Regional output feedback stabilization of semilinear time‐fractional diffusion systems in a parallelepipedon with control constraints

This article is concerned with the regional output feedback stabilization problems for semilinear time‐fractional diffusion systems in a 1≤n?dimensional parallelepipedon with control inequality constraints. For this, the spectrum decomposition method is used to derive a finite‐dimensional fractional ordinary differential equation (ODE) system that captures the dominant dynamics of the considered system. With this ODE system, we propose a finite‐dimensional fractional compensator to guarantee that the constrained closed‐loop semilinear systems are Mittag‐Leffler stable in some subregions of their evolution domains. An example is finally included to illustrate our results.     

带边界非线性干扰的反稳定波方程的镇定

本文研究了一类具有边界控制匹配非线性干扰的反稳定波动方程的镇定问题. 本文只用了两个量测, 构造了一个无限维干扰估计器来实时估计状态和总干扰, 该估计器既不要求干扰的导数有界, 也不需要高增益. 基于估计的总干扰和估计的状态, 本文设计了输出反馈控制律稳定原系统. 此外, 本文还证明了闭环系统的其他状态是有界的. 为了说明理论结果, 下文给出了一些数值模拟.     

Sliding mode control and active disturbance rejection control to the stabilization of one‐dimensional Schrödinger equation subject to boundary control matched disturbance

In this paper, we are concerned with the boundary stabilization of a one‐dimensional anti‐stable Schrödinger equation subject to boundary control matched disturbance. We apply both the sliding mode control (SMC) and the active disturbance rejection control (ADRC) to deal with the disturbance. By the SMC approach, the disturbance is supposed to be bounded only. The existence and uniqueness of the solution for the closed‐loop system is proved and the ‘reaching condition’ is obtained. Considering the SMC usually requires the large control gain and may exhibit chattering behavior, we develop the ADRC to attenuate the disturbance for which the derivative is also supposed to be bounded. Compared with the SMC, the advantage of the ADRC is not only using the continuous control but also giving an online estimation of the disturbance. It is shown that the resulting closed‐loop system can reach any arbitrary given vicinity of zero as time goes to infinity and high gain tuning parameter goes to zero. Copyright © 2013 John Wiley & Sons, Ltd.     

Boundary stabilization of a cascade of ODE‐wave systems subject to boundary control matched disturbance

In this paper, we are concerned with a cascade of ODE‐wave systems with the control actuator‐matched disturbance at the boundary of the wave equation. We use the sliding mode control (SMC) technique and the active disturbance rejection control method to overcome the disturbance, respectively. By the SMC approach, the disturbance is supposed to be bounded only. The existence and uniqueness of solution for the closed‐loop via SMC are proved, and the monotonicity of the ‘reaching condition’ is presented without the differentiation of the sliding mode function, for which it may not always exist for the weak solution of the closed‐loop system. Considering that the SMC usually requires the large control gain and may exhibit chattering behavior, we then develop an active disturbance rejection control to attenuate the disturbance. The disturbance is canceled in the feedback loop. The closed‐loop systems with constant high gain and time‐varying high gain are shown respectively to be practically stable and asymptotically stable. Then we continue to consider output feedback stabilization for this coupled ODE‐wave system, and we design a variable structure unknown input‐type state observer that is shown to be exponentially convergent. The disturbance is estimated through the extended state observer and then canceled in the feedback loop by its approximated value. These enable us to design an observer‐based output feedback stabilizing control to this uncertain coupled system. Copyright © 2016 John Wiley & Sons, Ltd.     

Output feedback stabilization of an unstable wave equation with general corrupted boundary observation

We consider boundary output feedback stabilization for an unstable wave equation with boundary observation subject to a general disturbance. We adopt for the first time the active disturbance rejection control approach to stabilization for a system described by the partial differential equation with corrupted output feedback. By the approach, the disturbance is first estimated by a relatively independent estimator; it is then canceled in the feedback loop. As a result, the control law can be designed almost as that for the system without disturbance. We show that with a time varying gain properly designed, the observer driven by the disturbance estimator is convergent, and that all subsystems in the closed-loop are asymptotically stable in the energy state space. We also provide numerical simulations which demonstrate the convergence results and underline the effect of the time varying gain estimator on peaking value reduction.     

Output‐based disturbance rejection control for 1‐D anti‐stable Schrödinger equation with boundary input matched unknown disturbance

In this paper, we are concerned with the output feedback control design for a system (plant) described by a boundary controlled anti‐stable one‐dimensional Schrödinger equation. Our output measure signals are the displacements at both side. An untraditional infinite‐dimensional disturbance estimator is developed to estimate the disturbance. Based on the estimator, we propose a state observer that is exponentially convergent to the original system and then design a stabilizing control law consisting of two parts: The first part is to compensate the disturbance by using its approximated value and the second part is to stabilize the observer system by applying the classical backstepping approach. The resulting closed‐loop system is shown to be exponentially stable with guaranteeing that all internal systems are uniformly bounded. An effective output‐based disturbance rejection control algorithm is concluded. An application, namely, a cascade of ODE–wave systems, is investigated by the developed control algorithm. Numerical experiments are carried out to illustrate the effectiveness of the proposed control law. Copyright © 2017 John Wiley & Sons, Ltd.     

自抗扰控制对边界带有干扰的非线性sine-Gordon方程的镇定

本文讨论边界具有内部不确定和外部扰动的非线性sine-Gordon方程的镇定问题. 为处理sine-Gordon方程中的非线性项, 文章给出一个新的总扰动观测器在线估计未知扰动, 并通过自抗扰控制方法, 设计一个控制器使得在反馈控制中实时补偿(消除)总扰动. 闭环系统被证明适定的并且受控系统是指数稳定而扰动观测器是有界的. 数值模拟说明提出方法的有效性.     

自抗扰控制对具边界扰动和区间内反阻尼的波动方程的镇定

本文讨论边界具有外部扰动和区域内具有反阻尼的一维波动方程的的镇定问题. 主要的方法是后退反演变换和自抗扰控制方法. 即通过扩张状态观测器将扰动在线估计并在反馈控制中实时消除. 本文在扩张状态观测器中使用了两种增益调整策略——常数高增益与时变增益. 为避免常数高增益带来的峰值问题, 在控制环节中使用了饱和方法. 时变的增益可以在很大程度上减少扩张状态观测器中由于常数高增益引起的峰值问题同时可以达到完全消除干扰的镇定效果.     

Robust disturbance rejection for repetitive control systems with time‐varying nonlinearities

This paper presents a disturbance‐rejection method for a modified repetitive control system with a nonlinearity. Taking advantage of stable inversion, an improved equivalent‐input‐disturbance (EID) estimator that is more relaxed for system design is developed to estimate and cancel out the influence of the disturbance and nonlinearity in the low‐frequency domain. The high‐frequency influence is filtered owning to the low‐pass nature of the linear part of the closed‐loop system. To avoid the restrictive commutative condition and choose a Lyapunov function of a more general form, a new design algorithm, which takes into account the relation between the feedback control gains and the observer and improved EID estimator gains, is developed for the nonlinear system. Furthermore, comparisons with the generalized extended‐state observer (GESO) and conventional EID methods are conducted. A clear relation between the developed estimator and the GESO is also clarified. Finally, simulations show the effectiveness and the advantage of the developed method.     

Guaranteed-cost active disturbance rejection control for uncertain systems with an integral controller

Guaranteed-cost active disturbance rejection control (ADRC) for uncertain systems is investigated in this study. Firstly, an integral action is introduced in the framework of ADRC to measure and reduce the tracking error. Then, a robust stability condition is presented, and a quadratic cost function where the tracking error is appearing explicitly is used for ADRC performance assessment. The cost bound is formulated by linear matrix inequality and optimised to obtain controller parameters. Full-dimension extended state observer is used, and thus, the proposed strategy is applicable to an uncertain system that allows relative-degree varying or right-half-plane zero. Finally, the validity of the proposed method and its advantages is demonstrated through the simulations of comparative examples and experiments on a motor speed control system.     

具有多层感知器力矩补偿的机器人自抗扰控制

本文针对机器人系统的控制特性,提出了一种基于自抗扰控制(ADRC)的关节控制算法,该算法可以克服传统控制算法中存在的如系统抗干扰能力弱,控制性能受限于建模精度,动态性能与稳态性能难以兼顾,控制律设计较为复杂等问题.针对受控系统特性给出了一套实际控制器的完整设计方法与参数整定方法,并根据控制性能指标设计优化函数完成了最优控制参数的优化,在系统参数辨识的基础上利用多层感知器(MLP)设计了对建模不确定性的补偿网络.数值仿真和实验结果均表明该算法能够实现机器人快速稳定的轨迹跟踪,具有良好的控制精度与很强的抗干扰能力,此外该算法不依赖于精确的系统模型,降低了实际设计和应用的难度,具有很好的工程应用价值.     

Active disturbance rejection control for nonlinear fractional‐order systems

This paper investigates active disturbance rejection control involving the fractional‐order tracking differentiator, the fractional‐order PID controller with compensation and the fractional‐order extended state observer for nonlinear fractional‐order systems. Firstly, the fractional‐order optimal‐time control scheme is studied to propose the fractional‐order tracking differentiator by the Hamilton function and fractional‐order optimal conditions. Secondly, the linear fractional‐order extend state observer is offered to acquire the estimated value of the sum of nonlinear functions and disturbances existing in the investigated nonlinear fractional‐order plant. For the disturbance existing in the feedback output, the effect of the disturbance is discussed to choose a reasonable parameter in fractional‐order extended state observer. Thirdly, by this observed value, the nonlinear fractional‐order plant is converted into a linear fractional‐order plant by adding the compensation in the controller. With the aid of real root boundary, complex root boundary, and imaginary boot boundary, the approximate stabilizing boundary with respect to the integral and differential coefficients is determined for the given proportional coefficient, integral order and differential order. By choosing the suitable parameters, the fractional‐order active disturbance rejection control scheme can deal with the unknown nonlinear functions and disturbances. Finally, the illustrative examples are given to verify the effectiveness of fractional‐order active disturbance rejection control scheme. Copyright © 2015 John Wiley & Sons, Ltd.     

Active disturbance rejection and Lyapunov redesign approaches for robust boundary control of plate vibration

In this paper, two approaches, namely active disturbance rejection control (ADRC) and Lyapunov redesign, are utilised to stabilise the vibration of a boundary-controlled flexible rectangular plate in the presence of exogenous disturbances. Based on ADRC, an estimation/cancellation strategy is applied where disturbance is estimated online by an extended state observer (ESO) and cancelled by injecting the output of ESO into the feedback loop. By the Lyapunov redesign, on the other hand, the control law intended for a nominal system is redesigned by adding a (discontinuous) control component that makes the system robust to large uncertainties. Both control algorithms are designed directly based on partial differential equation model of the plate so that spillover instabilities that are a result of model truncation are avoided. The established control schemes are able to stabilise the plate vibration by actuating and sensing only along the plate boundary while accounting for the dynamical effects of Gaussian curvature integral, in-plane membrane force and actuator mass. The stability of each control approach is proven using Lyapunov analysis. The efficacy of each proposed control is illustrated by simulation results.     

Parameter estimation and stabilization for a wave equation with boundary output harmonic disturbance and non‐collocated control

This paper is concerned with the parameter estimation and stabilization of a one‐dimensional wave equation with harmonic disturbance suffered by boundary observation at one end and the non‐collocated control at the other end. An adaptive observer is designed in terms of measured velocity corrupted by harmonic disturbance with unknown magnitude. The backstepping method for infinite‐dimensional system is adopted in the design of the feedback law. It is shown that the resulting closed‐loop system is asymptotically stable. Meanwhile, the estimated parameter is shown to be convergent to the unknown parameter as time goes to infinity. Copyright © 2010 John Wiley & Sons, Ltd.     

Stabilization of ODE with hyperbolic equation actuator subject to boundary control matched disturbance

ABSTRACT

In this paper, we consider stabilisation for a cascade of ODE and first-order hyperbolic equation with external disturbance flowing to the control end. The active disturbance rejection control (ADRC) and sliding mode control (SMC) approaches are adopted in investigation. By ADRC approach, the disturbance is estimated through a disturbance estimator with both time-varying high gain and constant high gain, and the disturbance is canceled online in the feedback loop. It is shown that the resulting closed-loop system with time-varying high gain is asymptotically stable and is practically stable with constant high gain. By SMC approach, the existence and uniqueness of the solution for the closed loop via SMC are proved, and the monotonicity of the ‘reaching condition’ is presented. The resulting closed-loop system is shown to be exponentially stable. The numerical experiments are carried out to illustrate effectiveness of the proposed control law.     

双质量弹簧基准问题自抗扰控制研究

本文研究了自抗扰控制(ADRC)方法在双质量弹簧基准问题的应用.传统的自抗扰控制方法倾向于使用高增益控制来抑制扰动和模型不确定性,但是对于双质量弹簧基准问题,高增益在评分中受到较大惩罚,而且对于模型参数变化没有足够的鲁棒性.为解决这一问题,本文对ADRC设计提出了两种改进方案.首先,为了减小控制信号的幅度,将扩张状态观测器(ESO)的一个极点配置在原点.其次,采用阻尼比来调整带宽.结果表明,所提出的ADRC设计可以很好地解决双质量弹簧基准系统的控制问题.     

A composite fractional order active disturbance rejection control method for a class of uncertain systems with multiple disturbances

The existing active disturbance rejection control (ADRC) method may not provide sufficient disturbance rejection to multiple mismatched disturbances for the fractional order systems. In this paper, a composite disturbance rejection approach is developed for a class of fractional order uncertain systems, by synthesizing the fractional order ADRC (FOADRC) approach and a disturbance observer (DO)-based compensation scheme. Taking advantage of more disturbance information and a filter structure, an improved DO is developed to achieve precise estimation of disturbances in the presence of sensor noises. In addition, a state transformation is developed to convert the system into a simple integral chain model with only matched disturbances. Then a composite control law is designed to compensate the disturbances and provide satisfying dynamic performance. The efficiency of the proposed method is demonstrated by a numerical simulation and an actual servo control simulation, as well as the comparison with two kinds of the existing ADRC methods and the commonly used integral sliding mode control (I-SMC) method.     

压电定位系统的自抗扰控制设计

纳米定位系统中广泛采用的压电驱动器因存在非线性、多映射的迟滞特性而严重影响了纳米定位系统的定位精度.为消除迟滞对定位精度的影响,将其视为干扰,设计不基于迟滞及定位系统精确数学模型的自抗扰控制算法,利用扩张状态观测器实时估计迟滞,进而补偿其对定位精度的影响,获得了良好的定化系统控制仿真效果.仿真结果表明,自抗扰控制器能够仃效消除迟滞、提高纳米定位系统的定位精度.     

Sliding mode control for N‐coupled reaction‐diffusion PDEs with boundary input disturbances

This paper develops the sliding mode control (SMC) design for N‐coupled reaction‐diffusion parabolic PDEs with boundary input disturbances. In order to reject the disturbances, the backstepping‐based boundary SMC law is constructed to steer the system trajectory to a suitable sliding surface and then maintain sliding motion on the surface thereafter, resulting in the exponential convergence to the zero equilibrium state. The well‐posedness of the closed‐loop system is established based on a detailed spectral analysis and Riesz basis generation. Finally, a simulation example is provided to illustrate the effectiveness of the SMC design.