Optimal Local and Remote Controllers With Unreliable Uplink Channels: An Elementary Proof

Abstract

Recently, a model of a decentralized control system with local and remote controllers connected over unreliable channels was presented in [1]. The model has a non-classical information structure that is not partially nested. Nonetheless, it is shown in [1] that the optimal control strategies are linear functions of the state estimate (which is a non-linear function of the observations). Their proof is based on a fairly sophisticated dynamic programming argument. In this note, we present an alternative and elementary proof of the result which uses common information based conditional independence and completion of squares.

Optimal Local and Remote Controls of Multiple Systems with Multiplicative Noises and Unreliable Uplink Channels

“…Note that we can always find some ε < 0 such that J ε N −J N < 0, which contradicts with (13). Thus, τ k = 0.…”

“…It is stressed that Problem 1 has not been solved in the existing literatures. The previous works [13]- [17] mainly focused on additive noise systems, and their multiplicative noise counterpart remains less investigated. The existence of unreliable uplink channels for multiplicative noise systems may result in the failure of "separation principle", making the design of optimal control in Problem 1 difficult.…”

“…By using the common information approach, the optimal local and remote controls are derived. Following [16], an elementary proof of the common information approach was given in [13]. Furthermore, [17] studies the optimal local and remote control problem with only one subsystem by using maximum principle approach.…”

“…Furthermore, [17] studies the optimal local and remote control problem with only one subsystem by using maximum principle approach. It should be pointed out that previous works [13]- [18] focused on NCSs with additive noise. Different from the previous works, this paper investigates the local control and remote control problem for networked systems with multiplicative noise and multiple subsystems.…”

“…As far as we know, the obtained results are new and innovative in the following aspects: 1) Compared with the common information approach adopted in previous works, the structure of the optimal local controllers and remote controller is not assumed in advance, see Lemma 3 in [13]; 2) In this paper, the multiple subsystems case is solved by using the Pontragin Maximum principle, while previous works focused on the single subsystem case [17]; 3) The existence of multiplicative noise results in that the state estimate error and the error covariance are involved with the controls, which may cause the separation principle to fail. To overcome this, new asymmetric CREs are introduced; It is verified that the separation principle holds for the considered optimization problem, i.e., the optimal local controllers and optimal remote controller can be designed, and the control gain matrices and the estimation gain matrix can be calculated separately; 4) It is noted that the obtained results include the results in [13], [16], [17] as special cases.…”