Formal Stability Analysis of Control Systems

  • Asad AhmedEmail author
  • Osman Hasan
  • Falah Awwad
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 1008)


Stability of a control system ensures that its output is under control and thus is the most important characteristic of control systems. Stability is characterized by the roots of the characteristic equation of the given control system in the complex-domain. Traditionally, paper-and-pencil proof methods and computer-based tools are used to analyze the stability of control systems. However, paper-and-pencil proof methods are error prone due to the human involvement. Whereas, computer based tools cannot model the continuous behavior in its true form due to the involvement of computer arithmetic and the associated truncation errors. Therefore, these techniques do not provide an accurate and complete analysis, which is unfortunate given the safety-critical nature of control system applications. In this paper, we propose to overcome these limitations by using higher-order-logic theorem proving for the stability analysis of control systems. For this purpose, we present a higher-order-logic based formalization of stability and the roots of the quadratic, cubic and quartic complex polynomials. The proposed formalization is based on the complex number theory of the HOL-Light theorem prover. A distinguishing feature of this work is the automatic nature of the formal stability analysis, which makes it quite useful for the control engineers working in the industry who have very little expertise about formal methods. For illustration purposes, we present the stability analysis of power converter controllers used in smart grids.


Stability Control systems Polynomials HOL-light 



This work is supported by ICT Fund UAE, fund number 21N206 at UAE University, Al Ain, United Arab Emirates.


  1. 1.
    Ahmad, M., Hasan, O.: Formal verification of steady-state errors in unity-feedback control systems. In: Lang, F., Flammini, F. (eds.) FMICS 2014. LNCS, vol. 8718, pp. 1–15. Springer, Cham (2014). Scholar
  2. 2.
    Ahmed, A.: System Analysis and Verification (SAVe) Lab. Accessed 12 Sept 2018
  3. 3.
    Amin, S.M., Wollenberg, B.F.: Toward a smart grid: power delivery for the 21st century. IEEE Power Energ. Mag. 3(5), 34–41 (2005)CrossRefGoogle Scholar
  4. 4.
    Dyke, P.: An Introduction to Laplace Transforms and Fourier Series. SUMS. Springer, London (2014). Scholar
  5. 5.
    Ekanayake, J., Jenkins, N.: Comparison of the response of doubly fed and fixed-speed induction generator wind turbines to changes in network frequency. IEEE Trans. Energy Convers. 19(4), 800–802 (2004)CrossRefGoogle Scholar
  6. 6.
    Giordano, V., et al.: Smart grid projects in Europe. JRC Ref Rep Sy 8. Publications Office of the European Union, Luxembourg (2011).
  7. 7.
    Harrison, J.: HOL light: an overview. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 60–66. Springer, Heidelberg (2009). Scholar
  8. 8.
    Harrison, J.: Theorem Proving with the Real Numbers. Springer, London (2012)zbMATHGoogle Scholar
  9. 9.
    Hasan, O., Ahmad, M.: Formal analysis of steady state errors in feedback control systems using HOL-Light. In: Proceedings of the Conference on Design, Automation and Test in Europe, pp. 1423–1426. EDA Consortium, San Jose (2013)Google Scholar
  10. 10.
    Heck, A., Heck, A.: Introduction to MAPLE. Springer, New York (1993). Scholar
  11. 11.
    Hornik, T., Zhong, Q.C.: A current-control strategy for voltage-source inverters in microgrids based on H\(^\infty \) and repetitive control. IEEE Trans. Power Electron. 26(3), 943–952 (2011)CrossRefGoogle Scholar
  12. 12.
    MathWorks: Control System Toolbox. Accessed 12 Sept 2018
  13. 13.
    MathWorks: Simulink. Accessed 12 Sept 2018
  14. 14.
    Momoh, J.A.: Smart Grid: Fundamentals of Design and Analysis, vol. 63. Wiley, Hoboken (2012)CrossRefGoogle Scholar
  15. 15.
    Nise, N.S.: Control Systems Engineering. Wiley, Hoboken (2007)zbMATHGoogle Scholar
  16. 16.
    Rashid, A., Hasan, O.: Formal analysis of linear control systems using theorem proving. In: Duan, Z., Ong, L. (eds.) ICFEM 2017. LNCS, vol. 10610, pp. 345–361. Springer, Cham (2017). Scholar
  17. 17.
    Rashid, A., Siddique, U., Hasan, O.: Formal verification of platoon control strategies. In: Johnsen, E.B., Schaefer, I. (eds.) SEFM 2018. LNCS, vol. 10886, pp. 223–238. Springer, Cham (2018). Scholar
  18. 18.
    Sanwal, M.U., Hasan, O.: Formally analyzing continuous aspects of cyber-physical systems modeled by homogeneous linear differential equations. In: Berger, C., Mousavi, M.R. (eds.) CyPhy 2015. LNCS, vol. 9361, pp. 132–146. Springer, Cham (2015). Scholar
  19. 19.
    Siddique, U., Aravantinos, V., Tahar, S.: Formal stability analysis of optical resonators. In: Brat, G., Rungta, N., Venet, A. (eds.) NFM 2013. LNCS, vol. 7871, pp. 368–382. Springer, Heidelberg (2013). Scholar
  20. 20.
    Skogestad, S., Postlethwaite, I.: Multivariable Feedback Control: Analysis and Design, vol. 2. Wiley, New York (2007)zbMATHGoogle Scholar
  21. 21.
    Sourceforge: Maxima. Accessed 12 Sept 2018
  22. 22.
    Spong, M.W., Hutchinson, S., Vidyasagar, M., et al.: Robot Modeling and Control, vol. 3. Wiley, New York (2006)Google Scholar
  23. 23.
    Stoorvogel, A.A.: The H\(^\infty \) Control Problem: A State Space Approach. Citeseer (1992)Google Scholar
  24. 24.
    Taqdees, S.H., Hasan, O.: Formalization of Laplace transform using the multivariable calculus theory of HOL-light. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR 2013. LNCS, vol. 8312, pp. 744–758. Springer, Heidelberg (2013). Scholar
  25. 25.
    Varaiya, P.: Smart cars on smart roads: problems of control. IEEE Trans. Autom. Control 38(2), 195–207 (1993)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Wellin, P.R., Gaylord, R.J., Kamin, S.N.: An Introduction to Programming with Mathematica®. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  27. 27.
    Zhong, Q.C., Hornik, T.: Control of Power Inverters in Renewable Energy and Smart Grid Integration, vol. 97. Wiley, Hoboken (2012)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Electrical Engineering and Computer Science (SEECS)National University of Sciences and Technology (NUST)IslamabadPakistan
  2. 2.Electrical Engineering Department, College of EngineeringUnited Arab Emirates UniversityAl-AinUAE

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