Abstract
This chapter studies the PID robust tracking control for second-order systems with input time-varying delay and uncertain nonlinearity. An input-output approach is adopted to derive a sufficient condition on the existence of robust PID control, which can tolerant the time-varying delay and uncertain nonlinearity. First, the uncertain nonlinear system under PID control subject to time-varying delay can be transferred into two inter-connected subsystems through the input-output approach, which is prone to be analyzed by the scaled small-gain theorem. Then the three-term approximation method is used to approximate the time-varying delay. By constructing the Lyapunov–Krasovskii functional, a sufficient condition on how to design the robust PID controller against time-varying delay and uncertain nonlinearity is obtained in terms of LMIs. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method.
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
References
Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994)
Chen, J., Ren, Z.: A comparison of small gain versus Lyapunov type robust stability bounds. Int. J. Robust Nonlinear Control 11, 1407–1414, 12 (2001)
el Aiss, H., Hmamed, A., EL Hajjaji, A.: Improved stability and \({H}_\infty \) performance criteria for linear systems with interval time-varying delays via three terms approximation. Int. J. Syst. Sci. 48(16), 3450–3458 (2017)
Fridman, E., Shaked, U.: Delay-dependent stability and \({H}_\infty \) control: constant and time-varying delays. Int. J. Control 76, 48–60 (2003)
Fridman, E.: On robust stability of linear neutral systems with time-varying delays. IFAC Proc. Vol. 39(10), 12–17 (2006)
Fridman, E.: Tutorial on Lyapunov-based methods for time-delay systems. Eur. J. Control 20(6), 271–283 (2014)
Fridman, E., Shaked, U.: Input-output approach to stability and \({L}_2\)-gain analysis of systems with time-varying delays. Syst. & Control Lett. 55(12), 1041–1053 (2006)
Gu, K.: An integral inequality in the stability problem of time-delay systems. In: Proceedings of the 39th IEEE Conference on Decision and Control, vol. 3, pp. 2805–2810 (2000)
Gu, K., Kharitonov, V., Chen, J.: Stability of Time-Delay System. Birkhuser, Basel (2003)
Gu, K., Zhang, Y., Xu, S.: Small gain problem in coupled differential-difference equations, time-varying delays, and direct Lyapunov method. Int. J. Robust Nonlinear Control 21(4), 429–451 (2011)
Hmamed, A., EL Aiss, H., EL Hajjaji, A..: Stability analysis of linear systems with time varying delay: an input output approach. In: IEEE Conference on Decision and Control, pp. 1756–1761 (2015)
Huang, Y., Zhou, K.: Robust stability of uncertain time-delay systems. IEEE Trans. Autom. Control 45(11), 2169–2173 (2000)
Khalil, H.: Nonlinear Systems, 3rd edn. Prentice Hall, London (2002)
Lu, Z., Ma, D., Wang, J.: Delay robustness of second-order uncertain nonlinear delay systems under PID control. In: 2019 IEEE Conference on Control Technology and Applications, pp. 1068–1073 (2019)
Ma, D., Chen, J.: Delay margin of low-order systems achievable by PID controllers. IEEE Trans. Autom. Control 64(5), 1958–1973 (2019)
Ma, D., Chen, J., Liu, A., Chen, J., Niculescu, S.-I.: Explicit bounds for guaranteed stabilization by PID control of second-order unstable delay systems. Automatica 100, 407–411 (2019)
Ma, D., Chen, J., Chai, T.: Role of integral control for enlarging second-order delay consensus margin under PID protocols: none. IEEE Trans. Cybern. 1–11 (2021). https://doi.org/10.1109/TCYB.2021.3085952
Ma, D., Tian, R., Zulfiqar, A., Chen, J., Chai, T.: Bounds on delay consensus margin of second-order multiagent systems with robust position and velocity feedback protocol. IEEE Trans. Autom. Control 64(9), 3780–3787 (2019)
Park, P., Lee, W.I., Lee, S.Y.: Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems. J. Frankl. Inst. 352(4), 1378–1396 (2015)
Shao, H., Han, Q.: Less conservative delay-dependent stability criteria for linear systems with interval time-varying delays. Int. J. Syst. Sci. 43(5), 894–902 (2012)
Xie, X., Yin, S., Gao, H., Kaynak, O.: Asymptotic stability and stabilisation of uncertain delta operator systems with time-varying delays. IET Control Theory & Appl. 7(8), 1071–1078 (2013)
Xu, J., Zhang, H., Xie, L.: Input delay margin for consensusability of multi-agent systems. Automatica 49(6), 1816–1820 (2013)
Zhang, J., Knopse, C.R., Tsiotras, P.: Stability of time-delay systems: equivalence between Lyapunov and scaled small-gain conditions. IEEE Trans. Autom. Control 46(3), 482–486 (2001)
Zhao, C., Guo, L.: PID controller design for second order nonlinear uncertain systems. Sci. China 60(2), 022201 (2017)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Yaopo, L., Ma, D. (2022). Robust PID Controller Design of Second-Order Uncertain Nonlinear Time-Varying Delay System. In: Shi, P., Stefanovski, J., Kacprzyk, J. (eds) Complex Systems: Spanning Control and Computational Cybernetics: Foundations. Studies in Systems, Decision and Control, vol 414. Springer, Cham. https://doi.org/10.1007/978-3-030-99776-2_11
Download citation
DOI: https://doi.org/10.1007/978-3-030-99776-2_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-99775-5
Online ISBN: 978-3-030-99776-2
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)