Simultaneous stabilization of discrete-time delay systems and bounds on delay margin

This paper studies the delay margin problem of linear time-invariant (LTI) systems. For discrete-time systems, this problem can also be posed as one of simultaneous stabilization of multiple unstable systems with different lengths of delay. Our contribution is threefold. First, for general LTI plants with a number of distinct unstable poles and nonminimum phase zeros, we employ analytic function interpolation and rational approximation techniques to derive bounds on the delay margin. We show that readily computable and explicit lower bounds can be found by computing the real eigenvalues of a constant matrix, and LTI controllers, potentially of a low order, can be synthesized to achieve the bounds based on the H_∞ control theory. Second, we show that these results can be coherently extended to systems with time-varying delays, also resulting in bounds on the delay range and delay variation that ensure simultaneous stabilization. Finally, we investigate the delay margin problem in the context of PID control. For first-order unstable plants, we obtain bounds achievable by PID controllers. It is worth noting that unlike its continuous-time counterpart, the discrete-time delay margin problem is fundamentally more difficult, due to the intrinsic difficulty in simultaneously stabilizing several systems. Nevertheless, while previous works on the discrete-time delay margin led to largely negative results, the bounds developed in this paper provide instead conditions that guarantee the simultaneous stabilization of multiple delay plants, or ranges within which the delay plants can be robustly stabilized.