Abstract
This paper provides stability analysis results for discretised time delay control (TDC) as implemented in a sampled data system with the standard form of zero-order hold. We first substantiate stability issues in discrete-time TDC using an example and propose sufficient stability criteria in the sense of Lyapunov. Important parameters significantly affecting the overall system stability are the sampling period, the desired trajectory and the selection of the reference model dynamics.
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R. Morgan, U. Ozguner. A decentralized variable structure control algorithm for robotic manipulators. IEEE Journal on Robotics and Automation, 1985, 1(1): 57–65.
K. Youcef-Toumi, O. Ito. Controller design for systems with unknown nonlinear dynamics. Proceedings of the American Control Conference, Minneapolis: IEEE, 1987: 836–845.
T. C. Hsia. On a simplified joint controller design for robot manipulators. Proceedings of the IEEE Conference on Decision and Control, Los Angeles: IEEE, 1987: 1024–1025.
T. C. Hsia. A new technique for robust control of servo systems. IEEE Transactions on Industrial Electronics, 1989, 36(1): 1–7.
P. H. Chang, D. S. Kim, K. C. Park. Robust force/position control of a robot manipulator using time-delay control. Control Engineering Practice, 1995, 3(9): 1255–1264.
H-S. Kim, K-H. Kim, M-J. Youn. On-line dead-time compensation method based on time delay control. IEEE Transactions on Control Systems Technology, 2003, 11(2): 279–285.
S. E. Talole, A. Ghosh, S. B. Phadke. Proportional navigation guidance using predictive and time delay control. Control Engineering Practice, 2006, 14(12): 1445–1453.
R. P. Kumar, C. S. Kumar, D. Sen, et al. Discrete time-delay control of an autonomous underwater vehicle: Theory and experimental results. Ocean Engineering, 2009, 36(1): 74–81.
P. H. Chang, J. H. Jung. A systematic method for gain selection of robust PID control for nonlinear plants of second-order controller canonical form. IEEE Transactions on Control Systems Technology, 2009, 17(2): 473–483.
G. R. Cho, P. H. Chang, S. H. Park, et al. Robust tracking under nonlinear friction using time-delay control with internal model. IEEE Transactions on Control Systems Technology, 2009, 17(6): 1406–1414.
J. Lee, C. Yoo, Y-S. Park, et al. An experimental study on time delay control of actuation system of tilt rotor unmanned aerial vehicle. Mechatronics, 2012, 22(2): 184–194.
J. H. Lee, B. J. Kim, J. S. Kim, et al. Time-delay control of ionic polymer metal composite actuator. Smart Materials and Structures, 2015, 24(4): DOI https://doi.org/10.1088/0964-1726/24/4/047002.
J. Kim, H. Joe, S-C. Yu, et al. Time-delay controller design for position control of autonomous underwater vehicle under disturbances. IEEE Transactions on Industrial Electronics, 2016, 63(2): 1052–1061.
B. U. Rehman, M. Focchi, J. Lee, et al. Towards a multilegged mobile manipulator. Proceedings of the IEEE International Conference on Robotics and Automation, Stockholm: IEEE, 2016: 3618–3624.
K. Youcef-Toumi, O. Ito. A time delay controller for systems with unknown dynamics. Journal of Dynamic Systems, Measurement, and Control, 1990, 112(1): 133–142.
K. Youcef-Toumi, S-T. Wu. Input/output linearization using time delay control. Journal of Dynamic Systems, Measurement, and Control, 1992, 114(1): 10–19.
J. H. Jung, P. H. Chang, D. Stefanov. Discretisation method and stability criteria for non-linear systems under discrete-time time delay control. IET Control Theory & Applications, 2011, 5(11): 1264–1276.
J. Lee, G. A. Medrano-Cerda, J. H. Jung. Corrections for Discretisation method and stability criteria for non-linear systems under discrete-time time delay control. IET Control Theory & Applications, 2016, 10(14): 1751–1754.
H. K. Khalil. Nonlinear Systems. 3rd ed. Upper Saddle River: Prentice Hall, 2002.
S. Dayal. A converse of Taylor's theorem for functions on Banach spaces. Proceedings of the American Mathematical Society, 1977, 65(2): 265–273.
B. Friedland. Control System Design — An Introduction to State Space Methods. New York: McGraw-Hill, 1989.
D. Nešić, A. R. Teel, P. V. Kokotović. Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations. Systems & Control Letters, 1999, 38(4): 259–270.
D. Nešićc, A. R. Teel, E. D. Sontag. Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems. Systems & Control Letters, 199, 38(1): 49–609.
F. R. Gantmacher. The Theory of Matrices. New York: Chelsea Publishing Company, 1959.
P. H. Chang, J. W. Lee. A model reference observer for timedelay control and its application to robot trajectory control. IEEE Transactions on Control Systems Technology, 1996, 4(1): 2–10.
J. R. Dormand, P. J. Prince. A family of embedded Runge-Kutta formulae. Journal of Computational and Applied Mathematics, 1980, 6(1): 19–26.
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Jinoh LEE received the B.Sc. degree in Mechanical Engineering from Hanyang University, Seoul, South Korea, in 2003 (awarded Summa Cum Laude), and the M.Sc. and Ph.D. degrees in Mechanical Engineering from the Korea Advanced Institute of Science and Technology, Daejeon, South Korea, in 2012. He is currently a Research Scientist at the Department of Advanced Robotics, Istituto Italiano di Tecnologia (IIT), Genoa, Italy, where he held postdoctoral researcher position from 2012 to 2017. His professional is about robotics and control engineering, which include manipulation of highly redundant robots such as dual-arm and humanoids, robust control of nonlinear systems and compliant robotic system control for safe human-robot interaction.
Gustavo A. MEDRANO-CERDA received the B.Sc. degree in Electro-Mmechanical Engineering from the Universidad Nacional Autonoma de Mexico in 1977, and the M.Sc. and Ph.D. degrees in Control Systems from Imperial College, London, in 1979 and 1982, respectively. From 1982 to 1985 he was an associate professor at the Division de Estudios de Postgrado, Facultad de Ingenieria, Universidad Nacional Autonoma de Mexico. From 1985 to 1986 he was a research fellow at the Department of Engineering, University of Warwick. From 1986 to 2002 he was a lecturer at the Department of Electronic and Electrical Engineering at the University of Salford. During this period he set up the Advanced Control and Robot Locomotion Laboratory at the University of Salford. In 1999 he became a control systems consultant at Las Cumbres Observatory (formerly Telescope Technologies Ltd.) and later in 2002 he joined the company as a senior control engineer pioneering work in H-infinity control system design and implementation for astronomical telescopes. Since 2009 he has been a senior research scientist at the Advanced Robotics Department, Istituto Italiano di Tecnologia. His research interests are in the areas of robust control, adaptive control, modelling and identification, fuzzy systems and advanced robotic applications, in particular to walking robots.
Je Hyung JUNG received his Ph.D. degree in Mechanical Engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea in 2006. He was a Postdoctoral Fellow in the Biorobotics Institute of Scuola Superiore Sant’Anna, located in Pontedera (Pisa), Italy from 2006 to 2008. Since June 2008, he has been with Tecnalia, where he is currently a senior researcher in Health Division. His research interests include modelling and control of robotic system such as multi degree of freedom manipulators and mobile robots, and design and control of rehabilitation, assistive and wearable (exoskeleton) robots, and variable stiffness materials.
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Lee, J., Medrano-Cerda, G.A. & Jung, J.H. Stability analysis for time delay control of nonlinear systems in discrete-time domain with a standard discretisation method. Control Theory Technol. 18, 92–106 (2020). https://doi.org/10.1007/s11768-020-9125-2
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DOI: https://doi.org/10.1007/s11768-020-9125-2