Describing functions for information channels subject to packet loss and quantization

Abstract

underwater environments. Examples include autonomous underwater vehicle (AUV) coordination and navigation, Wide Area Measurement System (WAMS) control of power grids, and automobile networked subsystems control. As one would expect, demand on throughput grows to fill the available channel capacity; in a clean short-range RF setting, entire images may be transferred in each cycle, as part of a vision-based control system, whereas in the ocean, acoustic channels with perhaps twenty bits per second allow only the most basic sensor and command information to be shared regularly. Packet-based wireless communication systems like these that operate near their limits are necessarily quantized, and often prone to loss. These properties directly impact the overall system performance, and thus methodologies for understanding and designing feedback systems with quantization and packet loss are valuable. This thesis makes several contributions to feedback control of systems subject to quantization and stochastic packet loss in the sensor feedback channel. First, we derive and verify describing functions (DFs) for information channels subject to quantization and packet loss. The DFs represent the loss and quantization effects by frequency- and amplitude-dependent gains and phases, similar to transfer functions. These DFs are unique because, unlike most other DFs that describe hardware and physical elements, these describe stochastic information channels. DFs are presented for a general codec algorithm, and for four commonly-used sensor-feedback codecs: Zero-Output, Hold- Output, Linear Filter, and Modified Information Filter. These are each given as closed-form mathematical expressions of the provably optimal gains and phases for each case, with each decoder a specific case of the general codec algorithm. Gains and phases predicted by the models are verified by simulation for open-loop stable, open-loop unstable, minimum phase and non-minimum phase example systems. Second, we show how the DFs can be used as analysis tools to predict limit cycles in dynamic feedback control systems. Computation times using the DFs are shown to be orders of magnitude faster than those from simulation for these calculations. Third, we propose a synthesis method to use the DFs to design a codec for the sensor feedback channel that decreases limit cycle amplitudes induced by quantization and packet loss for a large class of systems. Up to three-fold reductions in limit cycle amplitudes are shown, with the tradeoff being slightly higher system sensitivity to disturbances and slightly higher steady state errors to step inputs. The designed codec is of the special and simple form of a constant times the sent signal if the signal is received and a different constant times the previous decoded signal if the sent signal is lost. This is the equivalent structure and computation complexity to both Zero-Output and Hold-Output decoders. A DF for this decoder allows the constants to be solved for as functions of target limit cycle amplitudes. The constants reduce to solutions of cubic equations, which are guaranteed to have a real root, and thus the codec is physically realizable. The codec allows for multiple limit cycle frequency solutions for the same amplitude solution. The analysis and synthesis tools are verified both by numerical examples, and by a physical experiment controlling heading of a small robotic raft where the designed decoder results in smaller limit cycles than does a linear-filter-based decoder.