Quantization and overflow effects in digital implementations of linear dynamic controllers

The stability of a class of single-input/single-output (SISO) digital-feedback control systems is investigated. The systems considered contain a linear dynamic plant, a digital controller, and suitable D/A (digital-to-analog) and A/D converters. The design of such systems is usually accomplished by ignoring the nonlinear effects caused by quantization and overflow truncation. Simulations of specific examples establish the quantization in such systems can lead to loss of asymptotic stability of the origin. Obtained results prove that when quantization is taken into account (but overflow is neglected), one only has convergence to some (small) neighborhood about the origin. The results also prove that for initial conditions sufficiently far from the origin, overflow effects can lead to unbounded solutions. In particular, this is the case for observable systems with at least one eigenvalue in the open right-half plane (RHP). The case of systems with no eigenvalues in the open RHP, but with eigenvalues on the jω-axis, is still unresolved