Composite Learning Sliding Mode Control of Flexible-Link Manipulator

This paper studies the control of a flexible-link manipulator with uncertainty. The fast and slow dynamics are derived based on the singular perturbation (SP) theory. The sliding mode control is proposed while the adaptive design is developed using neural networks (NNs) and disturbance observer (DOB) where the novel update laws for NN and DOB are designed. The closed-loop system stability is guaranteed via Lyapunov analysis. The effectiveness of the proposed method is verified via simulation test.


Introduction
Flexible manipulators own the good characteristics of light weight, fast motion, and low energy consumption.Thus flexible manipulators can be used in many applications [1].Due to the flexibility, the response of the manipulator shows oscillation while it is difficult to obtain high tracking precision.These are the two major challenges within the control of flexible manipulator [2].As a result, many works have been aimed at controlling flexible manipulator [3][4][5][6][7].
In the literature, to deal with the dynamics transformation, output redefinition and SP method can be used.In [3], to avoid the difficulty of nonminimum phase system, the output is redefined for system transformation.In [8,9], the observer based design is presented when the states are not available.In [10,11], NNs are constructed to approximate the whole system uncertainty.The singularly perturbed model [12] is proposed to obtain fast dynamics and slow dynamics.Some other works can be found in [13,14].
As discussed in [1], to achieve high tracking accuracy, based on output redefinition or SP method, efficient learning of system uncertainty and disturbance should be key factor.For system uncertainty, fuzzy logic system (FLS)/NNs can be employed.In the literature, there exist many results of intelligent control which employ intelligent system for approximation and then construct the controller [15][16][17][18][19][20][21].One concern is whether the FLS or NN has successfully fulfilled the task of approximation.To verify the effectiveness, the approximation error should be checked.However, usually it is not possible to derive the signal directly.In [1], with the output redefinition, the composite learning [22] is proposed with the serial-parallel estimation model.It is shown that the obtained predictor error can highly enhance the update of the learning system.
While the uncertainty commonly exists, disturbance might deteriorate the system tracking performance.With the upper bound for robust design, sliding mode control is studied.However, in this way, it brings energy consumption.One concern is to develop the efficient learning to follow the trend of the disturbance.The basic idea is that if the disturbance observer is fast enough compared with the system dynamics then to a great certain it can follow the trend of the disturbance.Some results can be referred to in [23,24].In [25], the attempt using composite learning with NN and DOB is developed for a flexible-link manipulator.
It is noted that in [25], the design is using backstepping scheme.To facilitate the design procedure, borrowing the idea of composite learning, the composite sliding mode control of  degrees of freedom flexible-link manipulators will be proposed.
The rest of the paper is arranged as follows.Section 2 presents the flexible-link manipulator dynamics and the 2 Complexity transformation with SP approach.The control of the slow subsystem and the fast subsystem is given in Sections 3 and 4, respectively.The simulation is shown in Section 5 while the conclusion is discussed in Section 6.

Composite Learning Sliding Mode Control of Slow Subsystem
Backstepping design is employed for controller design in [25] and during the analysis it will introduce the error signal in each step.In this paper, the sliding mode control will be proposed with composite learning design.
Define  1 = where  is positive matrix.Define  =    and  = [  1 ,   2 ]  , where   is a positive design constant.The following approximation exists: where π ∈  × is the weight matrix,  is the number of hidden nodes, and  is the NN basis vector.The derivative of   is calculated as where ϝ =  +  −1   and  is the optimal weights matrix of function  approximation.
Define the prediction error as where   > 0 is the design constant and ϝ is adaptive signal constructed as with  > 0 and   > 0 as design parameters.Define π =  − π, ϝ = ϝ − ϝ where π is the estimation of  and ϝ is the estimation of ϝ.Then we have Finally   is proposed as where  1 ∈  × and  2 ∈  × are positive definite symmetric matrices.
The NN update law is proposed as where  > 0 and  > 0 are design constants.
The error dynamics are obtained as and the derivative of   is obtained as Theorem 3.With the controller (13), NN update law (14), and nonlinear DOB design (11), then all the signals in (A.1) are bounded.
See appendix for proof.

Sliding Mode Control of Fast Subsystem
The control input   of the fast subsystem is designed as [25]   = − (  ) where   is the positive definite gain matrix.The control input is presented as

Simulation Example
To verify the effectiveness of the proposed method, simulation of the 2-DOF flexible-link manipulators is given.The parameter selection is selected as the same as [25].The reference signals are given as The approach in this paper is marked as "CL-SMC" which means composite learning sliding mode control while the design using tracking error to update NN is denoted as "NN-SMC." The control parameters are set as   = 0.5 4×4 ,   = 2,  = [0.14×4 ,  4×4 ],   (1) = 0.5 sin ,   (2) = 0.5 sin , and  = 0.5 2×2 .The system tracking of link 1 and link 2 is shown in Figures 1 and 2, respectively.It is observed that under "CL-SMC," much higher tracking accuracy can be obtained.Also the steady error is small for "CL-SMC" while for "NN-SMC" the error is large and chattering all the time.From Figures 3 and 4, the composite learning can closely follow the compound uncertainty while under "NN-SMC" the NN cannot fulfill the task.It confirms the rationales using composite learning.The responses of NN weights, control inputs, and sliding mode surface are shown in Figures 5, 6, and 7, respectively.The signal   is converging the small neighborhood of zero.

Conclusion
Considering the flexible-link manipulators, this paper proposed the sliding mode control with NN and DOB for compound estimation.The composite learning control scheme can greatly enhance the tracking performance.The simulation results confirms the design philosophy that the composite learning can efficiently fulfil the estimation task.

Appendix
Proof.The Lyapunov candidate is chosen as where    Using ( 15), ( 17), (18), and ( 16), the derivatives of   ,  = 1, . . ., 4 can be obtained as Then the derivative of  is calculated as The following inequalities exist: where ‖‖ 2 ≤  and V 1 and V 2 are positive scalars.