Abstract
The problem of parametric synthesis of PID controllers of integer and fractional orders is solved. PID-controllers are an integral part of many cyber-physical systems. Two well-known methods of synthesis are considered—for a relative stability margin and a maximum value of the sensitivity function. Algorithms have been developed for calculating the limiting values of the differential gain of the controller, at which the boundary of the region of a given stability margin has a cusp. In the case of using the criterion for low-frequency disturbance rejection (LFDR), the limiting values of the relative stability margin and the maximum value of the sensitivity function are determined.
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
Åström, K.J., Hägglund, T.: Advanced PID control, p. 460. ISA-The Instrumentation, Systems, and Automation Society, Research Triangle Park, NC (2006)
Ang, K.H., Chong, G., Li, Y.: PID control system analysis, design, and technology. IEEE Trans. Control Syst. Technol. 13, 559–576 (2005)
Wallén, A., Åström, K.J., Hägglun, T.: Loop-shaping design of PID controllers with constant Ti/Td ratio. Asian J. Control. 4, 403–409 (2008)
Åström, K.J., Hägglund, T.: The future of PID control. IFAC Proc. Vol. 33, 19–30 (2000)
Tan, W., Liu, J., Chen, T., Marquez, H.J.: Comparison of some well-known PID tuning formulas. Comput. Chem. Eng. 30, 1416–1423 (2006)
Tang, W., Wang, Q.G., Lu, X., Zhang, Z.: Why Ti = 4Td for PID controller tuning. Robotics and Vision 2006 9th International Conference on Control, pp. 1–2. Automation. IEEE, Singapore (2006)
O’Dwyer A.: An overview of tuning rules for the PI and PID continuous-time control of time-delayed Single-Input, Single-Output (SISO) Processes. In: Vilanova R., Visioli A. (eds.) PID Control in the Third Millennium: Lessons Learned and New Approaches Springer London, pp. 3–44. London (2012)
Yadaiah, N., Malladi, S. (2013) An optimized relation between Ti and Td in Modified Ziegler Nichols PID controller tuning. In: 2013 IEEE International Conference on Control Applications (CCA), pp. 1275–1280
Leva, A., Maggio, M.: Model-Based PI(D) Autotuning. In: Vilanova, R., Visioli, A. (eds.) PID Control in the Third Millennium: Lessons Learned and New Approaches, pp. 45–73. Springer, London, London (2012)
Pecharromán, R.R., Pagola, F.L.: Control design for PID controllers auto-tuning based on improved identification. IFAC Proc. Vol. 33, 85–90 (2000)
Smirnov, N.I., Sabanin, V.R., Repin, A.I.: Sensitivity and robust tuning of PID controllers with real differentiation. Therm. Eng. 54, 777–785 (2007)
Ozyetkin, M.M., Onat, C., Tan, N.: PID tuning method for integrating processes having time delay and inverse response. In: IFAC-PapersOnLine, 3rd IFAC Conference on Advances in Proportional-Integral-Derivative Control PID 2018, vol. 51, pp. 274–279 (2018)
Ayazyan, G.K., Novozhenin, A.Yu., Tausheva, E.V.: Parametric synthesis of PID regulators for a given degree of oscillation (in Russian). In: XII All-Russian Meeting on the Problems of Control (VSPU-2014). Russia, Institute of Control Sciences V.A. Trapeznikova RAS, pp. 147–159 (2014)
Ayazyan, G.K.: Calculation of automatic systems with typical control algorithms, p. 136. Publishing house UNI, Ufa (1989). (in Russian)
Ayazyan, G.K., Tausheva, E.V., Shaymuhametova, M.R.: Applying symbolic computing system maple for parametric synthesis of controllers (in Russian). In: VI International Conference. Mathematics, Its Applications and Mathematical Education (MAME’17), pp. 59–64. Russia, Ulan-Ude (2017)
Volgin, V.V.: Concerning determination of optimum adjustment of PID-regulators. Avtomat. i Telemekh. 23, 620–630 (1962)
Safronova, I.N., Volgin, V.V.: The method of multiple roots in optimizing a control-system with a PID-algorithm. Therm. Eng. 36, 580–582 (1989)
Åström, K.J., Panagopoulos, H., Hägglund, T.: Design of PI controllers based on non-convex optimization. Automatica 34, 585–601 (1998)
Bruce, J.W., Giblin, P.J.: Curves and singularities: a geometrical introduction to singularity theory, 2nd ed. ed. Cambridge University Press, Cambridge [England] ; New York, NY, USA, 321 p, (1992)
Lord, E.A.: The mathematical description of shape and form: E. A. Lord and C. B. Wilson, Ellis Horwood series in mathematics and its applications. (vol. 260). Halsted Press, Chichester, West Sussex, Ellis Horwood, New York (1984)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Ayazyan, G., Tausheva, E. (2020). Limiting Values of the Stability Margins in the Parametric Synthesis of PID-Controllers. In: Kravets, A., Bolshakov, A., Shcherbakov, M. (eds) Cyber-Physical Systems: Industry 4.0 Challenges. Studies in Systems, Decision and Control, vol 260. Springer, Cham. https://doi.org/10.1007/978-3-030-32648-7_9
Download citation
DOI: https://doi.org/10.1007/978-3-030-32648-7_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-32647-0
Online ISBN: 978-3-030-32648-7
eBook Packages: EngineeringEngineering (R0)