Boundary geometric control of a linear Stefan problem

This paper addresses the geometric control of the position of a liquid–solid interface in a melting process of a material known as Stefan problem. The system model is hybrid, i.e. the dynamical behavior of the liquid-phase temperature is modeled by a heat equation while the motion of the moving boundary is described by an ordinary differential equation. The control is applied at one boundary as a heat flux and the second moving boundary represents the liquid–solid interface whose position is the controlled variable. The control objective is to ensure a desired position of the liquid–solid interface. The control law is designed using the concept of characteristic index, from geometric control theory, directly issued from the hybrid model without any reduction of the partial differential equation. It is shown by use of Lyapunov stability test that the control law yields an exponentially stable closed-loop system. The performance of the developed control law is evaluated through simulation by considering zinc melting.     

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.     

Almost sure stability for uncertain differential equation with jumps

Uncertain differential equation with jumps, as a crucial tool to deal with a discontinuous uncertain system, is a type of differential equation driven by both canonical Liu process and uncertain renewal process. So far, a concept of stability in measure for an uncertain differential equation with jumps has been proposed. As a supplement, this paper proposes a concept of almost sure stability for an uncertain differential equation with jumps. A sufficient condition is derived for an uncertain differential equation with jumps being stable almost surely. As a corollary, a sufficient condition is also given for a linear uncertain differential equation with jumps being stable almost surely.     

热—板传输系统的稳定性分析

本文研究由处于相邻区域的板方程和热方程构成的耦合系统的稳定性质, 其中耦合来自两个区域的交界 面上的传输边界条件. 在该传输系统中, 热方程起着控制器的作用, 且耗散通过交界面传输并影响板方程. 文献[1] 证明了在板方程上施加额外的控制器时, 该二维传输系统的能量呈指数衰减. 通过应用频域方法, 椭圆方程的正则 性理论等, 可以得到: 仅由热方程的耗散即可使得闭环系统指数稳定. 这一指数稳定的结论与相应的一维传输系统 的性质吻合. 最后, 文章还分析了不同传输边界条件下的板–热耦合系统的稳定性.     

ODE–热方程级联系统的事件触发控制

本文针对常微分方程(ODE)耦合偏微分方程(PDE)建模的分布式参数多智能体系统进行研究, 针对一致性同步问题, 提出了事件触发的网络化ODE–热方程级联系统多智能体一致性边界交互协议. 本文考虑的热方程左边界为Neumann边界条件, 并且与ODE系统耦合, 右边界为绝热边界条件. 假设网络化多智能体系统的连接方式为全联通有向拓扑图, 给出ODE–热方程级联系统的多智能体的一致性控制协议. 另外针对现有数字式控制器, 设计了事件触发的一致性控制协议, 并利用李雅普诺夫函数验证了在事件触发条件下ODE–热方程级联系统的稳定性. 最后给出了由5个ODE–热方程级联的多智能体系统的仿真结果, 验证了事件触发控制器的有效性.     

Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties

This paper considers the stabilization problem of a one-dimensional unstable heat conduction system (rod) modeled by a parabolic partial differential equation (PDE), powered with a Dirichlet type actuator from one of the boundaries. By applying the Volterra integral transformation, a stabilizing boundary control law is obtained to achieve exponential stability in the ideal situation when there are no system uncertainties. The associated Lyapunov function is used for designing an infinite-dimensional sliding manifold, on which the system exhibits the same type of stability and robustness against certain types of parameter variations and boundary disturbances. It is observed that the relative degree of the chosen sliding function with respect to the boundary control input is zero. A continuous control law satisfying the reaching condition is obtained by passing a discontinuous (signum) signal through an integrator.     

具有自适应噪声边界的Tube可达集鲁棒预测控制

针对过程噪声设定边界与真实噪声边界失配的有界干扰离散线性不确定系统,提出一种具有自适应噪声边界的Tube可达集鲁棒模型预测控制方法.首先,该算法引入基于MIT规则的自适应集员滤波在线估计系统状态和噪声边界.其次,基于估计值,通过迭代自适应集员滤波的时间更新部分计算出预测时域内闭环不确定系统状态的可达集.最后,用可达集代替不变集并根据Tube鲁棒模型预测控制策略,给出了实际不确定系统的控制律,确保系统状态鲁棒渐近稳定,并收敛于终端干扰不变集.仿真结果验证了该控制方法的有效性.     

Stabilization of the Euler–Bernoulli equation via boundary connection with heat equation

In this paper, we are concerned with the stabilization of a coupled system of Euler–Bernoulli beam or plate with heat equation, where the heat equation (or vice versa the beam equation) is considered as the controller of the whole system. The dissipative damping is produced in the heat equation via the boundary connections only. The one-dimensional problem is thoroughly studied by Riesz basis approach: The closed-loop system is showed to be a Riesz spectral system and the spectrum-determined growth condition holds. As the consequences, the boundary connections with dissipation only in heat equation can stabilize exponentially the whole system, and the solution of the system has the Gevrey regularity. The exponential stability is proved for a two dimensional system with additional dissipation in the boundary of the plate part. The study gives rise to a different design in control of distributed parameter systems through weak connections with subsystems where the controls are imposed.     

网络化热方程的集中事件触发控制

本文对热方程建模的分布式参数化多智能体系统进行研究,设计了基于事件触发控制下的一致性边界控制器,将网络化热方程的状态驱动到相同的稳定状态.其中每一个子系统的边界信息能被测量,并且所有的智能体由无向静态拓扑连接.事件触发控制器由以下两部分组成:一是基于网络拓扑的边界局部交互作用,驱动所有子系统达到相同的状态;二是由事件触发条件建立的触发时刻.本文证明了在事件触发的边界一致性控制下两个连续触发时刻之间存在最小停留时间以避免Zeno现象;同时利用李雅普诺夫函数分析并保证了闭环系统的稳定性和适定性.最后,给出了由5个热方程组成的多智能体系统的仿真算例,结果证实了本文所设计事件触发控制器的真实性.     

Robust stabilization of stochastic systems based on the LQ controller

The robust exponential stability in mean square for a class of linear stochastic uncertain control systems is dealt with. For the uncertain stochastic systems, we have designed an optimal controller which guarantees the exponential stability of the system. Actually, we employed Lyapunov fimction approach and the stochastic algebraic Riccati equation (SARE) to have shown the robusmess of the linear quadratic(LQ) optimal control law.And the algebraic criteria for the exponential stability on the linear stochastic uncertain closed-loop systems are given.     

Robust stabilization of stochastic systems based on the LQ cont roller

The robust exponential stability in mean square for a class of linear stochastic uncertain control systems is dealt with. For the uncertain stochastic systems ,we have designed an optimal controller which guarantees the exponential stability of the system. Actually ,we employed Lyapunov function approach and the stochastic algebraic Riccati equation (SARE) to have shown the robustness of the linear quadratic (LQ) optimal control law. And the algebraic criteria for the exponential stability on the linear stochastic uncertain closed- loop systems are given.     

Control effort considerations in the stabilization of uncertain dynamical systems

This paper investigates the problem of designing a state feedback control to stabilize an uncertain nonlinear system. We focus attention on the amplitude (norm) of the controller which is used to achieve this end. The uncertain system is described by a state equation which contains uncertain parameters which are unknown but bounded. A Lyapunov function is used to establish the stability of the closed loop system. The paper gives necessary and sufficient concitions for the uncertain system to be stabilizable with a given Lyapunov function. Furthermore, a procedure is indicated for the construction of the desired feedback control law.     

Distributed control of nonlinear diffusion systems by input–output linearization

This paper addresses the distributed control by input–output linearization of a nonlinear diffusion equation that describes a particular but important class of distributed parameter systems. Both manipulated and controlled variables are assumed to be distributed in space. The control law is designed using the concept of characteristic index from geometric control by using directly the PDE model without any approximation or reduction. The main idea consists in the control design in assuming an equivalent linear diffusion equation obtained by use of the Cole–Hopf transformation. This framework helps to demonstrate the closed‐loop stability using some concepts from the powerful semigroup theory. The performance of the proposed controller is successfully tested, through simulation, by considering a nonlinear heat conduction problem concerning the control of the temperature of a steel plate modeled by a nonlinear heat equation with Dirichlet boundary conditions. Copyright © 2012 John Wiley & Sons, Ltd.     

一类不确定动态时滞系统的无记忆鲁棒镇定控制

针对状态和控制均存在滞后,同时具有未知且有界的一类时变不确定线性时滞系统,提出了一种无记忆鲁棒镇定控制器设计算法.给出了闭环系统二次稳定的充分条件,并利用一等价线性时不变系统的H∞标准问题综合方法来构造出所需的线性状态反馈控制律,即可通过求解一代数Riccati型方程来求得控制律静态增益阵,从而保证了解的存在性和可解性.     

A stabilization algorithm for a class of uncertain linear systems

This paper presents an algorithm for the stabilization of a class of uncertain linear systems. The uncertain systems under consideration are described by state equations which depend on time-varying unknown-but-bounded uncertain parameters. The construction of the stabilizing controller involves solving a certain algebraic Riccati equation. Furthermore, the solution to this Riccati equation defines a quadratic Lyapunov function which is used to establish the stability of the closed-loop system. This leads to a notion of ‘quadratic stabilizability’. It is shown that the stabilization procedure will succeed if and only if the given uncertain linear system is quadratically stabilizable.The paper also deals with a notion of ‘overbounding’ for uncertain linear systems. This procedure enables the stabilization algorithm to be applied to a larger class of uncertain linear systems. Also included in the paper are results which indicate the degree of conservativeness introduced by this overbounding process.     

基于内流动力学的海洋输油柔性立管鲁棒边界控制

海洋输油柔性立管的振动是引起立管疲劳破坏的主要原因,对其研究边界控制是消除振动疲劳、减少断裂的有效方法.本文引入内流动力学,完善了立管原始无穷维分布参数模型,更好地表达了柔性立管的动力学响应.为抑制柔性立管在内外流激励下的振动奠下基础,本文用Lyapunov直接法对柔性立管系统的稳定性和状态一致有界性进行了证明,设计了边界控制器调节柔性立管的振动,其中控制器使用了符号函数来消除不确定性环境扰动对振动控制效果的影响,提高了系统的鲁棒性.仿真实验表明本文所设计的控制算法有效地减少了柔性立管的振动偏移量.     

Boundary feedback stabilization of a Schr$\＂{o}$dinger equation interconnected with a heat equation

In this paper, we study stabilization for a Schoedinger equation, which is interconnected with a heat equation via boundary coupling. A direct boundary feedback control is adopted. By a detailed spectral analysis, it is found that there are two branches of eigenvalues： one is along the negative real axis, and the other is approaching to a vertical line, which is parallel to the imagine axis. Moreover, it is shown that there is a set of generalized eigenfunctions, which forms a Riesz basis for the energy state space. Finally, the spectrum-determined growth condition is held and the exponential stability of the system is then concluded.     

Absolute stability via boundary control of a semilinear parabolic PDE

We consider the problem of achieving global absolute stability of an unstable equilibrium solution of a semilinear dissipative parabolic partial differential equation (PDE) through boundary control. The state space of the system is extended in order to write the action of the boundary control as an unbounded operator in an abstract evolution equation. Absolute stability via boundary control is accomplished by analyzing a control Lyapunov function based on the infinite-dimensional dynamics and applying a finite-dimensional linear quadratic regulator (LQR) controller. Sufficient conditions for absolute stability of the infinite-dimensional system are established by the feasibility of two finite-dimensional linear matrix inequalities (LMIs). Numerical results are presented for a Dirichlet boundary controlled system, however the analysis in this work applies to Nuemann and Robin type boundary controllers as well.     

Determination of inner boundaries in modified Helmholtz inverse geometric problems using the method of fundamental solutions

In this paper, an inverse geometric problem for the modified Helmholtz equation arising in heat conduction in a fin, which consists of determining an unknown inner boundary (rigid inclusion or cavity) of an annular domain from a single pair of boundary Cauchy data is solved numerically using the method of fundamental solutions (MFS). A nonlinear minimisation of the objective function is regularised when noise is added into the input boundary data. The stability of numerical results is investigated for several test examples.     

考虑测量不确定和输入饱和的充液航天器自适应鲁棒控制

本文研究了三轴稳定充液航天器控制系统中同时存在测量不确定,外部未知干扰,参数不确定和控制输入饱和的鲁棒自适应姿态机动控制问题.建模过程中,将晃动液体燃料等效为粘性球摆模型,采用动量矩守恒定律推导出充液航天器的耦合动力学方程.提出了一种将反步控制方法结合非线性干扰观测器和指令滤波器的鲁棒饱和输出反馈复合控制策略,该控制策略不仅能继承反步控制方法的优点,而且通过引入非线性干扰观测器实现对未知外部干扰,参数不确定以及测量不确定的补偿,还能利用指令滤波器处理控制力矩输入饱和的不利影响.基于Lyapunov稳定性分析方法证明了系统状态变量的渐进稳定性.仿真结果验证了提出控制方法的有效性和鲁棒性.