Uppsala University
Department of Information Technology

### Technical Report 2014-009

# On a Limitation in Networked Flow Control

### Torbjörn Wigren

March 2014

**Abstract:**The paper analyzes a continuous time flow control system, with flow of a general quantity from a source node to a sink node. The flow is one-directional, meaning that there is a saturation between the nodes that limits the flow to be positive and below a maximum. The controlled plant is located in the sink node and the controller is located in the source node. The plant and the controller are modeled by linear filters parameterized with poles and zeros. Feed forward control from measured disturbances is included. Delays affect both the downlink control signal and the uplink measurement signals. The paper proves that for large delays, *L*_{2}-stability does not follow from the Popov criterion unless the quotient of the products of all zeros and the product of all poles is less than *1/kG*_{p}, where *k* is the slope of the saturation and *G*_{p} is the gain constant of the loop gain. In case the plant models a leaky reservoir, the conclusion is that the amount of low frequency gain of the controller cannot be arbitrarily high at the same time as the amount of leakage of the reservoir is arbitrarily low. In communications this means that an increased requirement to regulate static errors of the reservoir needs to be accompanied by a reduced flow capacity.

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