Robust stabilization of discrete-time parameter-dependent systems: the finite precision problem

Realization of digital filters or implementation of controllers in a digital computer may lead to unexpected instabilities resulting from the finite precision effects. Stability is usually ensured for an idealized discrete-time realization of the system. However, as soon as A/D and D/A conversions are involved, the quantization of the state of the system, due to adder overflow, magnitude truncation, finite-wordlength format, may introduce severe nonlinearities responsible for overflow oscillations, limit cycles or chaotic behavior, even under zero input. This paper considers a parameter-dependent, discrete-time system in the companion form. We derive linear matrix inequality (LMI) conditions ensuring stability for the uncertain system in spite of the finite precision effect. We also seek an LMI formulation for the synthesis of a static output-feedback controller that guarantees robust stability for the finite precision problem.