Active disturbance rejection control to MIMO nonlinear systems with stochastic uncertainties: approximate decoupling and output-feedback stabilisation

ABSTRACT

In this paper, we apply the active disturbance rejection control, an emerging control technology, to output-feedback stabilisation for a class of uncertain multi-input multi-output nonlinear systems with vast stochastic uncertainties. Two types of extended state observers (ESO) are designed to estimate both unmeasured states and stochastic total disturbance which includes unknown system dynamics, unknown stochastic inverse dynamics, external stochastic disturbance without requiring the statistical characteristics, uncertain nonlinear interactions between subsystems, and uncertainties caused by the deviation of control parameters from their nominal values. The estimations decouple approximately the system after cancelling stochastic total disturbance in the feedback loop. As a result, we are able to design an ESO-based stabilising output-feedback and prove the practical mean square stability for the closed-loop system with constant gain ESO and the asymptotic mean square stability with time-varying gain ESO, respectively. Some numerical simulations are presented to demonstrate the effectiveness of the proposed output-feedback control scheme.     

基于扩张状态观测器的动态抗饱和补偿器设计方法

针对一类受外界扰动以及执行器饱和影响的不确定非线性系统,提出一种基于扩张状态观测器的动态抗饱和补偿器设计方法.首先通过将系统的不确定项以及外部扰动作为扩张状态,设计线性扩张状态观测器(ESO)对系统的总扰动进行估计;然后,在控制器中引入对扩张状态的估计值,对系统的总扰动进行补偿,设计了动态抗饱和补偿器,将控制器、观测器以及动态抗饱和补偿器的参数求解问题转化为基于LMI不等式组约束的优化问题,确保系统具有尽可能大的收敛域;最后通过数值仿真验证所提出设计方法的有效性.     

On convergence of active disturbance rejection control for a class of uncertain stochastic nonlinear systems

In this paper, the practical mean-square convergence of active disturbance rejection control for a class of uncertain stochastic nonlinear systems modelled by the Itô-type stochastic differential equations with vast stochastic uncertainties is developed. We first design an extended state observer (ESO) to estimate both the unmeasured states and the stochastic total disturbance which includes unknown internal system dynamics, external stochastic disturbance without known statistical characteristics, unknown stochastic inverse dynamics, and uncertainty caused by the deviation of control parameter from its nominal value. The stochastic total disturbance is then cancelled (compensated) in the feedback loop. An ESO-based output-feedback control is finally designed analogously as for the system without uncertainties. The practical mean-square reference tracking and practical mean-square stability of the resulting closed-loop system are achieved. The numerical experiments are carried out to illustrate the effectiveness of the proposed approach.     

一种基于反推技术和ESO补偿的解耦控制方法

针对一类非线性多入多出系统,设计了一种新的解耦控制器.首先采用反推技术设计每一个子系统的等效输入;然后用扩张状态观测器(ESO)对各个子系统的耦合、外扰和参数时变等不确定作用进行动态估计,并把估计值引入到反推控制器中进行补偿,最终实现解耦和摔制.利用Lyapunov稳定性理论证明了该方法的收敛性;数例仿真表明了该方法可获得较快的响应速度及较好的解耦效果和鲁棒性.     

Extended state observer for uncertain lower triangular nonlinear systems subject to stochastic disturbance

The extended state observer (ESO) is the most important part of an emerging control technology known as active disturbance rejection control to this day, aiming at estimating “total disturbance” from observable measured output. In this paper, we construct a nonlinear ESO for a class of uncertain lower triangular nonlinear systems with stochastic disturbance and show its convergence, where the total disturbance includes internal uncertain nonlinear part and external stochastic disturbance. The numerical experiments are carried out to illustrate effectiveness of the proposed approach.     

Active disturbance rejection control approach to output‐feedback stabilization of lower triangular nonlinear systems with stochastic uncertainty

In this paper, we apply the active disturbance rejection control approach to output‐feedback stabilization for uncertain lower triangular nonlinear systems with stochastic inverse dynamics and stochastic disturbance. We first design an extended state observer (ESO) to estimate both unmeasured states and stochastic total disturbance that includes unknown system dynamics, unknown stochastic inverse dynamics, external stochastic disturbance, and uncertainty caused by the deviation of control parameter from its nominal value. The stochastic total disturbance is then compensated in the feedback loop. The constant gain and the time‐varying gain are used in ESO design separately. The mean square practical stability for the closed‐loop system with constant gain ESO and the mean square asymptotic stability with time‐varying gain ESO are developed, respectively. Some numerical simulations are presented to demonstrate the effectiveness of the proposed output‐feedback control scheme. Copyright © 2016 John Wiley & Sons, Ltd.     

基于扩张状态观测器的 连铸结晶器振动位移系统自适应滑模控制

针对伺服电机驱动的连铸结晶器振动位移系统中存在时变负载转矩、参数不确定性等问题,本文提出了一种基于扩张状态观测器(extended state observer, ESO)的自适应非奇异终端滑模(nonsingular terminal sliding mode,NTSM)控制方法.首先,设计ESO对系统存在的综合扰动和不可测状态进行估计.然后,采用分层设计的方法,分别对位移跟踪子系统和电流环子系统设计基于ESO的自适应NTSM控制器和滑模控制器.为削弱ESO估计误差对跟踪精度的影响,在NTSM控制器中引入了自适应增益.可以证明,所设计的控制器能够保证闭环系统所有信号有界,系统状态可渐近收敛到原点附近的小邻域内.最后,仿真结果验证了所提出控制方法的有效性.     

Active disturbance rejection control approach to stabilization of lower triangular systems with uncertainty

In this paper, we apply the active disturbance rejection control (ADRC) to stabilization for lower triangular nonlinear systems with large uncertainties. We first design an extended state observer (ESO) to estimate the state and the uncertainty, in real time, simultaneously. The constant gain and the time‐varying gain are used in ESO design separately. The uncertainty is then compensated in the feedback loop. The practical stability for the closed‐loop system with constant gain ESO and the asymptotic stability with time‐varying gain ESO are proven. The constant gain ESO can deal with larger class of nonlinear systems but causes the peaking value near the initial stage that can be reduced significantly by time‐varying gain ESO. The nature of estimation/cancelation makes the ADRC very different from high‐gain control where the high gain is used in both observer and feedback. Copyright © 2015 John Wiley & Sons, Ltd.     

An approach to improve active disturbance rejection control

ABSTRACT

In order to reduce the error and phase delay of the classical extended state observer (ESO) in estimating the system state and disturbance, in this paper, we combine ESO and tracking differentiator (TD) to construct a tracking differential extended state observer (TDESO). The observation error and observation speed of TDESO are also discussed. Then a nonlinear active disturbance rejection control system improved by TDESO for a linear plant is transformed into a Lurie system. Moreover, the circular criterion is used to analyse the absolute stability of the transformed Lurie system. Finally, TDESO is optimised and an improved linear state error feedback (PLSEF) is proposed to improve the rapidity of the system by using simulation and time domain analysis. And a second-order system is used to illustrate the performance of the proposed scheme. The simulation results show that our algorithm is effective.     

非仿射系统的自学习滑模抗扰控制

针对一类单输入单输出(single-input single-output,SISO)非仿射非线性系统的控制问题,提出了一种自学习滑模抗扰控制方法.该方法用非线性光滑函数设计扩张状态观测器,实现SISO非仿射非线性系统内部不确定性和外部扰动的扩张状态估计,并将扩张状态观测器(extended state observer,ESO)与自学习滑模控制技术融为一体,实现SISO非仿射非线性系统的自学习滑模抗扰控制.该方法不依赖受控对象的数学模型,可以快速跟踪任意给定的参考信号.数值仿真试验表明了该方法响应速度快、控制精度高,具有很强的抗扰动能力,因而是一种鲁棒稳定性很强的控制方法,在SISO非仿射非线性系统控制领域具有重要作用.     

On Finite‐Time Stabilization of Active Disturbance Rejection Control for Uncertain Nonlinear Systems

This paper designs the active disturbance rejection control (ADRC) to achieve finite‐time stabilization for a class of uncertain nonlinear systems. The proposed control incorporates both an extended state observer (ESO) as well as an adaptive sliding mode controller. The ESO is utilized to estimate the full system states and the total uncertainties, and the adaptive strategy is incorporated to deal with the estimation errors. It is proved that, with the application of the proposed control law, semi‐global finite‐time stabilization can be achieved. Effectiveness of the proposed method is illustrated with a numerical example.     

带有非匹配扰动的连铸结晶器振动位移系统自适应反步滑模控制

针对伺服电机驱动的连铸结晶器振动位移系统中存在非匹配的参数不确定性、干扰等问题,提出一种基于非线性扩张状态观测器(ESO)的自适应反步滑模控制方法.首先,基于双曲正切函数和高增益构造ESO对系统中的非匹配扰动进行估计.然后,利用反步法将非匹配扰动视为匹配扰动进行处理,并在每一步设计自适应积分滑模控制器,以消除非匹配扰动对系统的影响.另外,在反步设计的后两步引入了积分滑模滤波器用于估计前一步虚拟控制律的导数.理论分析表明,所设计的ESO能够保证估计误差收敛到原点附近的小邻域内;所设计的控制器能够保证闭环系统所有信号有界,并且,通过选取适当的参数能够保证系统状态可渐近收敛到原点.最后,通过仿真结果验证了所提出方法的有效性.     

结晶器多变量耦合系统的自抗扰控制

主要研究新颖实用非线性自抗扰控制算法,在结晶器多变量耦合系统中的应用.自抗扰控制主要特性是实时估计对象模型摄动和外扰的总和作用量,并在控制信号中补偿掉,实现不确定性强非线性对象的实时动态反馈线性化.结合控制对象,建立了结晶器多变量耦合自抗扰控制系统.数值仿真试验表明自抗扰耦合控制的协调性、自适应跟随性和抗干扰性优于传统的PID解耦控制.     

惯性串联系统的自抗扰控制

针对非线性不确定惯性串联系统的控制问题,提出了惯性串联型扩张状态观测器（Extended state observer,ESO）,使其可直接对惯性串联系统的扩张状态进行估计,同时把被控对象的极点配置到期望位置,在此基础上提出了适合惯性串联系统的自抗扰控制（Active disturbance rejection control,ADRC）方法,该惯性串联型ADRC方法可以充分利用被控对象的已有知识.论文还给出了惯性串联型ADRC和基于扰动观测器（Disturbance observer,DOB）的控制方法之间的联系,指出它们具有相同的三自由度（three-degree of freedom control,3-DOF）控制系统结构和模块功能,都能实现对系统期望模型以外的总扰动进行估计和补偿.仿真结果表明,所提出的方法是有效的,惯性串联型ESO能实现系统总扰动的估计,惯性串联型ADRC能使系统输出能很好地跟踪系统参考输入.     

Robust output regulation of a triaxial MEMS gyroscope via nonlinear active disturbance rejection

This paper presents a nonlinear disturbance rejection–based controller for the robust output regulation of a triaxial microelectromechanical system (MEMS) vibratory gyroscope. In a MEMS gyroscope, parameter variations, mechanical couplings, suspension system nonlinearities, thermal noise, and centripetal/Coriolis forces are the main uncertainty sources. In the dynamical equations of the gyroscope, these uncertainties appear as a matched total disturbance, which does not coincide with the required structure of a standard output regulation problem. More specifically, the total disturbance is not guaranteed to belong to the solution space of a fixed dynamical system. Therefore, we propose a control system that comprises a nominal output regulator equipped with a disturbance rejection loop. On the basis of a suitable reference dynamics of the gyroscope, the control system is developed as the stabilization of a zero‐error invariant manifold in the tracking error space. In the disturbance rejection loop, a nonlinear extended state observer (ESO) is designed to estimate the total disturbance. The convergence of the ESO is analyzed in a Lyapunov‐Lurie framework by linear matrix inequalities (LMIs). In the nominal output regulation loop, the stabilization problem of the desired manifold is tackled by introducing a suitable distance coordinate. Next, to achieve guaranteed attenuation of the ESO estimation errors, an energy‐to‐peak design is pursued. On the basis of the center manifold theory, the stability of the overall closed‐loop system is guaranteed. The efficacy of the proposed control method is assessed through software simulations.     

An analysis and design method for a class of nonlinear systems with nested saturations

This paper is concerned with the analysis and design of a class of nonlinear systems subject to nested saturations. The proposed controller incorporates both an extended state observer (ESO), which is utilised to estimate the nonlinear dynamics of the plant, as well as a set of observer-based feedbacks. We first present analysis results for systems with nonlinear ESOs and show that local stabilisation can be achieved in a region including the origin. Then, in the case that the ESO is in linear form (LESO), the conditions for determining the estimate of the domain of attraction of the resulting closed-loop system are formulated as a convex optimisation problem. A linear matrix inequality based algorithm is then established to design the feedback gains and the LESO gain. An illustrative example is given to show the effectiveness of the proposed approach.     

Quantitative analysis on the phase margin of ADRC

The paper analyzes and compares the phase margins of four active disturbance rejection control (ADRC) designs, which are based on different common-used extended state observers (ESOs) of various orders, for second-order systems. The quantitative results indicate that, besides good dynamic response, ADRC can guarantee enough phase margin. It is also proved that by decreasing the order of ESO, the phase margin can be increased. Furthermore, it is demonstrated that the phase margins of the ADRC-based systems can be almost uninfluenced by the uncertainties of the system parameters. Finally, the theoretical results are verified by both simulations and experiments on a motion control platform.     

线性自抗扰控制器的稳定性研究

研究了线性扩张状态观测器(Extended state observer, ESO)的估计能力,并且分析了在线性自抗扰控制(Linear active disturbance rejection control, LADRC)下闭环系统的稳定性. 对于系统模型未知的情形, 给出了线性扩张观测器估计误差有界的证明, 并通过分析得出了如下结论: 在扩张状态观测器跟踪误差趋于零的前提下, 在线性自抗扰控制下的闭环系统可以实现对设定信号的精确跟踪以及输入-输出有界(Bounded input and bounded output, BIBO)稳定.     

基于模型信息的电静液作动器降阶线性自抗扰控制

针对电静液作动器(electro-hydrostatic actuators, EHA)系统存在内外部扰动、参数不确定性和变控制增益等问题,提出一种基于模型信息的降阶线性自抗扰位置控制方法.首先,基于系统模型信息选取控制增益.其次,通过降阶线性扩张观测器对系统总扰动进行估计,并在控制器中加入扰动项进行补偿.利用奇异摄动理论证明所提控制器可使闭环系统是半全局最终一致有界的,并且当观测器带宽足够大时,所提出的控制器理论上可以使系统输出以所需精度跟踪期望轨迹.仿真结果表明,所提控制方法响应速度较快,控制精度较高,对外部扰动和参数不确定性具有较强的鲁棒性.     

基于扩张状态观测器的车辆横摆稳定模型预测控制器设计

针对车辆横摆稳定性控制问题,本文提出一种基于扩张状态观测器的线性模型预测控制器设计方法.首先,将非线性车辆模型线性化,建立带有模型误差干扰项的线性模型,其中线性化导致的模型误差采用扩张状态观测器估计得到,并证明了观测器的稳定性.然后基于此模型设计线性预测控制器,近似实现了非线性预测控制器的控制效果,同时降低了计算量.最后,通过不同路况下的仿真实验结果,验证了所提方法的计算性能和控制效果.