Stability of systems with state delay subjected to digital control

Stability of linear delayed systems subjected to digital control is analyzed. These systems can typically be written in the form ẋ(t) = Ax(t) + Bx(t− τ) + Cx(tj−1) , t ∈ [tj , tj+1) , where tj = j∆t with ∆t being the sampling period for the digital controller. The point-delay term x(t−τ) is assumed to be inherently present in the governing equation of the uncontrolled system, while the term x(tj−1) is present due to the digital controller. Since the term x(tj−1) can be represented as a term with a piecewise linearly varying time delay, the system is time-periodic at period ∆t. The stability analysis for the system is performed using the semi-discretization method. As case studies, the stability charts of the delayed oscillator and the turning process are determined for a digital PD controller.