Excitation Conditions for Uniform Exponential Stability of the Cooperative Gradient Algorithm Over Weakly Connected Digraphs

Distributed Hybrid Gradient Algorithm with Application to Cooperative Adaptive Estimation

“…Remark 4. Note that Assumption 4 is the hybrid counterpart of the distributed persistence of excitation condition used in [12], which is shown to be necessary and sufficient for convergence in the continuoustime setting.…”

“…where each L 𝑠 ∈ R 𝑚 𝑠 ×𝑚 𝑠 is the Laplacian matrix of the strongly connected component G 𝑠 , the lower-left block 𝑀 𝑙 ∈ R 𝑘×𝑚 𝑦 −𝑘 , 𝑘 := 𝑚 𝑦 − 𝑠 ∈S 𝑙 𝑚 𝑠 , is a non-positive matrix, and the lower-right block 𝑀 𝑟 ∈ R 𝑘×𝑘 is a non-singular M-matrix; namely, 𝑀 𝑟 := 𝛾 𝑟 𝐼 𝑘 − 𝐶 𝑟 , where 𝐶 𝑟 ∈ R 𝑘×𝑘 is a non-negative matrix, 𝛾 𝑟 > 𝜌 (𝐶 𝑟 ) > 0, and 𝜌 (𝐶 𝑟 ) is the spectral radius of 𝐶 𝑟 ; see [6,12] for more details.…”

“…In this paper, inspired by [5,12,19], we carry a step further the relaxation of the conditions for parameter estimation using the hybrid-gradient estimation algorithm. We propose an algorithm that relies on a network of estimators, each of which is designed using a hybrid gradient-based adaptation law and has access to only limited data, involving the parameters of the global process.…”

“…The considered distributed estimation framework has been studied in the literature for the continuous-time case only [5,12,19]. Indeed, under different graph conditions, distributed (continuoustime) persistence of excitation conditions are shown to be necessary and sufficient to guarantee UES of the set { θ = 0}.…”

“…To achieve the identification of 𝜃 , the different estimators are designed to exchange their estimates, in real (hybrid) time, according to a communication graph denoted by G. That is, similarly to the algorithm proposed in [12] for distributed estimation in continuous time, we design an update law for each θ𝑖 along the discrete-time evolution and along the continuous-time evolution of 𝜓 as follows.…”