Time-domain analysis of sensor-to-sensor transmissibility operators

In some applications, multiple measurements are available, but the driving input that gives rise to those outputs may be unknown. This raises the question as to whether it is possible to model the response of a subset of sensors based on the response of the remaining sensors without knowledge of the driving input. To address this issue, we develop time-domain sensor-to-sensor models that account for nonzero initial conditions. The sensor-to-sensor model is in the form of a transmissibility operator that is a rational function of the differentiation operator. The development is carried out for both SISO and MIMO transmissibility operators. These time-domain sensor-to-sensor models can be used for diagnostics and output prediction. The traditional concept of input–output modeling distinguishes between inputs that evoke response and outputs that capture the response. In some applications, multiple measurements are available , but the driving inputs that give rise to those outputs may be unknown. This raises the question as to whether it is possible to model the response of a subset of sensors based on the response of the remaining sensors without knowledge of the driving input. Models of this type, which are called transmissibil-ities, are widely used in structural modeling and health monitor-In structural vibration analysis, a transmissibility is a relation between a pair of sensor measurements of the same type, for example, displacements , accelerations, or forces (Da Silva, 2007). While the transmissibility literature is extensive, a common feature is that transmissibilities are modeled in the frequency domain. A transmissibility is not a transfer function in the usual sense, however, since neither sensor captures the input driving the system except in the special case that one of the sensors measures ✩ the driving input. Consequently, a transmissibility does not have a state space realization with physically meaningful states. The goal of the present paper is to develop sensor-to-sensor models that account for nonzero initial conditions and thus are necessarily defined in the time domain. These models, which we call transmissibility operators, are rational functions of the differentiation operator. Accordingly, a transmissibility operator defines a differential equation involving the sensor signals. The internal state of the underlying input–output system loses its meaning within the context of a transmissibility operator. What is essential in defining the transmissibility operator, however, is that it must be independent of both the initial condition and inputs of the underlying system, which is assumed to be time-invariant. Transmissibility …