Stabilization with disturbance attenuation over a Gaussian channel

Abstract

Abstract-We propose a linear control and communication scheme for the purposes of stabilization and disturbance attenuation when a discrete Gaussian channel is present in the feedback loop. Specifically, the channel input is amplified by a constant gain before transmission and the channel output is processed through a linear time invariant filter to produce the control signal. We show how the gain and filter may be chosen to minimize the variance of the plant output. For an order one plant, our scheme achieves the theoretical minimum taken over a much broader class of compensators.

Minimum variance control over a Gaussian communication channel

“…To illustrate the preceding remarks, we now review the results of [6], in which it is assumed that the channel input is a scalar multiple of the plant output, s k = λy k , and the control input is the response of a linear time invariant filter to the channel output, denoted in the transform domain by U (z) = −K(z)R(z), where K(z) is the transfer function of the filter. The scalar λ and the filter K(z) solve the infinite horizon minimum variance problem…”

“…We now present a lower bound on the disturbance response that is valid for the potentially nonlinear communication and control strategies (5)- (6). A proof is found in [5].…”

“…To choose a value of λ, we note from [6] that (a) E{s 2 k|k−1 } is monotonically increasing with λ, becoming unbounded as λ → ∞, and (b) E{ỹ 2 k|k−1 } is monotonically decreasing with λ. Hence E{y 2 k } can be set equal to P/λ * 2 , where λ * is the value of the λ for which the channel input satisfies the power constraint (4) with equality.…”

“…Lemma II.1 Consider the plant (1)-(2), channel (3), and the communication and control strategies (5)- (6). Then…”

“…In Section II we note that the variance of the plant output is bounded below by that of the prediction estimation error, and present a control law that achieves this bound with equality. We also review the results of [6] discussed above. In Section III we present a lower bound on the variance of the output estimation error that holds for potentially nonlinear communication and control strategies.…”