Stabilizing a class of time delay systems

PID Stabilization of Retarded‐Type Time‐Delay System

“…The stabilization of a control system involves finding the zero distribution of its characteristic polynomial. It is relatively easy to study a delay-free system because of its finite number of zeros [3][4][5][6], while that of a time-delay system is somewhat difficult on account of its infinite number of zeros [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. It was Pontryagin [22] who originally studied and presented the necessary and sufficient conditions for the stability of certain classes of quasi-polynomials.…”

“…So far, two categories of approaches have been proposed to compute such a stabilizing region. One category, which is based on Tsypkins results and the generalized Nyquist criterion [6][7][8][9], on the concept of singular frequencies [10][11][12], or on a generalized Hermite-Biehler theorem [13][14][15][16], calculates the feasible k p-intervals first, then constructs the (ki, kd) region for a fixed kp. The other category, based on the boundary crossing theorem and D-decomposition theory [17][18][19], constructs the (kp, kd) or (kp, ki) region of a fixed ki or kd in given intervals.…”