Distributed Optimal Control over Bit-rate Constrained Networks with Communication delay

Optimal power‐constrained control of distributed systems with information constraints

“…Remark 1. In practice, the validity of the assumption that the controllers communicate with a maximum delay of one sampling interval [18] depends on the communication technology and sampling rates. In case of wired communication between subsystems, communication delay is typically low and can be assumed within the range of one sampling interval for systems with low and high sampling rates.…”

“…Proposition 2. Suppose that Γ 2 (0 : T − 1) in Equation ( 11) are given such that the information constraints in (18) are satisfied. Problem ( 7) is reduced to…”

“…Proof. Given the gains Γ 2 (0 : T − 1) that satisfy (18), in the cost function (28) only the first two terms are dependent on V 1 (0 : T − 1). Such cost is to be minimized with respect to V 1 (0 : T − 1) with constraints on the evolution of the dynamics of the estimator given by Equation ( 15), and subject to power constraints in (27).…”

“…Additionally, structural constraints on Γ 2 as defined in (18) have to be preserved. As we want to minimize J 2 with respect to Γ 2 (0 : T − 1) we first express elements of V 2 as a function of it.…”

“…where the last term with multiplier G(T − 1) is introduced to account for sparsity on Γ 2 (T − 1) according to (18). In what follows a quadratic term is introduced to penalize infeasible points, yielding the augmented Lagrangian…”