## 2 Control and Estimation under Information Constraints
### Minimum Necessary Data Rate for Estimation and Control  
The limitation of the data transmission rate over the communication channel can be expressed in informational terms. Assume that coding alphabet \(\mathbb{S}\) consists of \(\mu\) elements. Then at each step \(k=0,1,\dots\) over the channel an amount of \(\tilde{R}=\log_{2}\mu\) bit can be transmitted per each step. Let the data transmission be carried out at discrete instants \(t_{k}=kT_{0}\), where \(T_{0}\) is the _sampling interval_. Then the data transmission rate in bits per second is as \(R=T_{0}^{-1}\log_{2}\mu\) bit/s. In this regard, one speaks of _\"information constraints\"_ in control and estimation problems.  
The problem of determining the minimum bandwidth of the communication channel, at which it is possible to provide the required estimation accuracy, is posed and partially solved by [PERSON] and [PERSON] [56]. The sufficient condition for the value obtained in [PERSON] and [PERSON] [56] was developed in subsequent works in the form of the _Data Rate Theorem_ which is a fundamental result establishing the smallest value for which the stabilization (estimation) problem for linear systems is solvable in principle. [PERSON] and [PERSON] [31] studied the exponential stabilizability of LTI plants in the sense of achieving an exponential moment stability. For a deterministic initial state case the result of [31] can be roughly presented in the following form [57].  
Let the LTI discrete-time plant be described by the difference equation:  
\[x[k+1]=Ax[k]+Bu[k],\quad y[k]=Cx[k],\;\;k=0,1,\dots \tag{1}\]  
where \(x[k]\in\mathbb{R}^{n}\), \(y[k]\in\mathbb{R}^{l}\), and \(u[k]\in\mathbb{R}^{m}\) are the state, output, and control vectors, respectively; \(A\), \(B\), and \(C\) are the matrices of the corresponding dimensions; and \(k\in\mathbb{Z}_{+}\) denotes the step number (the discrete time). It is assumed that pair \((A,B)\) is reachable and \((C,A)\) is observable. Let the sensor be connected to the controller over a digital communication channel, and no more than \(\tilde{R}\) bits of data can be transmitted at each step \(\[\tilde{R}>\sum_{\eta_{j}\geq\rho}\log_{2}\Big{|}\frac{\eta_{j}}{\rho}\Big{|}\quad \text{(bit per step)}, \tag{2}\]  
where \(\eta_{j}\) are the eigenvalues of matrix \(A\), \(j=1,\ldots,n\). The right-hand side of (2), denoted as \(\tilde{R}_{NE}=\sum\limits_{\eta_{j}\geq\rho}\log_{2}\left|\frac{\eta_{j}}{ \rho}\right|\) (the _Nair-Evans-_, or _NE_-number), gives a tight admissible bound when \(\rho\)-exponential stabilization can be achieved. For real-time systems with the constant sampling interval \(T_{0}\), NE-number \(R_{NE}\) in bits per second has a form:  
\[R_{NE}=\frac{1}{T_{0}}\tilde{R}_{NE}=\frac{1}{T_{0}}\sum_{\eta_{j}\geq\rho} \log_{2}\left|\frac{\eta_{j}}{\rho}\right|\quad\text{(bit per second)}. \tag{3}\]  
Extensions of this result for stabilizing nonlinear systems in the vicinity of the origin and observing nonlinear systems through finite capacity communication channels, including large networks, were obtained in the series of the subsequent papers, see [58; 59; 60; 61; 62] to mention a few.  
It is worth mentioning that on practice, the data transmission rate can not be taken as small as the NE-number (3) gives due to several reasons for the data bitrate being usually much greater than \(R_{NE}\), but the NE-number can serve as a measure showing the maximum available possibilities of estimation and control over the existing communication channel. Furthermore, a promising approach is to employ the event-triggered control instead of the control with constant sampling time, see e.g., [63; 64; 65].