# Active RIS-Assisted Uplink NOMA with MADDPG for Remote State Estimation in Wireless Sensor Networks
## 2 The System Model
### Local State Estimation  
The output observed by sensor \(n\) at time \(k\) for the associated linear time-invariant (LTI) process can be expressed as follows  
\[\mathbf{x}_{n}^{k+1} =\mathbf{A}_{n}\mathbf{x}_{n}^{k}+\mathbf{w}_{n}^{k}, \tag{1}\] \[\mathbf{y}_{n}^{k+1} =\mathbf{C}_{n}\mathbf{x}_{n}^{k}+\mathbf{v}_{n}^{k}, \tag{2}\]  
where \(\mathbf{x}_{n}^{k}\in\mathbb{R}^{l_{n}}\) and \(\mathbf{y}_{n}^{k}\in\mathbb{R}^{r_{n}}\) represent the state and measurement vectors, respectively [14], with \(l_{n}\) and \(r_{n}\) denoting the dimensions of the state and measurement vectors for sensor \(n\). The state evolution is governed by the state transition matrix \(\mathbf{A}_{n}\in\mathbb{R}^{l_{n}\times l_{n}}\), while the measurement process is characterized by the measurement matrix \(\mathbf{C}_{n}\in\mathbb{R}^{l_{n}\times l_{n}}\). The disturbances \(\mathbf{w}_{n}^{k}\in\mathbb{R}^{l_{n}}\) and \(\mathbf{v}_{n}^{k}\in\mathbb{R}^{r_{n}}\) are assumed to be zero-mean, independent, and identically distributed (i.i.d.) Gaussian noise processes, with covariance matrices \(\mathbf{Q}_{n}\) and \(\mathbf{R}_{n}\), respectively.  
Each sensor operates a local Kalman filter (KF) and transmits the corresponding minimum mean squared error (MMSE) state estimate to the remote estimator, along with the associated remote state estimation (RSE) error covariance \(\mathbf{P}_{n}^{k|k}\), which are defined as [15]  
\[\hat{\mathbf{x}}_{n}^{k|k} \triangleq\mathbb{E}\Big{[}\mathbf{x}_{n}^{k}\mid\mathbf{y}_{n} ^{1},\mathbf{y}_{n}^{2},\ldots,\mathbf{y}_{n}^{k}\Big{]}, \tag{3}\] \[\mathbf{P}_{n}^{k|k} \triangleq\mathbb{E}\bigg{[}\Big{(}\mathbf{x}_{n}^{k}-\hat{ \mathbf{x}}_{n}^{k|k}\Big{)}\Big{(}\mathbf{x}_{n}^{k}-\hat{\mathbf{x}}_{n}^{k| k}\Big{)}^{\top}\bigg{]}. \tag{4}\]  
The prior state estimate \(\hat{\mathbf{x}}_{n}^{k|k-1}\) and the posterior state estimate \(\hat{\mathbf{x}}_{n}^{k|k}\) at time \(k\) are updated through the following steps. The Kalman gain is denoted by \(\mathbf{K}_{n}^{k}\), and the error covariances are represented by \(\mathbf{P}_{n}^{k|k-1}\) and \(\mathbf{P}_{n}^{k|k}\), corresponding to the prior and posterior error covariances, respectively. The KF update process is given by  
\[\hat{\mathbf{x}}_{n}^{k|k-1} =\mathbf{A}_{n}\hat{\mathbf{x}}_{n}^{k-1|k-1}, \tag{5}\] \[\mathbf{P}_{n}^{k|k-1} =\mathbf{A}_{n}\mathbf{P}_{n}^{k-1|k-1}\mathbf{A}_{n}^{\top}+ \mathbf{Q}_{n},\] (6) \[\mathbf{K}_{n}^{k} =\mathbf{P}_{n}^{k|k-1}\mathbf{C}_{n}^{\top}\Big{(}\mathbf{C}_{n} \mathbf{P}_{n}^{k|k-1}\mathbf{C}_{n}^{\top}+\mathbf{R}_{n}\Big{)}^{-1},\] (7) \[\hat{\mathbf{x}}_{n}^{k|k} =\hat{\mathbf{x}}_{n}^{k|k-1}+\mathbf{K}_{n}^{k}\Big{(}\mathbf{y} _{n}^{k}-\mathbf{C}_{n}\hat{\mathbf{x}}_{n}^{k|k-1}\Big{)},\] (8) \[\mathbf{P}_{n}^{k|k} =\Big{(}\mathbf{I}_{n}-\mathbf{K}_{n}^{k}\mathbf{C}_{n}\Big{)} \mathbf{P}_{n}^{k|k-1}, \tag{9}\]  
where \(\mathbf{I}_{n}\) is the \(l_{n}\times l_{n}\) identity matrix, and \(l_{n}\) denotes the dimension of the state vector \(\mathbf{x}_{n}^{k}\).  
The local estimation error covariance converges exponentially to a steady-state value. Without loss of generality, we assume that each sensor's KF reaches this steady state. To simplify the following analysis, we denote the steady-state covariance as \(\mathbf{P}_{n}^{k}=\mathbf{P}_{n}\), \(k\geqslant 1\).