# Predefined-Time and Prescribed-Performance Control Methods Combined with Second-Order Terminal Sliding Mode Control for an Unmanned Planing Hull System with Input Delay and Unknown Disturbance

## Abstract

**:**

## 1. Introduction

- (1)
- Porpoising disturbance due to the pitching moment of UPH systems was assumed to enhance sensitivity to variations in the unmatched uncertainties of UPH systems.
- (2)
- A second-order TSMC system was designed to accommodate the control input delay in a hydraulic rudder actuator system.
- (3)
- A second-order TSMC system with predefined-time control was designed to achieve improved settling-time convergence and robustness to unmatched disturbances when compared with a conventional TSMC system. Furthermore, it aimed to achieve a faster and more stable heading-angle response by the UPH than conventional TSMC for a perturbed environment.
- (4)
- Predefined-time and prescribed-performance control methods were used simultaneously for the first time to maximize the rejection performance and obtain stable tracking performance for an unmatched pitching-moment disturbance without adopting a complex disturbance observer. Consequently, the tracking performance of the heading-angle axis was largely improved by utilizing the proposed hybrid disturbance rejector.
- (5)
- The proposed second-order TSMC system with a hybrid disturbance rejector facilitated by predefined-time and prescribed-performance control exhibited a simpler controller structure than the conventional robust control systems equipped with a complex disturbance observer to estimate the disturbance.
- (6)
- To the best of our knowledge, predefined-time and prescribed-performance control methods have not been applied simultaneously in the field of USV control to improve disturbance rejection performance. In fact, this concept is not demonstrated in other control systems.

## 2. Modeling the UPH

#### 2.1. Dynamics of UPH

#### 2.2. The Modeling of the Drag Force and Pitching Moment of the UPH

**Remark**

**1.**

#### 2.3. Modeling the Rudder Hydraulic Actuator

## 3. Controller Design of the UPH

#### 3.1. Fundamentals Regarding Predefined-Time Stability and Prescribed-Performance Function

**Definition**

**1**

**[28].**

**Definition**

**2**

**[28].**

**Definition**

**3**

**[28].**

**Definition**

**4**

**[28].**

**Lemma**

**1**

**[28].**

**Theorem**

**1.**

#### 3.2. Predefined-Time TSMC Controller Design for Surge Velocity

**Remark**

**2.**

#### 3.3. Predefined-Time Second-Order TSMC System Design for Heading Angle

**Remark**

**3.**

#### 3.4. Predefined-Time and Prescribed-Performance Second-Order TSMC System Design for Suppressing the Disturbance of the Heading Angle

## 4. Simulation Results and Discussion

#### 4.1. Simulation Results for the Step Command of Surge Velocity and Heading Angle

#### 4.2. Simulation Results for the Multi-Step Command of Surge Velocity and Sine Wave Heading Angle

#### 4.3. Simulation Results for the Multi-Step Command of Surge Velocity and Sine Wave Heading Angle for Pitching-Moment Disturbance

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Savitsky, D. Hydrodynamic design of planning hulls. Mar. Technol. SNAME News
**1964**, 1, 71–95. [Google Scholar] [CrossRef] - Utkin, V.I. Sliding mode control design principles and applications to electric devices. IEEE Trans. Ind. Electron.
**1993**, 40, 23–36. [Google Scholar] [CrossRef] - Chen, X.; Liu, Z.; Zhang, J.; Zhou, D.; Dong, J. Adaptive sliding mode path following control system of the underactuated USV under the influence of ocean currents. J. Syst. Eng. Electron.
**2018**, 29, 1271–1283. [Google Scholar] - Qiu, B.; Wang, G.; Fan, Y.; Mu, D.; Sun, X. Adaptive sliding mode trajectory control for unmanned surface vehicle with modeling uncertainties and input saturation. Appl. Sci.
**2019**, 9, 1240. [Google Scholar] [CrossRef] - Kim, H.W.; Lee, J. Robust sliding mode control for a USV water-jet system. Int. J. Nav. Archit. Ocean Eng.
**2019**, 11, 851–857. [Google Scholar] [CrossRef] - Liu, W.; Ye, H.; Yang, X. Model-free adaptive sliding mode control method for unmanned surface vehicle course control. J. Mar. Sci. Eng.
**2023**, 11, 1904. [Google Scholar] [CrossRef] - Khan, M.F.; Islam, R.U.; Iqbal, J. Control strategies for robotic manipulator. In Proceedings of the 2012 International Conference of Robotics and Artificial Intelligence, Rawalpindi, Pakistan, 22–23 October 2012. [Google Scholar] [CrossRef]
- Yu, S.; Yu, X.; Shrinzadeh, B.; Man, Z. Continuous finite-time control for robotic manipulators with terminal sliding mode. Automatica
**2005**, 41, 1957–1964. [Google Scholar] [CrossRef] - Zhao, D.; Li, S.; Gao, F. A new terminal sliding mode control for robotic manipulator. Int. J. Control
**2009**, 82, 1804–1813. [Google Scholar] [CrossRef] - Ahmad, S.; Uppal, A.A.; Azam, M.R.; Iqbal, J. Chattering free sliding mode control and state dependent Kalman filter design for underground gasification energy conversion process. Electronics
**2023**, 12, 876. [Google Scholar] [CrossRef] - Wan, L.; Su, Y.; Shi, B. Global fast terminal sliding mode control based on radial basis function neural network for course keeping of unmanned surface vehicle. Int. J. Adv. Robot. Syst.
**2019**, 24, 1729881419829961. [Google Scholar] [CrossRef] - Fan, Y.; Liu, B.; Wang, G.; Mu, D. Adaptive fast non-singular terminal sliding mode path following control for an underactuated unmanned surface vehicle with uncertainties and unknown disturbances. Sensors
**2021**, 21, 7454. [Google Scholar] [CrossRef] - Alejando, F.G.; Herman, C. Adaptive integral terminal sliding mode control for an unmanned surface vehicle against external disturbances. IFAC-PapersOnLine
**2021**, 54, 202–207. [Google Scholar] - Xu, D.; Liu, Z.; Song, J.; Zhou, X. Finite time trajectory tracking with full-state feedback of underactuated unmanned surface vessel based on nonsingular fast terminal sliding mode. J. Mar. Sci. Eng.
**2022**, 10, 1845. [Google Scholar] [CrossRef] - Wang, D.; Kong, M.; Zhang, G.; Liang, X. Adaptive second-order fast terminal sliding-mode formation control for unmanned surface vehicles. J. Mar. Sci. Eng.
**2022**, 10, 1782. [Google Scholar] [CrossRef] - Gong, X.W.; Yi, D.; Tezdogan, T.; Incecik, A. Adaptive neural network and extended state observer-based non-singular terminal sliding mode tracking control for an underactuated USV with unknown uncertainties. Appl. Ocean Res.
**2023**, 135, 103560. [Google Scholar] - Ghadimi, P.; Loni, A.; Nowruzi, H.; Dashtimanesh, A.; Tavakoli, S. Parametric study of the effects of trim tabs on running trim and resistance of planning hulls. Adv. Shipp. Ocean Eng.
**2014**, 3, 1–12. [Google Scholar] - Mansoori, M.; Fernades, A.C.; Ghassemi, H. Interceptor design of optimum trim control and minimum resistance of plaining boats. Appl. Ocean Res.
**2017**, 69, 100–115. [Google Scholar] [CrossRef] - Sakaki, A.; Ghassemi, H.; Keyvani, S. Evaluation of the hydraulic performance of plaining boat with trim tab and interceptor and its optimization using generic algorithm. J. Mar. Sci. Appl.
**2018**, 18, 131–141. [Google Scholar] [CrossRef] - Chen, Z.; Zhang, Y.; Zhang, Y.; Nie, Y.; Tang, J.; Zhu, S. Disturbance-observer-based sliding mode control design for nonlinear unmanned surface vessel with uncertainties. IEEE Access
**2019**, 2019, 148522–148530. [Google Scholar] [CrossRef] - Xu, D.; Liu, Z.; Zhou, X.; Yang, L.; Huang, L. Trajectory tracking of underactuated unmanned surface vessels: Nonsingular terminal sliding mode control with nonlinear disturbance observer. Appl. Sci.
**2022**, 12, 3004. [Google Scholar] [CrossRef] - Yao, Q. Fixed-time trajectory tracking control for unmanned surface vessels in the presence of model uncertainties and external disturbances. Int. J. Control
**2022**, 95, 1133–1143. [Google Scholar] [CrossRef] - Zhang, J.; Yu, S.; Yan, Y. Fixed-time extended observer-based trajectory tracking and point stabilization control for marine surface vessels with uncertainties and disturbances. Ocean Eng.
**2019**, 186, 106109. [Google Scholar] [CrossRef] - Li, M.; Guo, C.; Yu, H. Filtered extended state observer based line-of-sight guidance for path following of unmanned surface vehicles with unknown dynamics and disturbances. IEEE Access
**2019**, 7, 178401–178412. [Google Scholar] [CrossRef] - Wang, N.; Zhu, Z.; Qin, H.; Deng, Z.; Sun, Y. Finite-time extended state observer-based exact tracking control of an unmanned surface vehicle. Int. J. Robust Nonlinear Control
**2021**, 312, 1704–1719. [Google Scholar] [CrossRef] - Fan, Y.; Qiu, B.; Liu, L.; Yang, Y. Global fixed-time trajectory tracking control of underactuated USV based on fixed-time extended state observer. ISA Trans.
**2023**, 132, 267–277. [Google Scholar] [CrossRef] - Sanchez-Torres, J.D.; Sanchez, E.; Loukianov, A.G. Predefined-time stability of dynamical systems with sliding modes. In Proceedings of the 2015 American Control Conference (ACC), Chicago, IL, USA, 1–3 July 2015; pp. 5842–5846. [Google Scholar]
- Jimenez-Rodriguez, E.; Sanchez-Torres, J.D.; Loukianov, A.G. On optimal predefined-time stabilization. Int. J. Robust Nonlinear Control
**2017**, 27, 3620–3642. [Google Scholar] [CrossRef] - Sanchez-Torres, J.D.; Gomez-Gutierrez, D.; Lopez, E.; Loukianov, A.G. A class of predefined-time stable dynamical systems. IMA J. Math. Control Inf.
**2018**, 35, 1–29. [Google Scholar] [CrossRef] - Ye, D.; Zou, A.M.; Sun, Z. Predefined-time predefined-bounded attitude tracking control for rigid spacecraft. IEEE Trans. Aerosp. Electron. Syst.
**2022**, 58, 464–472. [Google Scholar] [CrossRef] - Wang, F.; Miao, Y.; Li, C.; Hwang, I. Attitude control of rigid spacecraft with predefined-time stability. J. Frankl. Inst.
**2020**, 357, 4212–4221. [Google Scholar] [CrossRef] - Xie, S.; Chen, Q. Adaptive nonsingular predefined-time control for attitude stabilization of rigid spacecrafts. IEEE Trans. Circuits Syst. II Express Briefs
**2022**, 69, 189–193. [Google Scholar] [CrossRef] - Sanchez-Torres, J.D.; Defoort, M.; Munoz-Vazquez, M. Predefined-time stabilization of a class of nonholonomic systems. Int. J. Control
**2020**, 93, 2941–2948. [Google Scholar] [CrossRef] - Pan, J.; Xiao, T.; Yan, H. Task-space multiple-biparite consensus for networked heterogenous Euler-Lagrange systems via hierarchical predefined-time control algorithm. Nonlinear Dyn.
**2023**, 111, 17095–17108. [Google Scholar] [CrossRef] - Li, K.; Hua, C.; You, X.; Ahn, K. Output feedback predefined-time bipartite consensus control for high-order nonlinear multiagent systems. IEEE Trans. Circuits Syst. I Regul. Pap.
**2021**, 68, 3069–3078. [Google Scholar] [CrossRef] - Mao, B.; Wu, X.; Lu, J.; Chen, G. Predefined-time bounded consensus of multiagent systems with unknown nonlinearity via distributed adaptive fuzzy control. IEEE Trans. Cybern.
**2022**, 53, 2622–2635. [Google Scholar] [CrossRef] [PubMed] - Liang, C.D.; Ge, M.F.; Liu, Z.W.; Ling, G.; Liu, F. Predefined-time formation tracking control of networked marine surface vehicles. Control Eng. Pract.
**2021**, 107, 104682. [Google Scholar] [CrossRef] - Jiang, T.; Yan, Y.; Yu, S.H. Adaptive sliding mode control for unmanned surface vehicles with predefined-time tracking performances. J. Mar. Sci. Eng.
**2023**, 11, 1244. [Google Scholar] [CrossRef] - Benchlioulis, C.P.; Rovithakis, G.A. Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance. IEEE Trans. Autom. Control
**2008**, 53, 2090–2099. [Google Scholar] [CrossRef] - Li, Y.; Shao, X.; Tong, S. Adaptive fuzzy prescribed performance control of nontriangular structure nonlinear systems. IEEE Trans. Fuzzy Syst.
**2020**, 28, 2416–2426. [Google Scholar] [CrossRef] - Li, K.; Li, Y. Fuzzy adaptive optimization prescribed performance control for nonlinear vehicle platoon. IEEE Trans. Fuzzy Syst.
**2023**. [Google Scholar] [CrossRef] - Han, S.I. Fuzzy supertwisting dynamic surface control for MIMO strict-feedback nonlinear dynamic systems with supertwisting nonlinear disturbance observer and a new partial tracking error constraint. IEEE Trans. Fuzzy Syst.
**2019**, 27, 2101–2114. [Google Scholar] [CrossRef] - Li, S.; Ma, T.; Luo, X.; Yang, Z. Adaptive fuzzy output regulation for unmanned surface vehicles with prescribed performance. Int. J. Control Autom. Syst.
**2020**, 18, 405–414. [Google Scholar] [CrossRef] - Jiang, K.; Mao, L.; Su, Y.; Zheng, Y. Trajectory tracking control for underactuated USV with prescribed performance and input saturation. Symmetry
**2021**, 13, 2208. [Google Scholar] [CrossRef] - Shen, Z.; Wang, Q.; Dong, S.; Yu, H. Prescribed performance dynamic surface control for trajectory-tracking of unmanned surface vessel with input saturation. Appl. Ocean Res.
**2021**, 113, 102736. [Google Scholar] [CrossRef] - Li, J.; Xiang, X.; Dong, D.; Yang, S. Saturated-command-deviation based finite-time adaptive control for dynamic positioning of USV with prescribed performance. Ocean Eng.
**2022**, 266, 112941. [Google Scholar] [CrossRef] - Qu, Y.; Zhao, W.; Yu, Z.; Xiao, B. Distributed prescribed performance containment control for unmanned surface vehicles based on disturbance observer. ISA Trans.
**2022**, 125, 699–706. [Google Scholar] [CrossRef] [PubMed] - Available online: https://www.simerics.com/simulation-gallery/planing-hull (accessed on 27 October 2023).
- Bhat, S.P.; Bernstein, D.S. Finite-time stability of homogenous systems. In Proceedings of the 1997 American Control Conference, Albuquerque, NM, USA, 6 June 1997; pp. 1073–1078. [Google Scholar]

**Figure 5.**Time-delayed output angle and rudder slew rate for rudder input. (

**a**) rudder angle; (

**b**) angle rate.

**Figure 6.**Time response for step surge velocity command: (

**a**) surge velocity output; (

**b**) surge velocity error.

**Figure 7.**Time response for step heading-angle command under input delay: (

**a**) heading-angle output of TSMC_1st and TSMC_2nd systems; (

**b**) heading-angle output of TSMC_2nd and PTSMC_2nd systems; (

**c**) heading-angle error of TSMC_1st, TSMC_2nd, and PTSMC_2nd systems.

**Figure 8.**Time response and control input for multi-step surge velocity command of PTSMC_2nd and P&PTSMC_2nd systems: (

**a**) surge velocity output; (

**b**) surge velocity error; (

**c**) propulsion force control input.

**Figure 9.**Time response and control input for sine heading-angle command of TSMC_2nd and PTSMC_2nd systems: (

**a**) sine angle output; (

**b**) sine angle error; (

**c**) rudder angle control input.

**Figure 10.**Disturbance response for multi-step surge velocity command of PTSMC_2nd and P&PTSMC_2nd systems: (

**a**) multi-step velocity output; (

**b**) multi-step velocity error.

**Figure 11.**Disturbance response for sine heading-angle command: (

**a**) generated pitching moment; (

**b**) sine heading-angle output; (

**c**) sine heading-angle error of TSMC_2nd, PTSMC_2nd, and P&PTSMC_2nd systems; (

**d**) sine heading-angle error of PTSMC_2nd and P&PTSMC_2nd systems for the prescribed error bound.

Nomenclature | Force and Moment | Linear and Rotary Velocity | Position and Euler Angle |
---|---|---|---|

Surge | X | u | x |

Sway | Y | v | y |

Heave | Z | w | z |

Roll | K | p | $\varphi $ |

Pitch | M | q | $\theta $ |

Yaw | N | r | $\psi $ |

Symbol | Semantics |
---|---|

LCG | the longitudinal distance from the transom to the center of gravity |

VCG | the perpendicular distance for the bottom from keel to the center of gravity |

$N,T$ | the vertical drag force and the propulsion force of the propeller |

${D}_{f}$ | the viscous friction drag of the body |

$f,c$ | the distance between T, N, and the center of gravity, respectively |

$d,\tau ,b$ | the draft of the keel in the transom, the trim angle of keel, and the chine width |

$a$, $V$ | The distance between ${D}_{f}$ and the center of gravity, and the speed of the boat |

$\epsilon $ | the angle between the propulsion force and trim line |

$\beta $, $\Delta $ | the inclination angle of the planing side and the weight of the body |

${L}_{C},{L}_{k}$ | the flooded length of the chine and the flooded length of keel |

Parameter | Value | Parameter | Value |
---|---|---|---|

$m$ | $20.8\mathrm{kg}$ | $f,c,d,a$ | $0.026\mathrm{m},0.1\mathrm{m},0.12\mathrm{m},0.07\mathrm{m}$ |

${I}_{x},{I}_{y},{I}_{z}$ | $0.2{\mathrm{kgm}}^{2},0.452{\mathrm{kgm}}^{2},0.275{\mathrm{kgm}}^{2}$ | $b,\lambda $ | $0.71\mathrm{m},0.41$ |

${x}_{g},{y}_{g},{z}_{g}$ | $0.694\mathrm{m},0,0.23\mathrm{m}$ | $\epsilon ,\beta $ | ${4}^{\circ},{10}^{\circ}$ |

LCG, VCG | $1.47\mathrm{m},0.11\mathrm{m}$ | $\tau ,\alpha $ | ${3}^{\circ},{27.5}^{\circ}$ |

$\rho ,{C}_{f}$ | $1026{\mathrm{kg}/\mathrm{m}}^{3},0.8$ | ${A}_{w}$ | $0.21{\mathrm{m}}^{2}$ |

Parameter (Surge) | Value | Parameter (Heading) | Value |
---|---|---|---|

${c}_{u1},{c}_{u2},{c}_{u3}$ | $2,1.5,1$ | ${c}_{r1},{c}_{r2},{c}_{r3},{c}_{r4},{\kappa}_{r}$ | $2,1,2,10,50$ |

${k}_{u1},{k}_{u2},{\epsilon}_{u}$ | $1,1.2,0.01$ | ${k}_{r1},{k}_{r2},{\epsilon}_{r}$ | $1,1.25,0.01$ |

${T}_{cu},{p}_{u},{\gamma}_{u}$ | $3,0.2,0.5$ | ${\alpha}_{1},{\alpha}_{2}$ | $9/16,9/23$ |

${T}_{cr},{p}_{r},{\gamma}_{r}$ | $3,0.05,0.5$ | ||

${\rho}_{0},{\rho}_{ss},{a}_{r}$ | ${30}^{\circ},{10}^{\circ},0.5$ |

System | PSMC_2nd | P&PTSMC_2nd |
---|---|---|

ITAE (sec-deg) | 4557 (100%) | 3574 (78%) |

Maximum absolute error (deg) | 7.02 (100%) | 2.58 (37%) |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Han, S.
Predefined-Time and Prescribed-Performance Control Methods Combined with Second-Order Terminal Sliding Mode Control for an Unmanned Planing Hull System with Input Delay and Unknown Disturbance. *J. Mar. Sci. Eng.* **2023**, *11*, 2191.
https://doi.org/10.3390/jmse11112191

**AMA Style**

Han S.
Predefined-Time and Prescribed-Performance Control Methods Combined with Second-Order Terminal Sliding Mode Control for an Unmanned Planing Hull System with Input Delay and Unknown Disturbance. *Journal of Marine Science and Engineering*. 2023; 11(11):2191.
https://doi.org/10.3390/jmse11112191

**Chicago/Turabian Style**

Han, Seongik.
2023. "Predefined-Time and Prescribed-Performance Control Methods Combined with Second-Order Terminal Sliding Mode Control for an Unmanned Planing Hull System with Input Delay and Unknown Disturbance" *Journal of Marine Science and Engineering* 11, no. 11: 2191.
https://doi.org/10.3390/jmse11112191