Linear active disturbance rejection control of underactuated systems: The case of the Furuta pendulum

An Active Disturbance Rejection Control (ADRC) scheme is proposed for a trajectory tracking problem defined on a nonfeedback linearizable Furuta Pendulum example. A desired rest to rest angular position reference trajectory is to be tracked by the horizontal arm while the unactuated vertical pendulum arm stays around its unstable vertical position without falling down during the entire maneuver and long after it concludes. A linear observer-based linear controller of the ADRC type is designed on the basis of the flat tangent linearization of the system around an arbitrary equilibrium. The advantageous combination of flatness and the ADRC method makes it possible to on-line estimate and cancels the undesirable effects of the higher order nonlinearities disregarded by the linearization. These effects are triggered by fast horizontal arm tracking maneuvers driving the pendulum substantially away from the initial equilibrium point. Convincing experimental results, including a comparative test with a sliding mode controller, are presented.     

On the centrality of disturbance rejection in automatic control

In this paper, it is shown that the problem of automatic control is, in essence, that of disturbance rejection, with the notion of disturbance generalized to symbolize the uncertainties, both internal and external to the plant. A novel, unifying concept of disturbance rejector is proposed to compliment the traditional notion of controller. The new controller–rejector pair is shown to be a powerful organizing principle in the realm of automatic control, leading to a Copernican moment where the model-centric design philosophy is replaced by the one that is control–centric in the following sense: the controller is designed for a canonical model and is fixed; the difference between the plant and the canonical model is deemed as disturbance and rejected.     

Robust disturbance rejection control of a biped robotic system using high-order extended state observer

This study addressed the problem of robust control of a biped robot based on disturbance estimation. Active disturbance rejection control was the paradigm used for controlling the biped robot by direct active estimation. A robust controller was developed to implement disturbance cancelation based on a linear extended state observer of high gain class. A robust high-gain scheme was proposed for developing a state estimator of the biped robot despite poor knowledge of the plant and the presence of uncertainties. The estimated states provided by the state estimator were used to implement a feedback controller that was effective in actively rejecting the perturbations as well as forcing the trajectory tracking error to within a small vicinity of the origin. The theoretical convergence of the tracking error was proven using the Lyapunov theory. The controller was implemented by numerical simulations that showed the convergence of the tracking error. A comparison with a high-order sliding-mode-observer-based controller confirmed the superior performance of the controller using the robust observer introduced in this study. Finally, the proposed controller was implemented on an actual biped robot using an embedded hardware-in-the-loop strategy.     

Predictive active disturbance rejection control for processes with time delay

Active disturbance rejection control (ADRC) has been shown to be an effective tool in dealing with real world problems of dynamic uncertainties, disturbances, nonlinearities, etc. This paper addresses its existing limitations with plants that have a large transport delay. In particular, to overcome the delay, the extended state observer (ESO) in ADRC is modified to form a predictive ADRC, leading to significant improvements in the transient response and stability characteristics, as shown in extensive simulation studies and hardware-in-the-loop tests, as well as in the frequency response analysis. In this research, it is assumed that the amount of delay is approximately known, as is the approximated model of the plant. Even with such uncharacteristic assumptions for ADRC, the proposed method still exhibits significant improvements in both performance and robustness over the existing methods such as the dead-time compensator based on disturbance observer and the Filtered Smith Predictor, in the context of some well-known problems of chemical reactor and boiler control problems.     

Generalized predictor based active disturbance rejection control for non-minimum phase systems

In this paper, a generalized predictor based control scheme is proposed to improve system performance of set-point tracking and disturbance rejection for non-minimum phase (NMP) systems. By using a generalized predictor to estimate the system output without time delay, a model-based extended state observer (MESO) is designed to simultaneously estimate the system state and disturbance. Accordingly, an active disturbance rejection control design is developed which consists of a state feedback control and a feedforward control for the disturbance rejection. The MESO and feedback controllers are analytically derived by specifying the desired characteristic roots of MESO and closed-loop system poles, respectively. To improve the output tracking performance, a pre-filter is designed based on a desired closed-loop transfer function for the set-point tracking. A sufficient condition guaranteeing robust stability of the closed-loop system against time-varying uncertainties is established in terms of linear matrix inequalities (LMIs). Three illustrative examples from the literature are used to demonstrate the effectiveness and merit of the proposed control scheme.     

Modified active disturbance rejection control for time-delay systems

Industrial processes are typically nonlinear, time-varying and uncertain, to which active disturbance rejection control (ADRC) has been shown to be an effective solution. The control design becomes even more challenging in the presence of time delay. In this paper, a novel modification of ADRC is proposed so that good disturbance rejection is achieved while maintaining system stability. The proposed design is shown to be more effective than the standard ADRC design for time-delay systems and is also a unified solution for stable, critical stable and unstable systems with time delay. Simulation and test results show the effectiveness and practicality of the proposed design. Linear matrix inequality (LMI) based stability analysis is provided as well.     

Limit cycle analysis of active disturbance rejection control system with two nonlinearities

Introduction of nonlinearities to active disturbance rejection control algorithm might have high control efficiency in some situations, but makes the systems with complex nonlinearity. Limit cycle is a typical phenomenon that can be observed in the nonlinear systems, usually causing failure or danger of the systems. This paper approaches the problem of the existence of limit cycles of a second-order fast tool servo system using active disturbance rejection control algorithm with two fal nonlinearities. A frequency domain approach is presented by using describing function technique and transfer function representation to characterize the nonlinear system. The derivations of the describing functions for fal nonlinearities and treatment of two nonlinearities connected in series are given to facilitate the limit cycles analysis. The effects of the parameters of both the nonlinearity and the controller on the limit cycles are presented, indicating that the limit cycles caused by the nonlinearities can be easily suppressed if the parameters are chosen carefully. Simulations in the time domain are performed to assess the prediction accuracy based on the describing function.     

A simplified predictive control algorithm for disturbance rejection

Model predictive control (MPC) offers several advantages for control of chemical processes. However, the standard MPC may do a poor job in suppressing the effects of certain disturbances. This shortcoming is mainly due to the assumption that disturbances remain constant over the prediction horizon. In this paper, a simple disturbance predictor (SDP) is developed to provide predictions of the unmodeled deterministic disturbances for a simplified MPC algorithm. The prediction is developed by curve fitting of the past information. A tuning parameter is employed to handle a variety of disturbance dynamics and a procedure is presented to find an optimum value of the tuning parameter online. A comparison is made with the commonly used disturbance prediction on three example problems. The results show that an improved regulatory performance and zero offset can be achieved under both regular and ramp output disturbances by using the proposed disturbance predictor.     

Direct energy balance based active disturbance rejection control for coal-fired power plant

The conventional direct energy balance (DEB) based PI control can fulfill the fundamental tracking requirements of the coal-fired power plant. However, it is challenging to deal with the cases when the coal quality variation is present. To this end, this paper introduces the active disturbance rejection control (ADRC) to the DEB structure, where the coal quality variation is deemed as a kind of unknown disturbance that can be estimated and mitigated promptly. Firstly, the nonlinearity of a recent power plant model is analyzed based on the gap metric, which provides guidance on how to set the pressure set-point in line with the power demand. Secondly, the approximate decoupling effect of the DEB structure is analyzed based on the relative gain analysis in frequency domain. Finally, the synthesis of the DEB based ADRC control system is carried out based on multi-objective optimization. The optimized ADRC results show that the integrated absolute error (IAE) indices of the tracking performances in both loops can be simultaneously improved, in comparison with the DEB based PI control and H_∞ control system. The regulation performance in the presence of the coal quality variation is significantly improved under the ADRC control scheme. Moreover, the robustness of the proposed strategy is shown comparable with the H_∞ control.     

Robust absolute stability analysis for interval nonlinear active disturbance rejection based control system

Active disturbance rejection control (ADRC) is a relatively new, quite different but very practical control technology, which shows much promise in replacement of proportion-integration-differentiation (PID) with unmistakable advantage in performance and practicality. This paper mainly concerns with the robust absolute stability of the ADRC based control system with parameter perturbations of the plant, i.e., ADRC based interval control system. Firstly, the system is transformed into a perturbed indirect Lurie system. Then, the Popov criterion and its robust version are presented and some new methods are developed to analyze the (robust) absolute stability for the interval control system. Furthermore, an example is presented to illustrate (robust) absolute stability analysis via the above methods, which verifies the convenience and practicability of these methods and shows the strong stability robustness of ADRC in the presence of parametric uncertainties.     

A hybrid disturbance rejection control solution for variable valve timing system of gasoline engines

A novel solution for electro-hydraulic variable valve timing (VVT) system of gasoline engines is proposed, based on the concept of active disturbance rejection control (ADRC). Disturbances, such as oil pressure and engine speed variations, are all estimated and mitigated in real-time. A feed-forward controller was added to enhance the performance of the system based on a simple and static first principle model, forming a hybrid disturbance rejection control (HDRC) strategy. HDRC was validated by experimentation and compared with an existing manually tuned proportional-integral (PI) controller. The results show that HDRC provided a faster response and better tolerance of engine speed and oil pressure variations.     

Digital current regulator for proportional electro-hydraulic valves with unknown disturbance rejection

Solenoid current regulation is well-known and standard in any proportional electro-hydraulic valve. The goal is to provide a wide-band transfer function from the reference to the measured current, thus making the solenoid a fast and ideal force actuator within the limits of the power supplier. The power supplier is usually a Pulse Width Modulation (PWM) amplifier fixing the voltage bound and the Nyquist frequency of the regulator. Typical analog regulators include three main terms: a feedforward channel, a proportional feedback channel and the electromotive force compensation. The latter compensation may be accomplished by integrative feedback. Here the problem is faced through a model-based design (Embedded Model Control), on the basis of a wide-band embedded model of the solenoid which includes the effect of eddy currents. To this end model parameters must be identified. The embedded model includes a stochastic disturbance dynamics capable of estimating and correcting the electromotive contribution together with parametric uncertainty, variability and state dependence. The embedded model which is fed by the measured current and the supplied voltage becomes a state predictor of the controllable and disturbance dynamics. The control law combines reference generator, state feedback and disturbance rejection to dispatch the PWM amplifier with the appropriate duty cycle. Modeling, identification and control design are outlined together with experimental result. Comparison with an existing analog regulator is also provided.     

Combined feedforward and model-assisted active disturbance rejection control for non-minimum phase system

Control of the non-minimum phase (NMP) system is challenging, especially in the presence of modelling uncertainties and external disturbances. To this end, this paper presents a combined feedforward and model-assisted Active Disturbance Rejection Control (MADRC) strategy. Based on the nominal model, the feedforward controller is used to produce a tracking performance that has minimum settling time subject to a prescribed undershoot constraint. On the other hand, the unknown disturbances and uncertain dynamics beyond the nominal model are compensated by MADRC. Since the conventional Extended State Observer (ESO) is not suitable for the NMP system, a model-assisted ESO (MESO) is proposed based on the nominal observable canonical form. The convergence of MESO is proved in time domain. The stability, steady-state characteristics and robustness of the closed-loop system are analyzed in frequency domain. The proposed strategy has only one tuning parameter, i.e., the bandwidth of MESO, which can be readily determined with a prescribed robustness level. Some comparative examples are given to show the efficacy of the proposed method. This paper depicts a promising prospect of the model-assisted ADRC in dealing with complex systems.     

Fractional active disturbance rejection control

A fractional active disturbance rejection control (FADRC) scheme is proposed to improve the performance of commensurate linear fractional order systems (FOS) and the robust analysis shows that the controller is also applicable to incommensurate linear FOS control. In FADRC, the traditional extended states observer (ESO) is generalized to a fractional order extended states observer (FESO) by using the fractional calculus, and the tracking differentiator plus nonlinear state error feedback are replaced by a fractional proportional-derivative controller. To simplify controller tuning, the linear bandwidth-parameterization method has been adopted. The impacts of the observer bandwidth ω_o and controller bandwidth ω_c on system performance are then analyzed. Finally, the FADRC stability and frequency-domain characteristics for linear single-input single-output FOS are analyzed. Simulation results by FADRC and ADRC on typical FOS are compared to demonstrate the superiority and effectiveness of the proposed scheme.     

Modified Smith predictor design for periodic disturbance rejection

In this paper, a modified Smith predictor control scheme is proposed for periodic disturbance rejection in both stable and unstable processes with time delay. Without affecting the superior setpoint response of Smith predictor control, the regulation performance under periodic disturbance can be enhanced significantly by the proposed design. Meanwhile, the asymptotical rejection for non-periodic disturbance will also be improved. Internal stability is investigated for a closed-loop control system. The effectiveness of the proposed scheme will be demonstrated by simulations as well as a test on an experimental thermal system.     

Modified reduced order observer based linear active disturbance rejection control for TITO systems

This paper proposes an observer based control approach for two input and two output (TITO) plant affected by the lumped disturbance which includes the undesirable effect of cross couplings, parametric uncertainties, and external disturbances. A modified reduced order extended state observer (ESO) based active disturbance rejection control (ADRC) is designed to estimate the lumped disturbance actively as an extended state and compensate its effect by adding it to the control. The decoupled mechanism has been used to determine the controller parameters, while the proposed control technique is applied to the TITO coupled plant without using decoupler to show its efficacy. Simulation results show that the proposed design is efficiently able to nullify the interactions within the loops in the multivariable process with better transient performance as compared to the existing proportional-integral-derivative (PID) control methods. An experimental application of two tanks multivariable level control system is investigated to present the validity of proposed scheme.     

Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems

The paper considers the tracking problem for a class of uncertain linear time invariant (LTI) systems with both uncertain parameters and external disturbances. The active disturbance rejection tracking controller is designed and the resulting closed-loop system׳s characteristics are comprehensively studied. In the time-domain, it is proven that the output of closed-loop system can approach its ideal trajectory in the transient process against different kinds of uncertainties by tuning the bandwidth of extended state observer (ESO). In the frequency-domain, different kinds of parameters׳ influences on the phase margin and the crossover frequency of the resulting control system are illuminated. Finally, the effectiveness and robustness of the controller are verified through the actuator position control system with uncertain parameters and load disturbances in the simulations.     

基于自抗扰重复控制的压电驱动器高精度跟踪控制

针对压电驱动器的高精度控制问题,提出一种自抗扰重复控制设计方法。首先,给出压电驱动系统的动力学模型;然后,在线性自抗扰控制(LADRC)中引入输出反馈积分控制器和一类插入式重复控制器,提出一种具有阶跃、斜坡和周期信号跟踪/抑制能力的自抗扰重复控制策略。进一步,结合小增益定理,分析闭环系统的稳定性及控制系统的设计方法。最后,将所提方法应用于一类压电驱动系统,实验结果表明该方法与LADRC相比,能显著提升控制效果,且高精度跟踪/抑制多种外部信号。