Complexity
Volume 2017 (2017), Article ID 9430259, 6 pages
https://doi.org/10.1155/2017/9430259

Research Article

Composite Learning Sliding Mode Control of Flexible-Link Manipulator

Bin Xu and Pengchao Zhang

Shaanxi Provincial Key Laboratory of Industrial Automation, Shaanxi University of Technology, Hanzhong, Shaanxi 723000, China

Correspondence should be addressed to Bin Xu; smileface.binxu@gmail.com

Received 29 May 2017; Accepted 6 July 2017; Published 7 August 2017

Academic Editor: Junpei Zhong

Copyright © 2017 Bin Xu and Pengchao Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. 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 [3–7].

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 [15–21]. 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 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.

2. Dynamics Model

The model of degrees of freedom flexible-link manipulators is as follows:

where the physical meanings of , , , , , , , and can be found in [25].

With modal order as , the vectors and are defined as and , where denotes the -th joint angle variable and is the -th link -th modal variable.

Let and denote and as and . Then the dynamics can be written as

Define control input

where and are the control inputs of the slow subsystem and the fast subsystem, respectively.

Supposing , define , , and . The slow subsystem can be obtained as

where , , , , , and .

Remark 1 ( is the nominal value of ). Define , , and . The fast dynamics are written as

and the following expression is obtained:

where , , and .

Remark 2. With the SP method in [12, 26], the fast dynamics (5) and the slow dynamics (4) are obtained. The detail to obtain (4) and (6) can be found in [25].

3. 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 and , where and are the desired joint angle trajectories.

Define the sliding surface

where is positive matrix.

Define and , where is a positive design constant. The following approximation exists:

where is the weight matrix, is the number of hidden nodes, and is the NN basis vector.

The derivative of is calculated as

where and is the optimal weights matrix of function approximation.

Define the prediction error as

where is the design constant and is adaptive signal constructed as

with and as design parameters.

Define , where is the estimation of and is the estimation of . Then we have

Finally is proposed as

where and are positive definite symmetric matrices.

The NN update law is proposed as

where and 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.

4. 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

5. 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 , , , , , , , , , , , , , and . 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.

(a)

(b)

Figure 1: Joint angle of link 1. (a) System response. (b) Tracking error.

(a)

(b)

Figure 2: Joint angle of link 2. (a) System response. (b) Tracking error.

(a)

(b)

(c)

Figure 3: Uncertainty estimation 1. (a) Estimation under “CL-SMC.” (b) Estimation under “NN-SMC.” (c) Estimation error.

(a)

(b)

(c)

Figure 4: Uncertainty estimation 2. (a) Estimation under “CL-SMC.” (b) Estimation under “NN-SMC.” (c) Estimation error.

(a)

(b)

Figure 5: NN response. (a) Trajectory of . (b) Trajectory of .

(a)

(b)

Figure 6: Control signals. (a) . (b) .

(a)

(b)

Figure 7: Sliding mode surface. (a) Signal . (b) Signal .

6. 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 , , , and .
Using (15), (17), (18), and (16), the derivatives of , can be obtained as

Then the derivative of is calculated as

The following inequalities exist:

where and and are positive scalars.
Then we have

By selecting appropriate parameters , , and to satisfy , , where

it is concluded that

where , .
Then

It is concluded when , and the signals included in (A.1) are bounded.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61622308), Aeronautical Science Foundation of China (2015ZA53003), and Natural Science Basic Research Plan in Shaanxi Province (2016KJXX-86).

References

B. Xu and Y. Yuan, “Two performance enhanced control of flexible-link manipulator with system uncertainty and disturbances,” Science China Information Sciences, vol. 60, no. 5, 2017.

M. Bodur and M. E. Sezer, “Adaptive control of flexible multilink manipulators,” International Journal of Control, vol. 58, no. 3, pp. 519–536, 1993.

M. Moallem, K. Khorasani, and R. V. Patel, “Inversion-based sliding control of a flexible-link manipulator,” International Journal of Control, vol. 71, no. 3, pp. 477–490, 1998.

S. Ulrich, J. Z. Sasiadek, and I. Barkana, “Nonlinear adaptive output feedback control of flexible-joint space manipulators with joint stiffness uncertainties,” Journal of Guidance, Control, and Dynamics, vol. 37, no. 6, pp. 1961–1975, 2014.

S. Ulrich, J. Z. Sasiadek, and I. Barkana, “Modeling and direct adaptive control of a flexible-joint manipulator,” Journal of Guidance, Control, and Dynamics, vol. 35, no. 1, pp. 25–39, 2012.

F. M. Caswara and H. Unbehauen, “A neurofuzzy approach to the control of a flexible-link manipulator,” IEEE Transactions on Robotics and Automation, vol. 18, no. 6, pp. 932–944, 2002.

Z. Liu, J. Liu, and W. He, “Adaptive boundary control of a flexible manipulator with input saturation,” International Journal of Control, vol. 89, no. 6, pp. 1191–1202, 2016.

M. Moallem, R. V. Patel, and K. Khorasani, “Nonlinear tip-position tracking control of a flexible-link manipulator: Theory and experiments,” Automatica, vol. 37, no. 11, pp. 1825–1834, 2001.

J. de Le\'on-Morales, J. G. Alvarez-Leal, R. Castro-Linares, and J. Alvarez-Gallegos, “Control of a flexible joint robot manipulator via a non-linear control-observer scheme,” International Journal of Control, vol. 74, no. 3, pp. 290–302, 2001.

Y.-G. Tang, F.-C. Sun, Z.-Q. Sun, and T.-L. Hu, “Tip position control of a flexible-link manipulator with neural networks,” International Journal of Control, Automation and Systems, vol. 4, no. 3, pp. 308–317, 2006.

Y. Tang, F. Sun, and Z. Sun, “Neural network control of flexible-link manipulators using sliding mode,” Neurocomputing, vol. 70, no. 1-3, pp. 288–295, 2006.

B. Siciliano and W. J. Book, “A singular perturbation approach to control of lightweight flexible manipulators,” International Journal of Robotics Research, vol. 7, no. 4, pp. 79–90, 1988.

M. J. Yazdanpanah, K. Khorasani, and R. V. Patel, “Uncertainty compensation for a flexible-link manipulator using nonlinear h control,” International Journal of Control, vol. 69, no. 6, pp. 753–771, 1998.

M. Salehi and G. Vossoughi, “Impedance control of flexible base mobile manipulator using singular perturbation method and sliding mode control law,” International Journal of Control, Automation, and Systems, vol. 6, no. 5, pp. 677–688, 2008.

C. Yang, K. Huang, H. Cheng, Y. Li, and C. Y. Su, “Haptic identification by elm-controlled uncertain manipulator,” IEEE Transactions on Systems Man & Cybernetics: Systems.

B. Luo, T. Huang, H.-N. Wu, and X. Yang, “Data-driven H∞ control for nonlinear distributed parameter systems,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 11, pp. 2949–2961, 2015.

B. Xu and P. Zhang, “Minimal-learning-parameter technique based adaptive neural sliding mode control of MEMS gyroscope,” Complexity, Article ID 6019175, 8 pages, 2017.

C. Yang, Y. Jiang, Z. Li, W. He, and C.-Y. Su, “Neural control of bimanual robots with guaranteed global stability and motion precision,” IEEE Transactions on Industrial Informatics.

W. Chen, S. Hua, and H. Zhang, “Consensus-based distributed cooperative learning from closed-loop neural control systems,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 2, pp. 331–345, 2015.

F. Sun, L. Li, H.-X. Li, and H. Liu, “Neuro-fuzzy dynamic-inversion-based adaptive control for robotic manipulators - discrete time case,” IEEE Transactions on Industrial Electronics, vol. 54, no. 3, pp. 1342–1351, 2007.

F.-C. Sun, H.-X. Li, and L. Li, “Robot discrete adaptive control based on dynamic inversion using dynamical neural networks,” Automatica, vol. 38, no. 11, pp. 1977–1983, 2002.

B. Xu, Z. Shi, C. Yang, and F. Sun, “Composite neural dynamic surface control of a class of uncertain nonlinear systems in strict-feedback form,” IEEE Transactions on Cybernetics, vol. 44, no. 12, pp. 2626–2634, 2014.

B. Xu, “Disturbance observer-based dynamic surface control of transport aircraft with continuous heavy cargo airdrop,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 47, no. 1, pp. 161–170, 2016.

B. Xu, “Composite learning finite-time control with application to quadrotors,” IEEE Transactions on Systems, Man, and Cybernetics: Systems.

B. Xu, “Composite learning control of flexible-link manipulator using NN and DOB,” IEEE Transactions on Systems, Man, and Cybernetics: Systems.

V. Etxebarria, A. Sanz, and I. Lizarraga, “Control of a lightweight flexible robotic arm using sliding modes,” International Journal of Advanced Robotic Systems, vol. 2, no. 2, pp. 103–110, 2005.