Resilient Packet Ring (RPR) is a new technology currently being standardized in the IEEE 802.17 working group. The existed bandwidth allocation algorithms for RPR networks are not able to provide satisfactory solutions to meet the performance requirements. In this paper we propose one fair bandwidth allocation algorithm, termed PID-RPR, which satisfies the performance goals of RPR networks, such as fairness, high utilization and maximal spatial reuse. The algorithm is operated at each RPR node in a distributive way; the proportional, integral and differential (PID) controller is used to allocate bandwidth on the outgoing link of the node for the flows over the link in a weighted manner. To achieve the global coordination, one control packet containing every node's message runs around the ring in order to update the relevant message for all nodes on the ring. When the packet reaches one node, this node adjusts its own rate according to its own message in the control packet; in the meantime it updates other nodes' control message in the control packet. As the control packet propagates around the ring, each node can eventually adjust its sending rate to reach its fair share according to the fairness criterion, and the buffer occupancy at each node is kept within the target value. Our algorithm is of distributed nature in the sense that upstream ring nodes inject traffic at a rate according to congestion and fairness criteria downstream. The simulation results demonstrate that satisfactory performance of RPR networks can be achieved under the proposed bandwidth allocation scheme.