This paper studies a new boundary control strategy for a flexible manipulator subject to unknown fast timevarying disturbances. The flexible manipulator essentially is an infinite dimensional continuum. Hence, a continuous function of space and time can be employed to describe the position of such a distributed parameter structure, the motion of which can be described by partial differential equations (PDEs). To cope with fast timevarying external disturbances, a high order disturbance observer is adopted. A control strategy based on such a disturbance observer is proposed for the restrest maneuvering of the flexible manipulator. Moreover, a smooth hyperbolic function is included in the controller to satisfy the requirement of input saturation. The stability of the boundary control is analyzed using LaSalle’s invariance principle. Finally, the performance of the presented boundary controller is verified through comparison with that of employing a constant disturbance observer via numerical simulations.
At present, flexible manipulators have increasingly wide applications in industrial, agricultural, medical, and aerospace fields. To meet the demand of higher performance, the trend of development of space manipulators is towards lightweight parts, low energy expenditure, and fast movement [
Significant attention has been attracted to the PDE modeling and the controller design of flexible manipulators during recent years; that is, the variational principle is allowed to derive the differential equations [
In practice, the performance of the flexible manipulator system is significantly affected by disturbances [
Notably, most of the previous studies have investigated the control issue with slowly timevarying disturbances. For example, Chen et al. derived an improved nonlinear disturbance observer that could effectively account for constant disturbances [
Although the boundary control issue of the flexible manipulator has been discussed extensively, there are few studies on the control problem of flexible manipulators described via PDEs with input saturation and fast timevarying disturbances. Thus, the objectives of this research effort can be summarized as follows:
The organization of this article is as follows: the flexible manipulator is described using PDEs in Section
The flexible manipulator of concern is shown in Figure
Configuration of a flexible manipulator.
The manipulator rotates in the horizontal plane at a low speed, driven by the input torque
For simplicity, the symbols are introduced as follows:
To facilitate the analysis, introduce the auxiliary variable
The total kinetic energy of the flexible manipulator is
The nonconservative work is given by
Thus, the PDE model of the flexible manipulator reads
Two assumptions about the dynamic model are made, as given below.
The unknown input disturbances
If the total kinetic energy of the flexible manipulator is bounded for
The unknown external disturbances have attracted the attention of many researchers in the field of the control engineering [
Considering the boundary conditions (
Given two sets of constants
According to the hypothesis
For the fast timevarying disturbances, it is expected that the performance of such constant disturbance observer will deteriorate because the assumption of the slowly varying characteristic on the disturbance is no longer valid [
Denoting
From (
Substituting
From (
The auxiliary variables
Combining (
The objective of the scheme is to drive the angle of the motor at the shoulder to the desired value and realize the vibration suppression of the elastic beam simultaneously. By employing the high order disturbance observer, the control laws with input saturation are constructed as follows:
The control laws are given by (
(1) The boundary control is asymptotically stable: for
(2) The inputs are bounded, such as
(1) See Appendix for the details of the proof.
(2) For arbitrary
Based on the above analysis, one has
Numerical simulations are presented in this section to verify the boundary control based on the high order disturbance observer with input saturation. For comparison purposes, the simulations under the control scheme employing the constant disturbance observer are also given in this section. The physical parameters of the manipulator are listed in Table
Physical parameters of the manipulator.
Parameter  Physical significance  Value 


Flexural rigidity of the beam 


Mass of per unit length 


Inertia of the motor 


Length of the flexible beam 


Mass at the tip 

The aim of all the simulations is to drive the angle of the motor to the desired value; that is,
The parameters in the control scheme proposed in this paper are chosen as
In the control strategy employing the constant disturbance observer, the parameters are chosen as
As indicated in Figure
Responses in motor angle and angular velocity for (a) the boundary control scheme with the high order observer and (b) the control scheme employing the constant disturbance observer.
Figure
Deflections at the tip and at the middle for (a) the boundary control scheme with the high order observer and (b) the control scheme employing the constant disturbance observer.
The disturbance estimates by the two types of disturbance observers are compared in Figures
Disturbance estimates for (a) the boundary control scheme with the high order observer and (b) the control scheme employing the constant disturbance observer.
Comparison of the disturbance estimations.
The motion of the flexible manipulator was described via PDEs to overcome the problems caused by model truncation, and a high order disturbance observer was proposed thereafter to estimate the external disturbances for counteracting the disturbance effects. The physical requirement of input saturation was considered in the proposed control law using smooth hyperbolic functions. The stability of the boundary control system was demonstrated using LaSalle’s invariance principle. Finally, numerical simulations illustrated that the boundary controller works notably well. By contrast with the constant disturbance observer, the high order disturbance observer can accurately estimate the fast timevarying disturbances or the transient signals. In the future, the vibration suppression of multilink flexible manipulators will be discussed.
The parameters
The Lyapunov function is defined as
According to hypothesis (
LaSalle’s invariance principle is applied to analyze the stability of the controller
The stability of the controller when
According to (
Substituting
The variable separation method is adopted as
Letting
Because
Because
Therefore, the PDE boundary control in this paper is asymptotically stable by applying the extended LaSalle’s invariance principle; that is, for
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (Project nos. 11372130 and 11290153), the Equipment PreResearch Foundation (Grant no. 6140210010202), and the Research Fund of the State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics) (Grant no. MCMS0116K01).