Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Controllability and observability analysis of continuous-time multi-order fractional systems

  • Published:
Multidimensional Systems and Signal Processing Aims and scope Submit manuscript

Abstract

This paper proposes some analytical criteria for controllability and observability analysis of fractional order systems describing by multi-order state space equations. The controllability and observability gramians are presented in which their non-singularity is equivalent to controllability and observability of the mentioned systems. Moreover, to overcome the gramian calculation complexities, the controllability and observability matrices are introduced. It is proved that the sufficient condition for controllability or observability of a multi-order fractional system is that its controllability or observability matrix is nonsingular. Some illustrative examples are given to show the efficiency of the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Balachandran, K., & Govindaraj, V. (2014). Numerical controllability of fractional dynamical systems. Optimization, 63(8), 1267–1279.

    Article  MathSciNet  MATH  Google Scholar 

  • Balachandran, K., Govindaraj, V., Rodriguez-Germa, L., & Trujillo, J. J. (2013). Controllability of nonlinear higher order fractional dynamical systems. Nonlinear Dynamics, 71(4), 605–612.

    Article  MathSciNet  MATH  Google Scholar 

  • Balachandran, K., & Kokila, J. (2013). Constrained controllability of fractional dynamical systems. Numerical Functional Analysis and Optimization, 34(11), 1187–1205.

    Article  MathSciNet  MATH  Google Scholar 

  • Balachandran, K., Park, J. Y., & Trujillo, J. J. (2012). Controllability of nonlinear fractional dynamical systems. Nonlinear Analysis: Theory, Methods and Applications, 75(4), 1919–1926.

    Article  MathSciNet  MATH  Google Scholar 

  • Bettayeb, M., & Djennoune, S. (2008). New results on the controllability and observability of fractional dynamical systems. Journal of Vibration and Control, 14(9–10), 1531–1541.

    Article  MathSciNet  MATH  Google Scholar 

  • Caponetto, R., Dongola, G., Maione, G., & Pisano, A. (2014). Integrated technology fractional order proportional-integral-derivative design. Journal of Vibration and Control, 20(7), 1066–1075.

    Article  MathSciNet  Google Scholar 

  • EI-Sayed, A. M. A., & Gaafar, F. M. (2003). Fractional calculus and some intermediate physical processes. Applied Mathematics and Computation, 144(1), 117–126.

    Article  MathSciNet  MATH  Google Scholar 

  • Galkowski, K., Bachelier, O., & Kummert, A. (2006). Fractional polynomials and nD systems: A continuous case. InProceedings of the 45th IEEE conference on decision and control, San Diego, CA, USA, pp. 2913–2917.

  • Jiang, C. X., Carletta, J. E., Hartley, T. T., & Veillette, R. J. (2013). A systematic approach for implementing fractional-order operators and systems. IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 3(3), 301–312.

    Article  Google Scholar 

  • Kaczorek, T. (2009). Reachability of positive 2D fractional linear systems. Physica Scripta, 2009(T136), 014039.

    Article  Google Scholar 

  • Kaczorek, T. (2011a). Positive linear systems consisting of \(n\) subsystems with different fractional orders. IEEE Transactions on Circuits and Systems I: Regular Papers, 58(6), 1203–1210.

    Article  MathSciNet  Google Scholar 

  • Kaczorek, T. (2011b). Selected problems of fractional systems theory. Lecture notes in control and information sciences (Vol. 411). Berlin: Springer.

  • Kaczorek, T., & Sajewski, L. (2014). The realization problem for positive and fractional fystems. Studies in systems, decision and control (Vol. 1). Berlin: Springer.

  • Krishnan, B., & Jayakumar, K. (2013). Controllability of fractional dynamical systems with prescribed controls. IET Control Theory and Applications, 7(9), 1242–1248.

    Article  MathSciNet  Google Scholar 

  • Magin, R. L. (2010). Fractional calculus models of complex dynamics in biological tissues. Computers and Mathematics with Applications, 59(5), 1586–1593.

    Article  MathSciNet  MATH  Google Scholar 

  • Matignon, D., & D’Andréa-Novel, B. (1996). Some results on controllability and observability of finite-dimensional fractional differential systems. In Proceedings of the IEEE conference on systems, man and cybernetics (pp. 952–956). Lille, France.

  • Mondal, D., & Biswas, K. (2011). Performance study of fractional order integrator using single-component fractional order element. IET Circuits Devices and Systems, 5(4), 334–342.

    Article  Google Scholar 

  • Mozyrska, D., & Pawluszewicz, E. (2015). Controllability of h-difference linear control systems with two fractional orders. International Journal of Systems Science, 46(4), 662–669.

    Article  MathSciNet  MATH  Google Scholar 

  • Petras, I. (2009). Stability of fractional-order systems with rational-orders: A survey. Fractional Calculus and Applied Analysis, 12(3), 269–297.

    MathSciNet  MATH  Google Scholar 

  • Podlubny, I. (1999). Fractional differential equations. San Diego: Academic Press.

    MATH  Google Scholar 

  • Sabatier, J., Agrawal, O. P., & Tenreiro Machado, J. A. (2007). Advances in fractional calculus: theoretical developments and applications in physics and engineering. Dordrecht: Springer.

    Book  MATH  Google Scholar 

  • Sajewski, L. (2014). Reachability, observability and minimum energy control of fractional positive continuous-time linear systems with two different fractional orders. Multidimensional Systems and Signal Processing. doi:10.1007/s11045-014-0287-2.

  • Stiassnie, M. (1979). On the application of fractional calculus for the formulation of viscoelastic models. Applied Mathematical Modelling, 3(4), 300–302.

    Article  MATH  Google Scholar 

  • Tavakoli-Kakhki, M., & Haeri, M. (2010). The minimal state space realization for a class of fractional order transfer function. SIAM Journal on Control and Optimization, 48(7), 4317–4326.

    Article  MathSciNet  MATH  Google Scholar 

  • Tavakoli-Kakhki, M., Tavazoei, M. S., & Haeri, M. (2011). Notes on the state space realizations of rational order transfer functions. IEEE Transactions on Circuits and Systems I: Regular Papers, 58(5), 1099–1108.

    Article  MathSciNet  Google Scholar 

  • Tavazoei, M. S., & Tavakoli-Kakhki, M. (2013). Minimal realizations for some classes of fractional order transfer functions. IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 3(3), 313–321.

    Article  Google Scholar 

  • Tenreiro Machado, J. A., Jesus, I. S., Galhano, A., & Cunha, J. B. (2006). Fractional order electromagnetics. Signal Processing, 86(10), 2637–2644.

    Article  MATH  Google Scholar 

  • Varshney, P., Gupta, M., & Visweswaran, G. S. (2008). Implementation of switched-capacitor fractional order differentiator \((PD^{\delta })\) circuit. International Journal of Electronics, 95(6), 531–547.

    Article  Google Scholar 

  • Xu, D., Li, Y., & Zhou, W. (2014). Controllability and observability of fractional linear systems with two different orders. The Scientific World Journal, 2014, 1–8.

    Google Scholar 

  • Zeng, Q., Cao, G., & Zhu, X. (2005). Research on controllability for a class of fractional-order linear control systems. Journal of Systems Engineering and Electronics, 16(2), 376–381.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Khomeinishahr Branch, Islamic Azad University, Isfahan, Iran

    Mohammad Tavakoli & Mohammad Tabatabaei

Authors
  1. Mohammad Tavakoli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohammad Tabatabaei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammad Tabatabaei.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tavakoli, M., Tabatabaei, M. Controllability and observability analysis of continuous-time multi-order fractional systems. Multidim Syst Sign Process 28, 427–450 (2017). https://doi.org/10.1007/s11045-015-0349-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11045-015-0349-0

Keywords

Search

Navigation