Bao-Zhu Guo

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.

Riesz Basis Approach to the Stabilization of a Flexible Beam with a Tip Mass

Using an abstract condition of Riesz basis generation of discrete operators in the Hilbert spaces, we show, in this paper, that a sequence of generalized eigenfunctions of an Euler--Bernoulli beam equation with a tip mass under boundary linear feedback control forms a Riesz basis for the state Hilbert space. In the meanwhile, an asymptotic expression of eigenvalues and the exponential stability are readily obtained. The main results of [ SIAM J. Control Optim., 36 (1998), pp. 1962--1986] are concluded as a special case, and the additional conditions imposed there are removed.

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.

Riesz Basis Property and Exponential Stability of Controlled Euler--Bernoulli Beam Equations with Variable Coefficients

This paper studies the basis property and the stability of a distributed system described by a nonuniform Euler--Bernoulli beam equation under linear boundary feedback control. It is shown that there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state Hilbert space. The asymptotic distribution of eigenvalues, the spectrum-determined growth condition, and the exponential stability are concluded. The results are applied to a nonuniform beam equation with viscous damping of variable coefficient as a generalization of existing results for the uniform beam.

Controllability and stability of a second-order hyperbolic system with collocated sensor=actuator

A second-order hyperbolic system with collocated sensor/actuator is considered. The semigroup generation is shown for the closed-loop system under the feedback of a generic unbounded observation operator. The equivalence between the exponential stability of the closed-loop system and exact controllability of the open-loop system is established in the general framework of well-posed linear systems. Finally, the conditions are weakened for the diagonal semigroups with finite dimensional inputs. Example of beam equation is presented to display the application.

Shear force feedback control of flexible robot arms

For flexible robots with rotational joints it has been shown previously by Luo (1993), that direct strain feedback can damp out vibrations very satisfactorily. In this paper, a simple sensor-based output feedback control law, called shear force feedback, is newly proposed to control vibrations arising from structural flexibility of robots of Cartesian or SCARA types. Closed-loop exponential stability of such shear force feedback system is proved. Experimental results on set point control and trajectory tracking control are reported. It is found that the simple PI+shear force feedback can yield good performance for both robot motion and vibration suppression. >

The Active Disturbance Rejection Control to Stabilization for Multi-Dimensional Wave Equation with Boundary Control Matched Disturbance

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.

RIESZ BASIS PROPERTY OF EVOLUTION EQUATIONS IN HILBERT SPACES AND APPLICATION TO A COUPLED STRING EQUATION

Suppose that

On convergence of tracking differentiator

\{\lambda _n\}

Arbitrary Decay Rate for Euler-Bernoulli Beam by Backstepping Boundary Feedback

is the set of zeros of a sine-type generating function of the exponential system