Abstract
In control theory, optimization is a process that makes some system behavior as likelihood, functional, or practical as possible (depending on the design requirements). From a mathematical point of view, optimization is a way to find the minimum or maximum of the mathematical function by which the system is described. Moreover, if there is a time delay in the feedback loops inside a controlled system, the system has an infinite spectrum – infinitely many roots (poles); this degrades the control's quality, especially in terms of stability, robustness, and oscillation response. For these reasons, it is necessary to optimize the system transient response. This paper will compare selected heuristic methods of optimization of linear-time invariant (LTI) systems with time delay in an integral part of the P/I-delayed controller. Some examples are chosen as benchmarks to verify the effectiveness of the optimization. In future research, we would like to focus on implementing selected modern optimization algorithms for solving non-smooth, non-convex, and non-Lipschitz optimization problems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Lyapunov, A.M.: The General Problem of the Stability of Motion, pp. 1–251. Kharkiv Mathematical Society, Kharkiv (1892)
Zadeh, L.A., Deoser, C.A.: Linear System Theory: The State Space Approach. Dover Publications (2008)
Khalil, H.K.: Nonlinear Systems. Prentice Hall (2014)
Barnett, S.: Polynomials and Linear Control Systems. Dekker (1983)
Hristuvarsakelis, D., Levine, W.S.: Handbook of Networked and Embedded Control Systems. Birkhäuser, Boston (2005)
Fridman, E.: Introduction to Time-Delay Systems: Analysis and Control. Springer, Cham (2014)
Xia, Y., Fu, M., Shi, P.: Analysis and synthesis of dynamical systems with time delays. Springer, Berlin (2009)
Michiels, W., Engelborghs, K., Vansevevant, P., Roose, D.: Continuous pole placement for delay equations. Automatica 38(5), 747–761 (2002)
Araújo, J.M.: Discussion on ‘state feedback control with time delay’. Mech. Syst. Sig. Process. 98 (2018). https://doi.org/10.1016/j.ymssp.2017.05.004
Wong, L., Lee, Y.E., Lee, H.J.: Optimal transmission of messages in computer networks—an optimal control problem involving control-dependent time-delayed arguments. J. Inequal. Appl. (2022)
Kumar, P., Erturk, V.S.: The analysis of a time delay fractional COVID-19 model via Caputo type fractional derivative. In: Mathematical Methods in the Applied Sciences (2020)
Oertel, R., Wagner, P.: Delay-time actuated traffic signal control for an isolated intersection. In: Proceedings 90st Annual Meeting Transportation Research Board (TRB)
Joelianto, E.: Networked control systems: time delays and robust control design issues. In: Proceedings of 2011 2nd International Conference on Instrumentation Control and Automation, ICA 2011. https://doi.org/10.1109/ICA.2011.6130121
Kennell, D., Talkad, V.: Messenger RNA potential and the delay before exponential decay of messages. J. Mol. Biol. (1976)
Dollevoet, T., Huisman, D., Schmidt, M., Schöbel, A.: Delay Propagation and Delay Management in Transportation Networks (2018)
Palpal-latoc, C., Bernardo, R.C., Vega, I.: Testing time-delayed cosmology. Preprint (2021)
Mirkin, L., Palmor, Z.J.: Control issues in systems with loop delays. In: Hristu-Varsakelis, D., Levine, W.S. (eds.) Handbook of Networked and Embedded Control Systems. Control Engineering. Birkhäuser, Boston (2005)
Jnifene, A.: Active vibration control of flexible structures using delayed position feedback. Syst. Control Lett. 56(3), 215–222 (2007)
Peng, J., Xiang, M., Wang, L., Xie, W., Sun, H., Yu, J.: Nonlinear primary resonance in vibration control of cable-stayed beam with time delay feedback. Mech. Syst. Signal Process. 137, 106488 (2020)
Elmadssia, S., Saadaoui, K., Zaguia, A., Ezzedine, T., Wang, Q.-G.: Stabilization domains for second order delay systems. IEEE Access 9, 53518–53529 (2021)
Busłowicz, M.: Sufficient conditions for instability of delay differential systems. Int. J. Control 37(6), 1311–1321 (1983)
Pekař, L., Strmiska, M., Song, M., Dostálek, P.: Numerical gridding stability charts estimation using quasi-polynomial approximation for TDS. In: Proceedings of the 23rd International Conference on Process Control (PC), pp. 290–295 (2021)
Insperger, T., Stépán, G.: Stability charts for fundamental delay-differential equations. In: Semi-Discretization for Time-Delay Systems. Applied Mathematical Sciences, vol. 178. Springer, New York, NY (2011). https://doi.org/10.1007/978-1-4614-0335-7_2
Szaksz, B., Stepan, G.: Stability charts of a delayed model of vehicle towing. IFAC Pap. Online 54(18), 64–69 (2021)
Lv, Z.-H., Zhang, J.-F., Ouyang, H.: Receptance-based computation of stability crossing curves for single-input-multiple-output second-order linear systems with two time-delays. Int. J. Struct. Stab. Dyn. 22(01) (2022). https://doi.org/10.1142/s021945542250002x
Strmiska, M., Pekař, L., Araújo, J.M.: Stabilization of second-order systems using a P/I-delayed controller. In: Proceedings of the 6th Computational Methods in Systems and Software (2022)
Floudas, C., Pardalos, P.: Encyclopedia of Optimization, 2nd edn. Springer. USA, 2009. ISBN: 978-0-387-74759-0
Araújo, J.M., Santos, T.L.M.: Special issue on control of second-order vibrating systems with time delay. Mech. Syst. Signal Process. 137, 106527 (2020)
Vahid-Araghi, O., Golnaraghi, F.: Negative damping instability mechanism. In: Friction-Induced Vibration in Lead Screw Drives. Springer, New York, NY (2010)
Novella-Rodriguez, D.F., Muro-Cuellar, B.: Control of second order strictly proper unstable systems with time delay. Rev. Mex. Ing. Quím [Online] 10(3), 551–559 (2011)
Smith, C.A., Corripio, A.B.: Principles and Practice of Automatic Process Control. Wiley (2006)
Acknowledgements
The paper was supported by the Internal Grant Agency of Tomas Bata University in Zlín under the project number IGA/FAI/2022/008.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Strmiska, M., Araújo, J.M., Pekař, L. (2023). Comparison of Stabilization with P/I-Delayed Controllers for Second-Order Systems Using Built-In MATLAB Heuristic Optimization Methods. In: Silhavy, R., Silhavy, P., Prokopova, Z. (eds) Software Engineering Application in Systems Design. CoMeSySo 2022. Lecture Notes in Networks and Systems, vol 596. Springer, Cham. https://doi.org/10.1007/978-3-031-21435-6_55
Download citation
DOI: https://doi.org/10.1007/978-3-031-21435-6_55
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-21434-9
Online ISBN: 978-3-031-21435-6
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)