Optimal linear control for channels with signal-to-noise ratio constraints

Abstract

Abstract-We consider the problem of stabilizing and minimizing the disturbance response of a SISO LTI plant, subject to a stochastic disturbance, over an analog communication channel with additive white noise and a signal-to-noise ratio (SNR) constraint. The controller is linear, based on output feedback and has a structure with two degrees of freedom: One part represents sensing and encoding operations and the other part represents decoding and issuing the control signal. It is shown that the problem of simultaneously designing the two optimal controller parts can be solved in two stages: First a functional depending both on the 1-and 2-norms of the Youla parameter is minimized. This minimization can be arbitrarily well approximated by a quasiconvex program. The second stage consists of a spectral factorization.

A Characterization of the Minimal Average Data Rate That Guarantees a Given Closed-Loop Performance Level

“…The bounds on R(D) presented in Corollary 3.1 and Theorem 4.1 are functions of the minimal SNR γ ′ (D) in (24). In this section, we show that the problem of finding γ ′ (D) is equivalent to an SNR constrained optimal control problem previously addressed in [31], [46], [48].…”

“…Whilst not tight in general, the gap between the derived upper and lower bounds is smaller than (approximately) 1.254 bits per sample. Our results are constructive and given in terms of the solution to a signalto-noise ratio (SNR) constrained optimal control problem (see also [44]- [46]). We also propose a specific randomized coding scheme that achieves the prescribed level of performance D, while incurring an average data rate that is strictly smaller than the derived upper bound on R(D).…”

Performance limitations for SISO LTI plants controlled over SNR constrained channels

“…Since (4) holds, the above discussion shows that λ opt satisfying (29) always exists and is positive and unique, as claimed. 2) Equation (30) follows, again, from Theorem 7 in [2] (compare (11) and (13a)-(13b), with (8) and (10)). To show that (31) holds, and that λ opt,f satisfying (31) always exist and is unique, one can start by noting that the optimization problem in (12) is also a special instance of the problem solved in Theorem 7 in [2].…”

“…Our goal here is however different: We aim at obtaining closed form characterizations for the optimal performance, i.e., for σ 2 y Γ and σ 2 y f Γ in (8) and (11). To that end, we will introduce an additional assumption on the plant model.…”

Rate-distortion in closed-loop LTI systems

“…To find γ(D), one can resort to the results in [4]. A case where an explicit solution is available is when D → ∞, i.e., when only stabilization is sought.…”