Systems & Control Letters 61 (2012) 55–61 Contents lists available at SciVerse ScienceDirect Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle Signal-to-noise ratio fundamental limitations in the discrete-time domain Alejandro J. Rojas ∗ Departamento de Ingeniería Eléctrica, Universidad de Concepción, Chile article info Article history: Received 2 February 2010 Received in revised form 25 February 2011 Accepted 12 September 2011 Available online 20 November 2011 Keywords: Fundamental limitations Signal-to-noise ratio Control over networks Stabilisability Plant modelling error abstract Fundamental limitations in feedback control are a well established area of research. In recent years it has been extended to the study of limitations imposed by the consideration of a communication channel in the control loop. Previous results characterised these limitations in terms of a minimal data transmission rate necessary for stabilisation. In this paper a signal-to-noise ratio (SNR) approach is used to obtain a tight condition for the linear time invariant output feedback stabilisation of a discrete-time, single-input- single-output (SISO) unstable, non-minimum phase (NMP) plant with arbitrary relative degree over an additive Gaussian coloured noise (ACGN) communication channel with memory. The obtained result gives a guideline in estimating the severity of the fundamental SNR limitation imposed by the plant unstable poles, NMP zeros and relative degree as well as the channel NMP zeros, bandwidth, and noise colouring. We then characterise the output feedback sensitivity function for the infimal SNR solution and follow up by quantifying the extra SNR imposed by suboptimal solutions (for example due to plant modelling errors). © 2011 Elsevier B.V. All rights reserved. 1. Introduction Fundamental limitations in control design have been an important area of research with early seminal results from [1,2]. For a linear time invariant (LTI) plant it is well understood that unstable poles, non-minimum phase (NMP) zeros and time-delay cause unavoidable limitations both in regulation and performance (see for example [3] and references therein). In recent years, the study of fundamental limitations has been extended to control over networks, [4, Theorem 4.6], [5], attracting a growing interest (see for example [6] and the recent survey by Nair et al. [7]). Most results in control over network use information theory arguments to obtain necessary and sufficient lower bounds on, for example, the transmission data rate required for stabilisability for noiseless channels [8–10] or noisy channels [11–14]. Another line of research, that does not make use of information theory arguments, introduces a framework to study stabilisability of a feedback loop over channels that have a signal to noise ratio (SNR) constraint [15] (as well as related work in [16,17] and similar work in [18]). A recognisable characteristic of the proposed SNR approach is that it is a linear formulation allowing the use of all the linear design techniques. There are many reasons to consider the full details of a discrete- time scenario including: ∗ Tel.: +56 041 2661227. E-mail address: arojasn@udec.cl. (1) A plant model can be continuous-time, but implementation will require sampling, for which discrete-time results can prove to be relevant. (2) Most results from communication theory are developed for discrete-time communication channel models. Furthermore, the presence of a memory constraint may be im- posed for several reasons, for example to allocate bandwidth and avoid interference between different channels in a communication system or to model communication hardware properties. Finally, coloured noise is a more flexible and realistic feature for a commu- nication channel. We are thus motivated to use the SNR approach to study the novel fundamental limitations imposed by the presence of an additive coloured Gaussian noise (ACGN) channel with memory on the stability of a linear single-input single-output (SISO) feedback control loop. We are also motivated to further study the SNR approach due to the fact that in some cases it is known that the infimal bounds obtained from the SNR approach (when paired with the appropriate channel capacity definitions) match results for channel capacity and transmission data rate obtained with information theoretic arguments (see for example [13,15] for the discussion on the additive white Gaussian noise (AWGN) channel case and more recently [19] for a class of ACGN channels). The main contribution of the present paper is to present the closed-form solution to the discrete-time LTI SNR constrained problem for the case of ACGN channels with memory; see Fig. 1. We then obtain the closed-form expression for the feedback sensitivity function, whenever the controller solution achieving the infimal channel SNR is in place. Finally, we use the 0167-6911/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2011.09.010