Transformation of nonlinear discrete-time system into the extended observer form

The paper addresses the problem of transforming discrete-time single-input single-output nonlinear state equations into the extended observer form, which, besides the input and output, also depends on a finite number of their past values. Necessary and sufficient conditions for the existence of both the extended coordinate and output transformations, solving the problem, are formulated in terms of differential one-forms, associated with the input–output equation, corresponding to the state equations. An algorithm for transformation of state equations into the extended observer form is proposed and illustrated by an example. Moreover, the considered approach is compared with the method of dynamic observer error linearisation, which likewise is intended to enlarge the class of systems transformable into an observer form.     

Transformation of Nonlinear MIMO Discrete‐Time Systems into the Extended Observer Form

The paper addresses the problem of transforming discrete‐time multi‐input multi‐output nonlinear state equations into the extended observer form, which, besides the inputs and outputs, also depends on a finite number of their past values. Necessary and sufficient conditions for the existence of the extended coordinate transformation are formulated in terms of differential one‐forms, associated with the input‐output equations, corresponding to the state equations. The difference between the single‐input single‐output and multi‐input multi‐output cases is described. The applicability of the conditions is illustrated by an example.     

Extended nonlinear observer canonical form depending on system output and auxiliary state

This paper deals with the problem of transforming a single output nonlinear system with an auxiliary dynamics into an extended nonlinear observer canonical form (ENOCF). The proposed ENOCF depends on system output and auxiliary state, and admits a kind of high-gain observer. We provide two necessary conditions and an equivalent condition for the existence of such a transformation, and then apply the results to the Rössler system as a case study.     

Output feedback regulation control for a class of cascade nonlinear systems by time‐varying Kalman observer

The global output feedback regulation problem is studied for a class of cascade nonlinear systems. The considered system represents more general classes of nonlinear uncertain systems, including the integral input‐to‐state stable (iISS) unmodeled dynamics, the unknown control direction, the parameter uncertainty, and the external disturbance additively in the input channel. Technically, we explore the changing supply rate technique for the iISS system to deal the iISS unmodeled dynamics and apply the Nussbaum‐type gain into the control design to overcome the unknown control direction. Additionally, a dynamic extended state observer in the form of a time‐varying Kalman observer is novelly constructed to overcome the unmeasured state components in the nonlinear uncertainties. It is shown that the global regulation problem is well addressed by the proposed method, and its efficacy is demonstrated by a fan speed control system.     

Stabilization of Uncertain Multi‐Order Fractional Systems Based on the Extended State Observer

The extended state observer (ESO) based controller has been used successfully with integer‐order systems involving large uncertainties. In this paper, the robust control of uncertain multi‐order fractional‐order (FO) systems based on ESO is investigated. First, we transform the multi‐order FO system into an equivalent system in the form of a same‐order state‐space equation. Then, the ESO for the new system is established for estimating both the state and the total disturbance. Sufficient conditions for bounded‐input and bounded‐output stability are derived, and the asymptotic stability of the closed loop system is analyzed, based on whether the states are available or not. Finally, numerical simulations are presented to demonstrate the validity and feasibility of the proposed methodology.     

A modified dynamic surface approach for control of nonlinear systems with unknown input dead zone

This paper focuses on the robust output precise tracking control problem of uncertain nonlinear systems in pure‐feedback form with unknown input dead zone. By designing an extended state observer, the states unmeasurable problem in traditional feedback control is solved, and the lumped uncertainty, which is caused by system unknown functions and input dead zone, is estimated. In order to apply separation principle, finite‐time extended state observer is designed to obtain system states and estimate the lumped uncertainty. Then, by introducing tracking differentiator, a modified dynamic surface control approach is developed to eliminate the ‘explosion of complexity’ problem and guarantee the tracking performance of system output. Because tracking differentiator is a fast precise signal filter, the closed‐loop control performance is significantly improved when it is used in dynamic surface control instead of first‐order filters. The L _∞ stability of the whole closed‐loop system, which guarantees both the transient and steady‐state performance, is shown by the Lyapunov method and initialization technique. Numerical and experiment examples are performed to illustrate our proposed control scheme with satisfactory results. Copyright © 2013 John Wiley & Sons, Ltd.     

Generalized extended state observer–based repetitive control for systems with mismatched disturbances

A generalized extended state observer (GESO) is devised to improve the disturbances rejection performance in a repetitive‐control system (RCS) for a class of single‐input, single‐output nonlinear plants with nonintegral chain form and mismatched disturbances. By appropriately choosing a disturbance compensation gain and incorporating the disturbance estimate into a repetitive control law, a GESO‐based RCS is established. In this system, the repetitive controller ensures tracking of a periodic reference input, and the incorporation of the disturbance compensation into the control input enables attenuating the lumped disturbance from the system output. Stability criteria and design algorithms have been developed for the system. A case study on the speed control of a rotational control system exhibits that the GESO‐based RCS delivers not only a promising disturbance rejection performance but also a superior property of tracking performance.     

直流降压变换器的降阶扩张状态观测器与滑模控制设计与实现

本文针对直流降压变换器的负载电阻扰动和输入电压变化等系统不确定因素对输出电压的影响,提出了基于降阶扩张状态观测器的滑模控制方法(SMC+RESO).首先设计降阶扩张状态观测器对系统状态,负载电阻扰动和输入电压变化进行估计,然后基于估计值利用滑模控制技术设计控制器,实现对直流降压变换器系统给定电压跟踪的快速性和准确性.值得注意的是,不同于文[1]所提出的基于扩张状态观测器的滑模控制方法(SMC+ESO),本文所提出的方法采用降阶扩张状态观测器,实现简单,且无需电流传感器,减小了实际应用的成本.利用Lyapunov稳定性定理从理论上证明了所设计的控制器可以保证闭环系统的稳定性.仿真和实验结果表明,与已有的基于扩张状态观测器的滑模控制方法相比,所提出的控制方法更好地改善了系统的跟踪性能和对干扰和不确定性的鲁棒性能,且减少了成本,但是牺牲了系统稳态性能.     

Time‐varying input delay compensation for nonlinear systems with additive disturbance: An output feedback approach

This paper addresses the output feedback tracking control of a class of multiple‐input and multiple‐output nonlinear systems subject to time‐varying input delay and additive bounded disturbances. Based on the backstepping design approach, an output feedback robust controller is proposed by integrating an extended state observer and a novel robust controller, which uses a desired trajectory‐based feedforward term to achieve an improved model compensation and a robust delay compensation feedback term based on the finite integral of the past control values to compensate for the time‐varying input delay. The extended state observer can simultaneously estimate the unmeasurable system states and the additive disturbances only with the output measurement and delayed control input. The proposed controller theoretically guarantees prescribed transient performance and steady‐state tracking accuracy in spite of the presence of time‐varying input delay and additive bounded disturbances based on Lyapunov stability analysis by using a Lyapunov‐Krasovskii functional. A specific study on a 2‐link robot manipulator is performed; based on the system model and the proposed design procedure, a suitable controller is developed, and comparative simulation results are obtained to demonstrate the effectiveness of the developed control scheme.     

基于迭代扩张状态观测器的数据驱动最优迭代学习控制

针对一类带扰动有限时间内重复运行的离散时间非线性非仿射不确定系统,本文提出了一种基于迭代扩张状态观测器的数据驱动最优迭代学习控制方法.首先,提出了改进的迭代动态线性化方法,将被控系统线性化为与控制输入有关的仿射形式,并将不确定性合并到一个非线性项中;然后,设计了迭代扩张状态观测器对非线性不确定项进行估计,作为对扰动的补偿;最后,设计了性能指标函数,通过最优技术,提出了参数迭代更新律和最优学习控制律.本文通过数学分析,证明了跟踪误差的有界收敛性.仿真结果验证了方法的有效性.所提出的新型迭代动态线性化方法可很大程度上降低线性化后的控制增益的动态复杂性,使其易于估计.所提出的迭代扩张状态观测器可以在重复中学习,对非重复扰动可进行有效的估计.此外,本文控制器的设计与分析是数据驱动的控制方法,除了被控系统的输入输出数据以外,不需要任何其他模型信息.     

Output regulation boundary control of first-order coupled linear MIMO hyperbolic PIDE systems

ABSTRACT

The work addresses the output regulation problem for coupled linear multiple input multiple output (MIMO) hyperbolic partial integro-differential equation systems with disturbances affecting the systems through the space and boundary input. The exosystems are extended to generate ramp signals and general family of polynomial signals. The system decomposition is applied through the state transformation and yields a decoupled equivalent system. Based on the decoupled form, the backstepping transformation is applied and then in the new coordinate, the full state and output-feedback regulators are designed, respectively. For the state feedback regulator, the corresponding regulator equation is obtained and its solvability conditions are provided to facilitate the regulator design and feasibility. The design of observer-based regulator is based on the decoupling of the observer error system into a PDE subsystem and an ODE subsystem so that the backstepping approach achieves stabilisation by eigenvalue assignment leading to design of observer stabilizing gains.     

A transfer function approach to the realisation problem of nonlinear systems

This article studies the nonlinear realisation problem, i.e. the problem of finding the state equations of a nonlinear system from the transfer function representation being easily computable from the higher order input–output differential equation. The realisation in both observer and controller canonical forms is studied. The results demonstrate a clear connection with those from linear theory. In the solution the concept of adjoint polynomials, adjoint transfer function and right factorisation of the transfer function play a key role. Finally, the results are applied for system linearisation up to input–output injection used in the observer design.     

Anti-Disturbance Control for Tethered Aircraft System With Deferred Output Constraints

In this paper, we investigate the peaking issue of extended state observers and the anti-disturbance control problem of tethered aircraft systems subject to the unstable flight of the main aircraft, airflow disturbances and deferred output constraints.Independent of exact initial values, a modified extended state observer is constructed from a shifting function such that not only the peaking issue inherently in the observer is circumvented completely but also the accurate estimation of the lumpe...     

Trigger‐based tracking control for a class of uncertain nonlinear systems via an event‐triggered extended state observer

An event‐triggered observer‐based output feedback control issue together with triggered input is investigated for a class of uncertain nonlinear systems subject to unknown external disturbances. Two separate event‐triggered conditions are located on the measurement channel and control channel, respectively. An event‐triggered extended state observer (ETESO) is employed to estimate unmeasurable states and compensate uncertainties and disturbances in real time while it is not required for real‐time output measurement. Then, combined with backstepping method and active disturbance rejection control, an output feedback control scheme is proposed, where an event‐triggered input is developed for reducing the communication rate between the controller and the actuator. The triggered instants are determined by a time‐varying event‐triggered condition. Two simulations, including a numerical example and an permanent‐magnet motor, are illustrated to verify the effectiveness of the proposed schemes.     

Observable state space realizations for multivariable systems

This paper derives two canonical state space forms (i.e., the observer canonical form and the observability canonical form) from multiple-input multiple-output systems described by difference equations. The state space model is expressed by the first-order difference equation and is equivalent to the input–output representation. More specifically, by setting the different state variables, the difference equations or the input–output representations can be transformed into two observable canonical forms and the canonical state space model can be also transformed into the difference equations. Finally, two examples are given.     

一类不确定对象的扩张状态观测器

利用观测器形式的跟踪一微分器,对形如ｘ＾（ｎ）＝ｆ（ｘ,ｘ,．．．Ｘ＾（ｎ－１）,ｔ）＋Ｗ（ｔ）的不确定系统给出了“扩张状态观测器”。只要适当选取观测器中的非线性函数和相应参数,它能很好地跟踪一批不确定对象的扩张状态。     

FTESO‐Based Finite Time Control for Underactuated System Within a Bounded Input

Finite time control problem is investigated for a class of underactuated systems with uncertainties and external disturbances. For the sake of expanding control region furthest within a bound input, finite time extended state observer (FTESO) and a novel adaptive terminal sliding mode (ATSM) controller are applied to improve the stability performance of system. Compared to the general extended state observer (ESO), FTESO makes use of fractional powers to reduce the estimation errors to zero in finite time. The coordinate transformation is made for more degrees of design freedom. Rigorous analysis of finite time convergence results has been performed through Lyapunov theory and sufficient conditions are provided for the observer/controller‐design. Finally, simulation results on the Rotating Inverted Pendulum are given to demonstrate the effectiveness of the proposed controller and observer.     

采用ESO的Boost变换器改进滑模控制

本文针对Boost变换器的负载电阻扰动和输入电压变化等系统不确定因素对输出电压的影响,提出了一种基于扩张状态观测器的改进滑模控制算法.首先,设计扩张状态观测器对系统状态、负载电阻和输入电压值进行估计,然后基于得到的估计值采用改进滑模控制算法设计系统控制器.通过理论分析,证明了Boost变换器闭环控制系统的稳定性.仿真和实验结果表明,与传统的滑模控制算法相比,本文所提出的控制算法更好地改善了系统的跟踪性能,提升了系统的鲁棒性.     

Discrete-time extended state observer-based model-free adaptive constrained sliding mode control with modified prescribed performance via CFDL and PFDL

To tackle the trajectory tracking problem and achieve high control accuracy in many actual nonlinear systems with unknown dynamics and input saturation, a novel discrete-time extended state observer-based model-free adaptive constrained sliding mode control with modified prescribed performance is investigated via compact-form dynamic linearization (CFDL) and partial-form dynamic linearization (PFDL). Firstly, the original non-affine system is turned into an affine one comprising an unknown nonlinear term and a linearly parametric term affine to the input via both PFDL and CFDL. Then, a discrete-time extended state observer (DESO) is used to estimate the lumped disturbance containing the unknown nonlinear time-varying term and the term relevant to the estimation error of pseudo partial derivative (PPD) parameter. Furthermore, a modified prescribed performance function is introduced in the model-free adaptive sliding mode control scheme to keep the output tracking error in the prescribed bound without causing any asymmetric offset error in the steady-state. Meanwhile, to suppress the influence of input saturation on the control system, an anti-windup compensator is used. Finally, rigorous theoretical analyses show the robust convergence of the tracking error via the proposed CFDL and PFDL-based methods under external disturbances. Simulations verify the superiority of the modified prescribed performance function, DESO, and anti-windup compensator in the proposed method. Also, the effects of the PFDL-based method and the CFDL-based one are compared during the simulation.