Controller design for linear time-periodic delay systems via a symbolic approach

This paper presents a symbolic method for the delayed state feedback controller (DSFC) design for the linear time-periodic delay (LTPD) systems that are open loop unstable. By using shifted Chebyshev polynomials, the closed loop "monodromy matrix" of the LTPD system is obtained symbolically in terms of controller parameters. The stability criterion is given in terms of the eigenvalues of such a symbolic monodromy matrix. This enables one to design a delayed state feedback controller (DSFC) to asymptotically stabilize the original unstable dynamic system. Two controllers designs are presented. The first design is a constant gain DSFC and the second one is a periodic gain DSFC. The periodic gain DSFC has a larger region of stability in the parameter space than the constant gain DSFC. The asymptotic stability of the LTPD system obtained by the proposed method is illustrated in numerical examples by stabilizing an open loop unstable delayed Mathieu equation.