Stability of shift-varying 2-D state-space digital filters

Stability conditions for 2-D shift-varying systems are presented. The shift-varying nature of such systems emerges in applications such as adaptive filtering or adaptive image processing, where the coefficients are neither static nor periodic. The forms considered in this paper are the Givone-Roesser and the Fornasini-Marchesini models, both of which are discrete 2-D state-space filters. The sufficient conditions for BIBO stability that are proven herein are an outgrowth of the 1-D time-varying state-space conditions that have been previously established. The nature of feedback in the 2-D space is explored and found to be much more complex than for the 1-D case. However, it is also shown that when every feedback path is guaranteed to satisfy a variation on exponential stability, then BIBO stability of these two models can be assured. Further conditions are also established which engage the Lyapunov equation and guarantee the exponential stability requirement.