Stabilization and disturbance rejection for the wave equation

We consider a system described by the one-dimensional linear wave equation in a bounded domain with appropriate boundary conditions. To stabilize the system, we propose a dynamic boundary controller applied at the free end of the system. The transfer function of the proposed controller is a proper rational function of the complex variable and may contain a single pole at the origin and a pair of complex conjugate poles on the imaginary axis, provided that the residues corresponding to these poles are nonnegative; the rest of the transfer function is required to be a strictly positive real function. We then show that depending on the location of the pole on the imaginary axis, the closed-loop system is asymptotically stable. We also consider the case where the output of the controller is corrupted by a disturbance and show that it may be possible to attenuate the effect of the disturbance at the output if we choose the controller transfer function appropriately. We also present some numerical simulation results which support this argument     

Stabilization and disturbance rejection for the beam equation

We consider a system described by the Euler-Bernoulli beam equation. For stabilization, we propose a dynamic boundary controller applied at the free end of the system. The transfer function of the controller is a marginally stable positive real function which may contain poles on the imaginary axis. We then give various asymptotical and exponential stability results. We also consider the disturbance rejection problem     

Nonlinear control of a coupled PDE/ODE system modeling a switched power converter with a transmission line

We consider an infinite dimensional system modeling a boost converter connected to a load via a transmission line. The governing equations form a system coupling the telegraph partial differential equation with the ordinary differential equations modeling the converter. The coupling is given by the boundary conditions and the nonlinear controller we introduce. We design a nonlinear saturating control law using a Lyapunov function for the averaged model of the system. The main results give the well-posedness and stability properties of the obtained closed loop system.     

Transfer functions of distributed parameter systems: A tutorial

In this tutorial article the rich variety of transfer functions for systems described by partial-differential equations is illustrated by means of several examples under various boundary conditions. An important feature is the strong influence of the choice of boundary conditions on the dynamics and on system theoretic properties such as pole and zero locations, properness, relative degree and minimum phase. It is sometimes possible to design a controller using the irrational transfer function, and several such techniques are outlined. More often, the irrational transfer function is approximated by a rational one for the purpose of controller design. Various approximation techniques and their underlying theory are briefly discussed.     

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.     

具有对称初始数据的二维反应扩散方程的边界镇定

研究了二维圆盘上具有对称初始数据的反应扩散方程的边界控制. 由于初始条件和边界条件关于圆心旋转对称, 系统可以转化为等价的极坐标系下的一维抛物方程. 此时, 极点的奇异性成为了控制器设计中的难点. 本文设计了一系列方程变换, 消除了核函数方程中极点奇异性的影响, 将其转化为修正的Bessel方程, 求出了显式的核函数表达式和精确的边界反馈控制律, 扩展了偏微分方程的backstepping方法. 系统的收敛速度可通过改变控制器中的一个参数来调节. 然后用Lyapunov函数法证明了闭环系统在H1范数下指数稳定, 表明了系统对初值的连续依赖. 最后用数值仿真验证了方法的有效性.     

具有Neumann边界的分布参数切换系统的容错控制

本文研究了一类具有Neumann边界条件的分布参数切换系统的容错控制问题.当执行器失效或部分失效时,运用Lyapunov函数法和Green公式,获得了闭环切换系统混杂状态反馈容错控制器存在的充分条件.然后运用线性矩阵不等式将容错控制器设计问题转化为一组线性矩阵不等式求可行解的问题,因而可以借助MATLAB中线性矩阵不等式工具箱来完成.同时,运用Poincare不等式减少控制系统设计的保守性.最后通过数值算例,验证所提出设计方法的有效性.     

Stability of a feedback controlled distributed system by modal representation

In this work a linear distributed parameter system controlled by a finite number of lumped linear feedback loops is considered. The spatial portion of the system is decomposed using generalized Fourier expansions, and a complex frequency domain transcendental characteristic determinant results.

The theory developed is then applied to the automatic regulation of water flow in a ‘ reach ’ of an open irrigation canal. The dynamics of water flow in the transient state are described by a set of two first-order non-linear partial differential equations. The boundary conditions, governed by the water-flow dynamics under the gate, are also non-linear equations. For this particular system the feedback controller signal originates at the downstream boundary and controls the gates at the ‘ head ’ of the canal.

The stability analysis of the complete closed-loop system is based on perturbation about some steady-state flow conditions which allows for the linearization of the system's non-linear partial differential equations and boundary conditions. Subsequent application of the Nyquist criterion to a modal representation of the distributed parameter portion of the control loop in cascade with the lumped parameter controller yields the stability conditions.     

Feedback stabilisation of an Euler–Bernoulli beam with the boundary time-delay disturbance

In this paper, we consider the feedback stabilisation of an Euler–Bernoulli beam with the boundary time-delay disturbance. Due to unknown time-delay coefficient, the system might be exponentially increasing at the lack of control. We design the feedback control law based on Lyapunov function method. Different from usual use of Lyapunov function method, our approach is to combine the construction of Lyapunov functionals with the controller design, which will guarantee the system energy function decays exponentially. In this procedure, we deduce the inequality equations satisfied by the system parameters. We prove the well-posedness of the corresponding closed-loop system by using semigroup theory and the inequality equations are solvable. Moreover, the exponential decay rate of the system is estimated. In addition, some numerical simulations are also presented to support the obtained results.     

Switching fuzzy controller design based on switching Lyapunov function for a class of nonlinear systems

This paper presents a switching fuzzy controller design for a class of nonlinear systems. A switching fuzzy model is employed to represent the dynamics of a nonlinear system. In our previous papers, we proposed the switching fuzzy model and a switching Lyapunov function and derived stability conditions for open-loop systems. In this paper, we design a switching fuzzy controller. We firstly show that switching fuzzy controller design conditions based on the switching Lyapunov function are given in terms of bilinear matrix inequalities, which is difficult to design the controller numerically. Then, we propose a new controller design approach utilizing an augmented system. By introducing the augmented system which consists of the switching fuzzy model and a stable linear system, the controller design conditions based on the switching Lyapunov function are given in terms of linear matrix inequalities (LMIs). Therefore, we can effectively design the switching fuzzy controller via LMI-based approach. A design example illustrates the utility of this approach. Moreover, we show that the approach proposed in this paper is available in the research area of piecewise linear control.     

Stability analysis of the Euler-Bernoulli beam with multi-delay controller

In this paper, we investigate the stabilization of an Euler-Bernoulli beam with time delays in the boundary controller. The boundary velocity feedback law is applied to obtain the closed-loop system. It is shown that this system generates a C0- semigroup of linear operators. Moreover, the stability of the closed-loop system is discussed for different values of the controller constants and time delays via using spectral analysis and a suitable Lyapunov function.     

Extremal pole placement in control systems with a low order controller

We develop an optimizational approach to the design of linear control systems with a low order controller. The objective function is given by the right boundary of the location of poles in the closed-loop system, which depends on the values of controller parameters. We specify the types of mutual pole placements (root diagrams) corresponding to singular manifolds of such functions in the space of parameters and, in particular, their critical points. We establish the exact number of critical root diagrams depending on the dimension of the parameter space. With the example of finding a stabilizing control for a triple mathematical pendulum, we demonstrate the algebraic approach to finding the global minimum of the objective function.     

滞后抛物型控制系统的变结构控制

讨论了滞后抛物型控制系统的变结构控制问题。通过对系统进行直接分析,给出了仅由状态函数描述的变结构控制器的设计方法。所得结果削弱了文献[7,10]中对系统的某些要求,减化了变结构控制器的设计步骤,并且给出了整个切换面为滑动模态区的条件以及轨线到达滑动模态区上的时间估计。     

球对称n维反应扩散方程的边界观测与输出反馈控制

许多实际系统可用n 维超球坐标系来描述, 并且系统有球对称的性质, 因而可通过研究半径方向的状态变化, 得到系统的全局动态过程. 通过将高维的对称系统转化为等价的径向一维方程, 本文采用边界Backstepping 方法设计了球对称反应扩散方程的输出反馈控制器. 使用容易测量的边界状态值, 设计了状态观测器来估计系统在空间域的所有状态, 从而实现输出反馈控制. 本文扩展了连续Backstepping 方法,提出了n维球坐标的Volterra 积分映射, 从而求出了显式表达的控制器和状态观测器. 论文用Lyapunov 函数法证明了输出反馈系统在H1范数下指数稳定, 表明状态对初值的连续依赖, 确保控制系统具有较好的性质, 不会在空间某点发散. 最后进行了数值仿真, 仿真结果表明系统在输出反馈控制律的作用下收敛到稳态值.     

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.     

A STABLE SELF-LEARNING OPTIMAL FUZZY CONTROL SYSTEM

The issue of developing a stable self-learning optimal fuzzy control system is discussed in this paper. Three chief objectives are accomplished: 1) To develop a self-learning fuzzy controller based on genetic algorithms. In the proposed methodology, the concept of a fuzzy sliding mode is introduced to specify the system response, to simplify the fuzzy rules and to shorten the chromosome length. The speed of fuzzy inference and genetic evolution of the proposed strategy, consequently, is higher than that of the conventional fuzzy logic control. 2) To guarantee the stability of the learning control system. A hitting controller is designed to achieve this requirement. It works as an auxiliary controller and supports the self-learning fuzzy controller in the following manner. When the learning controller works well enough to allow the system state to lie inside a pre-defined boundary layer, the hitting controller is disabled. On the other hand, if the system tends to diverge, the hitting controller is turned on to pull the state back. The system is therefore stable in the sense that the state is bounded by the boundary layer. 3) To explore a fuzzy rule-base that can minimize a standard quadratic cost function. Based on the fuzzy sliding regime, the problem of minimizing the quadratic cost function can be transformed into that of deriving an optimal sliding surface. Consequently, the proposed learning scheme is directly applied to extract the optimal fuzzy rulebase. That is, the faster the hitting time a controller has and the shorter the distance from the sliding surface the higher fitness it possesses. The superiority of the proposed approach is verified through simulations.     

失重环境下可控柔性臂的模态特性

在非重力场中,考虑控制器动态反馈的影响,对存在控制器定位约束的柔性臂系统进行动力分析,研 究其在相对平衡位置的模态特性．以具有柔性关节和弹性臂杆的可控柔性臂为研究对象,分析了控制器作用下的反 馈约束特性,将控制器位置和速度增益引入力边界条件,得到了耦合控制器参量的模态特征方程,证明了反馈约束 的存在使得系统特征频率为复频率,且模态主振型是复变函数．通过数值仿真,明确了可控柔性臂的模态特性与控 制器增益之间的关系,得到了不同于经典振动理论的结论．设计了可控柔性臂的仿失重实验平台,试验模态结果证 明了理论分析的有效性．     

A supervisory controller for fuzzy control systems that guaranteesstability

A supervisory controller is a controller which operates only when some undesirable phenomena occur, e.g., when the state hits the boundary of constraint set. In this note, the author develops a supervisory controller for a nonlinear fuzzy control systems. The supervisory controller works in the following way: if the fuzzy control system (without the supervisory controller) is stable in the sense that the state is inside the constraint set, the supervisory control is idle; if the state hits the boundary of the constraint set, the supervisory controller begins operation to force the state back to the constraint set. The author proves that the fuzzy control system equipped with this supervisory controller is globally stable in the sense that the state is guaranteed to be within the constraint set specified by the system designer. The author also proposes schemes by continuously switching between supervisory and nonsupervisory modes. Finally, the author applies a fuzzy controller with the supervisory controller to the inverted pendulum balancing problem where it is required that the state variables must be within a fixed bound     

Sliding Mode Control for Flexible-link Manipulators Based on Adaptive Neural Networks

This paper mainly focuses on designing a sliding mode boundary controller for a single flexible-link manipulator based on adaptive radial basis function (RBF) neural network. The flexible manipulator in this paper is considered to be an Euler-Bernoulli beam. We first obtain a partial differential equation (PDE) model of single-link flexible manipulator by using Hamiltons approach. To improve the control robustness, the system uncertainties including modeling uncertainties and external disturbances are compensated by an adaptive neural approximator. Then, a sliding mode control method is designed to drive the joint to a desired position and rapidly suppress vibration on the beam. The stability of the closed-loop system is validated by using Lyapunov’s method based on infinite dimensional model, avoiding problems such as control spillovers caused by traditional finite dimensional truncated models. This novel controller only requires measuring the boundary information, which facilitates implementation in engineering practice. Favorable performance of the closed-loop system is demonstrated by numerical simulations.     

基于反步法的耦合分数阶反应扩散系统边界输出反馈控制

针对具有空间依赖耦合系数的分数阶反应扩散系统, 利用反步法设计了基于观测器的边界输出反馈控制器, 证明了观测增益和控制增益核函数矩阵方程的适定性. 针对误差系统和输出反馈的闭环系统, 利用分数阶Lyapunov方法分析了系统的Mittag-Leffler稳定性, 且利用Wirtinger不等式改进了耦合系统稳定的条件. 当系统具有空间依赖的耦合系数时, 难以求得控制增益和观测增益核函数的解析解, 为此, 给出了核函数偏微分方程的数值解方法. 数值仿真验证了理论结果.