A Super-Twisting Extended State Observer for Nonlinear Systems

A linear extended state observer (LESO) can just estimate constant disturbances with no steady-state error. To estimate time-varying disturbances more actively and accurately, a phase optimisation law (POL) is proposed. Based on the POL, both a phase optimised extended state observer (POESO) and a phase optimised active disturbance rejection control (POADRC) are proposed. Estimation errors, estimation phases, and frequency response have been analysed and compared. The estimation phase of a POESO always leads the one of a LESO, and a POESO estimates slope disturbances with no steady-state error. Transfer functions of tracking, total disturbance and noise are also obtained. Ability of the POADRC and the linear active disturbance rejection control (LADRC) to cancel out the total disturbance or suppress noise, and their stable regions are also analysed and compared. Both frequency analyses and numerical results show that the POADRC estimates the time-varying disturbances and improve the closed-loop performance more effectively.

The active disturbance rejection control is used to design the control scheme for the vertical inverted pendulum system. Based on the original active disturbance rejection control algorithm, a linear/nonlinear switched extended state observer is proposed for such an unstable system with single input and double output, strong nonlinear and strong coupling. Firstly, the reason of the limited ability of nonlinear extended state observer to estimate large perturbations is analyzed, then a linear extended state observer is proposed to make up for the defects of NLESO, and the feasibility of the proposed linear/nonlinear auto disturbance rejection switching control is verified by numerical simulation.

Considering the strong nonlinearity, fast time-varying and strong coupling dynamic characteristics of the high-speed vehicles (HSV), combining with the fact that the extended state observer (ESO) can effectively observe the total disturbance of the system and the predictive sliding mode controller (PSMC) has strong robustness and the ability to reduce chattering, a predictive sliding mode control method with extended state observer is proposed in this paper. Firstly, the HSV model is simplified for the attack angle control; Secondly, the tracking sliding mode surface is predicted and the sliding mode control law is derived by using the optimized performance index, which will avoid the switching function and reduce chattering effectively; Thirdly, a linear extended state observer (LESO) is designed to observe the parameters perturbation and the total disturbances of the system, the estimated value is compensated into the PSMC. Fourthly, the numerical simulation shows that the PSMC with LESO can track the step instruction of angle of attack stably even if there are parameters deviations and external composite disturbances. In addition, the control amount does not exceed the rudder deflection angle of the actuator, and the chatter phenomenon is effectively alleviated.

This paper presents a sliding mode synchronization controller (SMSC) based on extended state observer (ESO) for the dual-drive system in gantry-type machine tools (GTMT). This controller achieves not only excellent tracking accuracy but also high synchronization performance. Firstly, a sliding mode synchronization controller is designed for the dual-drive system based on a simplified mode. Then, in order to handle the difficulties associated with the parameter uncertainties, external disturbances and coupling effects of the dual-drive system, a linear extended state observer (LESO) is designed. In this active disturbance rejection control (ADRC) technology, the total disturbances are treated as an extended state of the system and compensated with the disturbance observed value. Moreover, Rigorous theoretical analysis is performed to show that the proposed control method is able to achieve asymptotical tracking and synchronization performance for the dual-drive system in GTMT. Finally, Simulation studies are carried out and the results show that the proposed control strategy performs an improved tracking and synchronization motion control in comparison to the independent axis P-PI control approach.

Time-varying disturbance of drum water level is difficult to estimate and compensate, which leads to time-varying periodic error of the whole system. System model is reconstructed by using prior information of equivalent disturbance, and the components of known modes in equivalent disturbance are estimated and compensated by the generalized extended state observer. Thus, the error can be eliminated and the control accuracy of the system can be improved. Compared with the traditional active disturbance rejection control (ADRC) system based on linear extended state observer, the control performance under different types of external disturbances is studied. Simulation results show that the proposed generalized extended state observer can completely estimate and compensate the components of the known modes in the equivalent disturbance with small overshoot, short adjustment time, strong anti-interference ability and high control precision.KeywordsADRCState observerDrum water level

The suspension gap of the electromagnetic suspension maglev train is around 8 mm. In practice, it is found that the system gap fluctuations are amplified due to the inner coupling of the suspension module system in the maglev train. In addition, maglev trains are affected by load disturbances and parameter perturbations during operation. These uncertainties reduce the ride comfort. Therefore, it is necessary to propose a novel control strategy to suppress inner coupling while reducing the influence of uncertainties on the system. In this paper, a control strategy based on feedback linearization and extended state observer (ESO) is proposed to address this challenge. Firstly, the suspension module system model is established with parameter uncertainties and external disturbances. Additionally, the inner coupling of the suspension module is represented in this model. Subsequently, the feedback linearization method based on differential geometry theory is applied to reduce the effect of inner coupling. Meanwhile, the system uncertainties are transformed into equivalent disturbances by this method. Afterward, a linear ESO is designed to estimate the equivalent disturbances. Finally, a state feedback controller is used to achieve stable suspension and compensate for the disturbances. Simulation and experimental results show that the proposed decoupled control strategy significantly suppresses the influence of inner coupling and uncertainties on the system compared with the traditional PID control strategy.

This paper proposes a novel class of adaptive extended state observers (AESOs) that significantly expand the applications of extended state observers (ESOs) to nonlinear disturbed systems. An AESO is designed as a linear time-varying form that, as a result, combines both the advantages of theoretical completeness in a conventional linear ESO (LESO) and good practical performance in a conventional nonlinear ESO (NESO). To tune the time-varying observer gains, AESO error dynamics is first transformed into a canonical (phase-variable) form. Then, time-varying PD-eigenvalues are assigned for the canonical system based on differential algebraic spectral theory. Theorems for stability and estimate error bounds of the AESO are given in the presence of unknown disturbances. These theorems also offer some important guidelines for assigning the PD-eigenvalues. To demonstrate the effectiveness of this new observer, two representative applications, including a numerical single-input–single-output example and a practical multiple-input–multiple-output hypersonic vehicle application, are exemplified, and comparison simulations are conducted among AESO, LESO, and NESO. Future work is pointed out in the end.

A practical data-based third-order linear extended state observer (ESO) filter is proposed based on state estimated by ESO. The filter is simple to use with only one parameter to tune. Then its transfer function, parameters tuning method and discrete simulation realization are given respectively. The proposed ESO filter is applied to the tracking of sine signal and square signal, the filter of signal with white Gaussian noise, and the filter of the width data for hot strip rolling mill in real time with measurement noise. Simulation results show the proposed ESO filter is practical without signal model, noise model and statistical characteristic of the measurement noise. Only according to present moment of the measurement data, it could meet the requirement with appropriate parameter.

A nonlinear Third-Order Extended State Observer (TESO) is proposed for the estimation of the rotor position and speed for sensorless Interior Permanent Magnet (IPM) synchronous motor drives. An optimized parameter tuning method is deployed to reconcile the needs of expanding the disturbance observation bandwidth and suppressing the measurement noise effectively. Both the steady state and transient performance of the TESO are verified through the experimental tests. In comparison with the performance of a conventional Linear Extended State Observer (LESO), the proposed observer is proved to have better noise suppression ability and disturbance observation performance in wide operating range.

In sensorless control of interior permanent magnet synchronous motor (IPMSM) drives, extended state observers (ESOs) are preferred due to their advantages, e.g., flexible application, and simple implementation. However, the conventional ESOs may experience the degraded performance with the challenging issues of rapidly changing back electromotive force (BEMF) in a sinusoidal form and ramp disturbances in position estimation. To enhance the position estimation accuracy, this article proposes an estimation scheme based on a novel generalized integrator ESO (NGI-ESO) and an integrally compensated (IC)-third-order (3rd)-linear ESO (IC-3rd-LESO). In the proposed estimation scheme, with the modified mathematical model of the BEMF, the NGI-ESO is designed to achieve accurate estimations of the BEMF. Moreover, the IC-3rd-LESO with a compensation loop is used to mitigate position estimation errors caused by the ramp disturbances. Consequently, the overall performance of sensorless-controlled-IPMSM drives is enhanced. The performance of the proposed estimation scheme under different conditions is verified and compared to that of the conventional ESO schemes by using extensive experimental tests.

The electric propulsion system is composed of permanent magnet synchronous motor equipped with propeller load. Due to the strong coupling of propeller speed and load torque, they considerably characterize dynamic response and disturbance suppression capabilities of the speed controller. Traditional nonlinear active disturbance rejection control (NADRC) is less effective in dealing with a step change of command speed and large disturbance. In order to provide faster response and stronger disturbance rejection potential, an enhanced NADRC strategy is proposed in this article. Initially, a linear extended state observer is applied to preliminary estimate the total disturbance. Then, using the designed weight regular function, this estimation is introduced into the nonlinear extended state observer with a cascade structure. The convergence, stability, and disturbance estimation performance of the proposed strategy are then analyzed. Finally, experimental validation is performed based on a constructed experimental platform. The results show that the proposed method reasonably enhances the speed response and antidisturbance performances.

The pitching axis is the main axis of a satellite camera and is used to control the pitch posture of satellite cameras. A control strategy based on extended state observer (ESO) is designed to obtain a fast response speed and highly accurate pitching axis control system and eliminate disturbances during the adjustment of pitch posture. First, a sufficient condition of stabilization for ESO is obtained by analyzing the steady-state error of the system under step input. Parameter tuning and disturbance compensation are performed by ESO. Second, the ESO of speed loop is designed by the speed equation of the pitching axis of satellite cameras. The ESO parameters are obtained by pole assignment. By ESO, the original state variable observes the motor angular speed and the extended state variable observes the load torque. Therefore, the external load disturbances of the control system are estimated in real time. Finally, simulation experiments are performed for the system on the cases of nonload starting, adding external disturbances on the system suddenly, and the load of system changing suddenly. Simulation results show that the control strategy based on ESO has better stability, adaptability, and robustness than the PI control strategy.

For the purpose of realizing permanent magnet synchronous motor (PMSM) systems with strong robustness and smooth speed, the quasi-proportional-resonant (QPR) controller is combined with super-twisting extended state observer (STESO) in this paper. Although traditional active disturbance rejection control scheme generally selects a sufficiently large bandwidth of linear extended state observer (LESO) to enhance the time-varying disturbance rejection capacity, the bandwidth of LESO is limited by system stiffness. To solve this tough issue, the STESO using generalized super-twisting technique is developed to optimize the stability and anti-disturbance capacity of motor drive system. In addition, the QPR controller is combined with the proposed STESO to suppress the motor speed fluctuation. Experimental results show that the proposed STESO-based QPR control algorithm has satisfactory characteristics.

The dual observation of periodic and aperiodic interferences cannot be balanced by traditional linear extended state observer (T-LESO). Therefore, the speed fluctuations caused by uncertain periodic and aperiodic interferences in the interior permanent magnet synchronous motor control system cannot be suppressed by the traditional linear active disturbance rejection controller. To resolve the aforementioned problem, an improved linear active disturbance rejection controller (I-LADRC) in which multifrequency quasi-resonant filtering decoupling linear extended state observer and generalized linear extended state observer (G-LESO) are cascaded is proposed. In this controller, the interference feedforward control method of T-LESO is first changed, the observer bandwidth <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ω_o</i> and controller bandwidth <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ω_c</i> in the transfer function based on the interference term are decoupled from each other, reducing the difficulty of parameter tuning. On this basis, a multifrequency quasi-resonant filter is embedded in the decoupled linear extended state observer (LESO), so that periodic interferences can be accurately observed. For G-LESO, a fourth-order LESO is adopted to enable more accurate observation of aperiodic high-order interferences. The outer loop controller based on I-LADRC is designed, and the correctness and effectiveness of this method are verified through experiments.

The examples of controls in a high-precision scientific laboratory are countless and most scientific applications&#13;\nrely on controls of the powerful proportional-integral-derivative&#13;\n(PID), first introduced by Minorsky in 1922. In recent years&#13;\na novel Active Disturbance Rejection Control (ADRC) has been&#13;\nproposed. The ADRC is a linear, simple, intuitive, and inherently&#13;\nrobust form of control. It is based on a linear extended state&#13;\nobserver (LESO) that estimates and compensates the disturbance&#13;\nof the system in real time with a certain bandwidth. The&#13;\nbandwidths of the control and of the observer are the only&#13;\nparameters needed for tuning. The design of the ADRC is model&#13;\nindependent and requires only some approximate knowledge of&#13;\nthe fast frequency response of the system. We applied the ADRC&#13;\nto the stabilization of the temperature of optical ultra-stable&#13;\ncavities. Such cavities are used for laser frequency stabilization&#13;\nfor high-resolution spectroscopy and in optical clocks. In our case&#13;\na 578 nm yellow laser is locked to the cavities and it will be used&#13;\nfor the spectroscopy of ytterbium atoms. The temperature of the&#13;\ncavities has to be stabilized to prevent drift in their length and&#13;\nthus in the frequency of the laser. We will present the details&#13;\nof the ADRC and show the performance of the temperature&#13;\nstabilization. The ADRC shows attractive features that motivates&#13;\nour choice: it is easy to design and to implement digitally,&#13;\ninsensitive to environmental changes, and avoids typical problems&#13;\nencountered with PID like overshoots and integral windup.