Proportional-Integral Observer Design for Multi-Rate Systems Under Decode-and-Forward Relays: Tackling Power-Dependent Packet Dropouts

In this paper, the problem of proportional-integral observer (PIO) design is investigated for a class of discrete-time multi-rate systems with multiple sensors, with the sensor sampling periods being allowed to differ from the system updating periods. The facilitation of communication between sensors and the remote PIO through wireless networks, which are subject to probabilistic packet dropouts, is achieved through the utilization of a decode-and-forward relay-based strategy. The occurrence of packet dropouts is governed by a Bernoulli-distributed random variable whose probability is dependent on the available transmission power. A decode-and-forward relay-based strategy, developed based on different components, is capable of processing information from different encoders at different physical locations. For the convenience of observer design, the lifting technique is employed with aim to cast the multi-rate system into a single-rate one. By establishing sufficient conditions, the combined effect of external noises and relaying-aided communication on estimation performance is intuitively illustrated. Subsequently, a PIO with an adjustable parameter is designed by solving certain optimization problems. A simulation example is finally provided to validate the theoretical results.