Controllability of Higher Order Linear Systems with Multiple Delays in Control

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1196)


In the present chapter finite-dimensional dynamical control systems described by linear higher-order ordinary differential state equations with multiple point delays in control are considered. Using algebraic methods, necessary and sufficient conditions for relative controllability in a given time interval for linear dynamical system with multiple point delays in control are formulated and proved. This condition is generalization to relative controllability case some previous results concerning controllability of linear dynamical systems without multiple point delays in the control. Proof of the main result is based on necessary and sufficient controllability condition for linear systems without delays in control. Simple numerical example, which illustrates theoretical result is also given. Finally, some remarks and comments on the existing results for controllability of dynamical systems with delays in control are also presented.


Controllability Linear systems Higher order systems 



The work is supported by National Science Centre in Poland under grant: “Modelling, optimization and control for structural reduction of device noise”, DEC-2017/25/B/ST7/02236.


  1. 1.
    Kalogeropoulos, G., Psarrakos, P.: A note on the controllability of higher-order linear systems. Appl. Math. Lett. 17, 1375–1380 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Klamka, J.: Controllability of Dynamical Systems. Kluwer Academic Publishers, Dordrecht (1991)zbMATHGoogle Scholar
  3. 3.
    Klamka, J.: Controllability of dynamical systems - a survey. Arch. Control Sci. 2(3–4), 281–307 (1993)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Klamka, J.: Controllability of dynamical systems: a survey. Bull. Pol. Acad. Sci. Tech. Sci. 61(2), 221–229 (2013)Google Scholar
  5. 5.
    Klamka, J.: Constrained controllability of second order dynamical systems with delay. Control Cybern. 42(1), 111–121 (2013)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Silesian University of TechnologyGliwicePoland

Personalised recommendations