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Structural Controllability and Observability of Linear Systems Over Finite Fields With Applications to Multi-Agent Systems

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2 Author(s)
Sundaram, S. ; Dept. of Electr. & Comput. Eng., Univ. of Waterloo, Waterloo, ON, Canada ; Hadjicostis, C.N.

We develop a graph-theoretic characterization of controllability and observability of linear systems over finite fields. Specifically, we show that a linear system will be structurally controllable and observable over a finite field if the graph of the system satisfies certain properties, and the size of the field is sufficiently large. We also provide graph-theoretic upper bounds on the controllability and observability indices for structured linear systems (over arbitrary fields). We then use our analysis to design nearest-neighbor rules for multi-agent systems where the state of each agent is constrained to lie in a finite set. We view the discrete states of each agent as elements of a finite field, and employ a linear iterative strategy whereby at each time-step, each agent updates its state to be a linear combination (over the finite field) of its own state and the states of its neighbors. Using our results on structural controllability and observability, we show how a set of leader agents can use this strategy to place all agents into any desired state (within the finite set), and how a set of sink agents can recover the set of initial values held by all of the agents.

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Automatic Control, IEEE Transactions on  (Volume:58 ,  Issue: 1 )