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Structured Robust Stability Analysis Of An Inverted Pendulum System With A Fixedfeedback

Year 2005, Volume: 18 Issue: 2, 55 - 70, 31.12.2005

Ahmet Yazıcı Abdurrahman Karamancıoğlu

Abstract

Robust stability analysis of inverted pendulum system with a fixed linear quadratic feedback is carried out using μ analysis tools. A mathematical model of inverted pendulum system is presented and linearized about the desired equilibrium point. A linear quadratic control feedback matrix is used in the configuration of the nominal inverted pendulum system for its stabilization. After this, uncertainties in the inverted pendulum with a fixed feedback, is modeled in linear fractional transformation form , which is suitable for structured singular value computation. Both parametric and modeling uncertainties are considered in the inverted pendulum system. After deriving out uncertain system model, mixed μ analysis method which is a structured uncertainty analysis method, is used to compute the uncertainty bound that does not cause instability of the inverted pendulum under feedback. A simulation test for validity of the results is provided.

Keywords

Structured singular value Uncertainty Robustness Inverted pendulum

References

  • [1] P. Gaspar, I. Szaszi, and J. Bokor, ”Robust control design for mechanical systems using the mixed μ synthesis,” Periodica Polytechnica Ser. Transp. Eng., v. 30, pp. 37- 52, 2002.
  • [2] S. Dussy and L. El-Ghaoui, “Multiobjective bounded control of uncertain nonlinear systems: An inverted pendulum example”, European Control Conference, Brussels, July 1997.
  • [3] S. Cheang and W. J. Chen, “Stabilizing control of an inverted pendulum system based on H∞ loop shaping design procedure”, Proceedings of the 3rd World Congress on Intelligent Control and Automation, v. 5, pp. 3385 – 3388, 2000.
  • [4] G.A. Medrano-Cerda, “Robust computer control of an inverted pendulum”, IEEE Control Systems Magazine, v.19, pp.58 – 67, 1999.
  • [5] K. Ogata, “Modern Control Engineering”, Prentice-Hall Inc. 1990.
  • [6] A. Yazıcı, “ Application of sliding mode control on an inverted pendulum system”, Masters thesis, Osmangazi University, December 2000 (in Turkish).
  • [7] R. P. Braatz, P. M Young, J. C. Doyle, and M. Morari, “Computational-Complexity Of Mu-Calculation”, IEEE Transactions on Automatic Control, v. 39, pp.1000-1002, 1994.
  • [8] A. Nemirovskii, “Several NP-hard problems arising in robust stability analysis”, Mathematics of Control, Signals, and Systems, v. 6, pp. 99–105, 1993.
  • [9] M. K. H Fan, A. L Tits, and J. C. Doyle, “Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics”, IEEE Transactions on Automatic Control, v.36, pp. 25-38, 1991.
  • [10] P. M. Young, “Structured singular value approach for systems with parametric uncertainty”, Int. Journal of Robust and Nonlinear Control, v.11, pp.653-680, 2001.
  • [11] M. Fu and N. E. Barabanov, “Improved Upper Bounds for the Mixed Structured Singular Value”, IEEE Transactions on Automatic Control, v. 42, pp.1447-1452, 1997.
  • [12] P. M. Young, M. P. Newlin, and J. C. Doyle, “Computing Bounds For The Mixed- Mu-Problem”, Int. Journal of Robust and Nonlinear Control, v.5, pp. 573-590, 1995.
  • [13] K. Zhou and J. C. Doyle, “Essentials of Robust Control", Prentice-Hall, Inc, 1998.
  • [14] G. J. Balas, J. C. Doyle, K. Glover, A. Packard, and R. Smith, “ μ -Analysis and Synthesis Toolbox: User’s Guide, MUSYN Inc. and The MathWorks, Inc.”, 2001.

Structured Robust Stability Analysis Of An Inverted Pendulum System With A Fixedfeedback

Year 2005, Volume: 18 Issue: 2, 55 - 70, 31.12.2005

Ahmet Yazıcı Abdurrahman Karamancıoğlu

Abstract

Robust stability analysis of inverted pendulum system with a fixed linear quadratic feedback is carried out using μ analysis tools. A mathematical model of inverted pendulum system is presented and linearized about the desired equilibrium point. A linear quadratic control feedback matrix is used in the configuration of the nominal inverted pendulum system for its stabilization. After this, uncertainties in the inverted pendulum with a fixed feedback, is modeled in linear fractional transformation form , which is suitable for structured singular value computation. Both parametric and modeling uncertainties are considered in the inverted pendulum system. After deriving out uncertain system model, mixed μ analysis method which is a structured uncertainty analysis method, is used to compute the uncertainty bound that does not cause instability of the inverted pendulum under feedback. A simulation test for validity of the results is provided.

Keywords

Structured singular value Uncertainty Robustness Inverted pendulum

References

  • [1] P. Gaspar, I. Szaszi, and J. Bokor, ”Robust control design for mechanical systems using the mixed μ synthesis,” Periodica Polytechnica Ser. Transp. Eng., v. 30, pp. 37- 52, 2002.
  • [2] S. Dussy and L. El-Ghaoui, “Multiobjective bounded control of uncertain nonlinear systems: An inverted pendulum example”, European Control Conference, Brussels, July 1997.
  • [3] S. Cheang and W. J. Chen, “Stabilizing control of an inverted pendulum system based on H∞ loop shaping design procedure”, Proceedings of the 3rd World Congress on Intelligent Control and Automation, v. 5, pp. 3385 – 3388, 2000.
  • [4] G.A. Medrano-Cerda, “Robust computer control of an inverted pendulum”, IEEE Control Systems Magazine, v.19, pp.58 – 67, 1999.
  • [5] K. Ogata, “Modern Control Engineering”, Prentice-Hall Inc. 1990.
  • [6] A. Yazıcı, “ Application of sliding mode control on an inverted pendulum system”, Masters thesis, Osmangazi University, December 2000 (in Turkish).
  • [7] R. P. Braatz, P. M Young, J. C. Doyle, and M. Morari, “Computational-Complexity Of Mu-Calculation”, IEEE Transactions on Automatic Control, v. 39, pp.1000-1002, 1994.
  • [8] A. Nemirovskii, “Several NP-hard problems arising in robust stability analysis”, Mathematics of Control, Signals, and Systems, v. 6, pp. 99–105, 1993.
  • [9] M. K. H Fan, A. L Tits, and J. C. Doyle, “Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics”, IEEE Transactions on Automatic Control, v.36, pp. 25-38, 1991.
  • [10] P. M. Young, “Structured singular value approach for systems with parametric uncertainty”, Int. Journal of Robust and Nonlinear Control, v.11, pp.653-680, 2001.
  • [11] M. Fu and N. E. Barabanov, “Improved Upper Bounds for the Mixed Structured Singular Value”, IEEE Transactions on Automatic Control, v. 42, pp.1447-1452, 1997.
  • [12] P. M. Young, M. P. Newlin, and J. C. Doyle, “Computing Bounds For The Mixed- Mu-Problem”, Int. Journal of Robust and Nonlinear Control, v.5, pp. 573-590, 1995.
  • [13] K. Zhou and J. C. Doyle, “Essentials of Robust Control", Prentice-Hall, Inc, 1998.
  • [14] G. J. Balas, J. C. Doyle, K. Glover, A. Packard, and R. Smith, “ μ -Analysis and Synthesis Toolbox: User’s Guide, MUSYN Inc. and The MathWorks, Inc.”, 2001.
There are 14 citations in total.

Details

Subjects Electrical Engineering
Journal Section Research Articles
Authors

Ahmet Yazıcı

Abdurrahman Karamancıoğlu

Publication Date December 31, 2005
Acceptance Date July 1, 2005
Published in Issue Year 2005 Volume: 18 Issue: 2

Cite

APA Yazıcı, A., & Karamancıoğlu, A. (2005). Structured Robust Stability Analysis Of An Inverted Pendulum System With A Fixedfeedback. Eskişehir Osmangazi Üniversitesi Mühendislik Ve Mimarlık Fakültesi Dergisi, 18(2), 55-70.
AMA Yazıcı A, Karamancıoğlu A. Structured Robust Stability Analysis Of An Inverted Pendulum System With A Fixedfeedback. ESOGÜ Müh Mim Fak Derg. December 2005;18(2):55-70.
Chicago Yazıcı, Ahmet, and Abdurrahman Karamancıoğlu. “Structured Robust Stability Analysis Of An Inverted Pendulum System With A Fixedfeedback”. Eskişehir Osmangazi Üniversitesi Mühendislik Ve Mimarlık Fakültesi Dergisi 18, no. 2 (December 2005): 55-70.
EndNote Yazıcı A, Karamancıoğlu A (December 1, 2005) Structured Robust Stability Analysis Of An Inverted Pendulum System With A Fixedfeedback. Eskişehir Osmangazi Üniversitesi Mühendislik ve Mimarlık Fakültesi Dergisi 18 2 55–70.
IEEE A. Yazıcı and A. Karamancıoğlu, “Structured Robust Stability Analysis Of An Inverted Pendulum System With A Fixedfeedback”, ESOGÜ Müh Mim Fak Derg, vol. 18, no. 2, pp. 55–70, 2005.
ISNAD Yazıcı, Ahmet - Karamancıoğlu, Abdurrahman. “Structured Robust Stability Analysis Of An Inverted Pendulum System With A Fixedfeedback”. Eskişehir Osmangazi Üniversitesi Mühendislik ve Mimarlık Fakültesi Dergisi 18/2 (December 2005), 55-70.
JAMA Yazıcı A, Karamancıoğlu A. Structured Robust Stability Analysis Of An Inverted Pendulum System With A Fixedfeedback. ESOGÜ Müh Mim Fak Derg. 2005;18:55–70.
MLA Yazıcı, Ahmet and Abdurrahman Karamancıoğlu. “Structured Robust Stability Analysis Of An Inverted Pendulum System With A Fixedfeedback”. Eskişehir Osmangazi Üniversitesi Mühendislik Ve Mimarlık Fakültesi Dergisi, vol. 18, no. 2, 2005, pp. 55-70.
Vancouver Yazıcı A, Karamancıoğlu A. Structured Robust Stability Analysis Of An Inverted Pendulum System With A Fixedfeedback. ESOGÜ Müh Mim Fak Derg. 2005;18(2):55-70.
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