Convergence Of Nonlinear Active Disturbance Rejection Control

The active disturbance rejection control, as an unconventional design strategy, was first proposed by Jingqing Han of the Institute of Systems Science at the Chi-nese Academy of Sciences in1980s-1990s. After almost twenty years practice and developments, It has been now 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 engineer-ing practices such as motion control, tension control in web transport and strip precessing systems, DC-DC power converts in power electronics, continuous stirred tank reactor in chemical and process control, micro-electro-mechanical systems gy-roscope. Unfortunately, like some other important ideas in control, the theoretical study on active disturbance rejection control lags far behind the applications. The objective of this thesis is to establish the mathematical foundation for this powerful new control strategy.The active disturbance rejection control is composed mainly of three parts: tracking differentiator; extended state observer; and the closed-loop system under the extended state observer based feedback control.The aim of propose of the tracking differentiator can be traced back to the PID control. It is known that the powerful yet primitive proportional-integral-derivative (PID) control law developed in the period of the1920s-1940s in the last century is still playing very important role in modern engineering control practice. However, because of the noise sensitivity, the derivative control is not always physically im-plementable for most of control systems. The tracking differentiator in the active; disturbance rejection control is proved by numerous numerical simulation and en-gineering practice to be noise-tolerant. Although the first effort of the convergence proof for the tracking differentiator was made in1994by Jingqing Han and his stu-dent, it is indicated in the first part of this thesis that its proof is actually incorrect except for the constant signals. We give here a first correct proof of the convergence of the tracking differentiator under quite general conditions.It is known that for an observable control system, the design and convergence of the state observer have been a big issue in understanding the system behavior from the measurement as well as in the control design and fault diagnosis. The second yet the most important part of the active disturbance rejection control is the extended state observer. The extended state observer is the extension of the usual state observer. But the difference between state observer and the extended state observer is that the state observer is to recover the full state of the system only, while the extended state observer copes with the systems with the uncertainty coming from either the system itself or from the external disturbance. In this seminal idea, the uncertain part is considered as an augmented state and is estimated through the observer. The extended state observer is thus regarded as the major step toward the active disturbance rejection control. In the second part of this thesis, we give proof of the convergence for SISO systems and MIMO systems respectively.The third part of this thesis continues to build the last link of the nonlinear active disturbance rejection control for both SISO and MIMO systems. This is the closed-loop system under the extended state observer based feedback so that the to-tal disturbance is online canceled and the original nonlinear system is transformed into almost the canonical form of the linear one. This explains why the active dis-turbance rejection control is more economic control strategy. We establish sufficient conditions for the convergence of some nonlinear active disturbance rejection con-trol, thus providing a solution to this very theoretical problem of applicability of the active disturbance rejection control.The contents of the thesis are as follows:In Chapter1, we introduce some basic knowledge relevant to the active disturbance rejection control. These pre-liminaries include the controllability, observability, detectability. stabilizability of linear systems; Lyapunov stable theorem and Lyapunov converse theorem; weighted homogeneity of functions, vectors and systems; finite-time stability.In Chapter2, the convergence of tracking differentiator is presented. These include the convergence of the generalized nonlinear tracking differentiator and the finite-time stable tracking differentiator. Examples and numerical simulations are given for comparison of the effectiveness of the different differentiator. In addition, as an application, the estimation of the frequency of sinusoidal signals and boundary stabilization of one dimensional wave equation are studied numerically.In Chapter3, the nonlinear extended state observer for both SISO and MIMO nonlinear systems are proposed and the convergence is established. Examples are p-resented, and the numerical comparison of the linear and the weighted homogeneous extended state observer are presented.In Chapter4, the convergence of the closed-loop system under the extended s-tate observer, that is, the separation principles for both SISO and MIMO nonlinear systems are established. These include semi-global and global convergence. Finally, a class of regulation problem for linear uncertain systems that can be solved by the active disturbance rejection control are specified. The internal model principle method and the active disturbance rejection control method are compared numeri-cally.