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Research Papers

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

[+] Author and Article Information
Levi B. Wood

Department of Mechanical Engineering,  Massachusetts Institute of Technology, 77 Massachusetts Avenue, Room 3-351, Cambridge, MA 02139 e-mail: woodl@mit.edu

Fellow ASME Department of Mechanical Engineering,  Massachusetts Institute of Technology, 77 Massachusetts Avenue, Room 3-346, Cambridge, MA 02139 e-mail: asada@mit.edu

Broadcasting error signal et and broadcasting probabilities, p and q, are equivalent if p and q are uniquely determined at each cellular unit from et .

In each case, there is a time beyond which $et2$ is very rarely nonzero. This means that the estimates of the ensemble means are very small and based on unlikely events, and the estimation of $α^tmin$ is poorly scaled. The "time series" for each case is truncated at that time.

J. Dyn. Sys., Meas., Control 133(6), 061011 (Nov 11, 2011) (11 pages) doi:10.1115/1.4004582 History: Received August 13, 2008; Revised April 30, 2011; Published November 11, 2011; Online November 11, 2011

## 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.

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## Figures

Figure 1

Cellular systems

Figure 2

Input-output characteristics of hysteric materials

Figure 3

Coordinated control of multitude of cellular units

Figure 4

State transition diagram of nonhysteretic on-off cellular unit

Figure 5

Markov chain representation of single cell local control system

Figure 6

Distribution of cells and their transitions

Figure 7

Conceptual diagram of converging error distribution

Figure 8

Sketch of J(p,0;Xt)−et2 when NtFR>1, and (a) NtFR≤et/η and (b) NtFR>et/η

Figure 9

Deterministic regulator responding to a series of step inputs with and without preloading HLC

Figure 10

RMS error of deterministic HLC and probability broadcast HLC controllers responding to a series of step inputs

Figure 11

RMS error and best fractional convergence in the mean versus time for cases (a), (b), (c), and (d)

Figure 12

Fraction of 10,000 trials versus displacement for control to r=50 given each of the cases (a), (b), (c), or (d)

Figure 13

Figure 14

Total time integral of root mean square error

Figure 15

Figure 16

Control to r=50 using Eq. 36 with different gains, k

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