A Note About the Stability of a String of LTI Systems

This note concerns the stability of a string of LTI systems; it is shown here that, if the weak interaction condition among systems in the string is satisfied (i.e., the sum of the ∞-norms of the interaction/error propagation transfer functions is less than unity), then the string is L p stable for every p ≥ 1. Since the ∞-norm of a transfer function is the smallest of all its induced L p norms, the result presented here enables one to obtain a tighter estimate of the geometric rate of attenuation of the states of systems in the string.