Abstract
This paper introduces a novel ISS stability evaluation for a LNU based HONU-MRAC control loop where an LNU serves as a plant and a HONU as a non-linear polynomial feedback controller. Till now, LNUs have proven their advantages as computationally efficient and effective approximators, further optimisers of linear and weakly non-linear dynamic systems. Due to the fundamental construction of an HONU-MRAC control loop featuring analogies with discrete-time non-linear dynamic models, two novel state space representations of the whole LNU based HONU-MRAC control loop are presented. Backboned by the presented state space forms, the ISS stability evaluation is derived and verified with theories of bounded-input-bounded-state (BIBS) and Lyapunov stability on a practical non-linear system example.
Keywords
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Benes, P.M., Bukovsky, I.: Neural network approach to hoist deceleration control. In: 2014 International Joint Conference on Neural Networks (IJCNN), pp. 1864–1869 (2014)
Benes, P.M., Erben, M., Vesely, M., Liska, O., Bukovsky, I.: HONU and supervised learning algorithms in adaptive feedback control. In: Applied Artificial Higher Order Neural Networks for Control and Recognition, IGI Global, pp. 35–60 (2016)
Benes, P.M., Bukovsky, I., Cejnek, M., Kalivoda, J.: Neural network approach to railway stand lateral skew control. In: Computer Science & Information Technology (CS& IT), Sydney, Australia, vol. 4, pp. 327–339 (2014)
Bukovsky, I., Benes, P., Slama, M.: Laboratory systems control with adaptively tuned higher order neural units. In: Silhavy, R., Senkerik, R., Oplatkova, Z.K., Prokopova, Z., Silhavy, P. (eds.) Intelligent Systems in Cybernetics and Automation Theory, pp. 275–284. Springer International Publishing (2015)
Benes, P., Bukovsky, I.: On the intrinsic relation between linear dynamical systems and higher order neural units. In: Silhavy, R., Senkerik, R., Oplatkova, Z.K., Prokopova, Z., Silhavy, P. (eds.) Intelligent Systems in Cybernetics and Automation Theory. Springer International Publishing (2016)
Yadav, R.N., Kalra, P.K., John, J.: Time series prediction with single multiplicative neuron model. Appl. Soft Comput. 7(4), 1157–1163 (2007)
Zhao, H., Zeng, X., He, Z.: Low-complexity nonlinear adaptive filter based on a pipelined bilinear recurrent neural network. IEEE Trans. Neural Netw. 22(9), 1494–1507 (2011)
Zhao, H., Zhang, J.: A novel nonlinear adaptive filter using a pipelined second-order Volterra recurrent neural network. Neural Netw. 22(10), 1471–1483 (2009)
Bukovsky, I., Homma, N.: An approach to stable gradient-descent adaptation of higher order neural units. IEEE Trans. Neural Netw. Learn. Syst. 28(9), 2022–2034 (2017)
Barabanov, N.E., Prokhorov, D.V.: A new method for stability analysis of nonlinear discrete-time systems. IEEE Trans. Autom. Control 48(12), 2250–2255 (2003)
Zhao, W., Zhu, Q.: New results of global robust exponential stability of neural networks with delays. Nonlinear Anal. Real World Appl. 11(2), 1190–1197 (2010)
Liao, X., Chen, G., Sanchez, E.N.: LMI-based approach for asymptotically stability analysis of delayed neural networks. IEEE Trans. Circ. Syst. Fundam. Theory Appl. 49(7), 1033–1039 (2002)
Liao, X., Chen, G., Sanchez, E.N.: Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach. Neural Netw. 16(10), 1401–1402 (2003)
Yasuda, Y., Ito, H., Ishizu, H.: On the input-output stability of linear time-varying systems. SIAM J. Appl. Math. 38(1), 175–188 (1980)
Li, Z., Fei, Z., Gao, H.: Stability and stabilisation of Markovian jump systems with time-varying delay: an input-output approach. IET Control Theory Appl. 6(17), 2601–2610 (2012)
Lazar, M., Heemals, W.P.M.H., Teel, A.R.: Further input-to-state stability subtleties for discrete-time systems. IEEE Trans. Autom. Control 58(6), 1609–1613 (2013)
Sontag, E.D.: Input to state stability: basic concepts and results. In: Nonlinear and Optimal Control Theory, vol. 1932, pp. 163–220. Springer, Heidelberg (2008)
Angeli, A., Sontag, E.D., Wang, Y.: A characterization of integral input-to-state stability. IEEE Trans. Autom. Control 45(6), 1082–1097 (2000)
Jiang, Z.P., Wang, Y.: Input-to-state stability for discrete-time nonlinear systems. Automatica 37(6), 857–869 (2001)
Yang, Z., Zhou, W., Huang, T.: Exponential input-to-state stability of recurrent neural networks with multiple time-varying delays. Cogn. Neurodyn. 8(1), 47–54 (2014)
Ahn, C.K.: Robust stability of recurrent neural networks with ISS learning algorithm. Nonlinear Dyn. 65, 413–419 (2011)
Zhu, S., Shen, Y.: Two algebraic criteria for input-to-state stability of recurrent neural networks with time-varying delays. Neural Comput. Appl. 22(6), 1163–1169 (2013)
Acknowledgements
Authors acknowledges support from the EU Operational Programme Research, Development and Education, and from the Center of Advanced Aerospace Technology (CZ.02.1.01/0.0/0.0/16_019/0000826)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Benes, P., Bukovsky, I. (2019). An Input to State Stability Approach for Evaluation of Nonlinear Control Loops with Linear Plant Model. In: Silhavy, R. (eds) Cybernetics and Algorithms in Intelligent Systems . CSOC2018 2018. Advances in Intelligent Systems and Computing, vol 765. Springer, Cham. https://doi.org/10.1007/978-3-319-91192-2_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-91192-2_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-91191-5
Online ISBN: 978-3-319-91192-2
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)