Event-triggered sensor data transmission policy for receding horizon recursive state estimation

Abstract

We consider a sensor data transmission policy for receding horizon recursive state estimation in a networked linear system. A good tradeoff between estimation error and communication rate could be achieved according to a transmission strategy, which decides the transfer time of the data packet. Here we give this transmission policy through proving the upper bound of system performance. Moreover, the lower bound of system performance is further analyzed in detail. A numerical example is given to verify the potential and effectiveness of the theoretical results.

Introduction

Because of the progress of science and technology, wireless sensor networks (WSNs) ¹ have gained much interest in the past few years. They have a wide range of applications including industrial process, smart building and agriculture, etc. ² State estimation problem is frequently confronted in the WSNs. ³ Sensors are usually powered by batteries with limited capacity for sensing, computation, and transmission in the WSNs, among which the transmission energy dominates the total energy cost and sensors are difficult to replace. Moreover, the capacity of a wireless channel may vary due to changes in the external environment. Meanwhile, time-varying channel capacity may influence overall dynamic system performance. Considering the low power nature of sensor and the time-varying characteristic of wireless channel, the lifetime of the sensors can be prolonged by reducing the sensor-to-estimator communication rate, which may decrease the estimation quality at the same time. Event-based policies could provide an inspiring chance to make a good trade-off between sensor-to-estimator communication rate and estimation error.

Most of the existing state estimation problems have been formulated by time-based sampling methods.^4–8 We consider the sensor transmission policy, which has to decide the transfer time of the data packet. This kind of sensor transmission policy is first presented by Åström and Bernhardsson ⁹ and then has received considerable attention from among the control community, the network community and communication community.^10–12 The optimal event-based state estimation with limited measurements was studied in Imer and Basar ¹³ for scalar linear systems and then Li et al. ¹⁴ extended the results to the vector systems by dropping the zero mean initial condition. Xu and Hespanha ¹⁵ considered the problem of computing an optimal transmission policy can be addressed in the framework of Markov decision processes. A hybrid sensor scheduler that combines conventional time and event-based methods were proposed in Shi et al. ¹⁶ considered the tradeoff between the average estimation error and the sensor-to-estimator communication rate. Han et al. ¹⁷ designed an open-loop and a close-loop stochastic event-triggered sensor schedule that used real-time information to reduce the average error covariance. Sijs and Lazar ¹⁸ proposed a state estimator, which was applicable to various types of event sampling policies. The event-based state estimation problem under the maximum likelihood estimation framework was analyzed in Shi et al. ¹⁹ The variance-based event-triggering conditions were considered in Trimpe and D’Andrea ²⁰ for scalar linear systems and the asymptotic convergence properties of the iteration of the prediction variance was proved.

The standard choice for estimating the state of a linear system is the Kalman filter ²¹ developed in 1960 and named after its discoverer. It is very popular that Kalman filter possesses many important properties such as minimum mean square error and stability. In some cases, however, it is impossible to directly apply the standard Kalman filter to the WSNs unless the sensor in the network can receive all the information. Additionally, Kalman filter may diverge for persistent modeling errors ²² or numerical errors^23,24 since undesirable signals may accumulate in the internal state. Therefore, to solve the aforementioned problems, we introduce a receding horizon strategy to state estimation and the basic concept of receding horizon state estimation is to estimate the state by using a finite number of past measurements. This estimation method is more robust against numerical errors and has been successfully applied to some engineering problems.

In this article, the event-based state estimation problem is studied via receding horizon strategy. The state estimation of a process based on the measurements was provided by a battery-powered sensor. The remote estimator receives the measurements through a wireless channel. Moreover, we assume that transmission consumes more energy than estimator computation and thus event detector is considered on the process side to prolong the sensor lifetime and reduce the sensor-to-estimator communication rate. To our best knowledge, the design framework is novel. The main contributions of our work include:

We put forward an event-triggered decision rule based on a least squares state estimator via the receding horizon strategy. We provide a simple algorithm for computing a transmission policy in order to balance a tradeoff between estimation error and transmission rate.

For this transmission policy, we use semi-definite programming for determining the upper and lower bounds of communication cost in detail. Note that this event-based transmission scheme just requires the feedback loop transmission when an event occurs, rather than construct a copy estimator on the process side which increases the additional infrastructure and computational burden.

The rest of the article is organized as follows. “Problem statement” section provides the system description and problem statement. Main results about the event-triggered policy and bounds on system’s performance are presented in “Main results” section. “Simulations” section provides a numerical example to illustrate the efficiency of the proposed results, followed by some concluding remarks in the final section.

Notations

$\mathbb{N}$ and $ℝ$ denotes the sets of natural and real numbers, respectively. $ℝm\times n$ denotes the sets of m by n real-valued matrices, whereas $ℝn$ is short for $ℝn\times 1$ , ${ℝ}_{+}^{n\times n}$ and ${ℝ}_{++}^{n\times n}$ are the sets of $n\times n$ positive semi-definite and positive definite matrices, respectively. When $X\in {ℝ}_{+}^{n\times n}$ , we simple write $X\ge 0$ (or X >  $0$ if $X\in {ℝ}_{++}^{n\times n}$ ). For $X\in ℝm\times n$ , $X\text{T}$ denotes the transpose of X and $E[·]$ denotes the mathematical expectation. For $s\in \mathbb{N}$ and $X\in ℝm\times n$ , $Xs$ denotes the X to the power s.

Problem statement

Consider the following discrete linear time-invariant process:

$x_{k+1}={\mathrm{Ax}}_k$

(1)

$y_k={\mathrm{Cx}}_k$

(2) where $x_k\in ℝn$ is the state vector, $y_k\in ℝm$ is the sensor measurement. It is assumed that the constant matrices A and C are known. After y_k is taken, the event detector decides to send it to the estimator or not. Let ${\gamma }_k$ be the decision variable: ${\gamma }_k=1$ indicates that y_k is sent and ${\gamma }_k=0$ otherwise. As a result, only when ${\gamma }_k=1$ , the estimator knows the exact value of y_k.

Recall from the Batch Least Squares (BLS) estimator in Ling and Lim, ⁶ i.e. ${\gamma }_k=1$ for all k, the state estimate ${\stackrel{\wedge }{x}}_k$ are computed recursively as

${\stackrel{\wedge }{x}}_{k+1}=A{\stackrel{\wedge }{x}}_k+{\kappa }_k(y_{k+1}-\mathrm{CA}{\stackrel{\wedge }{x}}_k)$

(3)

${\kappa }_k=AN{P}_N(\mathrm{CA}N)\text{T}$

(4)

$P_N^{-1}=P_{N-1}^{-1}+(\mathrm{CA}N)T\mathrm{CA}N$

(5) where ${\kappa }_k$ represents the BLS estimator gain and N is the tuning parameter to be called the state estimation horizon. This is a receding horizon approach and the resulting state estimator is termed the receding horizon state estimator. Otherwise, the estimator updates its estimates as

${\stackrel{\wedge }{x}}_{k+1}=A{\stackrel{\wedge }{x}}_k$

(6)

In this article, we define the error cost as: then $e_{k+1}$ obeys the following form:

$e_{k+1}=x_{k+1}-{\stackrel{\wedge }{x}}_{k+1},={\mathrm{Ax}}_k-(1-{\gamma }_k)A{\stackrel{\wedge }{x}}_k-{\gamma }_k(A{\stackrel{\wedge }{x}}_k+{\kappa }_k(y_{k+1}-\mathrm{CA}{\stackrel{\wedge }{x}}_k)),={\mathrm{Ae}}_k-{\gamma }_k{\kappa }_k{\mathrm{CAe}}_k,=(I-{\gamma }_k{\kappa }_{k}C){\mathrm{Ae}}_k$

(8)

Let $T\in \mathbb{N}$ be the time horizon, and choose J as the average cost, i.e.

$J={\text{limsup}}_{T\to \infty }\frac{1}{T}\sum _{k=0}^{T-1}(c(e_k))$

(9) where

$c(e_k)=(1-{\gamma }_k)e_k^{\text{T}}{\mathrm{Me}}_k+\lambda {\gamma }_k$

(10)

A block diagram representing the architecture is shown in Figure 1. It is seen that an event detector requires the feedback loop transmission when an event occurs, i.e. ${\gamma }_k=1$ for all k. A copy local estimator in the process side won’t be constructed, because it may increase the additional infrastructure and computation burden. During the period of non-event occurs, a simple recursive step is only needed for this scheme.

Remark 1

See Arapostathis et al. ²⁵ for background on this choice of J. The cost given by equation (9) is called the average per-period communication cost. An upper bound of J is obtained for balancing a tradeoff between average cost of e_k, as measured by the quadratic form defined by M and the frequency with which we transmit by setting ${\gamma }_k=1$ . Before giving the main result, two lemmas are cited.

Figure 1.

The diagram of an event-triggered sensor transmission scheme for state estimation scenario.

Lemma 1

²⁶ Suppose that there exists a sequence $z_0,z_1,\dots$ satisfied with state space Z, supposed $f:Z\to ℝ$ , $c:Z\to ℝ$ . Define:

$\overline{J}={\text{limsup}}_{T\to \infty }\frac{1}{T}\sum _{k=0}^{T-1}(c(z_k))$

(11)

If there exists $a\in ℝ$ such that

$f(z)\ge a\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{forall}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}z\in Z$

(12) then we have

$\overline{J}\le {\text{sup}}_{\delta \in Z}(c(\delta )+(f(z_{k+1})|z_k=\delta )-f(\delta ))$

(13)

Lemma 2

²⁶ Suppose $x_0,x_1,\dots$ is a sequence with state space X, supposed ɛ: $X\to ℝ$ $f:X\to ℝ$ . Define:

$\underset{¯}{J}={\text{limsup}}_{T\to \infty }\frac{1}{T}\sum _{k=0}^{T-1}(c(x_k))$

(14)

If there exists $b\in ℝ$ such that

$f(x)\le b\text{forall}x\in X$

(15) then we have

$\underset{¯}{J}\ge {\text{inf}}_{\delta \in X}(c(\delta )+(f(x_{k+1})|x_k=\delta )-f(\delta ))$

(16)

Main results

In this section, we will present the main results of this article. We mainly focus on the tradeoff between communication rate and estimation performance.

Upper bound

We are now ready to give this transmission strategy through the following theorem proving the upper bound on system performance.

Theorem 1

Given the error weighted value M and the transmission weighted value λ, suppose that $H\ge 0$ is symmetric matrix, and $\varsigma \ge 0$ is constant. If

$(\lambda -\varsigma )Q-H+A\text{T}\mathrm{HA}\le 0,A\text{T}\mathrm{HA}-H+M-\varsigma Q\le 0,C\text{T}{\kappa }_k^{\text{T}}H{\kappa }_{k}C-H{\kappa }_{k}C-C\text{T}{\kappa }_k^{\text{T}}H\le 0,\varsigma -\lambda \le 0$

(17) where

$Q=C\text{T}\frac{\lambda }{\mathrm{traceM}}C$

(18) then a measurement is sent to the remote estimator by setting

${\gamma }_k=\{0\mathrm{if}(y_k-C{\stackrel{\wedge }{x}}_k)\text{T}\frac{\lambda }{\mathrm{traceM}}(y_k-C{\stackrel{\wedge }{x}}_k)\le 1,1\mathrm{otherwise}$

(19) and then the cost function can be derived

$J\le \varsigma$

(20)

Proof

Let the function f be

$f(e)=e\text{T}\mathrm{He}$

(21)

Clearly, $f(e)\ge 0$ for all e and the recursion of $e_{k+1}$ via (8) is

$e_{k+1}=\{{\mathrm{Ae}}_k{\gamma }_k=0,(I-{\kappa }_{k}C){\mathrm{Ae}}_k{\gamma }_k=1$

(22)

We can obtain

$f(e_{k+1}|e_k=\delta )=\{\delta \text{T}A\text{T}\mathrm{HA}\delta {\gamma }_k=0,\delta \text{T}A\text{T}\mathrm{HA}\delta -\delta \text{T}A\text{T}H{\kappa }_k\mathrm{CA}\delta -\delta \text{T}A\text{T}C\text{T}{\kappa }_k^{\text{T}}\mathrm{HA}\delta \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}+\delta \text{T}A\text{T}C\text{T}{\kappa }_k^{\text{T}}H{\kappa }_k\mathrm{CA}\delta {\gamma }_k=1$

(23)

Then define a function $g:ℝn\to ℝ$ as

$g(e_k|e_k=\delta )=f(e_{k+1}|e_k=\delta )+c(e_k|e_k=\delta )-f(e_k|e_k=\delta )$

(24)

Based on the Lemma 1 and $c(e_k|e_k=\delta )$ defined on (10), we can derive the following result:

In order to apply Lemma 1, we need to compute an upper bound for g. First, when $(y_k-C{\stackrel{\wedge }{x}}_k)\text{T}\frac{\lambda }{\mathrm{traceM}}(y_k-C{\stackrel{\wedge }{x}}_k)>1$ , we get

$g(\delta )=\eta -\delta \text{T}(H-A\text{T}\mathrm{HA})\delta +\delta \text{T}A\text{T}(C\text{T}{\kappa }_k^{\text{T}}H{\kappa }_{k}C-H{\kappa }_{k}C-C\text{T}{\kappa }_k^{\text{T}}H)A\delta \text{forall}\delta \in ℝn$

(26)

Considering condition in (17), we know that if $C\text{T}{\kappa }_k^{\text{T}}H{\kappa }_{k}C-H{\kappa }_{k}C-C\text{T}{\kappa }_k^{\text{T}}H\le 0$ , then the inequality

$g(\delta )\le \eta -\delta \text{T}(H-A\text{T}\mathrm{HA})\delta$

(27) holds. Due to $(\eta -\varsigma )Q\le H-A\text{T}\mathrm{HA}$ , we have

$(\eta -\varsigma )\delta \text{T}Q\delta \le \delta \text{T}(H-A\text{T}\mathrm{HA})\delta$

(28)

Since $(y_k-C{\stackrel{\wedge }{x}}_k)\text{T}\frac{\lambda }{\mathrm{traceM}}(y_k-C{\stackrel{\wedge }{x}}_k)>1$ , we can obtain $\delta \text{T}C\text{T}\frac{\lambda }{\mathrm{traceM}}C\delta >1$ .

For the condition $\delta \text{T}M\delta >1$ , we have

$\eta -\delta \text{T}(H-A\text{T}\mathrm{HA})\delta \le \varsigma$

(29) which implies that

$f(\delta )\le \varsigma$

(30)

Next, when $(y_k-C{\stackrel{\wedge }{x}}_k)\text{T}\frac{\lambda }{\mathrm{traceM}}(y_k-C{\stackrel{\wedge }{x}}_k)\le 1$ , we can obtain

$g(\delta )=\delta \text{T}(A\text{T}\mathrm{HA}-H+M)\delta \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{forall}\delta \in ℝn$

(31)

Due to the condition in (17), if $A\text{T}\mathrm{HA}-H+M\le \varsigma Q$ and $\varsigma \ge 0$ , then we have

$\delta \text{T}(A\text{T}\mathrm{HA}-H+M)\delta \le \varsigma \delta \text{T}Q\delta$

(32)

Knowing that $(y_k-C{\stackrel{\wedge }{x}}_k)\text{T}\frac{\lambda }{\mathrm{traceM}}(y_k-C{\stackrel{\wedge }{x}}_k)\le 1$ , we obtain $\delta \text{T}C\text{T}\frac{\lambda }{\mathrm{traceM}}C\delta \le 1$ . Since $\delta \text{T}Q\delta \le 1$ , we have the inequality $\delta \text{T}(A\text{T}\mathrm{HA}-H+M)\delta \le \varsigma$ , that is

$f(\delta )\le \varsigma \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{forall}\delta \in ℝn$

(33)

Therefore, if the hypothesis of (17) holds, we immediately have

$J\le {\text{sup}}_{\delta \in ℝn}g(\delta )\le \varsigma$

(34)

Lower bound

In this section, we provide a tighter low bound on J. The main results are presented by using the Lemma 2.

Theorem 2

Suppose there exists a symmetric matrix $G\ge 0$ and a constant $\beta \ge 0$ such that

$M-\beta G\ge 0,\frac{1}{2}\le e\text{T}\mathrm{Ge}\le 1,\eta -\frac{1}{2}\beta \ge 0$

(35)

Then a cost function satisfies

$J(\zeta )\ge \frac{1}{2}\beta -\frac{1}{4}$

(36)

Proof

Choose a function

$f(e)=(e\text{T}\mathrm{Ge})2-e\text{T}\mathrm{Ge}$

(37)

It is easy to see that $f(e)$ is bounded above and $f(e_{k+1}|e_k=\delta )$ can be described as

Then let the function g: $ℝn\to ℝ$ be

$g(\delta )=(f(x_{k+1})|x_k=\delta )+c(\delta )-f(\delta )$

(39) and we have

$g(\delta )=\{(\delta \text{T}A\text{T}\mathrm{GA}\delta )2-\delta \text{T}A\text{T}\mathrm{GA}\delta -(\delta \text{T}G\delta )2+\delta \text{T}G\delta +\delta \text{T}M\delta {\gamma }_k=0(\delta \text{T}A\text{T}(I-{\kappa }_{k}C)\text{T}G(I-{\kappa }_{k}C)A\delta )2-\delta \text{T}A\text{T}(I-{\kappa }_{k}C)\text{T}G(I-{\kappa }_{k}C)A\delta \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}-(\delta \text{T}G\delta )2+\delta \text{T}G\delta +\eta {\gamma }_k=1$

(40)

First, when ${\gamma }_k=1$ , we obtain the equality

$g(\delta )=((A\delta -{\kappa }_k\mathrm{CA}\delta )\text{T}G(A\delta -{\kappa }_k\mathrm{CA}\delta )-\frac{1}{2})2\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}+\delta \text{T}G\delta -(\delta \text{T}G\delta )2+\eta -\frac{1}{4}$

(41)

We get $g(\delta )\ge \eta -\frac{1}{4}$ in that $\frac{1}{2}\le \delta \text{T}G\delta \le 1$ for all $\delta \in ℝn$ . Therefore, we have $g(\delta )\ge \eta -\frac{1}{4}\ge \frac{1}{2}\beta -\frac{1}{4}$ through the hypothesis $\eta \ge \frac{1}{2}\beta$ .

Next, when ${\gamma }_k=0$ , we get

$g(\delta )=(\delta \text{T}A\text{T}\mathrm{GA}\delta -\frac{1}{2})2+\delta \text{T}G\delta -(\delta \text{T}G\delta )2+\delta \text{T}M\delta -\frac{1}{4}$

(42)

In a similar way, we also obtain $g(\delta )\ge \delta \text{T}M\delta -\frac{1}{4}$ .

Considering the assumptions $M\ge \beta G$ and $\delta \text{T}G\delta \ge \frac{1}{2}$ for all $\delta \in ℝn$ , we further have $\delta \text{T}M\delta \ge \frac{1}{2}\beta$ for all $\delta \in ℝn$ and hence if the inequality (35) holds, we immediately have the following conclusion:

$J\ge {\text{inf}}_{\delta \in ℝn}g(\delta )\ge \frac{1}{2}\beta -\frac{1}{4}$

(43)

Simulations

In this section, numerical simulations are provided to illustrate the benefits of the above theoretical results, we consider a linear discrete time-invariant state space model with

$A=(0.80.20.050.8),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}C=(11)$

(44)

For convenience, the horizon length is taken as $N=3$ and the error weighted value and transmission weighted value are specified by

$Q=[3103],\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\lambda =10$

(45)

Then we obtain the average cost by using (9–10):

$J_{\text{avg}}=\frac{1}{T}\sum _{k=0}^{T-1}(c(e_k)),=\frac{1}{T}\sum _{k=0}^{T-1}((1-{\gamma }_k)e_k^{\text{T}}{\mathrm{Me}}_k+\lambda {\gamma }_k),=\frac{1}{T}\sum _{k=0}^{T-1}((1-{\gamma }_k)(x_k-{\stackrel{\wedge }{x}}_k)\text{T}Q(x_k-{\stackrel{\wedge }{x}}_k)+\lambda {\gamma }_k)$

(46)

Under this average cost, by varying the transmission weighted value λ, Figure 2 plots the upper and lower bound obtained in Theorems 1 and 2. In Figure 3, the relationship between estimation performance and average communication rate is further analyzed. The proposed transmission policy achieves better performance compared with ${\gamma }_k$ is independently and identically distribute Bernoulli. It is shown that the performance of standard Kalman filter is similar to the receding horizon least squares estimation in Figure 4.

Figure 2.

The upper bound and lower bound of $J_{\text{avg}}$ .

Figure 3.

Compared with other method which ${\gamma }_k$ is IID Bernoulli.

Figure 4.

Compared with standard Kalman filter about $J_{\text{avg}}$ .

Conclusions

In this work, we consider an event-based state estimation problem under the receding horizon strategy for linear time-invariant systems. At the same time, we provide a transmission policy to decide when to send data in order to balance a tradeoff between estimation error and transmission rate. The upper and lower bounds of communication cost are analyzed. The future work is to extend this work to stochastic system and multi-sensor scheduling.