-Mode Boundary Control of an In-Line Heating System Governed by Coupled PDE/ODE Dynamics

This present article addresses the feedback control design problem of in-line liquid flow heating units used to process silicon wafers in the semiconductor industry. A physics-based model is developed to describe the thermal behavior of the heater and the liquid circulating in it. Under sensible approximations, process dynamics are shown to be governed by a coupled partial differential equation (PDE)/ordinary differential equation (ODE) system. A boundary control law capable of steering the fluid temperature to the desired set-point value while rejecting matched disturbances is developed. The proposed boundary control is sliding-mode-based and makes use of boundary measurements only. The convergence properties of the closed-loop system are formally demonstrated by means of Lyapunov-based analysis. In addition, the input-to-state stability (ISS) properties of the closed-loop system with respect to nonmatching boundary disturbances are investigated. To corroborate the theoretical findings, this article additionally presents the experimental results with the proposed controller implemented in the closed loop on a real in-line heating unit.

Deployment of second-order networked mobile agents over a smooth curve

In this paper, we investigate second-order deployment of multiple mobile agents over a C2 curve in 2D or 3D space. We assume that an informed agent in the group can measure its relative position with respect to the given curve and velocity, other agents have access to the local information about the given curve as well as the position and velocity in regard to their closest neighbors, where the agent dynamics is described by a strongly damped wave equation. Without delay, we establish a boundary control law for the informed agent. When taking into account time-varying delay, we assume that the informed agent transmits its measurements to its neighbors through a communication mechanism. Given appropriate controller gains, second-order deployment can be achieved with an exponential decay. Finally, some simulation results are presented to illustrate our theoretical results.

Output‐feedback boundary adaptive fault‐tolerant control for scalar hyperbolic partial differential equation systems with actuator faults

SummaryThis article studies the output‐feedback adaptive fault‐tolerant boundary control problem for scalar hyperbolic partial differential equation systems with actuator faults. All the coefficients of the controlled plant are unknown, and two types of actuator faults, that is, multiplicative faults and additive faults, are considered simultaneously. For the state estimation problem, two filters are constructed, based on which an observer is obtained to estimate the system state. A parametric model is established for actuator faults, based on which the parameter updating laws of gradient type are then developed to identify actuator faults and to estimate the unknown system coefficients. With the observer and the parameter updating laws, an output‐feedback adaptive fault‐tolerant boundary control law is developed via infinite‐dimensional backstepping method. The boundness of all the signals involved in the control design is guaranteed and the convergence of system states is also confirmed. Finally, the simulation results are given to testify the effectiveness of the proposed control scheme.

Boundary state feedback control for semilinear fractional-order reaction diffusion systems

This paper investigates the stabilization results for a semilinear time fractional-order reaction diffusion partial differential equations using backstepping method. The parabolic system is addressed with time fractional-order of (0, 1) in the Caputo sense. More general type of actuation setup which can be expressed as Dirichlet, Neumann and Robin type boundary actuation's is considered. The main aim is to achieve the stabilization of the considered system using an invertible transformation through the stability of target system which is verified by Mittag-Leffler stability based on (Q, S, R) dissipativity theory and linear matrix inequality (LMI) technique. The explicit solutions of kernel functions are found by method of successive approximation and to be design the boundary control law for the closed loop system. Finally, the proposed results are validated through numerical examples.

Robust non-fragile boundary control for non-linear parabolic PDE systems with semi-Markov switching and input quantization

In this paper, the robust boundary control problem for non-linear PDE systems is discussed. Specifically, the PDE under investigation is of parabolic type with semi-Markov jumping signals subject to non-linearities and parameter uncertainties. The main goal of this paper is to devise a non-fragile boundary control law which assures the robust stabilization of the addressed system in spite of gain fluctuations and quantization in its design. In particular, to reduce the burden in the communication medium, the input signals can be quantized before transmitted to the actuator by using logarithmic quantizers. Moreover, by utilizing the Lyapunov stability theory and vector-valued Wirtinger’s inequality, a set of sufficient conditions for robust stabilization of the underlying system is established in the form of linear matrix inequalities (LMIs). Based on the obtained conditions, the existence of the desired controller is confirmed and an explicit form of the controller gains are developed. Eventually, the efficacious of the extracted theoretical results are manifested by virtue of two simulation examples.