Partial differential equation boundary control of a flexible manipulator with input saturation

In this study, we consider the boundary control problem of a flexible manipulator in the presence of input saturation and input disturbances. The dynamics of the flexible system are represented by partial differential equations (PDEs). Based on disturbance observers, a boundary control scheme is designed to regulate angular position and suppress elastic vibration simultaneously. The proposed control scheme allows the application of smooth hyperbolic functions, which satisfy physical conditions and input restrictions, easily be realised. It is proved that the proposed control scheme can be guaranteed in handling input saturation and external disturbances. The stability is achieved through rigorous analysis without any simplification of the dynamics. Numerical simulations demonstrate the effectiveness of the proposed scheme.     

Modeling and boundary control of a flexible marine riser coupled with internal fluid dynamics

In this paper, vibration reduction of a flexible marine riser with time-varying internal fluid is studied by using boundary control method and Lyapunov’s direct method. To achieve more accurate and practical riser’s dynamic behavior, the model of marine riser with time-varying internal fluid is modeled by a distributed parameter system (DPS) with partial differential equations (PDEs) and ordinary differential equations (ODEs) involving functions of space and time. The dynamic responses of riser are completely different if the time-varying internal fluid is considered. Boundary control is designed at the top boundary of the riser based on original infinite dimensionality PDEs model and Lyapunov’s direct method to reduce the riser’s vibrations. The uniform boundedness and closed-loop stability are proved based on the proposed boundary control. Simulation results verify the effectiveness of the proposed boundary control.     

Model predictive control of axial dispersion chemical reactor

This paper discusses the development of model predictive control algorithm which accounts for the input and state constraints applied to the parabolic partial differential equations (PDEs) system describing the axial dispersion chemical reactor. Spatially varying terms arising from the nonlinear PDEs model are accounted for in model development. Finite-dimensional modal representation capturing the dominant dynamics of the PDEs system is derived for controller design through Galerkin's method and modal decomposition technique. Tustin's discretization and Cayley transform are used to obtain infinite-dimensional discrete-time dynamic modal representations which are used in subsequent constrained controller design. The proposed discrete-time constrained model predictive control synthesis is constructed in a way that the objective function is only based on the low-order modal representation of the PDEs system, while higher-order modes are utilized only in the constraints of the PDEs state. Finally, the MPC formulations are successfully applied, via simulation results, to the PDEs system with input and state constraints.     

Boundary Control of a Flexible Robotic Manipulator With Output Constraints

In this study, we consider a boundary control problem of a flexible manipulator with input disturbances and output constraints, achieving pre‐set performance attributes on position tracking error and the deflection error at the end of the beam. The dynamics of the system are represented by partial differential equations (PDEs). With the Lyapunov's direct method, a boundary controller with disturbance observer is designed to regulate the angular position and suppress elastic vibration simultaneously. The proposed control scheme allows the errors to converge to an arbitrarily small residual set, with convergence rate larger than a pre‐specified value. Numerical simulations demonstrate the effectiveness of the proposed scheme.     

Active vibration control of large space flexible slewing truss using cable actuator with input saturation

This paper proposes a composite approach to implementing attitude tracking and active vibration control of a large space flexible truss system. The system dynamic model is based on Hamilton's principle and discretized using the finite difference method. A nonlinear attitude controller for position tracking is developed based on the input‐output linearization of the discretized system, which can effectively improve system performance compared with a traditional proportional‐differential feedback controller. A taut cable actuator scheme is presented to suppress tip vibration because the mechanical model is a large large‐span spatial structure; furthermore, because the cable has the feature of unilateral input saturation constraint, which can provide only a pulling force, a nonlinear quadratic regulator controller is developed by introducing a piecewise nonquadratic cost function to suppress the vibration of the flexible structure. To investigate the factors that influence the damping effects of the cable, the parametrically excited instability of a cable under 2 supports is analyzed. Simulation results illustrate that the proposed attitude controller can implement the task of position tracking, and the vibration suppression control law is shown to be optimal for functional performance with input saturation.     

Adaptive boundary control of a flexible manipulator with input saturation

In this study, we consider the anti-windup design as one of the approaches for the boundary control problem of a flexible manipulator in the presence of system parametric uncertainties, external disturbances and bounded inputs. The dynamics of the system are represented by partial differential equations (PDEs). Using the singular perturbation approach, the PDE model is divided into two simpler subsystems. With the Lyapunov's direct method, an adaptive boundary control scheme is developed to regulate the angular position and suppress the elastic vibration simultaneously and the adaptive laws are designed to compensate for the system parametric uncertainties and the disturbances. The proposed control scheme allows the application of smooth hyperbolic functions, which satisfy physical conditions and input restrictions, be easily realised. Numerical simulations demonstrate the effectiveness of the proposed scheme.     

Vibration control of the flexible manipulator with input constraints and external disturbances based on Nussbaum function

In this paper, a boundary control scheme based on the partial differential equation (PDE) model is proposed for the vibration control problem of the flexible manipulator with input constraints and external disturbances. Based on the backstepping method, two boundary controllers are designed to stabilize the position loop subsystem and the attitude loop subsystem, respectively, and auxiliary systems based on the smooth hyperbolic tangent function and Nussbaum function are designed in the controllers to deal with the input saturation and external disturbances. The Nussbaum function can overcome the difficulties in controller design and stability analysis caused by the derivatives of smooth hyperbolic tangent functions. The well-posedness of the closed-loop system is proven by employing the semigroup theory, and the uniformly bounded stability is proved by Lyapunov direct method. Finally, the performance of the proposed control laws is verified by numerical simulations.     

Boundary Control for a Vibrating String System with Bounded Input

In this study, we deal with the control problem of a vibrating string system under the condition of restricted input and external disturbance. The major objectives are the development of a boundary vibration control scheme for globally stabilizing the string system and simultaneously for compensating the effect of the input saturation, and the design of a disturbance observer for tracking the external disturbance. To this end, a boundary control is proposed based on smooth hyperbolic function to suppress the vibration and eliminate the input restriction effect, and a disturbance observer is employed to mitigate the external disturbance. The asymptotic stability of the controlled system is demonstrated employing the extended LaSalle's invariance principle. In order to verify the control performance of the proposed control, simulation results are presented by the choice of proper control design parameters.     

Boundary control of flexible aircraft wings for vibration suppression

In this paper, we propose a boundary control strategy for vibration suppression of two flexible wings. As a basic approach, Hamilton's principle is used to ascertain the system dynamic model, which includes governing equations – four partial differential equations and boundary conditions – several ordinary differential equations. Considering the coupled bending and torsional deformations of flexible wings, boundary control force and torque act on the fuselage to regulate unexpected deformations of flexible wings. Then, we present the stability analysis of the closed-loop system through Lyapunov's direct method. Simulations are carried out by using finite difference method. The simulation experimental results illustrate the significant effect of the developed control strategies.     

大加减速轴向移动系统自适应反步边界控制

针对具有大加减速和系统参数不确定性的轴向移动系统振动主动控制问题,基于Lyapunov直接法、经典反步控制和自适应技术,提出自适应反步边界控制算法对系统振动进行主动控制,并以符号函数处理边界扰动.同时,考虑不连续符号函数在工程上易引起输入颤振,设计饱和函数替代控制器中符号函数,以提升振动控制品质.所设计的控制算法能够补偿系统参数不确定性,确保闭环系统稳定性.数字仿真结果验证了所提出的控制算法的有效性.     

Feedback control design with vibration suppression for flexible air-breathing hypersonic vehicles

This paper investigates the problem of feedback control design with vibration suppression for a flexible air-breathing hypersonic vehicle （FAHV）. FAHV includes intricate coupling between the engine and flight dynamics, as well as complex interplay between flexible and rigid modes, which results in an intractable system for the control design. In this paper, a longitudinal model, which is described by a coupled system of ordinary differential equations （ODEs） and partial differential equations （PDEs）, is adopted. Firstly, a linearized ODE model for the rigid part is established around the trim condition, while vibration of the fuselage is described by PDEs. Secondly, based on the Lyapunov direct method, a control law via ODE state feedback and PDE boundary output feedback is designed for the system such that the closed-loop exponential stability is ensured. Finally, simulation results are given to illustrate the effectiveness of the proposed design method.     

Nonlinear Robust Adaptive Deterministic Control for Flexible Hypersonic Vehicles in the Presence of Input Constraint

A nonlinear deterministic robust control scheme is developed for a flexible hypersonic vehicle with input saturation. Firstly, the model analysis is conducted for the hypersonic vehicle model via the input‐output linearized technique. Secondly, the sliding mode manifold is designed based on homogeneity theory. Then an adaptive high order sliding mode control scheme is proposed to achieve tracking for the step change in altitude and velocity for hypersonic vehicles where the uncertainty boundary is unknown. Furthermore, the control input constraint is investigated and another new adaptive law is proposed to estimate the uncertainties and to guarantee the stability of the system with input saturation. Finally, the simulation results are provided to demonstrate the effectiveness of the proposed method.     

Vibration suppression and fault-tolerant control of an aerial refueling hose with multiple actuators

In this paper, we propose an adaptive fault-tolerant boundary vibration control approach for the flexible aerial refueling hose with variable length, variable speed, and multiple actuators. A distributed parameter system (DPS) is utilized to represent the dynamic behavior of the flexible refueling hose more precisely and accurately. Based on the established DPS model, we present a boundary vibration controller to suppress the vibration of the flexible refueling hose. In the controller, fault-tolerant control with multiple actuators is considered to tackle the failure issues, and both multiplicative and additive failures are discussed in the fault situation. Then, by the Lyapunov direct method, we prove that the stability of the closed-loop system is guaranteed under the proposed control approach. Numerical examples are presented to support the theoretical derivation.     

Audio steganography with AES for real-time covert voice over internet protocol communications

As a popular real-time service on the Internet, Voice over Internet Protocol （VoIP） communication attracts more and more attention from the researchers in the information security field. In this study, we proposed a VoIP steganographic algorithm with variable embedding capacities, incorporating AES and key distribution, to realize a real-time covert VoIP communication. The covert communication system was implemented by embedding a secret message encrypted with symmetric cryptography AES-128 into audio signals encoded by PCM codec. At the beginning of each VoIP call, a symmetric session key （SK） was assigned to the receiver with a session initiation protocol-based authentication method. The secret message was encrypted and then embedded into audio packets with different embedding algorithms before sending them, so as to meet the real- time requirements of VolP communications. For each audio packet, the embedding capacity was calculated according to the specific embedding algorithm used. The encryption and embedding processes were almost synchronized. The time cost of encryption was so short that it could be ignored. As a result of AES-based steganography, observers could not detect the hidden message using simple statistical analysis. At the receiving end, the corresponding algorithm along with the SK was employed to retrieve the original secret message from the audio signals. Performance evaluation with state-of-the-art network equipment and security tests conducted using the Mann-Whitney-Wilcoxon method indicated that the proposed steganographic algorithm is secure, effective, and robust.     

Adaptive nonlinear information fusion preview control for autonomous surface vessels subject to measurement noises and unknown input saturations

This study presents an adaptive nonlinear information fusion preview control (NIFPC) method for trajectory tracking of autonomous surface vessels (ASVs) subject to system uncertainty, measurement noise, and unknown input saturations. The NIFPC is developed based on the nonlinear information fusion estimation methodology, in which the system's future reference trajectory information, noise information, performance index requirements, and system dynamic model are all transformed into information equations related to control input, and then the current control action is obtained by fusing these previewed future information via the nonlinear information fusion optimal estimation. In order to avoid the unknown input saturation constraints, a fuzzy asymmetric saturated approximator (FASA) is designed and integrated into the controller, where the fuzzy logic system (FLS) is used to adaptively adjust the key boundary parameters of the approximator. As a result, the negative effects caused by system uncertainty and measurement noise can be effectively suppressed, while the completely unknown input saturation constraints in the system actuator are guaranteed not to be violated. The convergence of the tracking errors of the closed-loop system is guaranteed via Lyapunov stability theory. Numerical simulation results have been provided to demonstrate the satisfactory performance of the proposed control scheme.     

PDE-based event-triggered containment control of multi-agent systems with input delay under spatial boundary communication

This article studies the containment control problem for multi-agent systems with input delay under spatial boundary communication by employing an event-based approach. Firstly, the collective dynamics of multi-agent systems are described as parabolic partial differential equations (PDEs). Applying the integral transformation method developed in PDEs, the delayed parabolic PDEs are transformed into a series of new coupled PDE-PDE systems. Then, by taking the spatial boundary communication scheme into account and using the local boundary information, two boundary containment control protocols together with a dynamic event-triggered scheme (DETS) are designed, such that the states of all followers converge to a convex hull formed by multiple leader agents with and without input delay. The optimal protocols are given by minimizing the 2-norm of the designed control gain matrix, and the exclusion of the Zeno behavior is analyzed. Finally, a numerical simulation example is provided to support the main results.     

柔性卫星系统的振动自适应边界控制

针对受外部干扰和具有结构参数不确定性的柔性卫星系统,为了抑制其振动和避免控制溢出问题,采用Hamilton变分原理和Euler-Bernoulli梁理论建立了结构无穷维偏微分方程模型,随后基于该无穷维模型设计了带有干扰自适应律的自适应边界控制对柔性卫星振动进行主动控制,并证明了闭环柔性卫星控制系统解的存在性、唯一性和收敛性.最后,仿真结果验证了所设计的自适应边界控制算法的有效性.     

Boundary Control for a Flexible Inverted Pendulum System Based on a PDE Model with Input Saturation

This research considers the control problem of a flexible inverted pendulum system (FIPS) in the presence of input saturation. The model for a flexible inverted pendulum system (FIPS) is derived via the Hamilton principle. The FIPS model is divided into a fast subsystem and a slow subsystem via the singular perturbation method. We introduce an auxiliary system to deal with the input saturation of a fast subsystem. To stabilize the fast subsystem, a boundary anti‐windup control force is applied at the free end of the beam. It is proven that the closed‐loop subsystem is stable. For the slow subsystem, a sliding mode control method is employed to design a controller and a new decoupling method to design the sliding surface. Then it is shown that the slow subsystem is stable. Finally, simulation results are provided to confirm the efficacy of the proposed controller.     

分布参数系统的时空ARX建模及预测控制

本文针对一类可由抛物型偏微分方程描述的分布参数系统,研究了一种基于输入输出数据的建模与控制方法.首先利用Karhunen-Loève（K-L）分解提取系统的一组主导空间基函数,并以此对系统输出进行时空分解,随后由时空分解得到的时间系数部分以及系统激励构成输入输出信息,利用最小二乘法辨识出时域ARX模型,最后针对该模型设计了广义预测控制器.仿真结果表明,上述控制方法能够对分布参数系统取得良好的控制效果.     

Robust adaptive control of a thruster assisted position mooring system

In this paper, robust adaptive control is developed for a thruster assisted position mooring system in the transverse direction. To provide an accurate and concise representation for the dynamic behavior of the mooring system, the flexible mooring lines are modeled as a distributed parameter system of partial differential equations (PDEs). The proposed control is applied at the top boundary of the mooring lines for station keeping via Lyapunov’s direct method. Adaptive control is designed to handle the system parametric uncertainties. With the proposed robust adaptive control, uniform boundedness of the system under the ocean current disturbance is achieved. The proposed control is implementable with actual instrumentations since all the signals in the control can be measured by sensors or calculated by using a backward difference algorithm. The effectiveness of the proposed control is verified by numerical simulations.