Abstract
The nonlinear trajectory tracking control problem for underactuated mechanical systems is considered on the example of a ship motion model. The control that implements the goal should be designed taking into account feedback delay as well as state and control constraints. The purpose is to propose a control law such that, on the one hand, it has the form of explicit feedback, on the other hand, its parameters can be found using standard software tools, provided that the numerical characteristics of the system are specified. In this case, uniform asymptotic stability of the desired motion of the system should be ensured.
The problem is solved by reducing to the stabilization problem for a triangular LPV system with the subsequent transformation into several optimization problems with standard numerical procedures for solving. The domain of attraction that fit into the prescribed region is estimated via sublevel sets of quadratic Lyapunov functions using the Razumikhin conditions and stability properties of cascaded systems. The developed algorithms are implemented based on standard procedures of computational software. The results obtained can be applied to other control problems.
Supported by Basic Research Program I.7 “New Developments in Perspective Areas of Energetics, Mechanics and Robotics” of the Presidium of RAS.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Aguiar, A.P., Hespanha J.P.: Position tracking of underactuated vehicles. In: Proceedings of the 2003 American Control Conference, Denver, CO, USA, June 2003, vol. 3, pp. 1988–1993 (2003)
Al-Hiddabi, S., McClamroch, N.: Tracking and maneuver regulation control for nonlinear nonminimum phase systems: application to flight control. IEEE Trans. Contr. Syst. Tech. 10(6), 780–792 (2002)
Balandin, D.V., Kogan, M.M.: Linear control design under phase constraints. Autom. Remote Control 70, 958–966 (2009)
Bao-Li, M., Wen-Jing, X.: Global asymptotic trajectory tracking and point stabilization of asymmetric underactuated ships with non-diagonal inertia/damping matrices. Int. J. Adv. Robotic Sy. 10(9), 336 (2013)
Berge, S., Ohtsu, K., Fossen, T.: Nonlinear control of ships minimizing the position tracking errors. Model. Identif. Control. 20, 141–147 (1999)
Do, K.D., Jiang, Z.P., Pan, J.: Underactuated ship global tracking under relaxed conditions. IEEE Trans. Automat. Contr. 47(9), 1529–1536 (2002)
Druzhinina, O.V., Sedova, N.O.: Analysis of stability and stabilization of cascade systems with time delay in terms of linear matrix inequalities. J. Comput. Syst. Sci. Int. 56, 19–32 (2017)
Fredriksen, E., Pettersen, K.Y.: Global \(k\)-exponential way-point maneuvering of ships: theory and experiments. Automatica 42, 677–687 (2006)
Fossen, T.I.: Guidance and Control of Ocean Vehicles. Wiley, Chichester (1994)
Fossen, T.I.: Handbook of Marine Craft Hydrodynamics and Motion Control. Wiley, Hoboken (2011)
Fossen, T.I., Pettersen, K.Y., Nijmeijer, H. (eds.): Sensing and Control for Autonomous Vehicles. LNCIS, vol. 474. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-55372-6
Jiang, Z.-P.: Global tracking control of underactuated ships by Lyapunovs direct method. Automatica 38, 301–309 (2002)
Kaminer, I., Pascoal, A., Hallberg, E., Silvestre, C.: Trajectory tracking controllers for autonomous vehicles: an integrated approach to guidance and control. J. Guid. Control Dyn. 21(1), 29–38 (1998)
Khlebnikov, M.V., Shcherbakov, P.S.: Optimal feedback design under bounded control. Autom. Remote Control 75(2), 320–332 (2014). https://doi.org/10.1134/S0005117914020118
Lefeber, A.A.J.: Tracking Control of Nonlinear Mechanical Systems. Universiteit Twente, Enschede (2000)
Loria, A., Fossen, T.I., Panteley, E.: A separation principle for dynamic positioning of ships: theoretical and experimental results. IEEE Trans. Control Syst. Technol. 8(2), 332–343 (2000)
Loria, A., Panteley, E.: 2 Cascaded nonlinear time-varying systems: analysis and design. In: Lamnabhi-Lagarrigue, F., Loria, A., Panteley, E. (eds.) Advanced Topics in Control Systems Theory. LNCIS, vol. 311, pp. 23–64. Springer, London (2005). https://doi.org/10.1007/11334774_2
Panteley, E., Loria, A.: Growth rate conditions for uniform asymptotic stability of cascaded time-varying systems. Automatica 37(3), 453–460 (2001)
Pesterev, A.V., Rapoport, L.B.: Construction of invariant ellipsoids in the stabilization problem for a wheeled robot following a curvilinear path. Autom. Remote Control 70, 219–232 (2009)
Pettersen, K.Y., Nijmeijer, H.: Global practical stabilization and tracking for an underactuated ship - a combined averaging and backstepping approach. In: Proceedings of IFAC Conference System Structure Control, Nantes, France, July 1998, pp. 59–64 (1998)
Pettersen, K.Y., Nijmeijer, H.: Underactuated ship tracking control: theory and experiments. Int. J. Control 74(14), 1435–1446 (2001)
Razumikhin, B.S.: On stability on systems with delay. Prikl. Mat. Mekh. 20, 500–512 (1956). [in Russian]
Strand, J.P., Fossen, T.I.: Nonlinear passive observer design for ships with adaptive wave filtering. In: Nijmeijer, H., Fossen, T. (eds.) New Directions in nonlinear observer design. LNCIS, vol. 244, pp. 113–134. Springer, London (1999). https://doi.org/10.1007/BFb0109924
Tanaka, K., Wang, H.O.: Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. Wiley, Hoboken (2001)
Veremey, E.I. Dynamical correction of positioning control laws. In: Proceedings of 9th IFAC Conference on Control Applications in Marine Systems (CAMS-2103). Japan, Osaka, pp. 31–36 (2013)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Druzhinina, O., Sedova, N. (2020). Optimization Problems in Tracking Control Design for an Underactuated Ship with Feedback Delay, State and Control Constraints. In: Olenev, N., Evtushenko, Y., Khachay, M., Malkova, V. (eds) Optimization and Applications. OPTIMA 2020. Lecture Notes in Computer Science(), vol 12422. Springer, Cham. https://doi.org/10.1007/978-3-030-62867-3_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-62867-3_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-62866-6
Online ISBN: 978-3-030-62867-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)