# A Model-Free Output Feedback Control Approach for the Stabilization of Underactuated TORA System with Input Saturation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dynamics and Control Problem of the TORA System

## 3. Model-Free Output Feedback Controller Design

**Theorem**

**1.**

**Remark**

**1.**

## 4. Simulation Results

#### 4.1. Comparison Study

#### 4.2. Robustness Test

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Bupp, R.T.; Bernstein, D.S.; Coppola, V.T. A benchmark problem for nonlinear control design. Int. J. Robust Nonlinear Control
**1998**, 8, 307–310. [Google Scholar] [CrossRef][Green Version] - Fradkov, A.; Tomchina, O.; Tomchin, D. Controlled passage through resonance in mechanical systems. J. Sound Vib.
**2011**, 330, 1065–1073. [Google Scholar] [CrossRef] - Sun, N.; Wu, Y.; Fang, Y.; Chen, H. Nonlinear Stabilization Control of Multiple-RTAC Systems Subject to Amplitude-Restricted Actuating Torques Using Only Angular Position Feedback. IEEE Trans. Ind. Electron.
**2017**, 64, 3084–3094. [Google Scholar] [CrossRef] - Shah, S.A.A.; Gao, B.; Ahmed, N.; Liu, C. Advanced robust control techniques for the stabilization of translational oscillator with rotational actuator based barge-type OFWT. Proc. Inst. Mech. Eng. Part M J. Eng. Marit. Environ.
**2021**, 235, 327–343. [Google Scholar] [CrossRef] - Zhang, Y.; Li, L.; Cheng, B.; Zhang, X. An active mass damper using rotating actuator for structural vibration control. Adv. Mech. Eng.
**2016**, 8, 1–9. [Google Scholar] [CrossRef][Green Version] - He, M.E.; Hu, Y.Q.; Zhang, Y. Optimization design of tuned mass damper for vibration suppression of a barge-type offshore floating wind turbine. Proc. Inst. Mech. Eng. Part M J. Eng. Marit. Environ.
**2017**, 231, 302–315. [Google Scholar] [CrossRef] - Wu, Y.; Sun, N.; Fang, Y.; Liang, D. An increased nonlinear coupling motion controller for underactuated Multi-TORA systems: Theoretical design and hardware experimentation. IEEE Trans. Syst. Man Cybern. Syst.
**2019**, 49, 1186–1193. [Google Scholar] [CrossRef] - Liu, Y.; Yu, H.N. A survey of underactuated mechanical systems. IET Control Theory Appl.
**2013**, 7, 921–935. [Google Scholar] [CrossRef][Green Version] - He, B.; Wang, S.; Liu, Y.J. Underactuated robotics: A review. Int. J. Adv. Robot. Syst.
**2019**, 16, 1729881419862164. [Google Scholar] [CrossRef][Green Version] - Chen, T.; Goodwine, B. Controllability and accessibility results for N-link horizontal planar manipulators with one unactuated joint. Automatica
**2021**, 125, 109480. [Google Scholar] [CrossRef] - Wang, L.; Chen, S.; Zhang, P.; She, J.; Lai, X. A Simple Control Strategy Based on Trajectory Planning for Vertical Acrobot. Actuators
**2021**, 10, 308. [Google Scholar] [CrossRef] - Chen, S.; Wang, Y.; Zhang, P.; Su, C.Y. Continuous Control Strategy of Planar 3-Linkage Underactuated Manipulator Based on Broad Neural Network. Actuators
**2021**, 10, 249. [Google Scholar] [CrossRef] - Yao, X.Y.; Park, J.H.; Ding, H.F.; Ge, M.F. Event-triggered consensus control for networked underactuated robotic systems. IEEE Trans. Cybern.
**2020**. early access. [Google Scholar] [CrossRef] [PubMed] - Wang, L.; Lai, X.; Meng, Q.; Wu, M. Effective Control Method Based on Trajectory Optimization for Three-Link Vertical Underactuated Manipulators With Only One Active Joint. IEEE Trans. Cybern.
**2021**. early access. [Google Scholar] [CrossRef] - Quan, Q.; Cai, K.Y. Additive-state-decomposition-based tracking control for benchmark. J. Sound Vib.
**2013**, 332, 4829–4841. [Google Scholar] [CrossRef][Green Version] - Wu, X.; Zhao, Y.; Xu, K. Nonlinear disturbance observer based sliding mode control for a benchmark system with uncertain disturbances. ISA Trans.
**2021**, 110, 63–70. [Google Scholar] [CrossRef] [PubMed] - Sun, N.; Wu, Y.; Fang, Y.; Chen, H.; Lu, B. Nonlinear Continuous Global Stabilization Control for Underactuated RTAC Systems: Design, Analysis, and Experimentation. IEEE/ASME Trans. Mechatron.
**2017**, 22, 1104–1115. [Google Scholar] [CrossRef] - Wu, X.; Xu, K.; Ma, M.; Ke, L. Output feedback control for an underactuated benchmark system with bounded torques. Asian J. Control
**2021**, 23, 1466–1475. [Google Scholar] [CrossRef] - Jiang, Z.P.; Kanellakopoulos, I. Global output feedback tracking for a benchmark nonlinear system. IEEE Trans. Autom. Control
**2000**, 45, 1023–1027. [Google Scholar] [CrossRef] - Yang, T.; Sun, N.; Fang, Y. Neuroadaptive control for complicated underactuated systems with simultaneous output and velocity constraints exerted on both actuated and unactuated states. IEEE Trans. Neural Netw. Learn. Syst.
**2021**. early access. [Google Scholar] [CrossRef] - Yang, T.; Chen, H.; Sun, N.; Fang, Y. Adaptive neural network output feedback control of uncertain underactuated systems with actuated and unactuated state constraints. IEEE Trans. Syst. Man Cybern. Syst.
**2021**. early access. [Google Scholar] [CrossRef] - Yang, T.; Chen, H.; Sun, N.; Fang, Y. Adaptive fuzzy control for a class of MIMO underactuated systems with plant uncertainties and actuator deadzones: Design and experiments. IEEE Trans. Cybern.
**2021**. early access. [Google Scholar] [CrossRef] [PubMed] - Lee, C.H.; Chang, S.K. Experimental implementation of nonlinear TORA system and adaptive backstepping controller design. Neural Comput. Appl.
**2012**, 21, 785–800. [Google Scholar] [CrossRef] - Liu, D.; Guo, W. Nonlinear backstepping design for the underactuated TORA system. J. Vibroengineering
**2014**, 16, 552–559. [Google Scholar] - Guo, W.; Liu, D. Nonlinear dynamic surface control for the underactuated translational oscillator with rotating actuator system. IEEE Access
**2019**, 7, 11844–11853. [Google Scholar] [CrossRef] - Wu, T.; Gui, W.; Hu, D.; Du, C. Adaptive fuzzy sliding mode control for translational oscillator with rotating actuator: A fuzzy model. IEEE Access
**2018**, 6, 55861–55869. [Google Scholar] [CrossRef] - Wu, X.; Xu, K. Global sliding mode control for the underactuated translational oscillator with rotational actuator system. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng.
**2021**, 235, 540–549. [Google Scholar] [CrossRef] - Zhang, A.C.; She, J.H.; Qiu, J.D. A new control method for global stabilisation of translational oscillator with rotational actuator. Int. J. Syst. Sci.
**2019**, 50, 954–960. [Google Scholar] [CrossRef] - Gao, B.T. Dynamic modeling and energy-based control design for TORA. Acta Autom. Sin.
**2008**, 34, 1221–1224. [Google Scholar] [CrossRef] - Wu, X.Q.; Gu, M.M. Adaptive control of the TORA system with partial state constraint. Trans. Inst. Meas. Control
**2019**, 41, 1172–1177. [Google Scholar] [CrossRef] - Burg, T.; Dawson, D. Additional notes on the TORA example: A filtering approach to eliminate velocity measurements. IEEE Trans. Control Syst. Technl.
**1997**, 5, 520–523. [Google Scholar] [CrossRef] - Escobar, G.; Ortega, R.; Sira-Ramrez, H. Output-feedback global stabilization of a nonlinear benchmark system using a saturated passivity-based controller. IEEE Trans. Control Syst. Technol.
**1999**, 7, 289–293. [Google Scholar] [CrossRef] - Gao, B.; Xu, J.; Zhao, J.; Huang, X. Stabilizing control of an underactuated 2-dimensional TORA with only rotor angle measurement. Asian J. Control
**2013**, 15, 1477–1488. [Google Scholar] [CrossRef] - Gao, B.; Bao, Y.; Xie, J.; Jia, L. Passivity-based control of two-dimensional translational oscillator with rotational actuator. Trans. Inst. Meas. Control
**2014**, 36, 111–118. [Google Scholar] [CrossRef] - Gao, B.; Ye, F. Fuzzy Lyapunov synthesis control of an underactuated 2DTORA system. J. Intell. Fuzzy Syst.
**2015**, 28, 581–589. [Google Scholar] [CrossRef] - Xu, K.; Wu, X.; Ma, M.; Zhang, Y. Energy-based output feedback control of the underactuated 2DTORA system with saturated inputs. Trans. Inst. Meas. Control
**2020**, 42, 2822–2829. [Google Scholar] [CrossRef] - Zheng, G.B.; Gao, B.T.; Liu, C.D. Fuzzy control design of periodic dynamic trajectory of TORA system. Control Eng.
**2019**, 26, 1029–1034. [Google Scholar] - Liu, C.; Gao, B.; Zheng, G.; Sun, G. Fuzzy control design of oscillating trajectory tracking for underactuated TORA. Electr. Mach. Control
**2018**, 22, 117–122. [Google Scholar] - Avis, J.M.; Nersesov, S.G.; Nathan, R. A comparison study of nonlinear control techniques for the RTAC system. Nonlinear Anal. Real World Appl.
**2010**, 11, 2647–2658. [Google Scholar] [CrossRef] - She, J.; Zhang, A.; Lai, X.; Wu, M. Global stabilization of 2-DOF underactuated mechanical systems-an equivalent-input-disturbance approach. Nonlinear Dyn.
**2012**, 69, 495–509. [Google Scholar] [CrossRef] - He, S.; Ji, H.; Yang, K. Semi-global output feedback tracking to reference system with input for a benchmark nonlinear system. Asian J. Control
**2019**, 21, 749–758. [Google Scholar] [CrossRef] - Qi, Z.; Li, H.; Wu, C. Adaptive Fuzzy Control of Nonlinear Systems With Unmodeled Dynamics and Input Saturation Using Small-Gain Approach. IEEE Trans. Syst. Man Cybern. Syst.
**2017**, 47, 1979–1989. [Google Scholar] - Zheng, Z.; Sun, L. Path following control for marine surface vessel with uncertainties and input saturation. Neurocomputing
**2016**, 177, 158–167. [Google Scholar] [CrossRef] - Bupp, R.T.; Bernstein, D.S.; Coppola, V.T. Experimental implementation of integrator backstepping and passive nonlinear controllers on the RTAC tested. Int. J. Robust Nonlinear Control
**1998**, 8, 435–457. [Google Scholar] [CrossRef] - Khalil, H.K. Nonlinear Systems, 3rd ed.; Prentice Hall: Upper Saddle River, NJ, USA, 2002. [Google Scholar]

**Figure 5.**Simulation results of the TORA system with external disturbances and parameter uncertainties.

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**MDPI and ACS Style**

Pan, C.; Cui, C.; Zhou, L.; Xiong, P.; Li, Z.
A Model-Free Output Feedback Control Approach for the Stabilization of Underactuated TORA System with Input Saturation. *Actuators* **2022**, *11*, 97.
https://doi.org/10.3390/act11030097

**AMA Style**

Pan C, Cui C, Zhou L, Xiong P, Li Z.
A Model-Free Output Feedback Control Approach for the Stabilization of Underactuated TORA System with Input Saturation. *Actuators*. 2022; 11(3):97.
https://doi.org/10.3390/act11030097

**Chicago/Turabian Style**

Pan, Changzhong, Chenchen Cui, Lan Zhou, Peiyin Xiong, and Zhijing Li.
2022. "A Model-Free Output Feedback Control Approach for the Stabilization of Underactuated TORA System with Input Saturation" *Actuators* 11, no. 3: 97.
https://doi.org/10.3390/act11030097