Convergence of a Distributed Least Squares

Distributed order estimation of ARX model under cooperative excitation condition

“…We know that for the case where the system orders p 0 , q 0 are known, the distributed LS algorithm is one of the most basic algorithms to estimate the unknown parameters, and it has wide applications because of its fast convergence rate, e.g., in the area of cloud technologies (e.g., [39]). The details of the distributed LS algorithm can be found in the following Algorithm 2.1 (see [36]).…”

“…For the case where the system orders p 0 , q 0 are known, Xie and Guo in [36] proved that the distributed LS algorithm can converge to the true parameters almost surely (a.s.) under a cooperative excitation condition. However, when the system orders p 0 , q 0 are unknown, the estimation for both the system orders and the parameters makes the design and analysis of the distributed algorithms quite complicated.…”

“…Lemma 3.2. [36] Under Assumptions 3.1 and 3.2, we have for p ≥ p 0 and q ≥ q 0 , n i=1 θ T t,i (p, q)P −1 t,i (p, q) θ t,i (p, q) = O(log r t (p, q)),…”

“…Then under Assumption 3.1 and Assumptions 4.1-4.3, we havemax m0≤l≤ht V t+1 (l) = O(h t log t) + o(η 2 (t) log log t),where h t and η(t) are defined in Assumption 4.3.Proof. the proof of Lemma 4.4 in[36], we have for l ≥ m 0V t+1 (l) = O(1) + t k=0 W T k+1 d k (l)Φ T k (l)P k (l)Φ k (l)W k+1 . (4.4)By (3.17) and Lemma 3.1, we havemax 1≤l≤ht t k=0 λ max {d k (l)Φ T k (l)P k (l)Φ k (l)} ≤ max 1≤l≤ht [log det(P −1 t+1 (l)) − log det(P j (l) 2 = O(h t log t), (4.5) where Assumption 4.2 is used in the last equation.…”

“…In order to avoid using the independency and stationarity conditions of the regressors, some attempts are made for some distributed estimation algorithms. For the time-invariant unknown parameter, Xie and Guo studied the diffusion LS algorithm and established the convergence result in [36]. For time-varying unknown parameter, they investigated the consensus-based and diffusion LMS algorithm, and proposed the corresponding cooperative information condition to guarantee the stability of the algorithm (e.g., [24] [37] ).…”