Stabilizability of Stochastic Linear Systems with Finite Feedback Data Rates

Abstract

Abstract. Feedback control with limited data rates is an emerging area which incorporates ideas from both control and information theory. A fundamental question it poses is how low the closed loop data rate can be made before a given dynamical system is impossible to stabilize by any coding and control law. Analagously to source coding, this defines the smallest error-free data rate sufficient to achieve "reliable" control, and explicit expressions for it have been derived for linear timeinvariant systems without disturbances. In this paper, the more general case of finite-dimensional linear systems with process and observation noise is considered, the object being mean square state stability. By inductive arguments employing the entropy power inequality of information theory, and a new quantizer error bound, an explicit expression for the infimum stabilizing data rate is derived, under very mild conditions on the initial state and noise probability distributions.Key words. stochastic control, communication theory, source coding, entropy AMS subject classifications. 93E15, 94A05, 94A17, 94A291. Introduction. Communications and control have traditionally been areas with little common ground. For the most part communications theory is concerned with the reliable transmission of information from one point to another, and is relatively indifferent to its specific purpose or whether it is eventually fed back to the source. Control theory, in contrast, is concerned mainly with using information in a feedback loop to achieve some performance objective, and usually assumes that limitations in the communications links are not significant enough to affect performance drastically.The reasons usually given for this mutual indifference are firstly, that a communications system is generally used for a broad range of purposes and can rarely be designed to match a particular objective and, secondly, that to explicitly model communication limitations would complicate controller synthesis. However, in recent years emerging applications such as micro-electromechanical systems, mobile telephone power control, and networked industrial control systems have begun to cross the boundary between these disciplines. In these applications, the aim is to control a dynamical system consisting of many separate components connected by a digital communication network. Although the total available capacity in bits per second may be large, each component is effectively allocated only a small portion. This can introduce significant quantization errors and delays, due to the low resolution and finite transmission time of each discrete-valued, digital symbol. Quantization resolution can be improved at the expense of delay and vice-versa, but nonetheless there remains an upper bound on the amount of information, in some sense, that may be exchanged per unit time. Clearly, by designing coders and decoders that are matched to the dynamical system and controllers, a more economical use of communication

Conditions for tracking in networked control systems

“…The results were obtained in terms of the mutual information rate between the feedback and the reference signals, as well as the channel capacity and the unstable eigenvalues of the LTI system. The lower bound for the channel capacity obtained was expected since it corresponds to the one obtained in previous literature as [12] and [7]. We also obtained a lower bound in terms of the entropy of the reference signal for the maximum achievable accuracy in a tracking system, in the absence of constraint channel.…”

Fundamental limitations in control over a communication channel

“…For example, it is shown in Nair and Evans (2004) that for a noiseless (error-free) channel the minimal data transmission rate necessary and sufficient for stabilisation must satisfy a lower bound expressed as a function of the open loop unstable poles of the plant. This result is obtained using fairly complex information theoretic arguments, but is valid for a large class of feedback controllers assumed only to be casual.…”

Capacity of Communication Channels

“…Due to the enormous growth in communication technology, there has been a significant interest in the problem of control and state estimation via limited capacity communication channels in recent years (see, e.g., Delchamps [1990], Brockett and Liberzon [2000], Elia and Mitter [2001], Petersen and Savkin [2001], Ishii and Francis [2002], Liberzon [2003], Savkin and Petersen [2003], De Persis and Isidori [2004], Nair and Evans [2004], Matveev and Savkin [2005] Savkin [2007, 2008]). Minimum capacity of the communication channels required for state estimation and control has been investigated in, e.g., Nair and Evans [2004], Savkin [2006].…”

“…Minimum capacity of the communication channels required for state estimation and control has been investigated in, e.g., Nair and Evans [2004], Savkin [2006].…”