Abstract
Considering any connected network with unknown initial states for all nodes, the nearest-neighbor rule is utilized for each node to update its own state at every discrete-time step. Distributed function calculation problem is defined for one node to compute some function of the initial values of all the nodes based on its own observations. In this paper, taking into account uncertainties in the network and observations, an algorithm is proposed to compute and explicitly characterize the value of the function in question when the number of successive observations is large enough. While the number of successive observations is not large enough, we provide an approach to obtain the tightest possible bounds on such function by using linear programing optimization techniques. Simulations are provided to demonstrate the theoretical results.
Highlights
Consider a network where each node i has the knowledge of its initial value x0[i] ∈ R
When minimal number of time-steps is not satisfied, we propose an approach to solve the upper and lower bounds on the initial values of all the nodes
Let all data and assumptions be as given in Problem 1; the exact value of the initial state x0 can be solved from (6) for any disturbance d if and only if there exists U ∈ RNy×m which spans the null space of DTyd such that rank [UTCy0] = n
Summary
Consider a network where each node i has the knowledge of its initial value x0[i] ∈ R. Reference [9] shows that consensus function calculation can be performed in a minimal number of time-steps, and, the minimal number of time-steps can be fully characterized algebraically and graphically [10] When it comes to the noisy network where each node only obtains a noisy (or uncertain) measurement of its neighbors’ values (e.g., the transmission channel between nodes is noisy (bounded noise) and quantized (given a certain quantization scheme)), the estimation on initial value becomes a very challenging problem. When minimal number of time-steps is not satisfied, we propose an approach to solve the upper and lower bounds on the initial values of all the nodes These bounds are the tightest possible given the system model, output measurements, and the rough bounds on the unknown initial state.
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