Correction to: Characterizing Trust and Resilience in Distributed Consensus for Cyberphysical Systems (IEEE Transactions on Robotics (2022) 38:1 (71–91) DOI: 10.1109/TRO.2021.3088054)

In this correspondence, we correct the following points in the above paper. Since its publication we have noticed the following typos and points to correct in the paper [1]. In this correspondence we point them out, clarify and correct them. I. TYPOS Definition of a nonnegative matrix (pp. 73):”We say that a matrix A is nonnegative if its entries are nonnegative, i.e., A_ij > 0 for all i, j, and we write A ≥ 0 to denote that A is nonnegative.” The inequality > should be replaced with ≥, i.e., A_ij ≥ 0 for all i, j. The definition of the time T_f(T₀, t) and its upper bound (pp. 80): The equality sign in the term sup{l + 1: W_L(l + T₀ − 1) = W_L, l ∈ [0, t − T₀]} should be replaced with an inequality, i.e., sup{l + 1: W_L(l + T₀ − 1) ≠ W_L, l ∈ [0, t− T₀]}. Additionally, the relation T_f(T₀, t) = t should be replaced with T_f(T₀, t) < ∞. Theorem 3 (pp. 83): Equation (36) should replace the coefficient (m − T₀ + 1) with (m − T₀ + 2). Corollary 3 (pp. 83): The term 1v^′x_L(0) should be replaced with the term z(T₀)1. Theorem 5 - Bennet’s inequality (pp. 84): The citation of the Bennet’s inequality includes the following bound on the nontriviality of the theorem: 0 ≤ t < nM. The notation t should be replaced with the threshold notation b, i.e., 0 ≤ b < nM. We note that these typos are clear from our descriptive text in [1] and do not affect the results presented in the paper. II. CORRECTIONS Dependence of the consensus value on the initial legitimate values and malicious inputs: Here the deviation from the original text will be underlined and bolded. Corollary 2: Due to the substochasticity of the matrices W_L(t), the consensus point y(T₀) is in the convex hull of the legitimate agents’ initial values x_i(0), i ∈ L and 0. Additionally, z(T₀) is in the convex hull of the initial legitimate agent values x_i(0), i ∈ L and the malicious inputs x_i(t), i ∈ M, t ≥ T₀ − 1 such that malicious agent i is misclassified by a legitimate agent at time t, and its deviation from the nominal consensus value (the case with no malicious agents) depends on the starting time T₀ of the consensus algorithm. The analysis of the deviation is given in Theorem 2. The same correction should be applied in Theorem 1. The bound on the contribution of the malicious agents: The interpretation of ϕ_i(T₀, t) in Eq. (27) (correction is underlined): We denote by (Formula presented) the worst case effect on legitimate agent i due to incorrect malicious agents’ classification (labeling an untrustworthy agent as trustworthy) committed by any legitimate agent at some time t ≥ T₀ − 1. We note that the above Eq. (1) replaces (27) in [1]. Consequently, we revise the proof of Proposition 5. We utilize similar arguments to the ones utilized in the original line proof, with the following minor modifications. We remark that all other uses of the modified definition of ϕ_i(T₀, t), outside of the proof of Proposition 5, hold. Proof of Proposition 5: By the substochasticity of W_L(k) and the nonnegativity of W_M(k) for all k ≥ 0, we have for every i ∈ L and t ≥ T₀ ≥ 1, (Formula presented) where (Formula presented). Since the definition of ϕ_i(T₀, t) is identical for all i ∈ L, we denote ϕ(T₀, t) ≡ ϕ_i(T₀, t). We can recover the original bound elegantly. First, observe that by (2) we have that (Formula presented) Consequently, (Formula presented). We can now apply Markov’s inequality on the right-hand-side to deduce that (Formula presented). It is easy to verify by definition that ϕ(T₀, t) is a monotonically non-decreasing function of t, for every T₀. Additionally, for any realization of trust observations {β_ij(k)}_{i∈L,j∈M,k≥T0−1} we have the pointwise convergence (Formula presented) note that this limit may be ∞. Thus, by the Monotone Convergence Theorem [2] we can replace the order of expectation and limsup_t→∞ as follows: (Formula presented) Now, (Formula presented) Thus, (Formula presented) where (Formula presented) as Proposition 5 states.