Iterative Joint Detection of Kalman Filter and Channel Decoder for Sensor-to-Controller Link in Wireless Networked Control Systems

Wireless networked control systems (WNCSs) are control systems with the components, i.e., controllers, sensors and actuators, distributed and connected via wireless communication channels, where transmission error tightly relates to the control stability and performance [1, 2]. Kalman filter (KF) plays a fundamental role in estimating system states when transmission error occurs [3]. The major researches of KF model the wireless channels as independent and identically distributed (i.i.d.) Bernoulli processes [4, 5, 6] or Markov processes [7, 8, 9] to reduce the influence of transmission error on control performance.

From the view of communication, channel codes can effectively reduce transmission errors[10] and a simple on-off error control coding scheme can improve the control quality [11]. To ensure a wide range of application scenarios, channel codes are generally designed assuming that the transmitted bits obey uniform distribution and no prior information is considered. To improve the control and communication performance with the prior information of control system, [12] proposes a maximum a posteriori (MAP) receiver for each element of system states with cyclic redundancy check (CRC) codes, and exhibits potential in optimizing the block error rate (BLER) performance of communications and the root mean square error (RMSE) performance of controls with the prior information. However, the prior information directly calculated by the system states cannot be used in channel decoding.

To utilize the prior information for channel decoding, we propose an iterative joint detection algorithm in this letter, which exchanges the prior information of quantized bits and the decoding probabilities of outputs between KF and channel decoder. In the algorithm, we first use the KF to estimate the probability density of the predicted system states. Then, the probability density is transformed into the prior logarithmic likelihood ratios (LLRs) of quantized bits to assist decoder and obtain the decoded probabilities of outputs. Finally, the possible outputs are traversed to update the prior LLRs of quantized bits in order to implement iterative detection. The simulation results show that the prior information and the iterative structure can reduce the BLER performance of communications while improving the RMSE performance of controls, which shows the advantage of the joint design of controls and communications.

Notation Conventions: In this letter, the lowercase letters, e.g., $x$, are used to denote scalars. The bold lowercase letters, e.g., ${\mathbf{x}}$, are used to denote vectors. Notation $x_{i}$ denotes the $i$-th element of ${\mathbf{x}}$. The sets are denoted by calligraphic characters, e.g., $\mathcal{X}$, and the notation $|\mathcal{X}|$ denotes the cardinality of $\mathcal{X}$. The bold capital letters, e.g., $\mathbf{X}$, are used to denote matrices. Throughout this paper, $\mathbf{0}$ means an all-zero vector. ${\mathbb{R}}$ represents the real number field. $\left[\!\left[N\right]\!\right]$ denotes the set $\left\{1,2,\cdots,N\right\}$.

The considered system structure of WNCSs in this letter is shown in Fig. 1, which consists of control layer and communication layer.

The structure of iterative joint detection is also provided in Fig. 1, which consists of KF and channel decoder. The prior LLR vector ${\bm{\theta}}_{\text{pri}}\left[k\right]$ is sent from the KF to the channel decoder and the decoded LLR vector ${\bm{\lambda}}_{\text{dec}}\left[k\right]$ is sent oppositely, where

The channel decoding algorithm, such as the belief propagation (BP) decoding of LDPC code [10], is abstracted as a function $\left({\bm{\lambda}}_{\text{dec}}\left[k\right],f_{\text{crc}}\right)={\sf ChannelDecoding}\left({\bm{\theta}}\left[k\right]\right)$, where

The proposed detection algorithm iteratively update ${{\theta}_{\text{pri},i}}\left[k\right]$ by utilizing the prior information of $\mathbf{\tilde{x}}\left[k-1\right]$ to improve the control and communication performance, which is divided into the three steps as follows.

Step 1: Initializing ${\bm{\theta}}_{\text{pri}}\left[k\right]$. We first use the KF to predict the system states $\mathbf{\hat{x}}\left[k\right]$ and the covariance matrix $\mathbf{\hat{P}}_{x}\left[k\right]$ as

Then, the predicted sensor output is $\mathbf{\hat{y}}\left[k\right]=\mathbf{C}\mathbf{\hat{x}}\left[k\right]$ and the corresponding covariance matrix is $\mathbf{\hat{P}}_{y}\left[k\right]=\mathbf{C}\mathbf{\hat{P}}_{x}\left[k\right]\mathbf{C}^{T}+\mathbf{V}$. Thus, $\mathbf{y}\left[k\right]$ obeys Gaussian distribution with mean $\mathbf{\hat{y}}\left[k\right]$ and covariance matrix $\mathbf{\hat{P}}_{y}\left[k\right]$, i.e.,

Since ${\mathbf{y}}\left[k\right]$ is transformed into ${\mathbf{q}}\left[k\right]$ and ${\mathbf{b}}\left[k\right]$ by quantizer, the probability of quantized bit $b_{i_{2}}^{\beta\left(y_{i_{1}}\left[k\right]\right)}\in\left\{0,1\right\}$, ${i_{1}}\in\left[\!\left[N_{y}\right]\!\right]$, ${i_{2}}=n-1,\cdots,1,0$, is

where

where $l=0,1,\cdots,{{2}^{n-{i_{2}}-1}}-1$ and $\sigma_{{\hat{y}}_{i_{1}}\left[k\right]}^{2}$ is the ${i_{1}}$-th element of the diagonal of $\mathbf{\hat{P}}_{y}\left[k\right]$. (11) can be easily obtained by (9). For (12), since $b_{i_{2}}^{\beta\left(y_{i_{1}}\left[k\right]\right)}$ is decided by quantizing $y_{i_{1}}\left[k\right]$ directly, we have $P\left(\left.b_{i_{2}}^{\beta\left(y_{i_{1}}\left[k\right]\right)}\right|{{y}_{i_{1}}\left[k\right]}\right)\in\left\{0,1\right\}$. An example of $P\left(\left.b_{i_{2}}^{\beta\left(y_{i_{1}}\left[k\right]\right)}=0\right|{{y}_{i_{1}}\left[k\right]}\right)$ with $n=4$ is provided in Fig. 2. In Fig. 2, $P\left(\left.b_{i_{2}}^{\beta\left(y_{i_{1}}\left[k\right]\right)}=0\right|{{y}_{i_{1}}\left[k\right]}\right)$, $i_{2}=3,2,1,0$, changes from $0$ to $1$ or from $1$ to $0$ every ${{2}^{{{i}_{2}}}}$ quantized interval.

Then, we use $b_{j}\left[k\right]$ to represent the $j$-th element in ${\mathbf{b}}\left[k\right]$. There is a mapping between $b_{j}\left[k\right]$ and $b_{i_{2}}^{\beta\left({{y}_{i_{1}}\left[k\right]}\right)}$, i.e.,

Thus, $P\left({{b_{j}}\left[k\right]}\right)$ can be calculated by (10) and the prior LLR of $b_{j}\left[k\right]$ is

Given ${\mathbf{G}}=\mathbf{G}_{\text{out}}\mathbf{G}_{\text{in}}$, the $j$-th row and the $i$-th column element $g_{j,i}$ of $\mathbf{G}$ and ${\mathcal{G}}\left(i\right)=\left\{\left.j\right|{{g}_{j,i}}=1,j\in\left[\!\left[K\right]\!\right]\right\},i\in\left[\!\left[{{N}_{\text{in}}}\right]\!\right]$, we have ${{c}_{i}}\left[k\right]=\sum\nolimits_{j\in{\mathcal{G}}\left(i\right)}{{{b}_{j}\left[k\right]}}$ and the prior LLR of $c_{i}\left[k\right]$ is

Step 2: Decoding ${\bm{\theta}}\left[k\right]$. ${\theta}_{i}\left[k\right]$ is calculated by (7), where $\theta_{\text{rec},i}\left[k\right]$ is decided by (4) and ${\theta}_{\text{pri},i}\left[k\right]$ is initialized by step 1 and updated by step 3. Then, the decoded LLR vector is obtained by $\left({\bm{\lambda}}_{\text{dec}}\left[k\right],f_{\text{crc}}\right)={\sf ChannelDecoding}\left({\bm{\theta}}\left[k\right]\right)$, the estimation ${\mathbf{\hat{b}}}\left[k\right]$ of ${\mathbf{b}}\left[k\right]$ is decided by ${\bm{\lambda}}_{\text{dec}}\left[k\right]$ and $\mathbf{\hat{q}}\left[k\right]$ is recovered from ${\mathbf{\hat{b}}}\left[k\right]$. If $f_{\text{crc}}$ is $0$, the state vector $\mathbf{\tilde{x}}\left[k\right]$ and the input vector ${\mathbf{u}}\left[k\right]$ are calculated by Algorithm 1 and (3), respectively. If $f_{\text{crc}}$ is $1$, we use ${\bm{\lambda}}_{\text{dec}}\left[k\right]$ and KF to update ${\bm{\theta}}_{\text{pri}}\left[k\right]$ in step 3 to implement iterative detection.

When continuous decoding errors occur, the estimated probability density of ${\mathbf{y}}\left[k\right]$ is away from the real value, which further deteriorates the control and communication performance. To avoid the error propagation, if the maximum iteration number $I$ is reached and $f_{\text{crc}}$ is $1$, a conventional channel decoding $\left({\bm{\lambda}}_{\text{dec}}\left[k\right],f_{\text{crc}}\right)={\sf ChannelDecoding}\left({\bm{\theta}}_{\text{rec}}\left[k\right]\right)$ is used, since ${\bm{\theta}}_{\text{rec}}\left[k\right]$ is related to the i.i.d. AWGN and received signals.

Step 3: Updating ${\bm{\theta}}_{\text{pri}}\left[k\right]$. We have the probability $P\left(\left.b_{j}\left[k\right]\right|{\mathbf{r}}\left[k\right]\right)=\left(e^{\left(2b_{j}\left[k\right]-1\right)\lambda_{\text{dec},j}\left[k\right]}+1\right)^{-1}$. Given the set ${\mathcal{Z}}=\left\{\left.{\hat{z}}_{l}\right|l=0,1,\cdots,2^{n}-1\right\}$ of the midpoint ${\hat{z}}_{l}$, the probability of ${\hat{\mathbf{q}}}\left[k\right]\in{\mathcal{Z}}^{N_{y}}$ given ${\mathbf{r}}\left[k\right]$ is

Given $\hat{\mathbf{q}}\left[k\right]$, the estimated system states $\mathbf{\hat{x}}\left[k\right]$ and the covariance matrix $\mathbf{\hat{P}}_{x}\left[k\right]$ are

The estimated sensor output is $\mathbf{\hat{y}}\left[k\right]=\mathbf{C}\mathbf{\hat{x}}\left[k\right]$ and the corresponding covariance matrix is $\mathbf{\hat{P}}_{y}\left[k\right]=\mathbf{C}\mathbf{\hat{P}}_{x}\left[k\right]\mathbf{C}^{T}+\mathbf{V}$. Thus, $\mathbf{y}\left[k\right]$ given $\hat{\mathbf{q}}\left[k\right]$ obeys Gaussian distribution with mean $\mathbf{\hat{y}}\left[k\right]$ and covariance matrix $\mathbf{\hat{P}}_{y}\left[k\right]$, i.e.,

The probability of quantized bit $b_{i_{2}}^{\beta\left(\hat{q}_{i_{1}}\left[k\right]\right)}\in\left\{0,1\right\}$ is

where $P\left(\hat{\mathbf{q}}\left[k\right]\right)$ is set as $P\left(\left.\hat{\mathbf{q}}\left[k\right]\right|{\mathbf{r}}\left[k\right]\right)$ by (16). The calculation processes of $p\left({\left.{{y_{{i_{1}}}}\left[k\right]}\right|{\hat{\mathbf{q}}\left[k\right]}}\right)$ and $P\left({\left.{b_{{i_{2}}}^{\beta\left({{y_{{i_{1}}}\left[k\right]}}\right)}}\right|{y_{{i_{1}}}\left[k\right]}}\right)$ are similar to (11) and (12), respectively. With (19), we calculate ${{\lambda}_{\text{pri},j}}\left[k\right]$ by (14) and update ${\theta}_{\text{pri},i}\left[k\right]$ by (15) to implement iterative detection in step 2. To reduce the complexity, we traverse $n_{\lambda}$ quantized bits with the least $\left|\lambda_{\text{dec},j}\right|$, $j\in\left[\!\left[K\right]\!\right]$, and use the $2^{n_{\lambda}}$ results to decide $P\left(\left.\hat{\mathbf{q}}\left[k\right]\right|\mathbf{r}\left[k\right]\right)$ with normalization and calculate (19).

The whole process of the proposed iterative joint detection algorithm is summarized in Algorithm 2. In Algorithm 2, we first initialize ${{\bm{\theta}}_{\text{pri}}}\left[k\right]$ with (15) and calculate ${\bm{\theta}}\left[k\right]$ by (7). Then, channel decoding is used as $\left({\bm{\lambda}}_{\text{dec}}\left[k\right],f_{\text{crc}}\right)={\sf ChannelDecoding}\left({\bm{\theta}}\left[k\right]\right)$. If $f_{\text{crc}}$ is $0$, we assume the decoded results are correct and estimate $\mathbf{\tilde{x}}\left[k\right]$ and $\mathbf{\tilde{P}}\left[k\right]$ by KF directly. If $f_{\text{crc}}$ is $1$, we use ${\bm{\lambda}}_{\text{dec}}\left[k\right]$ and $\mathbf{\tilde{x}}\left[k-1\right]$ to update ${{\bm{\theta}}_{\text{pri}}}\left[k\right]$ in order to implement iterative detection until the maximum number of iteration $I$ is reached. When $I$ is reached and $f_{\text{crc}}$ is $1$, a conventional channel decoding $\left({\bm{\lambda}}_{\text{dec}}\left[k\right],f_{\text{crc}}\right)={\sf ChannelDecoding}\left({\bm{\theta}}_{\text{rec}}\left[k\right]\right)$ is used to avoid error propagation.

The complexity of the proposed iterative joint detection is shown in TABLE I. The complexity of step 1 is $O\left(N_{x}^{3}+2^{n}K\right)$, where the complexity of KF is $O\left(N_{x}^{3}\right)$ and the complexity of calculating the probability of quantized bits is $O\left(2^{n}K\right)$ when lookup table is used for integrals. Given the decoding complexity $O\left(D\right)$, the complexity of step 2 is $O\left(N_{x}^{3}+D\right)$. For step 3, the complexity is $O\left(2^{n_{\lambda}}\left(N_{x}^{3}+2^{n}K\right)\right)$. Hence, the complexity of the iterative joint detection is $O\left(I\left(D+2^{n_{\lambda}}\left(N_{x}^{3}+2^{n}K\right)\right)\right)$.

Though the iterative joint detection has high complexity, it can be implemented in parallel for real-time systems. The main time steps are shown in TABLE I. Assume $\tau_{\sf kf}$ and $\tau_{\sf dec}$ are the time steps of KF and decoding, respectively. For step 1, since the time step of calculating the probability of one quantized bit is $n$ and $K$ bits can be calculated in parallel, the time step of step 1 is $\tau_{\sf kf}+n$. The time step of step 2 is $\tau_{\sf kf}+\tau_{\sf dec}$. Then, since the time step of (16) is $n_{\lambda}$ and the $2^{n_{\lambda}}$ results can be calculated in parallel, the time step of step 3 is $\tau_{\sf kf}+n+n_{\lambda}$. Thus, the time step of the iterative joint detection is $2\tau_{\sf kf}+n+\tau_{\sf dec}+I\left(\tau_{\sf kf}+\tau_{\sf dec}+n+n_{\lambda}\right)$, which can satisfy the latency constraint of real-time systems with small $I$.

For control layer, a rotary inverted pendulum with $n=16$, $Z=\pi$ is employed as the plant. The system parameter matrices $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$, the controller gain matrix $\mathbf{K}_{\text{con}}$ and the reference state vector ${{\mathbf{x}}_{{\text{ref}}}}\left[k\right]$ are identical to those in [12]. $\mathbf{W}$ and $\mathbf{V}$ are diagonal matrices with the diagonal elements $\sigma_{W}^{2}$ and $\sigma_{V}^{2}$, respectively. The sampling interval is $10$ms, the simulation time is $100$s and the number of simulation runs is $100$. Once the pendulum falls, the simulation run is terminated. The performance of control layer is evaluated by the RMSE of the state vector against the ideal control case. For communication layer, $\left(48,32\right)$ 16-bit CRC code [13] and $\left(96,48\right)$ LDPC code [14] are used. The traversed bit number $n_{\lambda}$ is 4. The iteration number of BP decoding is 50.

Fig. 3 provides the BLER performance of iterative joint detection with $I=1$ and different system disturbances. In Fig. 3, as $\sigma_{W}^{2}$ and $\sigma_{V}^{2}$ decrease, the calculated prior probability is more accurate to improve the BLER performance of the iterative joint detection and [12]. Then, the BLER performance of [14] is same under different system disturbances, since the prior probability is not used in the BP decoding. Specifically, the iterative joint detection with $\sigma_{W}^{2}=10^{-6}$ and $\sigma_{V}^{2}=10^{-6}$ has about $0.78$dB and $2.5$dB performance gain at BLER $10^{-1}$ compared with [14] and [12], respectively. Hence, the prior information obtained from the system model can improve the BLER performance of communication layer.

Fig. 4 provides the RMSE performance of the iterative joint detection with $I=1$ and different system disturbances. In Fig. 4, we observe that the RMSE with different system disturbances is reduced and converged as the SNR increases. Then, the converged SNR of the iterative joint detection is less than those of [12] and [14]. Specifically, the performance gap between the iterative joint detection and [14] is about $0.67$dB with $\sigma_{W}^{2}=10^{-6}$ and $\sigma_{V}^{2}=10^{-6}$ at RMSE $10^{-1}$. Thus, the iterative joint detection can enhance the BLER performance of communications while improving the RMSE performance of controls, which shows the advantage of the joint design of controls and communications with prior information.

Fig. 5 shows the BLER performance of the iterative joint detection with $\sigma_{W}^{2}=10^{-8}$, $\sigma_{V}^{2}=10^{-8}$ and different $I$. In Fig. 5, we observe that as $I$ increases, the BLER performance is improved and converged. The performance gap between $I=1$ and $I=128$ is about $0.61$dB at BLER $10^{-3}$. Thus, updating the prior information can further improve the BLER performance of the iterative joint detection.

In this letter, an iterative joint detection algorithm of KF and channel decoder is proposed by utilizing the prior information of control system to improve the control and communication performance. In the algorithm, the prior information is initialized by KF and updated by traversing the possible outputs of control system in order to implement iterative detection. The simulation results show that the prior information and the iterative structure can improve the control and communication performance.