Boundary control for a constrained two-link rigid–flexible manipulator with prescribed performance

In this paper, we consider a boundary control problem for a constrained two-link rigid–flexible manipulator. The nonlinear system is described by hybrid ordinary differential equation–partial differential equation (ODE–PDE) dynamic model. Based on the coupled ODE–PDE model, boundary control is proposed to regulate the joint positions and eliminate the elastic vibration simultaneously. With the help of prescribed performance functions, the tracking error can converge to an arbitrarily small residual set and the convergence rate is no less than a certain pre-specified value. Asymptotic stability of the closed-loop system is rigorously proved by the LaSalle's Invariance Principle extended to infinite-dimensional system. Numerical simulations are provided to demonstrate the effectiveness of the proposed controller.     

Static output feedback control design for linear MIMO systems with actuator dynamics governed by diffusion PDEs

This paper deals with the problem of static output feedback (SOF) control design for a class of diffusion partial differential equation (PDE) and ordinary differential equation (ODE) cascades, where the ODE model is used to describe the dynamics of the multi-input and multi-output (MIMO) plant and the diffusion PDE model is employed to represent the dynamics of actuators. The objective of this paper is to develop a simple as well as effective SOF controller via the Lyapunov's direct method such that the resulting closed-loop system is globally exponentially stable. By constructing a quadratic Lyapunov function, the sufficient condition on the globally exponential stability of the closed-loop cascaded system is presented in terms of linear matrix inequality (LMI). Then, an LMI-based design method of the SOF controller is developed on the basis of the obtained stability analysis result. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed design method.     

具有外部干扰的不稳定剪切梁的输入–状态稳定

本文针对一端受到范德华力的不稳定剪切梁方程,考虑其输入–状态稳定性问题.通过可逆变换把方程等价地变成一个具有反馈循环的2×2的一阶运输方程与常微分方程的耦合系统.通过自抗扰控制方法,给出具有时变增益的扩张状态观测器来估计干扰.应用Backstepping变换和干扰估计量,设计系统的反馈控制来补偿系统本身的不稳定以及消除匹配干扰.通过C₀–半群方法证明闭环系统的适定性,以及Lyapunov方法证明闭环系统的输入–状态稳定性.数值仿真验证理论结果的正确性.     

Static output feedback control via PDE boundary and ODE measurements in linear cascaded ODE–beam systems

This paper addresses the problem of feedback control design for a class of linear cascaded ordinary differential equation (ODE)–partial differential equation (PDE) systems via a boundary interconnection, where the ODE system is linear time-invariant and the PDE system is described by an Euler–Bernoulli beam (EBB) equation with variable coefficients. The objective of this paper is to design a static output feedback (SOF) controller via EBB boundary and ODE measurements such that the resulting closed-loop cascaded system is exponentially stable. The Lyapunov’s direct method is employed to derive the stabilization condition for the cascaded ODE–beam system, which is provided in terms of a set of bilinear matrix inequalities (BMIs). Furthermore, in order to compute the gain matrices of SOF controllers, a two-step procedure is presented to solve the BMI feasibility problem via the existing linear matrix inequality (LMI) optimization techniques. Finally, the numerical simulation is given to illustrate the effectiveness of the proposed design method.     

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.     

Finite-dimensional constrained fuzzy control for a class of nonlinear distributed process systems.

This correspondence studies the problem of finite-dimensional constrained fuzzy control for a class of systems described by nonlinear parabolic partial differential equations (PDEs). Initially, Galerkin's method is applied to the PDE system to derive a nonlinear ordinary differential equation (ODE) system that accurately describes the dynamics of the dominant (slow) modes of the PDE system. Subsequently, a systematic modeling procedure is given to construct exactly a Takagi-Sugeno (T-S) fuzzy model for the finite-dimensional ODE system under state constraints. Then, based on the T-S fuzzy model, a sufficient condition for the existence of a stabilizing fuzzy controller is derived, which guarantees that the state constraints are satisfied and provides an upper bound on the quadratic performance function for the finite-dimensional slow system. The resulting fuzzy controllers can also guarantee the exponential stability of the closed-loop PDE system. Moreover, a local optimization algorithm based on the linear matrix inequalities is proposed to compute the feedback gain matrices of a suboptimal fuzzy controller in the sense of minimizing the quadratic performance bound. Finally, the proposed design method is applied to the control of the temperature profile of a catalytic rod.     

Analysis and control of parabolic PDE systems with input constraints

This paper develops a general framework for the analysis and control of parabolic partial differential equations (PDE) systems with input constraints. Initially, Galerkin's method is used for the derivation of ordinary differential equation (ODE) system that capture the dominant dynamics of the PDE system. This ODE systems are then used as the basis for the synthesis, via Lyapunov techniques, of stabilizing bounded nonlinear state and output feedback control laws that provide an explicit characterization of the sets of admissible initial conditions and admissible control actuator locations that can be used to guarantee closed-loop stability in the presence of constraints. Precise conditions that guarantee stability of the constrained closed-loop parabolic PDE system are provided in terms of the separation between the fast and slow eigenmodes of the spatial differential operator. The theoretical results are used to stabilize an unstable steady-state of a diffusion-reaction process using constrained control action.     

A galerkin/neural-network-based design of guaranteed cost control for nonlinear distributed parameter systems.

This paper presents a Galerkin/neural-network- based guaranteed cost control (GCC) design for a class of parabolic partial differential equation (PDE) systems with unknown nonlinearities. A parabolic PDE system typically involves a spatial differential operator with eigenspectrum that can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement. Motivated by this, in the proposed control scheme, Galerkin method is initially applied to the PDE system to derive an ordinary differential equation (ODE) system with unknown nonlinearities, which accurately describes the dynamics of the dominant (slow) modes of the PDE system. The resulting nonlinear ODE system is subsequently parameterized by a multilayer neural network (MNN) with one-hidden layer and zero bias terms. Then, based on the neural model and a Lure-type Lyapunov function, a linear modal feedback controller is developed to stabilize the closed-loop PDE system and provide an upper bound for the quadratic cost function associated with the finite-dimensional slow system for all admissible approximation errors of the network. The outcome of the GCC problem is formulated as a linear matrix inequality (LMI) problem. Moreover, by using the existing LMI optimization technique, a suboptimal guaranteed cost controller in the sense of minimizing the cost bound is obtained. Finally, the proposed design method is applied to the control of the temperature profile of a catalytic rod.     

Robust boundary control of an axially moving string by using a PR transfer function

In this note, a scheme for the vibration suppression of a translating string using a positive real (PR) transfer function is investigated. The transverse vibration of the string is controlled by hydraulic touch-rolls located at the right end of the string. The mathematical model of the system, which consists of a hyperbolic partial differential equation (PDE) describing the dynamics of the moving string and an ordinary differential equation (ODE) for the actuator dynamics, is derived by using Hamilton's principle for translating continua. The transfer function of the proposed boundary controller is a nonproper but PR function. The asymptotic stability of the closed-loop system in the presence of output disturbance is proved.     

一类具有复杂执行器动态的双曲线型偏微分方程输出调节

本文研究了一类具有边界执行器动态特性的双曲线型偏微分方程(Partial differential equation, PDE)系统的输出调节问题. 特别地, 执行器由一组非线性常微分方程(Ordinary differential equation, ODE)描述, 控制输入出现在执行器的一端而非直接作用在PDE系统上, 这使得控制任务变得相当困难. 基于几何设计方法和有限维与无限维反步法, 本文提出了显式表达的输出调节器, 实现了该类系统的扰动补偿及跟踪控制. 并且我们采用Lyapunov稳定性理论严格证明了闭环系统及跟踪误差在范数意义上的指数稳定性. 仿真实例对比验证了所提出控制方法的有效性.     

Adaptive output-feedback stabilisation for hybrid PDE–ODE systems with uncertain input disturbances

This paper is concerned with the adaptive output-feedback stabilisation for a class of hybrid partial differential equation (PDE)–ordinary differential equation (ODE) systems with uncertain input disturbances. Remarkably, in this paper, only two boundary measurements of the considered system are available for feedback. Moreover, the unknown parameters involved in the system are allowed to belong to an unknown interval. These two aspects make the considered system essentially different from those in the closely related literature. Inspired by the existing results, an observer is first introduced to estimate the unmeasured states of the original system. Then, by adaptive technique and backstepping method, an adaptive output-feedback controller is successfully constructed, which guarantees that the entire closed-loop system is asymptotically stable, and moreover, the parameter estimates converge to their own real values ultimately. Besides, by the semigroup approach, the well-posedness of the entire closed-loop system is achieved. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed method.     

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.     

Adaptive mesh generation for singularly perturbed fourth-order ordinary differential equations

In this paper, we consider a singularly perturbed boundary-value problem for fourth-order ordinary differential equation (ODE) whose highest-order derivative is multiplied by a small perturbation parameter. To solve this ODE, we transform the differential equation into a coupled system of two singularly perturbed ODEs. The classical central difference scheme is used to discretize the system of ODEs on a nonuniform mesh which is generated by equidistribution of a positive monitor function. We have shown that the proposed technique provides first-order accuracy independent of the perturbation parameter. Numerical experiments are provided to validate the theoretical results.     

Reduction of distributed system identification complexity using intelligent sensors

The identification problems of distributed-parameter systems (DPS) is dealt with, A special method of data collection by using moving sensors is proposed. The sensors are ‘intelligent’ in the sense that they are able to track the positions of the exiremum values of a system state at each time instant. It is shown how to reduce the identification problem for DPS to that of identifying a certain system described by ordinary differential equations (ODE). The proposed approach is applicable to the DPS described by quasi-linear partial differential equations of the hyperbolic type. Attention is focused on the problem of reduction, while the identification of the resulting ODE is not considered in detail, since it is an easier task.     

Stabilization of a class of nonlinear uncertain ordinary differential equation by parabolic partial differential equation controller

This article is concerned with stabilization for a class of uncertain nonlinear ordinary differential equation (ODE) with dynamic controller governed by linear 1?d heat partial differential equation (PDE). The control input acts at the one boundary of the heat's controller domain and the second boundary injects a Dirichlet term in ODE plant. The main contribution of this article is the use of the recent infinite‐dimensional backstepping design for state feedback stabilization design of coupled PDE‐ODE systems, to stabilize exponentially the nonlinear uncertain systems, under the restrictions that (a) the right‐hand side of the ODE equation has the classical particular form: linear controllable part with an additive nonlinear uncertain function satisfying lower triangular linear growth condition, and (b) the length of the PDE domain has to be restricted. We solve the stabilization problem despite the fact that all known backstepping transformation in the literature cannot decouple the PDE and the ODE subsystems. Such difficulty is due to the presence of a nonlinear uncertain term in the ODE system. This is done by introducing a new globally exponentially stable target system for which the PDE and ODE subsystems are strongly coupled. Finally, an example is given to illustrate the design procedure of the proposed method.     

Distributed learning and cooperative control for multi-agent systems

This paper presents an algorithm and analysis of distributed learning and cooperative control for a multi-agent system so that a global goal of the overall system can be achieved by locally acting agents. We consider a resource-constrained multi-agent system, in which each agent has limited capabilities in terms of sensing, computation, and communication. The proposed algorithm is executed by each agent independently to estimate an unknown field of interest from noisy measurements and to coordinate multiple agents in a distributed manner to discover peaks of the unknown field. Each mobile agent maintains its own local estimate of the field and updates the estimate using collective measurements from itself and nearby agents. Each agent then moves towards peaks of the field using the gradient of its estimated field while avoiding collision and maintaining communication connectivity. The proposed algorithm is based on a recursive spatial estimation of an unknown field. We show that the closed-loop dynamics of the proposed multi-agent system can be transformed into a form of a stochastic approximation algorithm and prove its convergence using Ljung’s ordinary differential equation (ODE) approach. We also present extensive simulation results supporting our theoretical results.     

Boundary Value Technique for Finding Numerical Solution to Boundary Value Problems for Third Order Singularly Perturbed Ordinary Differential Equations

A class of singularly perturbed two point boundary value problems (BVPs) for third order ordinary differential equations is considered. The BVP is reduced to a weakly coupled system of one first order Ordinary Differential Equation (ODE) with a suitable initial condition and one second order singularly perturbed ODE subject to boundary conditions. In order to solve this system, a computational method is suggested in this paper. This method combines an exponentially fitted finite difference scheme and a classical finite difference scheme. The proposed method is distinguished by the fact that, first we divide the domain of definition of the differential equation into three subintervals called inner and outer regions. Then we solve the boundary value problem over these regions as two point boundary value problems. The terminal boundary conditions of the inner regions are obtained using zero order asymptotic expansion approximation of the solution of the problem. The present method can be extended to system of two equations, of which, one is a first order ODE and the other is a singularly perturbed second order ODE. Examples are presented to illustrate the method.     

On-line estimation of the time delay via orthogonal collocation

The objective of this work is to develop a new real-time time-delay estimator for the input delay of the otherwise finite-dimensional single-input single-output linear differential system. The method applies also to tracking of possible time-variations in the delay. The key idea in the construction of the estimator is to transform the delay part of the system into a transport system described by a linear partial differential equation (PDE). The PDE is linearly parametrized by the delay argument. The PDE is approximated by a finite-dimensional ordinary differential equation (ODE) system. The approximation is obtained by applying orthogonal collocation in the spatial descretization of the PDE. A fixed-interval exponentially weighted cost functional is then applied in the ODE system in formulating the delay estimation problem in mathematical terms. The data forgetting issue allows tracking of changes in the delay. Minimization of the cost functional is carried out approximately to obtain a finite-dimensional recursive delay estimator for increasing values of the final time of the integral in the cost. Simulations for a fixed as well as for a time-varying delay are given to demonstrate estimation and tracking properties of the identifier.     

基于特征线法的双曲型分布参数模型预测控制及其应用

本文针对双曲型分布参数系统提出基于特征线法的模型预测控制算法. 通过特征线变换将描述分布参数模型的偏微分方程转化为常微分方程; 进而求解得到分布参数系统状态变量的解析式; 离散化后作为预测模型用于模型预测控制. 以循环流化床烟气脱硫系统中SO2浓度控制为例, 进行仿真研究, 结果表明基于特征线法的模型预测控制算法可以实现对双曲型分布参数系统的有效控制, 并且该算法的控制效果优于目前工程应用的前馈反馈 控制策略.