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The active disturbance rejection control (ADRC), first proposed by Jingqing Han in the 1980s is an unconventional design strategy. It has been acknowledged to be an effective control strategy in the absence of proper models and in the presence of model uncertainty. Its power was originally demonstrated by numerical simulations, and later by many engineering practices. For the theoretical problems, namely, the convergence of the tracking differentiator which extracts the derivative of reference signal; the extended state observer used to estimate not only the state but also the "total disturbance", by the output; and the extended state observer based feedback, progresses have also been made in the last few years from nonlinear lumped parameter systems to distributed parameter systems. The aim of this paper is to review the origin, idea and development of this new control technology from a theoretical perspective. Emphasis will be focused on output feedback stabilization for uncertain systems described by partial differential equations.

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... In the past two decades, ADRC has received much attention from the control community and industry. Its detailed survey can be found in [7][8][9]. ...

... The linear ESO (7) and the controller (8) are the FOLADRC, we call it as first-order because (4) is an integrator with order one. ...

... Furthermore, it is possible to replace several PI controllers with one FOLADRC controller. FOLADRC (7)- (8) with parameters (23) is an example that can control FC01-FC14. ...

Proportional-integral (PI) control is widely used in turbofan-engine control, while first-order linear active disturbance rejection control (FOLADRC) is a possible approach to update it. This paper investigates FOLADRC. In methodology, it proposes a new block diagram of FOLADRC, which shows that FOLADRC can be viewed as a PI controller, a low-pass feedback filter, and a pre-filter. The low-pass filter helps to reject high-frequency measurement noise, while the pre-filter can attenuate overshoot in step response. In simulation, 14 published linearized model matrices of NASA’s CMAPSS-1 90k engine model are used to verify the above theory. Simulations show one FOLADRC controller can be simultaneously used for the 14 linear models and guarantee that all the 14 low-pressure turbine speed control loops have enough phase margin and no overshoot. Thus, replacing several PI controllers with one FOLADRC controller is possible, and FOLADRC can be used to simplify the control system design of turbofan engines.

... The ADRC contains an important feature called the extended state observer (ESO), which can estimate the system states and disturbances in real time, followed by the total disturbance compensation via the feedback loop [6]. In this way, ADRC rejects different forms of disturbances, such as parameter perturbations, unknown system dynamics, and internal or external disturbances for both linear and nonlinear systems [7,9]. Over time, the ADRC controller having an -order design has been used for -state systems (for ≥1) [10][11][12]. ...

... Figures 6, 8, and 10 showed that the modified time-delay-based ADRC model did compensate for the input disturbance using its mechanism without the need for the ESPO. However, for the same cases, when step output disturbance was applied, it was noted that the novel proposed controller performed much better by attaining both quick output disturbance and noise compensation with time delay (refer to Figures 7,9,and 11). This is because the internal dynamics that were modelled during the time-delay with the noise present were estimated, compensated, and removed significantly using the predictive idea. ...

... However, the proposed method experiences a jump due to a shorter recovery time after applied output disturbance. This is visible in Figures 7,9,and 11. This jump can be further reduced by tuning the proposed controller parameters. ...

The latest research on disturbance rejection mechanisms has shown active disturbance rejection control (ADRC) to be an effective controller for uncertainties and nonlinear dynamics em-bedded in systems to be controlled. The significance of the ADRC controller is its model-free nature, as it requires minimal knowledge of the system model. In addition, it can actively estimate and compensate for the impact of internal and external disturbances present, with the aid of its crucial subsystem called the extended state observer (ESO). However, ADRC controller design becomes more challenging owing to different system disturbances, such as output disturbances, measure-ment noise, and varying time-delays persistent in the system’s communication channels. Most dis-turbance rejection techniques aim to reduce internal perturbations and external disturbances (input and output disturbance). However, output disturbance rejection with measurement noise under time-delay control is still a challenging problem. This paper presents a novel predictive ESO-based ADRC controller for time-delay systems by employing predictive methods to compensate for the disturbances originating from time delay. The prediction mechanism of the novel (proposed) con-troller design is greatly attributed to the extended state predictor observer (ESPO) integrated with the delay-based ADRC inside the proposed controller method. Thus, the proposed controller can predict the unknown system dynamics generated during the delay and compensate for these dy-namics via disturbance rejection under time-delay control. This approach uses the optimization mechanism to determine controller parameters, where the genetic algorithm (GA) is employed with the integral of time-weighted absolute error (ITAE) as the fitness function. The proposed controller is validated by controlling second-order systems with time delay. Type 0, Type 1, and Type 2 sys-tems are considered as the controlled plants, with disturbances (unknown dynamics due to delay and external disturbance), along with measurement noise present. The proposed controller method is compared with state-of-the-art methods, such as the modified time-delay-based ADRC method and the ESPO-based controller method. The findings indicate that the method proposed in this pa-per outperforms its existing competitors by compensating for the dynamics during the time delay and shows robust behaviour, improved disturbance rejection, and a fair extent of resilience to noise.

... Owing to its great potential for dealing with a wide range of disturbance structures, ESO based robust control, including the ADRC original version, has presented an unmistakable viability to address a large set of practical control applications before even having a rigorous proof of theoretical fundamental questions such as the ESO convergence or the closed loop stability which came several years later [39][40][41][42][43]. Moreover, it has shown a high lexibility to handle many more applications than PID control, such as time delayed systems control, multivariable decoupled control, cascade control, and parallel system control [10]. ...

... Moreover, it has shown a high lexibility to handle many more applications than PID control, such as time delayed systems control, multivariable decoupled control, cascade control, and parallel system control [10]. Also, ESO based control has known some major advances in the context of its generalization to more complex problems in the last few past years, such as stochastic systems control [44] and distributed parameter control systems [43]. ...

In the chemical and petrochemical industries, the Continuous Stirred Tank Reactor (CSTR) is, without doubt, one of the most popular processes. From a control point of view, the mathematical model describing the temporal evolution of the CSTR has a strongly nonlinear cross-coupled character. Moreover, modeling errors such as external disturbances, neglected dynamics, and parameter variations or uncertainties make its control task a very difficult challenge. This problem has been the subject of a wide number of control strategies. This article attempts to propose a viable, robust nonlinear decoupling control scheme. The idea behind the proposed approach lies in the design of two nested control loops. The inner loop is responsible for the compensation of the nominal model's nonlinear cross-coupled terms via a static nonlinear feedback; while the outer loop, designed around an Extended State Observer (ESO), which the additional state gathers the global effect of modeling errors, is charged with instantaneously estimating and then compensating the ESO extended state. This way, the CSTR complex dynamics are reduced to a series of decoupled linear subsystems easily controllable using a simple Proportional-Integral (PI) linear control to ensure the robust pursuit of reference signals respecting the desired performance. The presented control validation was performed numerically by an objective comparison to a classical PID controller.

... This paper is concerned with the stabilization of a linear hyperbolic system with a boundary control and subject to a disturbance (see e.g, 1 for a review on this class of system). To be more precise, we aim at designing an active disturbance rejection control 2,3 for that purpose. ...

... ) ∈ and for any ∈ 2 loc (0, ∞), there exists a unique mild 2 ...

This paper deals with the stabilization of a linear hyperbolic system subject to a boundary disturbance. Our feedback design relies on the strategy called active disturbance rejection control (ADRC). The unknown disturbance is estimated by an extended state observer. We prove the existence of solutions of the closed-loop system and the global asymptotic stability of the closed-loop system. A numerical example is given to illustrate the efficiency of our strategy.

... Theorem 1 and Fig. 8 establish a connection between double integrator (17) and LADRC (5)- (8). When b 0 = b, the controller (10) and the double integrator (17) make an ideal controller-plant pair. ...

... Theorem 3 establishes the locality of the second-order LADRC, which means the second-order LADRC (5)-(8) can be tuned based on local nominal model (17) and be used to control the plant P(s). Suppose P(s) is a typical engineering plant controlled by (5)- (8). The Bode plots of P(s)C(s) and P 0 (s)C(s) are drawn in Fig. 13. ...

Treating plant dynamics as an ideal integrator chain disturbed by the total disturbance is the hallmark of active disturbance rejection control (ADRC). To interpret its effectiveness and success, to explain why so many vastly different dynamic systems can be treated in this manner, and to answer why a detailed, accurate, and global mathematical model is unnecessary, is the target of this paper. Driven by a motivating example, the notions of normality and locality are introduced. Normality shows that, in ADRC, the plant is normalized to an integrator chain, which is called local nominal model and locally describes the plant’s frequency response in the neighborhood of the expected gain crossover frequency. Locality interprets why ADRC can design the controller only with the local information of the plant. With normality and locality, ADRC can be effective and robust, and obtain operational stability discussed by T. S. Tsien. Then viewing proportional-integral-derivative (PID) control as a low-frequency approximation of second-order linear ADRC, the above results are extended to PID control. A controller design framework is proposed to obtain the controller in three steps: (1) choose an integrator chain as the local nominal model of the plant; (2) select a controller family corresponding to the local nominal model; and (3) tune the controller to guarantee the gain crossover frequency specification. The second-order linear ADRC and the PID control are two special cases of the framework.

... In [11] a disturbance observer survey for practitioners is presented. Improvements on the efficiency of ESO and ADRC observers are reviewed in [12] and [13]. ...

... Using (13) and (14), the upper bound of e∆(t) is calculated as ...

A new active anti-disturbance control strategy based on a combination of Proportional Integral Observer, an Uncertainty and Disturbance Estimator, and a conventional predictor is developed for a class of unknown delayed LTI systems with disturbances and uncertainties in state and control input matrices. <br

... In [11] a disturbance observer survey for practitioners is presented. Improvements on the efficiency of ESO and ADRC observers are reviewed in [12] and [13]. ...

... Using (13) and (14), the upper bound of e∆(t) is calculated as ...

A new active anti-disturbance control strategy based on a combination of Proportional Integral Observer, an Uncertainty and Disturbance Estimator, and a conventional predictor is developed for a class of unknown delayed LTI systems with disturbances and uncertainties in state and control input matrices. <br

... The ADRC incorporates a nonlinear feedback mechanism and combines the state observer and error driven control law concepts from the current control theory and PID. Initially, the ADRC was developed by Han [42][43][44]; it has nonlinear functions and their tuning is complex. Gao in [44,45] formulated a linear framework of the ADRC as linear active disturbance rejection control (LADRC), which is superior to the NLADRC in parameter tuning and theoretical analysis [44]. ...

... LADRC is a useful design technique that has been successfully used in a variety of engineering domains, particularly high-speed and high precision control, such as aerospace [46], electric motors [47][48][49], power plants [50], robotics [51][52][53][54][55], and quad rotors [56]. The LADRC has been demonstrated as a strong alternative to PID in terms of performance and viability, offering a new approach to solving engineering problems with high precision and powerful disturbance rejection capabilities [41,42,[57][58][59]. ...

Functional metal parts with complicated geometry and internal features for the aerospace and automotive industries can be created using the laser powder bed fusion additive manufacturing (AM) technique. However, the lack of uniform quality of the produced parts in terms of strength limits its enormous potential for general adoption in industries. Most of the defects in selective laser melting (SLM) parts are associated with a nonuniform melt pool size. The melt pool area may fluctuate in spite of constant SLM processing parameters, like laser power, laser speed, hatching distance, and layer thickness. This is due to heat accumulation in the current track from previously scanned tracks in the current layer. The feedback control strategy is a promising tool for maintaining the melt pool dimensions. In this study, a dynamic model of the melt pool cross-sectional area is considered. The model is based on the energy balance of lumped melt pool parameters. Energy coming from previously scanned tracks is considered a source of disturbance for the current melt pool cross-section area in the control algorithm. To track the reference melt pool area and manage the disturbances and uncertainties, a linear active disturbance rejection control (LADRC) strategy is considered. The LADRC control technique is more successful in terms of rapid reference tracking and disturbance rejection when compared to the conventional PID controller. The simulation study shows that an LADRC control strategy presents a 65% faster time response than the PID, a 97% reduction in the steady state error, and a 98% reduction in overshoot. The integral time absolute error (ITAE) performance index shows 95% improvement for reference tracking of the melt pool area in SLM. In terms of reference tracking and robustness, LADRC outperforms the PID controller and ensures that the melt pool size remains constant.

... Existing perturbation observers can be classified as follows: 1) methods that require to know both the system parameters and the perturbation model and parameters (extended Luenberger observer) [6], 2) methods that require knowledge of the system parameters and the disturbance model with parametric uncertainty [7][8][9][10][11], 3) methods that require the disturbance model to be known, while the system and disturbance parameters can be unknown [12][13][14][15][16], 4) methods requiring only knowledge of the system parameters [1,[17][18][19][20], 5*) methods that require neither knowledge of the disturbance model nor the system parameters [21][22][23][24][25][26]. The algorithms that belong to group 5* are the subject of interest of this study since, compared to other solutions, they require minimum amount of a priori information about the system and the perturbation. ...

The problem of output-feedback compensation of bounded additive perturbations affecting a minimum-phase linear system with unknown parameters is considered. An adaptive auxiliary loop is developed, which does not require to know the perturbation model and allows one to: a) separate the processes of estimation of parametric and additive perturbations, b) estimate and compensate for the additive perturbation with any given accuracy if the conditions of the parametric identifiability are met. The above-mentioned separated estimation of two disturbances of different nature is achieved by augmentation of the A.M. Tsykunov auxiliary loop method with the law to identify the unknown parameters, which is based on the instrumental variables approach and the procedure of dynamic regressor extension and mixing (DREM). The obtained system of the additive perturbations compensation has a certain potential to be used together with the conventional industrial PI-, PID-controllers. The theoretical results of this study are validated via mathematical modelling.

... Electro-hydraulic servo systems (EHSSs) occupy a very important position in our industry, due to their advantages, such as large outputs (force/torque), high power-toweight ratios, quick response, large stiffness, and less space occupation [1]. EHSSs are widely used in various fields, such as vehicle active suspensions [2], manipulators [3], aircraft actuators [4], steel manufacturing equipment [5], load simulators [6], engineering machineries [7], and some recent applications also can be seen in [8][9][10]. However, EHSSs, particularly high-order EHSSs, are highly nonlinear systems that exhibit many uncertainties, including uncertain fluid parameters, unmodeled dynamics, and unknown external disturbances [11]. ...

In our industry, active disturbance rejection control already has been used to enhance the performance of the electro-hydraulic servo systems, despite the fact that electro-hydraulic servo systems are usually reduced to first-order and second-order systems. The aim of this paper is to extend the application of active disturbance rejection control to high-order electro-hydraulic servo systems by introducing a new tuning method. Active disturbance rejection control is transformed into two separate parts in the frequency domain: a pre-filter H(s) and a controller T(s). The parameters of the pre-filter and controller can be tuned to satisfy the performance requirements of high-order electro-hydraulic servo systems using quantitative feedback theory. To assess the efficacy of the proposed tuning approach, simulations and an application of a third-order electro-hydraulic servo system have been carried out and the stability of the application with an improved active disturbance rejection controller is analyzed. The results of simulations and experiments reveal that the new tuning method for high-order electro-hydraulic servo systems can obtain a better performance than the bandwidth tuning method and other methods.

... Ramlavi and Chidan [24] studied the linear active disturbance rejection control tracking control approach to enhance the controller's tracking performance and robustness and solved the model uncertainty and environmental disturbance of a single-wheeled robot. Drawing on Partovibakhsh and Liu's research [25], Guo and Zhao [26] enhanced the tracking capabilities of mobile robots through the compensation mechanism of ADRC. Lv [27] proposes a fuzzy auto disturbance rejection control (Fuzzy-ADRC) method for a three-phase four-arm inverter for suppressing motor torque pulsations under complex operating conditions. ...

To improve the control performance of a permanent magnet synchronous motor (PMSM) under external disturbances, an improved active disturbance rejection control (IADRC) algorithm is proposed. Since the nonlinear function in the conventional ADRC algorithm is not smooth enough at the breakpoints, which directly affects the control performance, an innovative nonlinear function is proposed to effectively improve the convergence and stability. On this basis, the proposed IADRC is constructed, and comparative simulation results with ADRC and other IADRC show that faster response speed, higher accuracy and stronger robustness are obtained.

... Anti-disturbance (AD) control techniques effectively attenuate unpredictable disturbances [7][8][9]. Many existing AD control methods have been proposed so far to achieve efficient disturbance suppression, such as output regulation theory [10,11], adaptive robust control theory [12,13] and active disturbance rejection control [14][15][16]. The EID approach, as an effectively active disturbance rejection method, is introduced to counteract both anticipated and unanticipated disruptions [17,18]. ...

This paper concentrates on the issue of anti-disturbance bumpless transfer (ADBT) control design for switched systems. The ADBT control design problem refers to designing a continuous controller and a switching rule to ensure the switched system satisfies the ADBT property. First, the concept of the ADBT property is introduced. Then, via a switched equivalent-input-disturbance (EID) methodology, a switched EID estimator is formulated to estimate the impact of external disturbances within the switched system. Second, a bumpless transfer control is then constructed via a compensator integrating an EID estimation. Finally, the effectiveness of the presented control scheme is verified by controlling a switching resistor–inductor–capacitor circuit on the Matlab platform. Above all, a new configuration for ADBT control of switched systems is established via a switched EID methodology.

... When the synchronization studies in the literature are examined, there is a lack of Active Disturbance Rejection Control (ADRC) application. ADRC is a cutting-edge control strategy that has gained significant attention and popularity in the field of control systems engineering (Feng and Guo, 2017;Huang and Xue, 2014). It represents a paradigm shift in the way control systems are designed and implemented, offering a robust and versatile approach to handling complex and dynamic processes in a wide range of applications. ...

In this paper, a synchronization study is proposed by using a 4D hyperchaotic system model to be used in secure data transfer applications. Active Disturbance Rejection Control (ADRC) method is used for synchronization process. To prove the success of ADRC method, it is compared with Proportional-Integral-Derivative (PID) control method. The coefficients of both control methods are optimized with Particle Swarm Optimization (PSO) algorithm. Synchronization system is modelled and tested in Matlab/Simulink environment. ADRC and PID methods are tested in simulation environment for the cases without disturbance and under disturbance. It can be seen from the test results that the ADRC method managed to keep the system synchronous without being affected by any disturbances. On the other hand, it is clearly seen that the PID method cannot maintain the synchronization of system under disturbance effects.

... Due to the need of practical application, a variety of disturbance rejection control methods have been proposed to deal with such problems. Effective processing can typically be achieved through passive disturbance rejection control [14,15] and active disturbance rejection control [16,17] as well as disturbance observer [18,19] and adaptive Kalman filter [20]. ...

For a class of chain non‐holonomic systems with external disturbance and function constraints, the tan‐type barrier Lyapunov function is used to solve the constraints of the system, and then the non‐linear disturbance observer is used to deal with the disturbance so that the disturbance error eventually converges exponentially. The control strategy designed by the backstepping method can effectively ensure that signals are bounded without violating the respective constraints. Through the simulation design of a three‐stage wheeled mobile robot, the effectiveness of the control scheme is verified again by the results.

... Refs. [31][32][33][34][35] also explored nonlinear ADRC and presented stability proofs. Since nonlinear ADRC is sometimes too complicated for actual applications, Ref. [36] presented linear ADRC. ...

Flying cars offer huge advantages due to their deformable structure, which can adapt to external environments and mission requirements. They represent a novel system that can realize vertical takeoff and landing. However, the structure of a flying car is complicated, placing higher requirements on modeling accuracy and control effectiveness. Thus, in this paper, a dynamic model of a flying car is proposed by combining a car body, motor, and propellers. Then, a double-loop controller based on active disturbance rejection control is proposed to accurately control its flight altitude. Utilizing the extended state observer, external wind and other disturbances are regarded as an extended state, which can be dynamically observed and compensated to significantly improve tracking accuracy. The effectiveness of the proposed controller is validated through detailed simulations and flight experiments. The proposed controller significantly improves control accuracy and disturbance rejection capability.

... Zhao and Guo introduced the novel ADRC, which has the capacity to track reference signals, reject disturbances, and maintain closed-loop stability for a group of singleinput single-output systems (23). Also, Feng and Guo have researched the output feedback stability for indefinite structures described by partial differential equations (24). Hosseini and Keighobadi developed an extended state observer-based robust active control to approximate both the speed and perturbation trajectories of the gyro's dynamics using the location signs (25). ...

... Compared with the conventional ADRC, GADRC performs better either in disturbance rejection property for low-frequency disturbances or in suppression performance for high-frequency measurement noise [10][11][12], and thus is attracting more attention in research on electric drives. Motivated by one of the original ideas in ADRC [13,14], i.e., establishing a direct connection between the classical PID control and other modern control techniques [15,16], the relationship between GADRC and generalized PID control was analysed in [17]. It was found that the GADRC can be interpreted as the generalized PID with low-pass filters or the so-called TDOF generalized PID. ...

In this paper, to achieve auto-setting of PI controller gains when mechanical parameters are unknown, two adaptive PI controllers for speed control of electric drives are developed based on model reference adaptive identification. The adaptive linear neuron (ADALINE) neural network is used to interpret the proposed adaptive PI controller. The effect of the low-pass filter used for the feedback speed and the Coulomb friction torque on parameter identification is analysed, and a new motion equation using filtered speed is given. Additionally, a parameter identification method based on unipolar speed reference is provided. The two proposed adaptive PI controllers and the conventional PI controller are compared based on the high-precision digital simulation using MATLAB/Simulink (version R2023a). The simulation results show that both of the two proposed adaptive PI controllers are able to identify mechanical parameters, but the adaptive PI-1 controller outperforms the adaptive PI-2 controller due to its better noise attenuation performance.

... where H i is the same as in (4). Adding ±β i e 2 i to (29), it becomeṡ ...

In this paper, an enhanced adaptive nonlinear extended state observer (EANESO) for single-input single-output pure feedback systems in the presence of external time-varying disturbances is proposed. In this paper, a nonlinear system with matched and mismatched disturbances is considered. The conventional extended state observer (ESO) can only be applied to systems that are in the form of integral chains. Moreover, this method has limitations in the face of mismatched disturbances. In the presence of time-varying disturbances, the traditional ESOs cannot estimate the disturbances accurately. To overcome this limitation, an EANESO is proposed in this paper. The main idea is to design the nonlinear ESO (NESO) to estimate the states of the system and multiple disturbances simultaneously. The observer gains are considered time-varying and adjusted with adaptation laws to improve the estimation accuracy and overcome the peaking phenomenon. Next, the proposed controller is designed based on output feedback to eliminate the effects of multiple disturbances and stabilize the closed-loop system. Subsequently, the stability analysis of the closed-loop system and convergence of the observer error are discussed. Finally, the proposed method is applied to the inverted pendulum system. The simulated results show good performance of the proposed method as compared with a recently published scheme in the related literature.

... It accomplishes this by employing an Extended State Observer (ESO) to estimate the system's current state, including not only its internal dynamics but also the disturbances affecting it. This disturbance estimation is used to generate a control signal that compensates for the disturbances, effectively canceling out their effects and ensuring that the system follows the desired trajectory or setpoint [12]. One of the key advantages of ADRC is its adaptability to a wide range of systems, both linear and nonlinear, making it a versatile choice for various applications, from mechanical and electrical systems to chemical processes and robotics [13]. ...

This study employs a Raspberry Pi to control the DC motor's position. The position control is achieved using the Active Disturbance Rejection Control (ADRC) method. Furthermore, wireless communication is established between the external control panel created within the Matlab/Simulink program and the Raspberry Pi. This control panel facilitates the system's activation and deactivation, permits adjustments to the ADRC gains, and enables real-time display of the motor's position data on the scope.

... In the realm of control engineering, where precision and stability are paramount, the pursuit of innovative control strategies has led to the development of various advanced techniques. Among these, Active Disturbance Rejection Control (ADRC) stands out as a promising and dynamic approach that has gained considerable attention and recognition in recent years [3]. ADRC represents a departure from traditional control methods by addressing disturbances in real-time with an adaptive and robust framework. ...

A coupled tank system is designed for liquid level control application with Raspberry Pi 4B board. To control of the liquid level, Active Disturbance Rejection Control (ADRC) method is used. Control algorithm is prepared on Simulink environment by using Simulink Support Package for Raspberry Pi hardware tool of Matlab. Then, the algorithm is loaded to Raspberry Pi board to run as microcontroller. The level control system is tested in real-time and results are obtained. It is seen that the ADRC method works stably in the system.

... Active disturbance control improve system stability by tracking reference signals in real time and estimating and compensating disturbance terms, which involves real-time estimation and compensation of disturbances, require high computational and real-time requirements [51]. ...

Quadrotor play an increasingly important role in emergency rescue and other comparable missions. However, external disturbances such as wind gusts and turbulence have a great impact on the stability of quadrotor, which means the control system of the vehicle needs to have a strong capability of anti- disturbance. This paper discuss the latest research on quadrotor control methods, and analyzes their characteristics and application. Traditional methods are improved and enhanced in order to attain better control performance. Meanwhile, artificial intelligence and hybrid control methods have garnered extensive research in the realm of quadrotor.

... It is a significant issue to ensure desired performance of control systems with various uncertainties in control science and control engineering. 1 Because of its ability in dealing with uncertainties effectively, simplicity in engineering implementation, and superior performance in practical application, active disturbance rejection control (ADRC) is becoming a popular control method in control engineering. [2][3][4] It has been proved to be successful in many industrial applications in last two decades, such as fan control in server, 5 position tracking and attitude control of quadrotors 6 and fuel cell temperature control. 7 Compared to its broad industrial applications, the rigorous theoretical analysis for ADRC ever faced great difficulties and lagged far behind for quite some time. ...

This paper studies performance recovery problem of a class of uncertain nonaffine strict-feedback systems with mismatched uncertainties. First, the strict-feedback nonaffine system with mismatched system dynamics is transformed into an integrator-chain nonaffine system by a diffeomorphism. Second, the transformed system is converted to an affine one by constructing an auxiliary control input term, and dynamical uncertainties and external disturbances are viewed as total disturbance of the affine system. Then, a two-time-scale active disturbance rejection controller (ADRC) is designed for the transformed affine system. Under some mild assumptions, it can be proved that the proposed ADRC-based closed-loop system can achieve performance recovery and weak performance recovery in different cases of mismatched system dynamics, respectively. Experimental results on a magnetic levitation ball system demonstrate the effectiveness of the proposed control scheme.

... EMA use the mechanical transmission mechanism driven by the motor to control the movement of the actuating cylinder, which makes the response faster and more accurate, and further improves the actuating efficiency. In the foreseeable future, a large number of EMA will be used in modern aircraft to realize full electrification and digitalization [1][2][3][4][5]. ...

Electromechanical actuators (EMA) are becoming more and more widely used. As the core technology of EMA, servo control technology determines their performance. In this paper, an active disturbance rejection control (ADRC) method with an improved extended state observer (ESO) is proposed to design a cascade controller of EMA based on permanent magnet synchronous motor (PMSM). The mathematical model of PMSM in a two-phase rotating coordinate system is established, then it is decoupled by an id=0 current control method to realize the vector control of the motor. In a three closed-loop vector control system, a PID controller including current loop, speed loop and position loop is designed. To solve the problems caused by measurement noise, the filter link and system are modeled as a whole, and an improved ESO is constructed. On this basis, an ADRC controller of the speed loop and position loop of PMSM is designed and simulated based on Simulink. Based on the physical test platform, a load step test and load disturbance test of ADRC are completed. The results show that, in comparison to the PID method, the ADRC method shortens the response time by 25% on average, and reduces the overshoot by 60% on average. So, it can be concluded that ADRC has good static and dynamic performance, which has a good guiding role for engineering practice.

... Recently, many used techniques incorporate detailed mathematical modeling to control these types of converters, but the need of the hour is to explore some other techniques as well that can predict, estimate, and reject the future disturbances (Chen et al., 2016) as ADRC. It was first introduced and discovered by Han (1999) and Han (2009) to work on both internal and external disturbances by estimating their mutual effect via an ESO (extended state observer), and much literature is presented on the said technique (Zhou et al., 2009;Zheng et al., 2012;Huang and Xue, 2014;Madoński and Herman, 2015;Feng and Guo, 2017). ADRC is also used recently to control these types of converters, and there have been different ADRC techniques from linear to high control gain and generalized ADRC (Saif and Ahmad, 2019). ...

This article compares the conventional model predictive control (MPC) and active disturbance rejection control (ADRC) with a novel MPADRC technique for controlling a non-minimum phase behavior in the DC–DC boost converter. The control of the boost converter is challenging as it is nonlinear, and it shows non-minimum phase behavior in a continuous conduction mode (CCM). Moreover, in this article, the comparison is presented for the boost converter and the two-phase interleaved boost converter using MPC and ADRC, and the effectiveness of the interleaving technique is shown. Finally, it is proved that the interleaving method has much more efficiency and less output ripple than the simple boost converter. To conclude, a novel technique has been introduced that combines both the techniques, that is, MPC and ADRC, in the outer and inner loop with a boost converter, respectively, and the response is clearly the best when compared to the said techniques individually. The overall impact of this technique includes the advantages of both the techniques, that is, the use of MPC allows us to optimize the current value by predicting the future values, and the use of ADRC ensures that the disturbance factor is well tackled and cancels the effect caused by all the disturbances including ignored quantities as well.

... However, in practical applications, the disturbance may be rather general. The paper [18] considered general external disturbance, and an infinite-dimensional ESO developed by Guo and Feng [37,38] was designed to estimate the external disturbance. Observe that nonlinear internal uncertainty is also a very important factor for the analysis of uncertainty [17,39]; in this case, the verifications of exponential stability and boundedness of the state are most sophisticated [17,29,39,40]. ...

We study boundary output tracking for a flexible beam with tip payload and boundary nonlinear disturbance. By constructing an infinite-dimensional extended state observer (ESO), the total disturbance is estimated on line. A servomechanism corresponding to the reference signal and servomechanism-based output feedback control law are designed. Under such control law, we prove that the performance boundary output tracks exponentially the reference signal. By virtue of the Riesz basis approach, the Riesz basis generation and exponential stability of an coupled system are verified, and thereby the boundedness of closed-loop system is obtained. Some numerical simulations are presented to illustrate the effectiveness.

... To the best of the authors' knownedge, very few papers considered output regulation problems with general reference signal and disturbance. In order to deal with systems with general disturbances, active disturbance rejection control (ADRC) was first introduced by Han 21 and then was applied to various systems with external disturbance and /or internal uncertainty [22][23][24][25][26][27][28] ; the core idea behind is to design an extended state observer (ESO) to estimate the state and disturbance simultaneously. Among those, we have to mention Feng and Guo 22 where instead of conventional ESO, an infinite dimensional disturbance estimator without high gain was introduced. ...

This paper is concerned with performance output tracking for an Euler–Bernoulli beam equation with moment boundary control and shear boundary disturbance. An infinite‐dimensional disturbance estimator is designed to estimate the total disturbance. By compensating the total disturbance, a servomechanism corresponding to the reference signal and servomechanism‐based output feedback control law are designed. It is proved that under such control law, the performance output tracks exponentially the reference signal and the involved states of closed‐loop system are bounded. The most important contribution is to deal with the shear boundary term stemmed from the error system between the disturbance estimator and the original system. The admissibility does not hold for such shear boundary term, while the corresponding boundary terms in the existing literature was proved to be admissible. Two key steps are presented to cope with such problem: First, the semigroup generation and exponential stability for a coupled beam system are verified by Riesz basis approach; second, the admissibility of a control operator for semigroup governed by such coupled beam system is proved. Moreover, Sobolev embedding theorem is introduced to simplify the proof of the boundedness of the closed‐loop systems with respect to the available literature. Some numerical simulations are presented to illustrate the effectiveness.

... The representative examples are Disturbance Observer (DOB) and Active Disturbance Rejection Control (ADRC). However, the observation of disturbance by DOB [13][14][15][16] or ADRC [17][18][19][20] is essentially the reverse solution of the disturbance value, according to the result information caused by the disturbance. The necessary condition is that the disturbance has affected the system. ...

At present, the cogging torque of permanent magnet synchronous motors (PMSM) seriously limits the Los pointing accuracy of aviation photoelectric stabilization platforms based on PMSM, which also restricts the requirements of ultra-long-distance and high-precision aviation reconnaissance and detection. For this problem, an off-line iterative learning control (ILC) was designed, and on this basis, a control method of negative effect compensation of disturbance (NECOD) is proposed. Firstly, the “dominant disturbance torque” in the system, that is, the cogging torque with the characteristics of position periodicity, was suppressed by off-line ILC according to different positions. Then, for the “residual disturbance” after compensation, NECOD was used to suppress it. In the constant speed scanning experiment of the aviation photoelectric stabilization platform, the method of combining the off-line iterative learning controller and the negative effect compensation of disturbance (NECOD + ILC) proposed in this paper significantly improved the Los control accuracy of the platform when compared with the classical active disturbance rejection control (ADRC) and ADRC + ILC methods, and the Los pointing error of the constant speed scanning process had only increased by less than 5% when the system had ±15% parameter perturbation. In addition, NECOD + ILC has fewer parameters and is easy to adjust, which is conducive to engineering application and promotion.

Disturbances and uncertainties, ubiquitous in real‐world systems, complicate model‐based control design. A key challenge is developing a precise control strategy that takes all disturbances into account. The automatic dry‐type clutch is a nonlinear, uncertain, and continuous system, where influencing factors are regarded as total disturbances. An active disturbance rejection control (ADRC) framework with an adaptive extended state observer (AESO) is proposed. The AESO's key strength is its adaptability, adjusting gains over time to minimize state estimation errors and noise impacts. This optimizes disturbance estimation, enhancing closed‐loop system performance. Furthermore, we prove the stability of the AESO through the Lyapunov method and, based on this, set the controller parameters as a Hurwitz matrix to ensure the stability of the closed‐loop system. Experimental data from clutch control demonstrates the efficacy of this controller. It ensures clutch position accuracy with a mean value of 13.5255 mm and a root mean square error of 28.7368 mm. Notably, the AESO's observation errors are minimal, averaging at 0.4365 mm with a root mean square error of 0.7068 mm, even under conditions of uncertain dynamics and measurement noise. This performance surpasses traditional ADRC methods relying on linear extended state observers (ESO) and PI control, thereby advancing the practical applicability of ADRC. Finally, through simulations comparing with fuzzy ESO and nonlinear ESO, the results show that the proposed AESO can achieve the same effect as them.

This paper presents a novel control strategy that provides active disturbance rejection predictive control on constrained systems with no nominal identified model. The proposed loop relaxes the modelling requirement to a fixed discrete-time state-space realisation of a first-order plus integrator plant despite the nature of the controlled process. A third-order discrete Extended State Observer (ESO) estimates the model mismatch and assumed plant states. Moreover, the constraints handling is tackled by incorporating the compensation term related to the total perturbation in the definition of the optimisation problem constraints. The proposal merges in a new way state-space Model Predictive Control (MPC) and Active Disturbance Rejection Control (ADRC) into an architecture suitable for the servo-regulatory operation of linear and non-linear systems, as shown through validation examples.

This paper considers the cloud‐based event‐triggered coordination problem of uncertain high‐order nonlinear multi‐agent systems. The communication among agents is realized by indirect and asynchronous access to a shared cloud database. A self‐triggered extended state observer (ESO) is first designed to estimate the total uncertainty. Based on the output of the ESO, an event‐triggered controller is proposed. It is proved that under the proposed output feedback controller, the multi‐agent system can track a virtual leader. We also show that Zeno behavior from output sampling or controller update will never happen. Finally, a numerical example and a vehicle platoon example are provided to corroborate the obtained theoretical results.

We are concerned with the dynamic stabilization for a cascaded Euler–Bernoulli beam (EBB) partial differential equation (PDE)–ordinary differential equation (ODE) system subject to boundary control and matched internal uncertainty and external disturbance. State feedback stabilization of such system without disturbance has been recently discussed by X.H. Wu, H. Feng (Sci China Inf Sci, 2022, 65(5): 159202). An infinite‐dimensional disturbance estimator is constructed in order to estimate the total disturbance. By compensating the total disturbance, we design a state observer to trace the state and then an estimated state and estimated total disturbance‐based output feedback control law. It is proved that the original system is exponentially stable, and other states of the closed‐loop are bounded. Some numerical simulations are presented.

This brief presents a simple proportional-type position regulator for servo systems independent from the actual servo system parameter information without speed and current feedback requirements. First, the extended state observer embedded in the modified gain structure implements the parameter-independent speed and its acceleration estimation mechanism, making it possible to design the output-feedback system structure. Second, the inner-loop controller consists of the speed-loop adaptation algorithm and pole-zero cancellation acceleration error stabilizer to maintain the closed-loop positioning performance at the desired level. The experimental validation is included to demonstrate the feasibility of the proposed solution using the experimental platform consisting of QUBE-servo2 and myRIO-1900.

This article aims to describe a novel design method of disturbance compensator in ADRC (Active Disturbance Rejection Control) based on Lyapunov stability theory. Error-based state equation is used to simplify the ADRC structure. The state feedback controller is designed using LQ (Linear Quadratic) method based on a linearized model. The generalized ESO (Extended State Observer) estimates the nonlinear part, model uncertainty, and external disturbance regarded as "total disturbance." Based on the estimate of the disturbance, disturbance compensator is designed using Lyapunov stability theory. In conventional ADRC, after estimating the disturbance compensator rejects the disturbance. But in this paper, it was not simply rejected. Instead the estimate of the disturbance is used in compensator design to make the Lyapunov candidate decrease more quickly. Our method allowed improved the convergence rate of ADRC. A numerical example and IWP (Inertia Wheel Pendulum) stabilization simulation confirmed the new method for effectiveness evaluation.

In this paper, active disturbance rejection control (ADRC) with higher convergence rate is designed by using linear quadratic (LQ) method and Lyapunov stability theory. Firstly, the state equation of the system is transformed into error-based form so as to simplify the ADRC structure, and it is linearized around its equilibrium point. Then the state feedback controller of ADRC is designed by using LQ method. The non-linearized part, model uncertainty, and external disturbances are regarded as ‘total disturbance’ and estimated by generalized extended state observer (ESO). In the conventional ADRC design, the disturbance estimated by ESO was fully cancelled by disturbance compensator. In this paper, the disturbance compensator is designed not to simply cancel the estimated disturbance, but to make the derivative of Lyapunov function smaller so as to increase the convergence rate. Proposed controller is adopted to stabilize inertia wheel pendulum (IWP) to verify its effectiveness.

Active disturbance-rejection methods are effective in estimating and rejecting disturbances in both transient and steady-state responses. This paper presents a deep observation on and a comparison between two of those methods: the generalized extended-state observer (GESO) and the equivalent input disturbance (EID) from assumptions, system configurations, stability conditions, system design, disturbance-rejection performance, and extensibility. A time-domain index is introduced to assess the disturbance-rejection performance. A detailed observation of disturbance-suppression mechanisms reveals the superiority of the EID approach over the GESO method. A comparison between these two methods shows that assumptions on disturbances are more practical and the adjustment of disturbance-rejection performance is easier for the EID approach than for the GESO method.

The ability of the autonomous farming vehicle to accurately and quickly track complex reference trajectories in the field is critical to crop yields and farmer incomes. The novelty of this paper is to design a new adaptive finite-time trajectory tracking control strategy with an adaptive extended state observer to improve the autonomous farming vehicle’s trajectory tracking accuracy and convergence speed in the complex farm operating environment. The adaptive extended state observer is constructed to deal with the slip disturbance and parameter uncertainty to improve the tracking accuracy. The finite-time control strategy is adopted to improve the convergence speed of the trajectory tracking of the autonomous farming vehicle. The effectiveness and advantages of the proposed control strategy are verified by comparing it with the active disturbance rejection control and traditional PID control strategies under MATLAB/Simulink environment.

States estimation of the nuclear reactor often plays a critical role in accomplishing load-following control and operation monitoring. This paper investigates the states (relative density of delayed neutron precursor (Cr), average temperature of reactor core (TR), and average temperature of helium in the primary loop (TH)) estimation of the modular high-temperature gas-cooled reactor (MHTGR) in the absence and presence of noise (continuous, random Gaussian white, random non-Gaussian colored). A real-time comparison of the Kalman filter (KF), particle filter (PF), and linear extended state observer (ESO) is performed under the load-following operation of the MHTGR. To make the comparison reasonable, the estimation performance comparison of the KF, the PF, and the linear ESO under the same conditions is realized, and four different simulation cases are taken into account to compare their estimation performance. Finally, numerical simulation results show that the KF provides better estimation performance for states Cr, TR and TH in comparison with the PF and the linear ESO.

In this paper, we propose a new method, by designing an unknown input type state observer, to stabilize an unstable 1-d heat equation with boundary uncertainty and external disturbance. The state observer is designed in terms of a disturbance estimator. A stabilizing state feedback control is designed for the observer by the backstepping transformation, which is an observer based output feedback stabilizing control for the original system. The well-posedness and stability of the closed-loop system are concluded. The numerical simulations show that the proposed scheme is quite effectively. This is a first result on active disturbance rejection control for a PDE with both boundary uncertainty and external disturbance.

In this paper, we apply active disturbance rejection control, an emerging control technology, to achieve practical output tracking for a class of nonlinear systems in the presence of vast matched and mismatched uncertainties including unknown internal system dynamic uncertainty, external disturbance, and uncertainty caused by the deviation of control parameter from its nominal value. The total disturbance influencing the performance of controlled output is refined first and then estimated by an extended state observer (ESO). Under the assumption that the inverse dynamics of the uncertain systems are bounded-input-bounded-state stable, a constant high gain ESO based output feedback is constructed to guarantee that the state is bounded and the output tracks practically a given reference signal. A time-varying gain ESO is also discussed to
reduce the peaking value near the initial stages of ESO caused by constant high gain. Numerical simulations are presented to demonstrate the effectiveness of the proposed output-feedback control scheme.

The disturbance estimate is the central idea of active disturbance rejection control (ADRC), where the disturbance is estimated via extended state observer (ESO). However, the conventional ESO requires the disturbance to have slow variation and in order to counteract such disturbance, the ESO must use high gain or discontinuous function. In this paper, we demonstrate for the first time, through a one-dimensional anti-stable wave system, the online disturbance estimation by designing an infinitedimensional disturbance estimator. We go back to the starting point of ADRC that the exactly observable output allows identification of the disturbance which can thereupon be estimated. This realizes the estimation/cancellation strategy of ADRC in a very different way without resorting conventional ESO. We do not regard the disturbance as an “extra-state” variable in the ESO, and hence avoid the two limitations of the ESO. Since the disturbance is estimated, it can be compensated in the feedback loop. It is shown that the disturbance estimator-based feedback control can successfully stabilize the PDE system and at the same time guarantees that all subsystems involved are uniformly bounded. A numerical simulation is presented to illustrate the effectiveness of the the proposed scheme.

This paper introduces an emerging technology called active disturbance rejection control (ADRC), and a story that spans East to West in geography and centuries in time. Emerges from ADRC is the idea that shadowed the ancient Chinese invention of the south-pointing chariot, the idea embedded in a Frenchman' invention of isochronous governor from the early 19 th Century, and the very same idea of invariance formalized in the 1939 Soviet Union. The idea was again revived in the 1990s in the form of ADRC and was soon spread to the U.S., where it took roots and gradually matured into a viable industrial control technology, as evident in the recent adoptions by major industrial concerns. Also seen in ADRC is the clashing of ideas in the history of control science, the ideas that go back to H.S. Tsien and R. Kalman in the 1950s. The origin and the making of ADRC can only be understood from the interplay of the big ideas as presented in this paper. Subject to scrutiny were the basic problems, the fundamental assumptions, and the very conception of control science itself. It is the disagreement on the very foundation of our craft that makes a paradigm-shift to ADRC seem inevitable, in theory and in practice.

This paper articulates, from a theoretical perspective, a new emerging control technology, known as active disturbance rejection control to this day. Three cornerstones toward building the foundation of active disturbance rejection control, namely, the tracking differentiator, the extended state observer, and the extended state observer based feedback are expounded separately. The paper tries to present relatively comprehensive overview about origin, idea, principle, development, possible limitations, as well as some unsolved problems for this viable PID alternative control technology .

The active disturbance rejection control (ADRC) was proposed by Jingqing Han in the late 1990s, which offers a new and inherently robust controller building block that requires very little information of the plant. Originally, the proposal was based largely on experiments with numerous simulations on various systems of different nature. Later, the effectiveness of the control strategy has also been demonstrated in many engineering applications such as motion control, web tension regulation, and chemical processes. However, many theoretical issues, including its applicability in stabilization, output regulation remain unanswered. In this paper, we consider the nonlinear ADRC for general single input single output nonlinear systems subject to dynamical and external uncertainties. We establish conditions that guarantee the ADRC achieving closed-loop system practical stability, disturbance attenuation, and practical reference tracking. Rigorous proofs are given to show the convergence of the variables. The peaking value reduction and high-frequency noise filtering by combination of the time-varying gain in the initial stage and the constant high gain afterwards are explained by linear ADRC. Illustrative examples are also provided.

In this paper, we give an abstract condition of Riesz basis generation for discrete operators in Hilbert spaces, from which we show that the generalized eigenfunctions of a Euler-Bernoulli beam equation with boundary linear feedback control form a Riesz basis for the state Hilbert space. As an consequence, the asymptotic expression of eigenvalues together with exponential stability are readily presented.

This paper proposes the adaptive extended state observer (AESO)-based active disturbance rejection control (ADRC) to deal with the uncertainties, both in the plant and in the sensors. The gain of ESO is automatically timely tuned to reduce the estimation errors of both states and “total disturbance” against the measurement noise. Furthermore, the satisfactory performance of the closed-loop system is achieved by compensation for uncertainties. This novel controller is applied to the air–fuel ratio (AFR) control of gasoline engine, which has large nonlinear uncertainties due to the unknown speed change, fuel film dynamics, etc. In addition, the measurement of AFR is polluted by sensor noise. The experimental results demonstrate that the proposed controller can ensure high deviation precision of AFR despite both uncertain dynamics and measurement noise. Moreover, the experimental comparison validates the effectiveness of the AESO's gain by which the performance of ADRC on mitigating uncertainties can be improved.

We consider boundary output feedback stabilization for a one-dimensional anti-stable wave equation subject to general control matched disturbance. The active disturbance rejection control (ADRC) approach is adopted in investigation. Using the output of the system, we first design a variable structure unknown input type state observer which is shown to be exponentially convergent. The disturbance is estimated, in real time, through
an extended state observer for an ODE reduced from the PDE observer. The disturbance is then canceled in the feedback loop by its approximated value. The stability of the resulting closed-loop system is proven. Simulation results are presented to validate the theoretical conclusions and to exhibit the peaking value reduction by time varying gain instead of constant high gain.

In this paper, we consider boundary stabilization for a multi-dimensional wave equation with boundary control matched disturbance that depends on both time and spatial variables. The active disturbance rejection control (ADRC) approach is adopted in investigation. An extended state observer is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multi-dimensional system by infinitely many test functions. The disturbance is canceled in the
feedback loop together with a collocated stabilizing controller. All subsystems in the closed-loop are shown to be asymptotically stable. In particular, the time varying high gain is first time applied to a
system described by the partial differential equation for complete disturbance rejection purpose and the peaking value reduction caused by the constant high gain in literature. The overall picture of the ADRC in dealing with the disturbance for multi-dimensional partial differential equation is presented through this system. The numerical experiments are carried out to illustrate the convergence and effect of peaking value reduction.

We consider boundary output feedback stabilization for a one-dimensional anti-stable wave equation subject to general control matched disturbance. The active disturbance rejection control (ADRC) approach is adopted in investigation. Using the output of the system, we first design a variable structure unknown input type state observer which is shown to be exponentially convergent. The disturbance is estimated, in real time, through an extended state observer for an ODE reduced from the PDE observer. The disturbance is then canceled in the feedback loop by its approximated value. The stability of the resulting closed-loop system is proven. Simulation results are presented to validate the theoretical conclusions and to exhibit the peaking value reduction by time varying gain instead of constant high gain.

We consider boundary stabilization for a one-dimensional Euler-Bernoulli equation with boundary moment control and disturbance. The active disturbance rejection control (ADRC) and sliding mode control (SMC) approaches are adopted in investigation.By the ADRC approach, a state feedback disturbance estimator with time-varying gain is designed to estimate the disturbance. It is shown that the closed-loop system is asymptotically stable by canceling the disturbance in the feedback loop with its online estimation.In the second part, the SMC is applied to reject the disturbance. The well-posedness of the closed-loop system via SMC is proven, and the monotonicity of the “reaching condition” is presented without differentiating the sliding mode function which may not always exist for the weak solution. The numerical experiments are presented to illustrate the convergence and the peaking value reduction caused by the constant high gain. In addition, the control energies are compared numerically for two approaches.

In this paper, an algorithm is developed to reject time and spatially varying boundary disturbances from a multidimensional Kirchhoff plate via boundary control. The disturbance and control input are assumed to be matched. The active disturbance rejection control approach is adopted for developing the algorithm. A state feedback scheme is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multidimensional system using infinitely many time-dependent test functions. The proposed control law cancels the disturbance using its estimated value. All subsystems in the closed loop are shown to be asymptotically stable. Simulation results are presented to validate the theoretical conclusions and to exhibit the reduction in the peaking phenomenon due to the use of time varying gains instead of constant high gains.

In this paper, we consider boundary stabilization for a multi-dimensional wave equation with boundary control matched disturbance that depends on both time and spatial variables. The active disturbance rejection control (ADRC) approach is adopted in investigation. An extended state observer is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multi-dimensional system by infinitely many test functions. The disturbance is canceled in the feedback loop together with a collocated stabilizing controller. All subsystems in the closed-loop are shown to be asymptotically stable. In particular, the time varying high gain is first time applied to a system described by the partial differential equation for complete disturbance rejection purpose and the peaking value reduction caused by the constant high gain in literature. The overall picture of the ADRC in dealing with the disturbance for multi-dimensional partial differential equation is presented through this system. The numerical experiments are carried out to illustrate the convergence and effect of peaking value reduction.

In this paper, we are concerned with the output feedback stabilization of a one-dimensional wave equation with an unstable term at one end, and the observation suffered by a general harmonic disturbance with unknown magnitudes at the other end. An adaptive observer is designed in terms of the corrupted observation. The backstepping method for infinite-dimensional systems is adopted in the design of the feedback law. It is shown that the resulting closed-loop system is asymptotically stable. Meanwhile, the estimated parameters are shown to be convergent to the unknown parameters as time goes to infinity.

we consider boundary stabilization for a one dimentional Euler-Bernoulli equation with boundary moment control

In this paper, the global and semiglobal convergence of the nonlinear Active Disturbance Rejection Control (ADRC) for a class of multi-input multi-output nonlinear systems with large uncertainty that comes from both dynamical modeling and external disturbance are proved. As a result, a class of linear systems with external disturbance that can be dealt with by the ADRC is classified. The ADRC is then compared both analytically and numerically to the well-known internal model principle. A number of illustrative examples are presented to show the efficiency and advantage of the ADRC in dealing with unknown dynamics and in achieving fast tracking with lower overstriking.

This paper is focused on the design and factory testing of a disturbance decoupling control (DDC) approach for hose extrusion processes. A unique dynamic DDC strategy, based on the active disturbance rejection control (ADRC) framework, is designed and implemented in programmable logic control (PLC) code for temperature regulation in the volumetric flow of a polymer single-screw extruder. With the DDC method, it is shown that a largely unknown square multivariable system is readily decoupled by actively estimating and rejecting the effects of both the internal plant dynamics and external disturbances. The proposed DDC approach requires very little information on plant model and has the inherent disturbance rejection ability, and it proves to be a great fit for the highly nonlinear and multivariable extrusion processes. Recently, the DDC design strategy has been put under rigorous test at Parker Hannifin Parflex hose extrusion plant. Across multiple production lines for over eight months, the product performance capability index (Cpk) was improved by 30 percent and energy consumption is reduced over 50 percent. The production line data demonstrates that ADRC is a transformative control technology with great potentials in streamline factory operations, saving energy and improving quality, all at the same time.

In this technical note, we are concerned with the boundary stabilization of a one-dimensional anti-stable wave equation subject to boundary disturbance. We propose two strategies, namely, sliding mode control (SMC) and the active disturbance rejection control (ADRC). The reaching condition, and the existence and uniqueness of the solution for all states in the state space in SMC are established. The continuity and monotonicity of the sliding surface are proved. Considering the SMC usually requires the large control gain and may exhibit chattering behavior, we then develop an ADRC to attenuate the disturbance. We show that this strategy works well when the derivative of the disturbance is also bounded. Simulation examples are presented for both control strategies.

In this study, the convergence of non-linear extended state observer (ESO) for a class of multi-input multi-output non-linear systems with uncertainty is studied. The unknown part that comes from either the system itself or the external disturbance is considered as an augmented state. The state variable and augmented state are estimated simultaneously through the ESO. It is shown that with the pertinent choice of non-linear functions for observer, the error between the state and observer can be as small as desired when the high-gain tuning parameter is sufficiently small. The current control for permanent-magnet synchronous motor is applied.

In this technical note, the weak convergence of a nonlinear high-gain tracking differentiator based on finite-time stable system is presented under some easy checkable conditions. An example is constructed by using homogeneity. Numerical simulation shows that this tracking differentiator takes advantages over the existing ones. This result relaxes the strict conditions required in existing literature that the Lyapunov function satisfies the global Lipschitz condition and the setting-time function is continuous at zero, both of them seem very restrictive in applications.

Observability and Controllability for Finite-dimensional Systems.- Operator Semigroups.- Semigroups of Contractions.- Control and Observation Operators.- Testing Admissibility.- Observability.- Observation for the Wave Equation.- Non-harmonic Fourier Series and Exact Observability.- Observability for Parabolic Equations.- Boundary Control Systems.- Controllability.- Appendix I: Some Background on Functional Analysis.- Appendix II: Some Background on Sobolev Spaces.- Appendix III: Some Background on Differential Calculus.- Appendix IV: Unique Continuation for Elliptic Operators.

Many flexible structures consist of a large number of components coupled end to end in the form of a chain. In this paper, we consider the simplest type of such structures which is formed by N serially connected Euler-Bernoulli beams, with N actuators and sensors co-located at nodal points. When these N beams are strongly connected at all intermediate nodes and their material coefficients satisfy certain properties, uniform exponential stabilization can be achieved by stabilizing at one end point of the composite beam. We use finite elements to discretize the partial differential equation and compute the spectra of these boundary damped operators. Numerical results are also illustrated.

For linear systems described by dx(t)/dt = Ax(t) + Bu(t), where A generates a semigroup on the state space X and B is an unbounded operator, some necessary as well as sufficient conditions are given for B to be admissible, i.e., for any t, the state x(t) should be in X and should depend continuously on the input u in L^p. This approach begins with an axiomatic description of such a system in terms of a functional equation. The results are applied to the wave equation on a bounded interval.

The tracking differentiator was first proposed by Han in 1989 and the proof of convergence was presented the first time in Han and Wang (Han, J.Q., and Wang, W. (1994), ‘Nonlinear Tracking-differentiator’, Journal of Systems Science and Mathematical Science, 14, 177–183 (in Chinese)). Unfortunately, the proof there is incomplete. This problem has been open for over two decades. In this article, we give a rigorous proof under some additional conditions. An application for online estimation of the unknown frequencies for the finite sum of the sinusoidal signals is presented. The numerical simulations illustrate the effectiveness of the estimation for both linear and nonlinear tracking differentiators.

AbstractThe classical regulator problem is posed in the context of linear, time-invariant, finite-dimensional systems with deterministic disturbance and reference signals. Control action is generated by a compensator which is required to provide closed loop stability and output regulation in the face of small variations in certain system parameters. It is shown, using the geometric approach, that such a structurally stable synthesis must utilize feedback of the regulated variable, and incorporate in the feedback path a suitably reduplicated model of the dynamic structure of the disturbance and reference signals. The necessity of this control structure constitutes the Internal Model Principle. It is shown that, in the frequency domain, the purpose of the internal model is to supply closed loop transmission zeros which cancel the unstable poles of the disturbance and reference signals. Finally, the Internal Model Principle is extended to weakly nonlinear systems subjected to step disturbances and reference signals.

This paper is concerned with the boundary output feedback stabilization for a Kirchho�-type
nonlinear beam equation with boundary observation subject to a general disturbance. The active
disturbance rejection control approach is adopted to design the controller and the disturbance
estimator. By this approach, the disturbance is estimated by a relatively independent estimator
and then canceled in the feedback loop. The existence and uniqueness of the classical solution
to the closed-loop system are proved. The asymptotically convergence of the closed-loop system
is obtained.

This chapter discusses the design of an extended state observer (ESO) for a class of lower triangular nonlinear systems with vast uncertainty. The uncertainty may contain unmodeled system dynamics and external disturbance. The objective of ESO is to estimate, in real time, both states and total disturbance by the output. The constant gain and the time-varying gain are used in ESO design separately. 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 a larger class of nonlinear systems but causes the peaking value near the initial stage, which can be reduced significantly by the time-varying gain ESO. The nature of estimation/cancelation makes the active disturbance rejection control (ADRC) very different from high gain control where the high-gain is used both in the observer and feedback.

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.

We consider boundary output feedback stabilization for a multi-dimensional Kirchhoff plate with boundary observation suffered from a general external disturbance. We adopt for the first time the active disturbance rejection control approach to stabilization of multi-dimensional partial differential equations under corrupted output feedback. In terms of this approach, the disturbance is estimated by a relatively independent estimator, based on (possibly) an infinite number of ordinary differential equations reduced from the original PDEs by infinitely many time-dependent test functions. This gives a state observer, an additional result via this approach. The disturbance is compensated in the feedback-loop. As a result, the control law can be designed almost as the same 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. We also provide numerical simulations which demonstrate the convergence results and underline the effect of the time varying high gain estimator. © 2015 European Control Association. Published by Elsevier Ltd. All rights reserved.

In this paper, we design, in a systematic way, an infinite-dimensional disturbance estimator by the active disturbance rejection control approach. The proposed disturbance estimator can be used to extract real signal from corrupted velocity signal. Its variant form can also be served as a tracking differentiator. The result is applied to stabilization for a multi-dimensional wave equation with position and corrupted velocity measurements.

Overview of Sliding Mode Control.- Overview of Active Disturbance Rejection Control.- Overview of Flight Vehicle Control.- The Descriptions of Flight Vehicle.- SMC for Missile Systems Based on Back-Stepping and ESO Techniques.- Adaptive SMC for Attitude Stabilization in Presence of Actuator Saturation.- Adaptive Nonsingular Terminal Sliding Mode Control for Rigid Spacecraft.- Attitude Tracking of Rigid Spacecraft with Uncertainties and Disturbances.- SMC for Attitude Tracking of Rigid Spacecraft with Disturbances.- Missile Guidance Law Based on ESO Techniques.- Missile Guidance Laws Based on SMC and FTC Techniques.- Cooperative Attack of Multiple Missiles Based on Optimal Guidance Law.

The active disturbance rejection control (ADRC) is now considered as a powerful control strategy in dealing with large uncertainty covering unknown dynamics, external disturbance, and unknown part in coefficient of the control. However, all theoretical works up to present are limited to deterministic uncertainty. In this technical note, we generalize the ADRC to uncertain nonlinear systems subject to external bounded stochastic disturbance described by an uncertain stochastic differential equation driven by white noise. We first design an extended state observer (ESO) that is used to estimate both state, and total disturbance which includes the internal uncertain nonlinear part and the external uncertain stochastic disturbance. It is shown that the resulting closed-loop system is practically stable in the mean-square topology. The numerical experiments are carried out to illustrate effectiveness of the proposed approach.

In this paper,we propose an modified nonlinear extended state observer (ESO)with a time-varying gain
in active disturbance rejection control (ADRC) to deal with a class of nonlinear systems which are
essentially normal forms of general affine nonlinear systems.The total disturbance which includes
unknown dynamics of the system, external disturbance, and unknown part of the control coefficient is
estimated through ESO and is canceled in nonlinear feedback loop.The practical stability for the
resulting closed-loop is obtained. It is shown that the “peaking value” occurred often in the constant
high gain design can be significantly reduced by the time-varying gain approach.

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 closedloop
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.

Topological Space.- Differentiable Manifold.- Algebra, Lie Group and Lie Algebra.- Controllability and Observability.- Global Controllability of Affine Control Systems.- Stability and Stabilization.- Decoupling.- Input-Output Structure.- Linearization of Nonlinear Systems.- Design of Center Manifold.- Output Regulation.- Dissipative Systems.- L 2 -Gain Synthesis.- Switched Systems.- Discontinuous Dynamical Systems.

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.

In this paper, an algorithm is developed to reject time and spatially varying boundary disturbances from a multidimensional Kirchhoff plate via boundary control. The disturbance and control input are assumed to be matched. The active disturbance rejection control approach is adopted for developing the algorithm. A state feedback scheme is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multidimensional system using infinitely many time-dependent test functions. The proposed control law cancels the disturbance using its estimated value. All subsystems in the closed loop are shown to be asymptotically stable. Simulation results are presented to validate the theoretical conclusions and to exhibit the reduction in the peaking phenomenon due to the use of time varying gains instead of constant high gains.

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.

In this paper, an output feedback nonlinear control is proposed for a hydraulic system with mismatched modeling uncertainties in which an extended state observer (ESO) and a nonlinear robust controller are synthesized via the backstepping method. The ESO is designed to estimate not only the unmeasured system states but also the modeling uncertainties. The nonlinear robust controller is designed to stabilize the closed-loop system. The proposed controller accounts for not only the nonlinearities (e.g., nonlinear flow features of servovalve), but also the modeling uncertainties (e.g., parameter derivations and unmodeled dynamics). Furthermore, the controller theoretically guarantees a prescribed tracking transient performance and final tracking accuracy, while achieving asymptotic tracking performance in the absence of time-varying uncertainties, which is very important for high-accuracy tracking control of hydraulic servo systems. Extensive comparative experimental results are obtained to verify the high-performance nature of the proposed control strategy.

The active disturbance rejection control, as a new control strategy in dealing with the large uncertainties, has been developed rapidly in the last two decades. Basically, the active disturbance rejection control is composed of three main parts: the differential tracking; the extended state observer; and the extended observer-based feedback control. In these three parts, the extended state observer plays a crucial role toward the active disturbance rejection control. The most of the extended state observers are based on the constant high gain parameter tuning which results inherently in the peaking problem near the initial time, and at most the attenuation effect for the uncertainty. In this paper, a time-varying-gain extended state observer is proposed for a class of nonlinear systems, which is shown to reject completely the disturbance and to avoid effectively the peaking phenomena by the proper choice of the gain function. The convergence of the extended state observer for the open-loop system is independently proved. The convergence for the closed-loop system which is based on the extended state observer feedback is also presented. Examples and numerical simulations are used to illustrate the convergence and the peaking diminution.

The present monograph (by 4 authors!) is apparently an important and very interesting presentation of the abstract Cauchy problem treated especially by means of the (vector-valued) Laplace and Laplace-Stieltjes transforms. It appears as a worthy continuation of the classical books by it Hille [Am. Math. Soc. (1948; Zbl 0033.06501)] and it Hille and it Phillips [Am. Math. Soc. (1957; Zbl 0078.10004)] about functional analysis and semigroups.par The basic idea reads: If $A$ is a closed linear operator on a Banach space $X$, one considers the Cauchy problem ("initial value problem") $u'(t)= Au(t),quad tge 0,quad u(0)= x,$ where $xin X$ is given. If $u(cdot)$ is an exponentially bounded continuous function, which is also a (mild) solution, that is: $int^t_0 u(s) dsin D(A)quadtextandquad u(t)= x+ Aint^t_0 u(s) ds,quad tge 0,$ and if one considers the Laplace transform: $widehat u(lambda)= int^infty_0 e^-lambda tu(t) dt,$ which converges for large $lambda$, then $(lambda- A)widehat u(lambda)= x$ ($lambda$ large), and conversely. Thus, if $lambdain rho(A)$ -- the resolvent set of $A$ -- then $widehat u(lambda)= (lambda- A)^-1x$.par This fundamental relationship indicates that the Laplace transform is the link between solutions and resolvents, between Cauchy problems and spectral properties of operators.par A further study concerns criteria to decide whether a given function is a Laplace transform. Such results -- in the vector-valued case -- when applied to the resolvent of an operator, would give information on the solvability of the Cauchy problem.par Finally, let us note that our aim here is not to provide a summary -- or appreciation -- of the wealth of concepts contained in this 500 pages book. We invite all interested mathematicians to study this monograph, or any part of it. The reward should be considerable, as always is when reading great mathematics.

In this paper, we are concerned with the boundary feedback stabilization of a one-dimensional Euler-Bernoulli beam equation with the external disturbance flowing to the control end. The active disturbance rejection control (ADRC) and sliding mode control (SMC) are adopted in investigation. By the ADRC approach, the disturbance is estimated through an extended state observer and canceled online by the approximated one in the closed-loop. It is shown that the external disturbance can be attenuated in the sense that the resulting closed-loop system under the extended state feedback tends to any arbitrary given vicinity of zero as the time goes to infinity. In the second part, we use the SMC to reject the disturbance by removing the condition in ADRC that the derivative of the disturbance is supposed to be bounded. The existence and uniqueness of the 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. The numerical simulations validate the effectiveness of both methods.

This paper addresses the control problem of extending the travel range of a MEMS electrostatic actuator through a closed-loop voltage control scheme. Since the electrostatic actuator is inherently unstable due to its pull-in limit, one of the major control goals is to stabilize the actuator system beyond the limit. In addition, the controller has to be robust against external disturbances and noise. After comparing and analyzing the advanced controllers being reported, an active disturbance rejection controller (ADRC) is originally employed to the micro actuator to solve the control problem. The ADRC mainly consists of an extended state observer (ESO) and a PD controller. The ESO is used to estimate system states and the external disturbance, which can be taken as an augmented state of the ESO. The PD controller based on the observed states drives the displacement output of the actuator to a desired level. The ADRC is successfully simulated onto a parallel-plate electrostatic actuator. The simulation results verified the effectiveness of the controller through extending the travel range of the actuator to 99% of the initial gap between two plates in the presences of noise and disturbance.

In the paper, methods are presented which permit the main system variables of the Whitaker-type model-reference adaptive control system to be calculated for input signals which are essentially sinusoidal or ramp functions. The optimum adapting parameters (as the input signal varies) are calculated very accurately, by means of a simple criterion. The approximate equations for the adapting system are formed by using a parameter-perturbation technique, and the resulting time-varying coefficients are linearised for a given input signal. The equations are then used to give an estimate of the stability of the adaptive loops, and of the system responses to step changes in the adapting parameters. It is also shown that it may be necessary to limit the adaptive parameters, in order to prevent the system from driving itself unstable in its efforts to match the model at input frequencies which lie outside the designed range. The whole of the theoretical work is supported by extensive analogue-simulation studies.

Past papers on adaptive control of unstable PDEs with unmatched parametric uncertainties have considered only parabolic PDEs and first-order hyperbolic PDEs. In this note we introduce several tools for approaching adaptive control problems of second-order-in-time PDEs. We present these tools through a benchmark example of an unstable wave equation with an unmatched (non-collocated) anti-damping term, which serves both as a source of instability and of parametric uncertainty. The key effort in the design is to avoid the appearance of the second time derivative of the parameter estimate in the error system.

Active disturbance rejection control (ADRC) can be summarized as follows: it inherits from proportional-integral-derivative (PID) the quality that makes it such a success: the error driven, rather than model-based, control law; it takes from modern control theory its best offering: the state observer; it embraces the power of nonlinear feedback and puts it to full use; it is a useful digital control technology developed out of an experimental platform rooted in computer simulations. ADRC is made possible only when control is taken as an experimental science, instead of a mathematical one. It is motivated by the ever increasing demands from industry that requires the control technology to move beyond PID, which has dominated the practice for over 80 years. Specifically, there are four areas of weakness in PID that we strive to address: 1) the error computation; 2) noise degradation in the derivative control; 3) oversimplification and the loss of performance in the control law in the form of a linear weighted sum; and 4) complications brought by the integral control. Correspondingly, we propose four distinct measures: 1) a simple differential equation as a transient trajectory generator; 2) a noise-tolerant tracking differentiator; 3) the nonlinear control laws; and 4) the concept and method of total disturbance estimation and rejection. Together, they form a new set of tools and a new way of control design. Times and again in experiments and on factory floors, ADRC proves to be a capable replacement of PID with unmistakable advantage in performance and practicality, providing solutions to pressing engineering problems of today. With the new outlook and possibilities that ADRC represents, we further believe that control engineering may very well break the hold of classical PID and enter a new era, an era that brings back the spirit of innovations.