Stabilisation of the second order system with a time delay controller

In this paper stabilisation of a second order system $\ddot{x}(t)+Ax(t)=Bu(t)$ with a time delay output feedback $u(t)=Ky(t-h)$ is analysed. Considered class of second order systems is described, that can be physically modeled as $\mathrm{LC}$ ladder networks, and at the same time can be used as an approximation of of a distributed parameter system with undamped oscillations. It is followed with stability analysis of resulting infinite dimensional system. It is shown that application of transfer functions is justified and apply the Pad\'e approximation in order to obtain approximated stability regions via constrained optimisation. Then the formulas for derivatives are given, along with their numerical effectiveness comparison. Finally  the obtained stability regions are used to optimise the impulse response of closed loop system determining the appropriate values of performance index with James-Nichols-Philips theorem. All these results are illustrated with simulations and optimisation results for different sizes of $\mathrm{LC}$ ladder.Also, the merits and limitations of  Pad\'e approximation are briefly discussed.