Receding Horizon Estimation for Linear Discrete-time Systems with Multi-channel Observation Delays

Ⅰ. INTRODUCTION

At present, there are many methods to solve the state estimation problems of the discrete systems. Receding horizon estimation (RHE) or moving horizon estimation (MHE) is one of the most famous methods in the field of control and signal processing. RHE is an optimization-based state estimation technique which has received an increasing amount of attention in recent years. At each time, the current state estimate is determined by solving an optimization problem taking a number of past measurements into account. The advantage of the RHE is that it has good capacity to achieve rolling optimization and online application, and it has good robustness and stability. In practice, RHE can be applied widely to system state tracking, process monitoring, and equipment leak detection.

In recent years, a number of results have been reported about RHE. In [1], the problem of estimating the state of a dynamic system has been solved by introducing a receding horizon objective with a weighted penalty term, and a complete stability analysis has been explored. In [2], a receding horizon state observer, which used a deterministic least squares framework, was obtained. The state estimation horizon N was introduced as a tuning parameter for the proposed state observer, and a stability result concerning the choice of N was established. In addition, a robust receding-horizon estimation was addressed for a class of uncertain discrete-time linear systems in the presence of bounded uncertainty in [3]. For nonlinear systems, the receding horizon state estimation as well as the convergence problem were discussed in [4]-[6]. It was shown that one of the most advanced algorithms, which was especially applicable to nonlinear state estimation problem, was the moving horizon estimation [7]. In [8], the receding horizon estimator for nonlinear discrete-time systems affected by disturbances was designed according to a sliding-window strategy. In [9], a new receding-horizon nonlinear Kalman filter formulation for state estimation was proposed. It has been shown that the approach was able to handle constraints and provide immunity against poor initialization.

In [10], a new moving horizon estimator for nonlinear detectable systems was proposed, where the corresponding cost function contained an additional max term. The robust global asymptotic stability in case of bounded disturbances and convergence of the estimation error in case of vanishing disturbances were established. In [11], [12], the consensus-based distributed receding horizon estimation problems of sensor networked systems were discussed in detail. And, in [13], the basic design methods for ensuring stability of MHE were summarized and the relationships of full information and MHE to other state estimation methods such as Kalman filtering and statistical sampling were discussed. It can be found that seldom results have been reported for the receding horizon estimation subject to transmission delays.

There usually exist time delays in practical applications. For example, communication through band-limited linear filter channels may result in time delay phenomenon, and the low Earth orbit satellite communication systems usually have multiple channel delays. Because the presence of a time-delay often causes serious deterioration of the stability and performance of the system, considerable research has been devoted to the estimation of transmission delay systems [14]-[18] under different criteria and modeling methods. There exist several techniques for handling the delay terms, such as the classical state augmentation method [14], the polynomial approach [16], the linear matrix inequality algorithm [15], and the reorganization innovation analysis method [17], [18]. For linear systems with transmission time-delay, there also exist many papers proposing novel research in the literature. To handle time-varying communication delays, an open-loop state predictor was designed for each subsystem to provide predictions of unavailable subsystem states in [19]. In [20], a receding horizon filter for discrete-time linear system with uncertainties and time delays was proposed, where the filter gain was derived by solving a partial Riccati equation. In this paper, we will employ the tool of Riccati equation to derive the receding horizon estimation, which can reduce the calculation complexity for the design process.

This paper considers the receding horizon state estimation for the linear discrete-time system with multi-channel observation delays. The reorganized observation technique is employed for the design of the receding horizon estimation with time delays, and the filter gains are obtained by solving a set of Riccati equations. Two forms of the receding horizon estimation are developed, and the stability analysis is given. The main novelty of this paper is that we first apply the reorganized observation technique to solve the optimal receding horizon estimation subject to measurement delays. The proposed estimator has the same dimension as that of the original system, which reduces the computational burden greatly.

The remainder of this paper is organized as follows. The estimation problem statement for discrete-time linear systems is stated in Section Ⅱ. Section Ⅲ mainly concerns with the design of the receding horizon estimation and its stability analysis. In Section Ⅳ, attention is paid to a simulation example to evaluate the performance of the receding horizon estimation. Finally, conclusions are drawn in Section Ⅴ.

Ⅱ. PROBLEM DESCRIPTION

Consider the following discrete-time linear system with multi-channel observation delays:

where $x(k)\in \mathbb{R}^{n}$ is the state, $w(k)\in \mathbb{R}^{r}$ is the input noise, $y_{l}(k)\in \mathbb{R}^{p_l}$ is the measurement and $v_{l}(k)\in \mathbb{R}^{p_l}$ is the measurement noise. Throughout the paper, it is assumed that the constant matrices $A$, $C$, $H_{l}$ are known, $[C, A]$ is observable, $A$ is nonsingular.

Assumption 1: $w(k)$ and $v_l(k)$ are white noises with covariance matrices E$\{w(k)w(s)^{\text T}\} = Q\delta_{ks}$, $\mathit{\boldsymbol{E}}\{v_l(k)v_l(s)^{\text T}\} = R_l\delta_{ks},$ respectively. $x_0$, $w(k)$, and $v_l(k)$ are mutually independent.

Denote

then the optimal filtering problem considered in this paper can be stated as follows:

Optimal receding horizon estimation: Given the observation $\{{\mathbf{y}}(s)|_{0 \leq s \leq k}\}$, find a linear minimum mean square error receding horizon estimator $\hat{{\mathbf{x}}}(k)$ of the state ${\mathbf{x}}(k)$ with the finite horizon ${N}$, such that ${\rm{E}}_{w, v}[\hat{{\mathbf{x}}}(k)] = {\rm{E}}_{w, v}[{\mathbf{x}}(k)]$.

Ⅲ. MULTI-CHANNEL RECEDING HORIZON ESTIMATION

In this section, we will firstly introduce the observation reorganization technique to transform the measurement delay system into a delay-free one. Then, the receding horizon estimation with a batch form and an iterative form are structured, as well as the relevant proofs are given.

A Observation Reorganization

Since the multiple observation channels have different time delays, it is difficult to consider state estimation directly. All the observation channels need to be seen as a whole. Therefore, we can construct a new observation equation by the observation reorganization technique, which is without delay.

For the given time $k$, the received observations can be rearranged into a set of delay-free sequences as follows:

For $0\leq s \leq k-L$,

For $s = k-n$ ($n = 0, 1, \ldots, L-1$)

B Receding Horizon Estimation

The problem considered here is how to obtain a receding horizon estimate ${\hat{\mathit{\boldsymbol{x}}} }(k|k-1)$ of the state vector ${\mathit{\boldsymbol{x}}}(k)$ by using a finite number of measurements of the system output $\bar{\mathit{\boldsymbol{y}}}(k)$ with weighted matrix. A receding horizon estimation is derived from the following two theorems.

Theorem 1: For systems (1), (3) and (4), when $(C, A)$ is observable, the receding horizon filter ${\hat{\mathit{\boldsymbol{x}}} }(k|k-1)$ with a batch form on the horizon $[k-N, k]$ is derived by the following steps:

Step 1: For $0\leq s \leq k-L$

where the optimal gain matrix $\mathit{\boldsymbol{F}}_{L}$ is determined by

with

Step 2: For $s = k-n; n = 0, 1, \ldots, L-1$

where the optimal estimate $\hat{\mathit{\boldsymbol{x}}}(k-n|k-n-1)$ is determined by the estimate of the time $s = k-n-1$, and the optimal gain matrix $\mathit{\boldsymbol{F}}_{n}$ is determined by

with

Proof: For $0\leq s \leq k-L$, the finite number of measurements on the horizon $[s-N, s]$ can be expressed in terms of the state $\mathit{\boldsymbol{x}}(s)$

where ${\bar{\mathcal{H}}}_{L, N}$ is as in (8), and

$\hat{\mathit{\boldsymbol{x}}}(s)$ can be indicated as a linear function of the finite measurements $\mathit{\boldsymbol{Y}}_{L}(s-1)$ on the horizon [s-N, s] as follows:

where

Taking the expectation on both sides of (13), the following relations are obtained

To satisfy the unbiased condition, ${\rm{E}} \hat{\mathit{\boldsymbol{x}}} = {\rm{E}} { \mathit{\boldsymbol{x}}}$, so

Deriving,

Thus, the estimation error can be represented as

It can be shown that $\bar{\mathit{\boldsymbol{C}}}_{L, N}{ \mathit{\boldsymbol{W}}}(s-1)+\bar{ \mathit{\boldsymbol{V}}}_{L}(s-1)$ is with zero-mean and covariance matrix $\boldsymbol{\Xi}_{L, N}$, which is given as

where

The objective now is to obtain the optimal gain matrix $\mathit{\boldsymbol{F}}_{L}$, subject to the unbiasedness constraint (14), in such a way that the error $\mathit{\boldsymbol{e}}(s)$ of the estimate $\hat{\mathit{\boldsymbol{x}}}(s|s-1)$ has minimum variance as follows

Before obtaining the solution to (16), we obtain the result on constraint optimization in the first place. To simplify the calculation, use $\mathit{\boldsymbol{F}}_{L}$ as a temporary replacement of $\mathit{\boldsymbol{F}}_{L}(s)$. Now, suppose that the following trace optimization problem is given

subject to

For convenience, partition the matrix $\mathit{\boldsymbol{F}}_{L}$ in (18) as

From (18), the $s$-th unbiasedness constraint can be written as

In terms of the partitioned vector $\mathit{\boldsymbol{f}}_{j}$, the cost function (17) is represented as

Thus, the optimization problem (17) is reduced to $N$ independent optimization problems

subject to

Obtaining the solutions to each optimization problem (20) and putting them together, we can finally get the solution to (17).

By the way of solving the optimization problem (21), we can establish the cost function

where $\boldsymbol{\lambda}_j$ is the $s$-th vector of a Lagrange multiplier, which is associated with the $s$-th unbiased constraint.

In order to minimize $\boldsymbol{\Phi}$, two necessary conditions

are satisfied, which give

Thus

then

From (23) and (24), we have

and

So put the above equation together, we can obtain

Substitute (25) into (13), we can reach the batch form of receding horizon estimation

This completes the proof of Step 1. The solution of Step 2 is similar to the solution of Step 1.

The filter with a batch form is represented in an iterative form for computational advantage. So we can get Theorem 2 by Theorem 1.

Theorem 2: Assume that $(\mathit{\boldsymbol{C}}, \mathit{\boldsymbol{A}})$ is observable. Then, the receding horizon filter $\hat{\mathit{\boldsymbol{x}}}(k)$ with an iterative form is given on the horizon $[k-N, k]$ by the following steps

Step 1: For $0\leq s \leq k-L$,

where

and $\mathit{\boldsymbol{P}}_{L, N}$ can be obtained from

with $\mathit{\boldsymbol{P}}_{L, 0} = {\pmb{\mathit 0}}$.

Step 2: For $s = k-n;n = 0, 1, \ldots, L-1$

where

and $\mathit{\boldsymbol{P}}_{n, N}$ can be obtained from

with $\mathit{\boldsymbol{P}}_{n, 0} = {\pmb{\mathit 0}}$.

Proof: Firstly, for $0\leq s\leq k-L$, define

Then, it can be expressed by the following Riccati Equation

where

$\mathit{\boldsymbol{P}}_{L, 0} = {\pmb{\mathit 0}}$ should be satisfied to obtain the above $\mathit{\boldsymbol{P}}_{L, 1}$ in Riccati (30).

Similarly, it is available that

On the basis of Theorem 1, we have

where

Now, we suppose that

From (33) and (34), we can reach

where $\check{\mathit{\boldsymbol{x}}}(s-N) = 0$ should be satisfied to obtain the same $\check{\mathit{\boldsymbol{x}}}(s-N+1)$ in Riccati (35). From (32) and (35), an iterative form for receding horizon estimation is derived.

We are able to get iterative form for receding horizon estimation of Step 2 according to Step 1.

Remark 1: In this paper, the reorganized observation technique is employed for the design of the receding horizon estimation with multi-channel observation delays. Compared with the general state augmentation method [21], the approach employed in this paper reduces the computational burden greatly since the filter developed in Theorem 2 has the same dimension as that of the original system. This advantage is more evident in the case of systems with large time delays and high dimensions.

Remark 2: Due to the regular horizon term $N$ is introduced in the optimized estimation, RHE can turn full information estimation problem into regular horizon estimation problem. And RHE is obtained by directly solving an optimization problem with the unbiasedness or deadbeatness constraint. Based on data from only the recent finite past (receding horizon), evidence shows that the proposed algorithm developed in Theorem 2 is more robust against dynamic model uncertainties compared with Kalman filters [20].

Ⅳ. STABILITY ANALYSIS

The stability of the receding-horizon filter will be investigated below. According to (28) and (30), it can be found that if the filter for $0\leq s \leq k-L$ is stable, the filter for $s = k-L+1$ is also stable since it is given by a recurrence Riccati equation of the previous sub-observer. Following a similar reasoning, all sub-observers will be stable. Then the stability of the overall filter will be reached. Thus we just need to analyze the stability of the filter developed in (27) and (28). This needs to require consideration of the filter's transfer matrix. From Theorem 2, we define the transfer matrix for $0\leq s \leq k-L$ as

Under the given assumption, the necessary and sufficient condition subject to the asymptotic stability of the proposed filter is that the transfer matrix $\boldsymbol{\Omega}_{L, N}$ of the filter is a stable matrix. It means that all of its eigenvalues are located in the unit circle. The stability of the observer is ensured by the following theorem.

Theorem 3: If $(\mathit{\boldsymbol{C}}, \mathit{\boldsymbol{A}})$ is observable, $\mathit{\boldsymbol{A}}$ nonsingular, then the matrix $\boldsymbol{\Omega}_{L, N}$ has all its eigenvalues strictly within the unit circle for all finite ${\rm{N}} \geq t-1$ where $t$ is the dimension of the state vector.

Proof: The proof process is similar to [2], and thus is omitted here.

Remark 3: Conditions for the stability of the proposed MHE are proposed for time-invariant systems. The advantage of this filter is that it is easy to implement since the filter gain can be computed off-line.

Ⅴ. ILLUSTRATIVE EXAMPLE

In this section, a simulation example with two channels is considered to illustrate the efficiency of the receding horizon estimation. A comparison to the Kalman filter is supplied in the simulation result. In this simulation, we set the time horizon $T = 100$, the filter horizon size $N = 5$, the time-delay $d = 1$. The parameters of the system are as follows

where $\{\mathit{\boldsymbol{w}}(k)\}, \{\mathit{\boldsymbol{v}}_0(k)\}$ and $\{\mathit{\boldsymbol{v}}_1(k)\}$ are mutually independent zero-mean Wiener processes with covariance matrices $\mathit{\boldsymbol{Q}} = \mathit{\boldsymbol{I}}_2, \mathit{\boldsymbol{R}}_0 = \mathit{\boldsymbol{I}}_2,$ and $\mathit{\boldsymbol{R}}_1 = \mathit{\boldsymbol{I}}_2$, respectively. The initial state $\mathit{\boldsymbol{x}}(0)$ is a random variable with $\mbox{E}(\mathit{\boldsymbol{x}}(0)) = 0$ and $\mbox{E}(\mathit{\boldsymbol{x}}(0)\mathit{\boldsymbol{x}}(0)') = I_2$. In the actual system we use $\mathit{\boldsymbol{x}}(0) = [0\ 0]'$ for the simulation.

Following the design procedures as in Theorem 2 and Kalman filter, we obtain the simulation results as follows. Fig. 1 shows the trace of the real state value $\mathit{\boldsymbol{x}}_{1}(k)$ and its estimate values according to the two algorithms. Fig. 2 shows the trace of the real state value $\mathit{\boldsymbol{x}}_{2}(k)$ and its estimate values according to the two algorithms. Fig. 3 shows the root of mean square estimation errors of $\mathit{\boldsymbol{x}}_{1}(k)$ according to the two algorithms, while Fig. 4 shows the root of mean square estimation errors of $\mathit{\boldsymbol{x}}_{2}(k)$ according to the two algorithms. It can be seen from the simulation results that the obtained receding horizon estimation for systems with multiple-channel observation delays track better than Kalman filter and the estimation scheme proposed in this paper produces good performance.

Figure  1.  State trajectories of the first state component: real state value (solid line), its RHE estimation (dash-dotted line), and Kalman filter estimation (discontinuous line).

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Figure  2.  State trajectories of the second state component: real state value (solid line), its RHE estimation (dash-dotted line), and Kalman filter estimation (discontinuous line).

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Figure  3.  The root mean square estimation errors of the first state component: RHE estimation (dash-dotted line), and Kalman filter estimation (discontinuous line).

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Figure  4.  The root mean square estimation errors of the second state component: RHE estimation (dash-dotted line) and Kalman filter estimation (discontinuous line).

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Ⅵ. CONCLUSION

In this note, we have studied the multi-channel receding horizon estimation for linear discrete-time system with observation delays. Firstly, the original multi-channel observation equations with time delays have been integrated by observation of reorganization. On the basis of observation of reorganization, a batch form and a recursive form for receding horizon estimation with some weighting gains were designed based on the measured data on the recent finite horizon $[s-N, s]$. And the stability of state estimation has been proven.

In this paper, the study mainly involves discrete linear systems. In fact, the solution to receding horizon problem for linear systems can be extended to infer receding horizon estimation of nonlinear systems. This work is worth researching further.