Analysis and Control of Linear Time Periodic System using Normal Forms

Susheelkumar Cherangara Subramanian, Sangram Redkar

Research output: Contribution to journalArticlepeer-review


Multiple techniques have been developed in the past towards stability and control of linear time periodic systems. Though the method of normal forms was predominantly applied to nonlinear equations, in this work, it is utilized to transform a linear time varying system with periodic coefficients to a time-invariant system (similar to a Lyapunov-Floquet transformation). The direct application of time independent normal forms is facilitated by a combination of an intuitive state augmentation technique and modal transformation. Additionally, this approach yields a closed form analytical expression for the Lyapunov–Floquet (L-F) transformation and state transition matrix. The transition curves and stability bounds are identified and multiple feedback control strategies are also discussed in this work. Furthermore, the authors demonstrate the control of an unstable periodic system to a stable point, desired periodic orbit and optimally controlled system states using the normal forms approach. The theoretical framework and controller implementation are illustrated using numerical simulations for the case of a linear Mathieu equation.

Original languageEnglish (US)
Pages (from-to)1626-1636
Number of pages11
JournalInternational Journal of Dynamics and Control
Issue number5
StatePublished - Oct 2022


  • Control system
  • Linear feedback controllers
  • Linear time periodic system
  • Normal forms
  • State transition matrix

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Civil and Structural Engineering
  • Modeling and Simulation
  • Mechanical Engineering
  • Control and Optimization
  • Electrical and Electronic Engineering


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