New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation

Uwe Nuss

doi:doi:10.11648/j.acis.20241201.11

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New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation

Uwe Nuss^*

Published in Automation, Control and Intelligent Systems (Volume 12, Issue 1)

Received: 5 March 2024 Accepted: 23 March 2024 Published: 12 April 2024

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Abstract

In this paper, a simple, generally valid stability proof for an anti-windup method for PI-state controlled systems is presented, with which it is possible to directly conclude the stability of the PI-state controlled system from a stable P-state controlled system with constraints in the manipulated variables, i.e. without having to perform a separate stability investigation of the anti-windup measures. The technique presented is based on the system description by means of state equations and Lyapunov's Direct Method using quadratic Lyapunov functions. Furthermore, the PI-state controller is designed in such a way that it provides the same command response as the P-state controller, for which a stability statement is already available. Both continuous-time and discrete-time systems are considered, which, apart from the saturation of the manipulated variables, show linear, time-invariant behavior. In addition, a general stability proof is given for discrete-time systems, which makes it possible to establish stable anti-windup methods for P- and PI-state controlled systems, which contain dead time elements in the path of the manipulated variables, without having to carry out separate stability investigations for them. For this purpose, the state controller design for the system with dead time elements in the manipulated variable paths is based on the principle that the same characteristics of the control behavior should be achieved as for the system without such dead time elements, but delayed by the dead time. The effectiveness of the presented methods is illustrated by an example from the field of electrical drives.

Published in	Automation, Control and Intelligent Systems (Volume 12, Issue 1)
DOI	10.11648/j.acis.20241201.11
Page(s)	1-14
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Pi-State Controller, Actuator Saturation, Anti-Windup Methods, Continuous-Time and Discrete-Time Controllers, Controlled System with Dead Time Elements

References

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[2]	Adamy, J. Nonlinear Systems and Controls. Berlin, Heidelberg: Springer Vieweg; 2022. https://doi.org/10.1007/978-3-662-65633-4
[3]	Tarbouriech, S., Garcia, G., Goes da Silva Jr., J. M., Queinnec, I. Stability and Stabilization of Linear Systems with Saturating Actuators. London: Springer; 2011. https://doi.org/10.1007/978-0-85729-941-3
[4]	Vidyasagar, M. Nonlinear Systems Analysis. 2nd edition. Philadelphia: SIAM; 2002 (unabridged republication). https://doi.org/10.1137/1.9780898719185
[5]	Lerch, S., Dehnert, R., Damaszek, M., Tibken, B. Anti Windup PID Control of Discrete Systems Subject to Actuator Magnitude and Rate Saturation: An Iterative LMI Approach. Proceedings of the 25th International Conference on System Theory, Control and Computing (ICSTCC). Iasi, Romania, 2021, pp. 413–418. https://doi.org/10.1109/ICSTCC52150.2021.9607157
[6]	Chen, Y., Yang, M., Liu, K., Long, J., Xu, D., Blaabjerg, F. Reversed Structure Based PI-Lead Controller Antiwindup Design and Self-Commissioning Strategy for Servo Drive Systems. IEEE Transactions on Industrial Electronics. 2022, 69 (7), pp. 6586–6599. https://doi.org/10.1109/TIE.2021.3097602
[7]	Wang, K., Wu, F., Sun, X.-M. Switchung Anti-windup Control for Aircraft Engines. IEEE Transactions on Industrial Electronics. 2023, 70 (2), pp. 1830–1840. https://doi.org/10.1109/TIE.2022.3163464
[8]	Nuss, U. Stabilitätsverhalten von zweistufig entworfenen zeitdiskreten PI-Zustandsreglern bei Stellgrößenbegrenzungen [Stability properties of two-stage designed discrete-time PI state controllers considering the limitation of input variables]. at – Automatisierungstechnik. 2017, 65 (10), pp. 705 – 717. https://doi.org/10.1515/auto-2016-0136 (in German)
[9]	Nuss, U. Zeitdiskrete Regelung [Discrete-time control]. Berlin, Offenbach: VDE; 2020 (in German)
[10]	Aström, K., Wittenmark, B. Computer-Controlled Systems. 2nd edition. Englewood Cliffs: Prentice-Hall; 1990
[11]	Quang, N.P., Dittrich, J.-A. Vector Control of Three-Phase AC Machines. Berlin, Heidelberg: Springer; 2008. https://doi.org/10.1007/978-3-540-79029-7
[12]	March, P., Turner, C.. Anti-Windup Compensator Designs for Nonsalient Permanent-Magnet Synchronous Motor Speed Regulators. IEEE Transactions on Industry Applications. 2009, 45 (5), pp. 1598–1609. https://doi.org/10.1109/tia.2009.2027157
[13]	Nuss, U. Ein einfacher Zustandsreglerentwurf im Zuge der Erweiterung der Systemstruktur um Reglerintegratoren und Rechentotzeiten [A simple state space controller design in the context of expanding the control structure by integrators and calculation dead time]. at – Automatisierungstechnik. 2016, 64 (1), pp. 29–40. https://doi.org/10.1515/auto-2015-0058 (in German)
[14]	Nuss, U. Regelungstechnik einfach verpackt [control theory simply presented]. forschung im fokus. Offenburg: University of Applied Science Offenburg. 2016, pp. 19 – 22. (in German)
[15]	Grabmair, G., Gahleitner, R. PI-Zustandsregler – eine methodische Neubetrachtung [PI statespace control revisited]. at – Automatisierungstechnik. 2019, 67 (9), pp. 727–738. https://doi.org/10.1515/auto-2019-0037 (in German)
[16]	Zurmühl, R., Falk, S. Matrizen und ihre Anwendungen [Matrices and their applications]. 5th edition. Berlin, Heidelberg, New York, Tokyo: Springer; 1984. (in German)
[17]	Nuss, U. Hochdynamische Regelung elektrischer Antriebe [Highly dynamic control of electrical drives]. 2nd edition. Berlin, Offenbach: VDE; 2017. (in German)

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APA Style

Nuss, U. (2024). New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation . Automation, Control and Intelligent Systems, 12(1), 1-14. https://doi.org/10.11648/j.acis.20241201.11

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ACS Style

Nuss, U. New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation . Autom. Control Intell. Syst. 2024, 12(1), 1-14. doi: 10.11648/j.acis.20241201.11

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AMA Style

Nuss U. New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation . Autom Control Intell Syst. 2024;12(1):1-14. doi: 10.11648/j.acis.20241201.11

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@article{10.11648/j.acis.20241201.11,
  author = {Uwe Nuss},
  title = {New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation
},
  journal = {Automation, Control and Intelligent Systems},
  volume = {12},
  number = {1},
  pages = {1-14},
  doi = {10.11648/j.acis.20241201.11},
  url = {https://doi.org/10.11648/j.acis.20241201.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acis.20241201.11},
  abstract = {In this paper, a simple, generally valid stability proof for an anti-windup method for PI-state controlled systems is presented, with which it is possible to directly conclude the stability of the PI-state controlled system from a stable P-state controlled system with constraints in the manipulated variables, i.e. without having to perform a separate stability investigation of the anti-windup measures. The technique presented is based on the system description by means of state equations and Lyapunov's Direct Method using quadratic Lyapunov functions. Furthermore, the PI-state controller is designed in such a way that it provides the same command response as the P-state controller, for which a stability statement is already available. Both continuous-time and discrete-time systems are considered, which, apart from the saturation of the manipulated variables, show linear, time-invariant behavior. In addition, a general stability proof is given for discrete-time systems, which makes it possible to establish stable anti-windup methods for P- and PI-state controlled systems, which contain dead time elements in the path of the manipulated variables, without having to carry out separate stability investigations for them. For this purpose, the state controller design for the system with dead time elements in the manipulated variable paths is based on the principle that the same characteristics of the control behavior should be achieved as for the system without such dead time elements, but delayed by the dead time. The effectiveness of the presented methods is illustrated by an example from the field of electrical drives.
},
 year = {2024}
}

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TY  - JOUR
T1  - New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation

AU  - Uwe Nuss
Y1  - 2024/04/12
PY  - 2024
N1  - https://doi.org/10.11648/j.acis.20241201.11
DO  - 10.11648/j.acis.20241201.11
T2  - Automation, Control and Intelligent Systems
JF  - Automation, Control and Intelligent Systems
JO  - Automation, Control and Intelligent Systems
SP  - 1
EP  - 14
PB  - Science Publishing Group
SN  - 2328-5591
UR  - https://doi.org/10.11648/j.acis.20241201.11
AB  - In this paper, a simple, generally valid stability proof for an anti-windup method for PI-state controlled systems is presented, with which it is possible to directly conclude the stability of the PI-state controlled system from a stable P-state controlled system with constraints in the manipulated variables, i.e. without having to perform a separate stability investigation of the anti-windup measures. The technique presented is based on the system description by means of state equations and Lyapunov's Direct Method using quadratic Lyapunov functions. Furthermore, the PI-state controller is designed in such a way that it provides the same command response as the P-state controller, for which a stability statement is already available. Both continuous-time and discrete-time systems are considered, which, apart from the saturation of the manipulated variables, show linear, time-invariant behavior. In addition, a general stability proof is given for discrete-time systems, which makes it possible to establish stable anti-windup methods for P- and PI-state controlled systems, which contain dead time elements in the path of the manipulated variables, without having to carry out separate stability investigations for them. For this purpose, the state controller design for the system with dead time elements in the manipulated variable paths is based on the principle that the same characteristics of the control behavior should be achieved as for the system without such dead time elements, but delayed by the dead time. The effectiveness of the presented methods is illustrated by an example from the field of electrical drives.

VL  - 12
IS  - 1
ER  -

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Author Information

Uwe Nuss

Department of Electrical Engineering, Medical Engineering and Computer Science, Offenburg University of Applied Science, Offenburg, Germany

Contact Email

http://orcid.org/0000-0001-8545-7269

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Plain Text BibTeX RIS

APA Style

Nuss, U. (2024). New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation . Automation, Control and Intelligent Systems, 12(1), 1-14. https://doi.org/10.11648/j.acis.20241201.11

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ACS Style

Nuss, U. New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation . Autom. Control Intell. Syst. 2024, 12(1), 1-14. doi: 10.11648/j.acis.20241201.11

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AMA Style

Nuss U. New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation . Autom Control Intell Syst. 2024;12(1):1-14. doi: 10.11648/j.acis.20241201.11

Copy | Download

@article{10.11648/j.acis.20241201.11,
  author = {Uwe Nuss},
  title = {New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation
},
  journal = {Automation, Control and Intelligent Systems},
  volume = {12},
  number = {1},
  pages = {1-14},
  doi = {10.11648/j.acis.20241201.11},
  url = {https://doi.org/10.11648/j.acis.20241201.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acis.20241201.11},
  abstract = {In this paper, a simple, generally valid stability proof for an anti-windup method for PI-state controlled systems is presented, with which it is possible to directly conclude the stability of the PI-state controlled system from a stable P-state controlled system with constraints in the manipulated variables, i.e. without having to perform a separate stability investigation of the anti-windup measures. The technique presented is based on the system description by means of state equations and Lyapunov's Direct Method using quadratic Lyapunov functions. Furthermore, the PI-state controller is designed in such a way that it provides the same command response as the P-state controller, for which a stability statement is already available. Both continuous-time and discrete-time systems are considered, which, apart from the saturation of the manipulated variables, show linear, time-invariant behavior. In addition, a general stability proof is given for discrete-time systems, which makes it possible to establish stable anti-windup methods for P- and PI-state controlled systems, which contain dead time elements in the path of the manipulated variables, without having to carry out separate stability investigations for them. For this purpose, the state controller design for the system with dead time elements in the manipulated variable paths is based on the principle that the same characteristics of the control behavior should be achieved as for the system without such dead time elements, but delayed by the dead time. The effectiveness of the presented methods is illustrated by an example from the field of electrical drives.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - New Findings in the Stability Analysis of PI-state Controlled Systems with Actuator Saturation

AU  - Uwe Nuss
Y1  - 2024/04/12
PY  - 2024
N1  - https://doi.org/10.11648/j.acis.20241201.11
DO  - 10.11648/j.acis.20241201.11
T2  - Automation, Control and Intelligent Systems
JF  - Automation, Control and Intelligent Systems
JO  - Automation, Control and Intelligent Systems
SP  - 1
EP  - 14
PB  - Science Publishing Group
SN  - 2328-5591
UR  - https://doi.org/10.11648/j.acis.20241201.11
AB  - In this paper, a simple, generally valid stability proof for an anti-windup method for PI-state controlled systems is presented, with which it is possible to directly conclude the stability of the PI-state controlled system from a stable P-state controlled system with constraints in the manipulated variables, i.e. without having to perform a separate stability investigation of the anti-windup measures. The technique presented is based on the system description by means of state equations and Lyapunov's Direct Method using quadratic Lyapunov functions. Furthermore, the PI-state controller is designed in such a way that it provides the same command response as the P-state controller, for which a stability statement is already available. Both continuous-time and discrete-time systems are considered, which, apart from the saturation of the manipulated variables, show linear, time-invariant behavior. In addition, a general stability proof is given for discrete-time systems, which makes it possible to establish stable anti-windup methods for P- and PI-state controlled systems, which contain dead time elements in the path of the manipulated variables, without having to carry out separate stability investigations for them. For this purpose, the state controller design for the system with dead time elements in the manipulated variable paths is based on the principle that the same characteristics of the control behavior should be achieved as for the system without such dead time elements, but delayed by the dead time. The effectiveness of the presented methods is illustrated by an example from the field of electrical drives.

VL  - 12
IS  - 1
ER  -

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