Cellular Stochastic Control of the Collective Output of a Class of Distributed Hysteretic Systems

Abstract

Stable stochastic feedback control of an aggregate output from a multitude of cellular units is presented in this paper. Similar to a skeletal muscle comprising a number of muscle fibers, the plant considered in this paper consists of many independent units (called cellular units), each of which contributes to an aggregate output of the whole system. The central controller regulates the aggregate output by stochastically recruiting as many cellular units as needed for producing a required output. Two challenges are considered. The first is how to deal with individual units having pronounced hysteresis and long latency time in transient response. It will be shown that slow response and poor stability due to the hysteresis and latency time can significantly be improved by coordinating the multitude of cellular units, which are in diverse phases in the hysteresis loop. The second challenge is how to build a central controller that coordinates the multitude of cellular units without knowing the state of individual units. Stochastic broadcast feedback is presented as a solution that meets those requirements. The central controller observes only the aggregate output value rather than the output and state of each unit, compares the aggregate output against a reference, and broadcasts an error signal to all the units, which are anonymous. In turn, each cellular unit makes a control decision stochastically with state transition probabilities that are modulated by the broadcast error signal from the central controller. Stability analysis based on supermatingale theory guarantees that this stochastic broadcast feedback is stable and robust against cell failures. The method is applied to the control of shape-memory-alloy muscle actuators with cellular architecture. Despite pronounced hysteresis and long latency time, stochastic broadcast feedback can achieve fast and stable control. Simulation experiments verify the theoretical results.