Skip to main content
SpringerLink
Log in

Determination of Periodic Trajectories of Dynamic Systems Subject to Switching Input Constraints

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The present paper is concerned with the determination of periodic state trajectories of discrete-time dynamic systems resulting from the application of bidirectional on/off input sequences subject to switching constraints. The main contribution consists of characterizing the feasible input sequences and associated state trajectories in terms of a set of linear constraints involving integer variables. For practical purposes, the resulting solutions obtained by an integer linear solver can be evaluated in terms of engineering criteria such as amplitude or fuel expense in order to choose an appropriate trajectory to be used as target in a control law. A numerical example involving a double-integrator system is presented to illustrate the use of the proposed constraint formulation for the determination of feasible periodic state trajectories.

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

Access this article

Log in via an institution

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. Indeed, from (41) it follows that \(x(i)=A^i x(0) + \sum \nolimits _{k=0}^{i-1} A^k B u(i-k-1)= \begin{bmatrix} x_1(0)\\0\end{bmatrix} + \begin{bmatrix} i x_2(0)\\x_2(0)\end{bmatrix} + \sum \nolimits _{k=0}^{i-1} A^k B u(i-k-1)\). Therefore, by changing \(x_1(0)\), the entire state trajectory is shifted along the \(x_1\)-axis.

References

  1. Egerstedt, M., Wardi, Y., Delmotte, F.: Optimal control of switching times in switched dynamical systems. In: Proceedings of the 42nd IEEE Conference on Decision and Control (2003)

  2. Wu, X., Zhang, K., Sun, C.: Parameter tuning of multi-proportional-integral-derivative controllers based on optimal switching algorithms. J. Optim. Theory Appl. 159(2), 454–472 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Liu, C., Loxton, R., Teo, K.L.: Switching time and parameter optimization in nonlinear switched systems with multiple time-delays. J. Optim. Theory Appl. 163(3), 957–988 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Loxton, R.C., Teo, K.L., Rehbock, V., Ling, W.K.: Optimal switching instants for a switched-capacitor DC/DC power converter. Automatica 45, 973–980 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Trecate, G.F., Gallestey, E., Letizia, P., Spedicato, M., Morari, M., Antoine, M.: Modeling and control of co-generation power plants: a hybrid system approach. IEEE Trans. Control Syst. Technol. 4(5), 694–705 (2004)

    Article  MATH  Google Scholar 

  6. Larsen, G.K.H., Foreest, N.D., Scherpen, J.M.A.: Distributed control of the power supply-demand balance. IEEE Trans. Smart Grid 4(2), 828–836 (2013)

    Article  Google Scholar 

  7. Larsen, G.K.H., van Foreest, N.D., Scherpen, J.M.A.: Distributed MPC applied to a network of households with micro-CHP and heat storage. IEEE Trans. Smart Grid 5(4), 2106–2114 (2014)

    Article  Google Scholar 

  8. Parisio, A., Rikos, E., Glielmo, L.: A model predictive control approach to microgrid operation optimization. IEEE Trans. Control Syst. Technol. 22(5), 1813–1827 (2014)

    Article  Google Scholar 

  9. Antropov, N.N., Diakov, G.A., Kim, V., Popov, G.A., Pokryshkin, A.I., Kazeev, M.N., Khodnenko, V.P.: Pulse plasma thrusters for spacecraft attitude and orbit control system. In: Proceedings of the 26th International Propulsion Conference, vol. 2, pp. 1129–1135 (1999)

  10. Kienitz, K.H., Bals, J.: Pulse modulation for attitude control with thrusters subject to switching restrictions. Aerosp. Sci. Technol. 9(7), 635–640 (2005)

    Article  MATH  Google Scholar 

  11. Kienitz, K.H.: Attitude stabilization with actuators subject to switching restrictions: an approach via exact relay control methods. IEEE Trans. Aerosp. Electron. Syst. 42(4), 1485–1492 (2006)

    Article  Google Scholar 

  12. Vieira, M.S., Galvão, R.K.H., Kienitz, K.H.: Attitude stabilization with actuators subject to switching-time constraints using explicit MPC. In: Proceedings of the IEEE Aerospace Conference (2011)

  13. Vieira, M.S.: Predictive control of systems with actuators subject to switching time constraints (in Portuguese). Ph.D. Thesis, Instituto Tecnológico de Aeronáutica, Brazil (2013)

  14. Rao, V.G., Bernstein, D.S.: Naive control of the double integrator. IEEE Control Syst. Mag. 21(5), 86–97 (2001)

    Article  Google Scholar 

  15. Li, S., Du, H., Lin, X.: Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics. Automatica 47, 1706–1712 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, J., Ren, W., Xu, S.: Distributed containment control with multiple dynamic leaders for double-integrator dynamics using only position measurements. IEEE Trans. Autom. Control 57(6), 1553–1559 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tian, B., Zuo, Z., Yan, X., Wang, H.: A fixed-time output feedback control scheme for double integrator systems. Automatica 80, 17–24 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the support of FAPESP (Grant 2011/17610-0), CNPq (Research Fellowships 303714/2014-0, 309331/2015-3) and Capes (Ph.D. Scholarships).

Author information

Authors and Affiliations

  1. Electronic Engineering Department, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, 12228-900, Brazil

    Matheus Henrique Marcolino, Roberto Kawakami Harrop Galvão & Karl Heinz Kienitz

  2. Embraer, Avenida Brigadeiro Faria Lima, 2170, São José dos Campos, SP, 12227-901, Brazil

    Márcio Santos Vieira

Authors
  1. Matheus Henrique Marcolino
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roberto Kawakami Harrop Galvão
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Karl Heinz Kienitz
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Márcio Santos Vieira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matheus Henrique Marcolino.

Additional information

Communicated by Alexander S. Strekalovsky.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Marcolino, M.H., Galvão, R.K.H., Kienitz, K.H. et al. Determination of Periodic Trajectories of Dynamic Systems Subject to Switching Input Constraints. J Optim Theory Appl 175, 848–864 (2017). https://doi.org/10.1007/s10957-017-1187-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-017-1187-z

Keywords

Mathematics Subject Classification

Search

Navigation