Tridimensional autonomous motion robust control of submersible ship based on averaged sub-gradient integral sliding mode approach

Abstract

The design of a state feedback and robust controller for regulating the tridimensional (3D) movement of autonomous underwater mobile crafts (AUMCs). The controller design is based on the application of Averaged Sub-Gradient Integral Sliding Mode Realisation (ASGISMR). The application of the ASGISMR yields the solution of the closed-loop extreme seeking control for a non-strictly convex functional depending on the tracking error between reference and 3D position of the AUMC. The design of reference trajectories was proposed to enforce the AUMC to follow a continued submersion and ellipsoidal detouring. The mechanical dynamical form of AUMC is well posed for applying the extended version of ASGISMR, considering that integral term represents an ASG associated to the euclidean norm of the tracking error. The time evolution of the functional over the controlled trajectories of the AUMC is compared with the corresponding functional enforced by a traditional state feedback controller with gravity effect compensation. The proposed controller exhibits better tracking and similar control magnitude than the considered reference state feedback realisation. These outcomes justify the potential contributions of the suggested ASGISMR to obtain the local minimisation of the evaluated functional, considering such controller as an extreme seeking realisation for the proposed AUMC.

Keywords:

• Autonomous vehicles
• sliding mode control
• averaged sub-gradient
• integral sliding mode
• robust control

Acknowledgements

Alejandra Hernandez-Sanchez thanks the Consejo Nacional de Ciencia y Tecnologia (CONACyT) for the economical support for the development of her PhD scholarship.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 For more details on the nature and mathematical description of $A_C$ and $A_D$ see to the review (da Silva et al., 2007).

2 By the proposition given in Def. 21.10 in Poznyak (2009), a vector $a\in R^n$, satisfying the following inequality $F(x+y)\ge F\left(x\right)+a^{⊺}y$for $y\in R^n,$ is known as the sub-gradient of the vector function $F\left(x\right)$ at the point $x\in R^n$ and it is denoted as $a=\mathrm{\partial }F\left(x\right)$. Assuming that $F\left(x\right)$ is differentiable at the given point x, therefore $\mathrm{\partial }F\left(x\right)=\mathrm{\nabla }F\left(x\right).$ A set of sub-gradient denoted by $\mathcal{D}F\left(x\right)$. At the extreme point $x^{\ast }$ we have $0\in \mathcal{D}F\left(x^{\ast }\right).$

Additional information

Funding

Alejandra Hernandez-Sanchez thanks the Consejo Nacional de Ciencia y Tecnologia (CONACyT) for the economical support for the development of her PhD scholarship.

Notes on contributors

Alejandra Hernandez-Sanchez

Alejandra Hernandez-Sanchez received the B.E. in automatization and control engineering from the Instituto Politecnico Nacional in 2016, the M.S. degree in Automatic Control from the CINVESTAV, MEXICO in 2018 respectively. Actually, she is a Ph. D. student in the same department. Her research interests include control theory, sliding mode control, Hamiltonian systems, mechanical systems, parameter identification and motion control.

Olga Andrianova

Olga Andrianova received an M.S. degree in Applied Mathematics from Bauman Moscow State Technical University in 2011, the Ph.D. in System Analysis, Automatic Control, and Information Processing from Trapeznikov Institute of Control Sciences of Russian Academy of Sciences (ICS RAS) in 2015. Actually she is a senior researcher at ICS RAS and an associate professor at the Department of Applied Mathematics of the Higher School of Economics. Her research interests include robust control, adaptive control, differential neural networks, convex optimization, stochastic system theory, control education.

Alexander Poznyak

Alexander S. Poznyak graduated from the Moscow Physical Technical Institute (MPhTI) in 1970. He earned his Ph.D. and Doctor Degrees from the Institute of Control Sciences of the Russian Academy of Sciences in 1978 and 1989, respectively. From 1973 up to 1993 he served this institute as a researcher and leading researcher, and in 1993 he accepted a post of full professor (3-E) at CINVESTAV of IPN in Mexico. He is a Regular Member of the Mexican Academy of Sciences and System of National Investigators. He is a Fellow of IMA (Institute of Mathematics and Its Applications, Essex, UK) and Associated Editor of the Oxford-IMA Journal on Mathematical Control and Information, as well as the Iberoamerican International Journal on ‘‘Computations and Systems”. He was also an Associated Editor of CDC, ACC and Member of the Editorial Board of IEEE CSS.

Isaac Chairez

Isaac Chairez earned the B.S. degree in biomedical engineering from the National Polytechnic Institute (IPN), Mexico City, Mexico, in 2002, and the master’s and Ph.D. degrees from the Department of Automatic Control, Center of Investigation and Advanced Researching (CINVESTAV), IPN, in 2004 and 2007, respectively, where he is currently pursuing the degree with CINVESTAV. He is currently with the Professional Interdisciplinary Unit of Biotechnology, IPN. His current research interests include neural networks, fuzzy control theory, nonlinear control, adaptive control, and game theory.