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Most broadband access networks such as passive optical networks (PONs) adopt the point-to-multipoint (P2MP) topology. One critical issue in such networks is the upstream resource management and allocation mechanism. Nonlinear predictor-based dynamic resource allocation (NLPDRA) schemes for improving the P2MP network upstream transmission efficiency have been investigated in an ad hoc manner. In this paper, we establish a general state space model to analyze the controllability and stability of the NLPDRA schemes from the P2MP network system's point of view and propose controller design guidelines to maintain the system stability under different scenarios. Analytical results show that NLPDRA maintains the P2MP network system controllability even when the loaded network traffic changes drastically. We further prove that a P2MP network system with NLPDRA is stable by proper pole placements as the traffic changes. Finally, we provide guidelines to design a optimal compensator to achieve system accuracy.