Zhi-Liang Zhao

On the convergence of an extended state observer for nonlinear systems with uncertainty

Abstract The extended state observer first proposed by Jingqing Han in [J.Q. Han, A class of extended state observers for uncertain systems, Control Decis. 10 (1) (1995) 85–88 (in Chinese)] is the key link toward the active disturbance rejection control that is taking off as a technology after numerous successful applications in engineering. Unfortunately, there is no rigorous proof of convergence to date. In this paper, we attempt to tackle this long unsolved extraordinary problem. The main idea is to transform the error equation of objective system with its extended state observer into a asymptotical stable system with a small disturbance, for which the effect of total disturbance error is eliminated by the high-gain.

ON CONVERGENCE OF THE NONLINEAR ACTIVE DISTURBANCE REJECTION CONTROL FOR MIMO SYSTEMS

In this paper, the global and semi-global convergence of nonlinear active disturbance rejection control (ADRC) for a class of multi-input multi-output (MIMO) nonlinear systems with large uncertainties that come from both dynamical modeling and external disturbance are proved. A class of linear systems with external disturbance that can be dealt with by ADRC is classified, from which a comparison with internal model principle is made both analytically and numerically. Numerical simulations illustrate the efficiency and advantage of ADRC in dealing with unknown dynamics, fast tracking, and lower overstriking.

On convergence of tracking differentiator

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.

Weak Convergence of Nonlinear High-Gain Tracking Differentiator

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.

On active disturbance rejection control for nonlinear systems using time-varying gain

Abstract 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.

On convergence of nonlinear active disturbance rejection for SISO systems

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 effective of the control strategy has also been demonstrated in many engineering applications such as motion control, web tension regulation, chemical processes. However, many theoretical issues, including its applicability in stabilization, output regulation remain unanswered. In this paper, we consider the 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 stability, disturbance rejection, and reference tracking.

Extended state observer for uncertain lower triangular nonlinear systems

Abstract The extended state observer (ESO) is a key part of the active disturbance rejection control approach, a new control strategy in dealing with large uncertainty. In this paper, a nonlinear ESO is designed for a kind of lower triangular nonlinear systems with large uncertainty. The uncertainty may come from unmodeled system dynamics and external disturbance. We first investigate a nonlinear ESO with high constant gain and present a practical convergence. Two types of ESO are constructed with explicit error estimations. Secondly, a time varying gain ESO is proposed for reducing peaking value near the initial time caused by constant high gain approach. The numerical simulations are presented to show visually the peaking value reduction. The mechanism of peaking value reduction by time varying gain approach is analyzed.

Active Disturbance Rejection Control for Nonlinear Systems: An Introduction

A concise, in-depth introduction to active disturbance rejection control theory for nonlinear systems, with numerical simulations and clearly worked out equations • Provides the fundamental, theoretical foundation for applications of active disturbance rejection control • Features numerical simulations and clearly worked out equations • Highlights the advantages of active disturbance rejection control, including small overshooting, fast convergence, and energy savings

Extended State Observer for Nonlinear Systems with Uncertainty

Abstract The extended state observer first proposed by Jingqing Han in [J. Q. Han, The extended state observer for a class of uncertain systems, Control and Decision, 10(1)(1995), 85-88] is the main interim loop of active disturbance rejection control that is taking off as a technology after numerous successful applications in engineering. Unfortunately, there is no a rigorous proof of convergence to date. In this paper, we attempt to tackle this long unsolved extraordinary problem. The main idea is to transform the error equation of objective system with its extended state observer into a disturbed system of a asymptotical stable system, with which the effect of total disturbance error is eliminated by high-gain.

A nonlinear extended state observer based on fractional power functions

Abstract In this paper, we investigate a nonlinear extended state observer (ESO) constructed from piece-wise smooth functions consisted of linear and fractional power functions. This structure of ESO was first proposed in the 1990’s and has been widely used in active disturbance rejection control for engineering controls. Its convergence, however, has remained an open problem up to this day. The main objective of this paper is to provide a convergence theory with explicit error estimation. The performances of this type ESO are studied by numerical simulation and compared with linear ESO. The numerical results show that the ESO proposed in this paper enjoys the advantages of smaller peaking value and better measurement noise tolerance.