Heterogeneous string stability of unidirectionally interconnected MIMO LTI systems

Abstract

One-dimensional formations of unidirectionally interconnected heterogeneous, multiple-input multiple-output (MIMO) linear time-invariant (LTI) systems are studied in terms of the spatial propagation of initial conditions, disturbances and reference inputs. The proposed mixed frequency-domain and complex state-space formulation of the string leads to the stability theory of switching and uncertain polytopic systems, and so the tools available there can be adopted for the analysis of interconnected systems. It is shown based on this analogy that in certain important cases string stability conditions for homogeneous and heterogeneous interconnected systems coincide. In addition to presenting general string (in)stability conditions, the necessity of introducing the notion of string performance is demonstrated. Special attention is devoted to interconnected rank one systems, i.e., systems whose transfer matrices are of rank one. The topic is motivated by car following problems for which the available analysis tools fail to provide appropriate heterogeneous string stability conditions. A cooperative adaptive cruise control (CACC) example is presented to illustrate the usefulness of the approach.