Linear State Estimation for Multi-rate NCSs With Multi-channel Observation Delays and Unknown Markov Packet Losses

1. Introduction

With the development and progress of the control science, the networked control system (NCS) has become a popular research topic due to its wide applications in various practical areas, such as fault warning [1], target tracking [2], power grids [3, 4], transportation [5], and energy storage [6]. However, since the NCS transmits information through the communication network, the limited bandwidth inevitably leads to unreliability of the transmission channel, such as transmission delays and data losses [7−11]. In recent years, state estimation for the NCSs has attracted much attention with fruitful results reported in the literature such as the $H_{\mathrm{\infty }}$ filter [12, 13], Kalman filter [14, 15], set membership filter [16, 17], and moving horizon estimator [18, 19].

As it is well known, in the practical industrial systems, the production process is always influenced by nonlinear factors. Nevertheless, in the modeling process of real industrial systems, the system model should be designed as simple as possible to maintain the essential system characteristics and facilitate system analysis and synthesis. Therefore, the linear system model is employed to characterize the basic features of practical systems such as companding signal processors [20], mixing tank systems [21] and direct current motors [22], where the influence of nonlinear factors on the system is not significant. In this paper, we focus on the optimal filtering for linear NCSs with delays and packet losses.

As for state estimation of discrete systems with time delays, some classical techniques have been developed such as the state augmentation method [23], polynomial method [24], reorganized innovation analysis method [25] and linear matrix inequality (LMI) algorithm [26, 27]. For state estimation of systems with packet losses, the packet loss is usually characterized by the Bernoulli process [28] or the Markov process [29], and the estimator design can borrow ideas from the estimator design for Markov jump linear systems (MJLSs) and multiplicative noise systems [30]. Actually, the NCSs with random transmission delays can also be transformed into systems with packet losses and transmission delays [31−37].

On the other hand, the multi-rate sampling system, which has shown great importance in improving estimation performance and energy conservation, has been applied to replace the traditional NCSs in a wide variety of fields, such as signal storage [38], transmission [39] and processing [40]. Up to now, there has been numerous results for the single-rate sampling systems, but such results cannot be extended to deal with the multi-rate sampling systems directly, which motivates the investigation of state estimation for the multi-rate sampled systems [41−44]. For the multi-rate uniform sampling system, the lifting technique and state augmentation method have been used in [45] to transform the multi-rate system into an equivalent single-rate system, and then an LMMSE has been designed for the augmented state. In [46], a novel “sampled output augmentation” method has been developed to deal with delayed and infrequent primary measurements by designing a multi-rate Kalman filter. The equivalence has been verified between the proposed method and the traditional fixed-lag smoothing method. For the random multi-rate sampling system [47], a new state model has been developed to describe the dynamics of the measurement sampling points during the state update period. By applying the innovation analysis methods, a non-augmented state estimator has been designed which is dependent on the data loss rate.

Unlike the filter design of single-rate sampling systems, the state estimation of multi-rate sampling systems becomes more complicated accounting for the different sampling periods of sensor components. In summary, so far, the state estimation has not been fully studied for NCSs with multi-channel observation delays and multiple unknown Markov packet losses, especially for the unreliable multi-rate sampling system, which is worthy of further studies.

Motivated by the above analysis, the purpose of this paper is to design a linear state estimator for multi-rate sampling systems with unknown multi-channel Markov packet losses and observation delays. A new set of two-state Markov chains is introduced to characterize the multi-channel packet losses. The original multi-rate sampling system is firstly transformed into a single-rate sampling system with packet losses and observation delays by the lifting technique, and further converted into a Markov jumping parameter system with multi-channel and delay-free observations by using the reorganization observation technique and introducing a new set of multi-mode Markov chains. Since the original packet loss is unknown, the realization of the Markov chain in the converted models is unknown. A new set of states is defined which contains both the original system state and the indicator functions of the new Markov chains. Based on such states, an LMMSE is derived on the basis of the innovation analysis method, where the estimator gain is given in light of a set of generalized coupled difference Riccati equations (GCDREs) with a set of coupled Lyapunov equations. So the original state estimation can be obtained through the characteristics of jump parameters. It can be proven that under appropriate assumptions, the GCDREs converge to a set of generalized coupled algebraic Riccati equations (GCAREs), resulting in a stationary filter. The stationary filter gains can be performed off-line. Finally, the effectiveness of this algorithm is verified through a simulation example. The main novelties of the current research work can be summarized as the following aspects.

1) The state estimation problem for multi-rate systems with multi-channel observation delays and unknown Markov packet losses is considered.

2) The lifting technique and the reorganization observation technique are used to transform the original system into a single-rate and delay-free one.

3) A new set of Markov chains is introduced to characterize the multi-channel packet losses.

4) The proposed algorithm is recursive and suitable for offline calculation.

The remainder of this paper is organized as follows. In Section 2, the problem statement and some preliminaries are given. In Section 3, the LMMSE is developed based on the innovation analysis method. In Section 4, the convergence analysis is given and a stationary filter is obtained. Finally, Section 5 provides a numerical example, and Section 6 draws some conclusions.

Notations: Throughout this paper, $R^n$ denotes the $n$ -dimensional Euclidean space, $R^{m\times n}$ represents the norm bounded linear space of all $m\times n$ matrices. Moreover, diag $\{\cdots \}$ means a block-diagonal matrix. For $L\in R^{n\times n}$, $L^T$ stands for the transpose of $L$. $E(.)$ indicates the mathematical expectation operator, $1_{[.]}$ stands for the Dirac measure, and $\mathrm{\Pi }(.)$ means the occurrence probability of an event.

2. Problem Formulation and Preliminaries

2.1. Problem Formulation

Consider the following multi-rate sampling data system with multi-channel Markov packet dropouts and observation delays:

where $x(k)\in R^n$ is the state sequence, ${\gamma }_l(kh)\in R^m,l=0,1,\cdots ,r$ is the measurement dropout sequence, $w_o(k)\in R^p$ is the process noise, and $v_l(kh)\in R^{q_l},l=0,1,\cdots ,r$, is the measurement noise in the $l$th channel. $A_o,B_o$ and $H_l,l=0,1,\cdots ,r$, are matrices of appropriate dimensions. The random variables ${\gamma }_l(kh)\in \{0,1\}$ ($l=0,1,\cdots ,r$) are employed to describe the observation dropout,, and each ${\gamma }_l(kh)$ is modeled as a time-homogeneous two-state Markov chain with the following transition probability matrix (TPM):

where $q_l=\mathrm{P}\mathrm{r}\{{\gamma }_l(kh+h)=1\mid {\gamma }_l(kh)=0\}$ is the recovery rate and $p_l=\mathrm{P}\mathrm{r}\{{\gamma }_l(kh+h)=0\mid {\gamma }_l(kh)=1\}$ is the failure rate. Denote

Assume that $0<p_l,q_l<1$. Then, there exists the limit probability distribution ${\pi }_l=\left[{\pi }_{l,1}{\pi }_{l,2}\right]$ with

Assumption 1: The initial state $x_0$, $w_o(k)$ and $v_l(kh)$, $l=0,1,\cdots ,r$, are null mean white noises with covariance matrices $P_0$, $Q_o$ and $R_l$$(l=0,1,\cdots ,r)$, respectively. $x_0$, $w_o(k)$ and $v_l(kh)$ ($l=0,1,\cdots ,r$) are mutually independent.

Assumption 2:${\{{\gamma }_l(kh)\}}_{l=0}^r$ are mutually independent.

The estimation problem considered in this paper can be formulated as follows.

Problem: Given system (1)-(2), we aim to derive an LMMSE estimator $\hat{x}(k\mid k)$ such that

is minimized, while the estimator gains are deterministic.

2.2. System Transformation

In this part, we first transform the multi-rate sampling data system into a single-rate system with multi-channel observation delays and packet losses by the lifting technique.

By iterating (1) after $h$ steps, the state update at the measurement sampling instant $k+1$ can be established as follows:

where

and the new system noise is defined as

The statistical properties of the system noise can be computed by

Next, we employ the reorganization observation method to transform the multi-channel observation delay system into a delay-free one. In view of the definition of $y(kh)$, we rearrange the received multi-channel observations to eliminate the hindrance of the observation delay. A new sequence without delays is constructed as

Then, ${\overline{y}}_r(sh)$ and ${\overline{y}}_{k-s}(sh)$ are the delay-free observations which satisfy

where

with ${\overline{v}}_r(sh)$ and ${\overline{v}}_{k-s}(sh)$ being white noises with zero means and following covariances:

Moreover, we introduce a set of new multi-mode Markov chains to convert the delay-free system (9) and (10) into a Markov jumping parameter system. This method is named as the reorganization Markov chain method.

For $0\le s<k-r$,

where the jumping variable is specified by the following equations:

with $N_r=2^{r+1}$.

The TPM for the Markov process $\{{\theta }_r(sh)\}$ is denoted as ${\overline{\mathrm{\Pi }}}_r=[p_{ij}^{(r)}]$ which is given by

Moreover, denote the distribution of ${\theta }_r(sh)$ as

${\theta }_r(sh)$ has the following unique stationary distribution:

which satisfies $\underset{k\to \mathrm{\infty }}{lim}{\mu }_{r,i}(kh)={\mu }_{r,i}$ for $i\in {\mathrm{\aleph }}_r$, and

For $k-r\le s\le k$,

where the jumping variables are specified by the following equations:

with $N_{k-s}=2^{k-s+1}$.

The TPM for the Markov process $\{{\theta }_{k-s}(s)\}$ is denoted as ${\overline{\mathrm{\Pi }}}_{k-s}=[p_{ij}^{(k-s)}]$ which satisfies

In addition, define the distribution of ${\theta }_{k-s}(sh)$ as

${\theta }_{k-s}(sh)$ has the following unique stationary distribution:

which satisfies $\underset{k\to \mathrm{\infty }}{lim}{\mu }_{k-s,i}(k)={\mu }_{k-s,i}$ for $i\in {\mathrm{\aleph }}_{k-s}$, and

3. Design of the LMMSE Filter

Note that the packet loss in each channel is unknown, so the jumping parameters are unknown at the present time. In this paper, we will estimate the state and Markov chains synchronously by introducing the following stochastic variables:

where $1_{[.]}$ represents the indicator function.

In view of (27) and (28), the new state equation can be written as the following compact form.

(i) For $0\le s<k-r$,

Then, we have

where

(ii) For $k-r\le s\le k$,

Then, we have

where

Therefore, the newly constructed state can be estimated directly.

The observations are further transformed into the following forms:

with

Thus, the LMMSE estimation problem for system (1)–(2) is transformed into the one for (1), (29) and (30). Before starting the design, some new definitions and notations are introduced.

Definition 3.1: Given the time $kh$ for $0\le s\le k-r$, the LMMSE predictor ${\hat{z}}_r(sh\mid sh-h)$ is defined as the projection of $z_r(sh)$ onto the following linear space:

For $k-r<s\le k$, the LMMSE predictor ${\hat{z}}_{k-s}(sh\mid sh-h)$ is defined as the projection of $z_{k-s}(sh)$ onto the following linear space:

Definition 3.2: Define the one-step prediction ${\hat{\overline{y}}}_r(sh\mid sh-h)$ as the projection of ${\overline{y}}_r(sh)$ onto the linear space of (31), and ${\hat{\overline{y}}}_{k-s}(sh\mid sh-h)$ is the projection of ${\overline{y}}_{k-s}(sh)$ onto the linear space of (32). Denote the corresponding innovation sequences as

then we can say that $\left\{{{\epsilon }_r(sh)|}_{0\le s<k-r}\right\}$ and $\left\{{{\epsilon }_{k-s}(sh)|}_{k-r\le s\le k}\right\}$ are in fact mutually independent zero mean processes, which span the same linear space as in (31) and (32). Furthermore, we define

Define

The second-order moment matrices of the augmented states and the corresponding innovation sequences are given below.

Lemma 3.1: The second-order moment matrices $Z_r(sh)$ and $Z_{k-s}(sh)$ are diagonal matrices, which can be calculated as follows.

For $0\le s<k-r$, the matrix $Z_r(sh)$ satisfies

where the initial value is

For $k-r\le s\le k$, the matrix $Z_{k-s}(sh)$ satisfies

Proof: See appendix A.

Denote

and define the operators

by

then the coupled Lyapunov equation can be rewritten as

Lemma 3.2: The second moment matrices $W_r(sh)$$(0\le s<k-r)$ and $W_{k-s}(sh)$$(k-r\le s\le k)$ satisfy the following equations:

Proof: See Appendix B.

Now, we are in the position to derive the main results of this paper. The covariance matrices of the estimation errors are given by the following theorem.

Theorem 3.1: The covariance matrices ${\tilde{Z}}_r(sh+h\mid sh)$$(0\le s<k-r)$ and ${\tilde{Z}}_{k-s-1}(sh+h\mid sh)$$(k-r\le s\le k)$ can be formulated as follows.

(i) For $0\le s<k-r$, ${\tilde{Z}}_r(sh+h\mid sh)$ satisfies the following GCDRE:

where the initial value is

and

(ii) For $k-r\le s\le k$, ${\tilde{Z}}_{k-s-1}(sh+h\mid sh)$ satisfies the following GDCRE: