text: We consider a joint sensor and controller design problem for linear Gaussian stochastic systems in which a weighted sum of quadratic control cost and the amount of information acquired by the sensor is minimized. This problem formulation is motivated by situations where a control law must be designed in the presence of sensing, communication, and privacy constraints. We show that the optimal joint sensor-controller design is relatively easy when the sensing policy is restricted to be linear. Namely, an explicit form of the optimal linear sensor equation, the Kalman filter, and the certainty equivalence controller that jointly solves the problem can be efficiently found by semidefinite programming (SDP). Whether the linearity assumption in our design is restrictive or not is currently an open problem.
  @cite_14: 
  @cite_22: 
  @cite_7: A current survey of the emerging field of networked control systems is provided. The aim is to introduce the fundamental issues involved in designing successful networked control systems, to provide a snapshot assessment of the current state of research in the field, to suggest useful future research directions, and to provide a broad perspective on recent fundamental results. Reflecting the goals of the Special Issue itself, this paper surveys relevant work from the areas of systems and control, signal processing, detection and estimation, data fusion, and distributed systems. We discuss appropriate network architectures, topics such as coding for robustly stable control in the presence of time-varying channel capacity, channels with fixed versus adaptively variable data width, issues in data rate problems in nonlinear feedback problems, and problems in routing for stability and performance. In surveying current research on networked control systems, we find that recent theoretical advances and target applications are intimately intertwined. The common goal of papers in the Special Issue which follows is to describe key aspects of this relationship. We also aim to provide a bridge between networked control systems and closely related contemporary work dealing with sensor networks and wireless communication protocols
  @cite_8: We show that the coarsest, or least dense, quantizer that quadratically stabilizes a single input linear discrete time invariant system is logarithmic, and can be computed by solving a special linear quadratic regulator problem. We provide a closed form for the optimal logarithmic base exclusively in terms of the unstable eigenvalues of the system. We show how to design quantized state-feedback controllers, and quantized state estimators. This leads to the design of hybrid output feedback controllers. The theory is then extended to sampling and quantization of continuous time linear systems sampled at constant time intervals. We generalize the definition of density of quantization to the density of sampling and quantization in a natural way, and search for the coarsest sampling and quantization scheme that ensures stability. Finally, by relaxing the definition of quadratic stability, we show how to construct logarithmic quantizers with only finite number of quantization levels and still achieve practical stability of the closed-loop system.
  @cite_21: In this paper, we deal with control systems where plants and controllers are connected with transmission channels of signals, and analyze their stabilizability under the constraints of the channel capacity. We show that this problem is resolved with a minimization problem of a kind of entropy of the related systems under some asymptotic conditions, and the stabilizability can be represented by the unstable poles and real zeros of the plants. This result corresponds to a synthesis version of Bode's theorem and an extended version of the strong stabilizability of plants. We also show a design method of the optimal controller.
  @cite_24: The average cost control problem for linear stochastic systems with Gaussian noise and quadratic cost is considered in the presence of communication constraints. The latter take the form of finite alphabet codewords, being transmitted to the controller with ensuing delay and distortion. It is shown that if instead of the state observations an associated “innovations process” is encoded and transmitted, then the separation principle holds, leading to an optimal control linear in state estimate. An associated “off-line” optimization problem for code length selection is formulated. Some possible extensions are also pointed out.
  @cite_19: The emerging area of control with limited data rates incorporates ideas from both control and information theory. The data rate constraint introduces quantization into the feedback loop and gives the interconnected system a two-fold nature, continuous and symbolic. In this paper, we review the results available in the literature on data-rate-limited control. For linear systems, we show how fundamental tradeoffs between the data rate and control goals, such as stability, mean entry times, and asymptotic state norms, emerge naturally. While many classical tools from both control and information theory can still be used in this context, it turns out that the deepest results necessitate a novel, integrated view of both disciplines
  @cite_23: 
  @cite_2: In this paper, we review how Shannon's classical notion of capacity is not enough to characterize a noisy communication channel if the channel is intended to be used as part of a feedback loop to stabilize an unstable scalar linear system. While classical capacity is not enough, another sense of capacity (parametrized by reliability) called "anytime capacity" is necessary for the stabilization of an unstable process. The required rate is given by the log of the unstable system gain and the required reliability comes from the sense of stability desired. A consequence of this necessity result is a sequential generalization of the Schalkwijk-Kailath scheme for communication over the additive white Gaussian noise (AWGN) channel with feedback. In cases of sufficiently rich information patterns between the encoder and decoder, adequate anytime capacity is also shown to be sufficient for there to exist a stabilizing controller. These sufficiency results are then generalized to cases with noisy observations, delayed control actions, and without any explicit feedback between the observer and the controller. Both necessary and sufficient conditions are extended to continuous time systems as well. We close with comments discussing a hierarchy of difficulty for communication problems and how these results establish where stabilization problems sit in that hierarchy
  @cite_5: This paper addresses feedback stabilization problems for linear time-invariant control systems with saturating quantized measurements. We propose a new control design methodology, which relies on the possibility of changing the sensitivity of the quantizer while the system evolves. The equation that describes the evolution of the sensitivity with time (discrete rather than continuous in most cases) is interconnected with the given system (either continuous or discrete), resulting in a hybrid system. When applied to systems that are stabilizable by linear time-invariant feedback, this approach yields global asymptotic stability.
  @cite_16: The problem of stabilizing an unstable, time-invariant, discrete-time, linear system by means of state feedback when the measurements of the state are quantized is addressed. It is found that there is no control strategy that stabilizes the system in the traditional sense of making all closed-loop trajectories asymptotic to zero. If the system is not excessively unstable, feedback strategies that bring closed-loop trajectories arbitrarily close to zero for a long time can be implemented. It is also found that when the ordinary linear feedback of quantized state measurements is applied, the resulting closed-loop system behaves chaotically. The asymptotic pseudorandom closed-loop system dynamics differ substantially from what would be predicted by a conventional signal-with-noise analysis of the quantization's effects. Probabilistic reformulations of the stability problem in terms of the invariant measure are considered. >
  @cite_25: Networked control systems (NCSs) are spatially distributed systems for which the communication between sensors, actuators, and controllers is supported by a shared communication network. We review several recent results on estimation, analysis, and controller synthesis for NCSs. The results surveyed address channel limitations in terms of packet-rates, sampling, network delay, and packet dropouts. The results are presented in a tutorial fashion, comparing alternative methodologies
  @cite_12: 
  @cite_11: We consider the problem of minimizing the response of a plant output to a stochastic disturbance using a control law that relies on the output of a noisy communication channel. We discuss a lower bound on the performance achievable at a specified terminal time using nonlinear time-varying communication and control strategies and show that this bound may be achieved using strategies that are linear. We also consider strategies that are defined over an infinite horizon that may achieve better transient response that those that are optimal for the terminal time problem.
summary: Feedback control over noisy channels has been a popular research topic in the past two decades. Most of the early contributions focus on stabilization of unstable dynamical systems using feeback control over band-limited communication channels. A very partial list of papers in this context is @cite_16 @cite_5 @cite_8 @cite_12 @cite_21 @cite_14 @cite_2 . This research direction naturally leads to trade-off studies between the achievable control performance and the required capacity of the sensor-controller communication channel. If the communication rate is finite, larger block length (achieving high resolution) is not necessarily preferred since the resulting delay leads to the loss of control performance @cite_24 . LQG control performance subject to capacity constraints is considered in @cite_23 , where a certain separation principle" between control design and communication design is reported. The authors of @cite_11 consider a fundamental performance limitation of the finite horizon minimum-variance control (MVC) over noisy communication channels in the LQG regime. More comprehensive literature surveys on control designs over communication channels are available in @cite_19 @cite_25 @cite_7 @cite_22 .