Event-triggered minimax state estimation with a relative entropy constraint

Automatica 110 (2019) 108592

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Event-triggered minimax state estimation with a relative entropy constraint✩ ∗

Jiapeng Xu a , Yang Tang a , , Wen Yang a , Fangfei Li b , Ling Shi c a

Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, East China University of Science and Technology, Shanghai 200237, China b Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China c Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong, China

article

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Article history: Received 6 January 2019 Received in revised form 14 June 2019 Accepted 28 August 2019 Available online xxxx Keywords: Event-triggered state estimation Minimax estimation Robustness Relative entropy constraint

a b s t r a c t In this paper, we consider an event-triggered minimax state estimation problem for uncertain systems subject to a relative entropy constraint. This minimax estimation problem is formulated as an equivalent event-triggered linear exponential quadratic Gaussian problem. It is then shown that this problem can be solved via dynamic programming and a newly defined information state. As the solution to this dynamic programming problem is computationally intractable, a one-step eventtriggered minimax estimation problem is further formulated and solved, where an a posteriori relative entropy is introduced as a measure of the discrepancy between probability measures. The resulting estimator is shown to evolve in recursive closed-form expressions. For the multi-sensor system scenario, a one-step event-triggered minimax estimator is also presented in a sequential fusion way. Finally, comparative simulation examples are provided to illustrate the performance of the proposed one-step event-triggered minimax estimators. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Event-triggered state estimation, whose aim is to maintain an acceptable estimation performance at a reduced communication cost, has gained significant attention in cyber–physical systems (CPSs). Different from traditional time-triggered systems in which the sensor data are taken by the estimator at every time instant, an event-triggered estimator receives the data via a network only when a predefined event-triggered condition is satisfied. The event-triggered condition defines some importance metric of the data and provides implicit information available to the remote estimator, so that a good tradeoff between communication cost and estimation performance is achieved (Battistelli, Benavoli, & ✩ This work was supported by the National Key Research and Development Program of China under Grant 2018YFC0809302, the National Natural Science Foundation of China (Grant Nos. 61751305, 61673176, 61973123, 61573143, 61773161), the Hong Kong RGC General Research Fund 16204218, the Science and Technology Commission of Shanghai Municipality under Grant 18ZR1409800, and the Programme of Introducing Talents of Discipline to Universities (the 111 Project) under Grant B17017. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Michele Basseville under the direction of Editor Torsten Söderström. ∗ Corresponding author. E-mail addresses: [email protected] (J. Xu), [email protected] (Y. Tang), [email protected] (W. Yang), [email protected], [email protected] (F. Li), [email protected] (L. Shi). https://doi.org/10.1016/j.automatica.2019.108592 0005-1098/© 2019 Elsevier Ltd. All rights reserved.

Chisci, 2012; Han et al., 2015; Liu, Wang, He, & Zhou, 2015; Shi, Chen, & Darouach, 2016; Shi, Chen, & Shi, 2014a; Sijs & Lazar, 2012; Trimpe & Campi, 2015; Wu, Jia, Johansson, & Shi, 2013). During the last decade, a number of event-triggered schedules and estimators have been proposed. In general, these estimators can be classified into two categories in terms of the designed event-triggered conditions: deterministic event-triggered (DET) estimators (Battistelli et al., 2012; He, Chen, & Qi, 2019; Liu et al., 2015; Shi et al., 2014a; Sijs & Lazar, 2012; Wu et al., 2013; Zhang & Han, 2015) and stochastic event-triggered (SET) estimators (Han et al., 2015; Huang, Shi, & Chen, 2017; Mohammadi & Plataniotis, 2017; Shi et al., 2016; Weerakkody, Mo, Sinopoli, Han, & Shi, 2016; Yang, Zhang, Chen, Yang, & Shi, 2019). In the DET conditions, a data packet is sent to the estimator only if it goes beyond a certain range. This is a natural and effective way to trigger the data transmission, since the so-called useless data defined by the importance metric should be definitely discarded. In this case, the exact minimum mean square error (MMSE) estimators, however, are computationally intractable to be obtained even for linear Gaussian systems. Thus, the Gaussian assumption (Shi et al., 2014a; Wu et al., 2013) and a sum of Gaussians approach (Sijs & Lazar, 2012) are commonly used to derive approximate MMSE estimators. More recently, in He et al. (2019), an exact MMSE estimation algorithm with tractable expressions was proposed for a based-innovation DET condition by introducing generalized closed skew normal distribution. In He et al. (2019), there still

2

J. Xu, Y. Tang, W. Yang et al. / Automatica 110 (2019) 108592

exists a computational issue since dimensions of some matrices in the algorithm increase linearly when a packet is not sent. In contrast to DET conditions, a SET condition, as the name suggests, allows every data packet to be sent in a stochastic way. The transmission probability involves a radial basis function kernel such that for linear Gaussian systems, the conditional state distribution maintains Gaussian. Thus, the exact MMSE estimators are Kalman-like estimators, which are easy to be implemented in practice (Han et al., 2015). Moreover, the precise estimation error can be analyzed based on the exact estimators. On the other hand, in CPSs it is usually difficult to characterize exactly the dynamics of the physical plant considered by a mathematical model due to the complexities of systems and some uncertainty factors (Gao & Li, 2014). If the exact probability model is known, it is well known that Kalman filter (KF) is the MMSE estimator. However, KF may suffer from poor performance from the perspective of robustness and it indeed does not consider the uncertainty information of systems. In this context, it is necessary to consider system uncertainty for event-triggered state estimation problems. Recently, several event-triggered estimation results have been developed for uncertain systems based on H∞ (Dong, Wang, Ding, & Gao, 2015; Meng & Chen, 2014; Zhang & Han, 2015) and risk-sensitive (Huang, Shi, & Chen, 2019; Xu, Ho, Li, Yang, & Tang, 2019) estimation frameworks. It is well known that H∞ and robust H∞ estimation methods are proposed for uncertain systems with deterministic noise inputs (see Dong et al., 2015; Gao & Li, 2014; Petersen & Savkin, 1999; Shaked & Theodor, 1992; Zhang & Han, 2015 and the references therein). The risksensitive criterion, which is an exponential index of cost function, has a potential robustness to uncertain systems with stochastic noises, but there are no specific class of uncertainties that are targeted. Hence, an alternative way is to consider a minimax state estimation problem for uncertain systems with stochastic noises. Since in information theory relative entropy is commonly used to measure the discrepancy between two probability distributions, a relative entropy constraint is naturally used to describe stochastic uncertain systems (Levy & Nikoukhah, 2013; Petersen, 2006; Petersen, James, & Dupuis, 2000; Xie, Ugrinovskii, & Petersen, 2008; Yoon, Ugrinovskii, & Petersen, 2004). One attractive feature of this approach is that the resulting minimax linear quadratic Gaussian (LQG) controller and minimax state estimator with a quadratic cost have closed-form expressions and generalize the standard LQG controller and KF, respectively. In addition, the worst-case quadratic performance can be achieved by using this minimax formulation. In consideration of the attractive features of the minimax formulation with a relative entropy description of uncertainty, we investigate an event-triggered state estimation problem with a general SET condition under this minimax framework. It is shown that this event-triggered minimax estimation problem can be solved via dynamic programming and a newly defined information state. However, unlike the case of linear Gaussian systems with SET conditions (Han et al., 2015) or with a relative entropy description of uncertainty (Levy & Nikoukhah, 2013; Yoon et al., 2004), we are not able to obtain an analytical solution to this event-triggered minimax estimation problem. The essential reason is that this problem is a dynamical programming (DP) with exponential quadratic cost and binary decision variables introduced by the event trigger. This motivates us to further consider a one-step event-triggered minimax state estimation problem, under which a closed-form solution is obtained. We also demonstrate by a simulation example that under admissible uncertainties introduced by a relative entropy constraint, the proposed one-step minimax estimator has a better robustness than the risk-sensitive estimator proposed in Huang et al. (2019).

The main results and contributions of this paper are summarized as follows. (1) The framework of the minimax state estimation problem with general SET measurements is formulated. We show that this problem can be solved via DP and a newly defined information state. It is worth mentioning that the information state in this paper is defined in a direct way, which does not need to introduce the concepts of change of measure and Radon–Nikodym derivative used in previous information state approaches (e.g., Collings, James, & Moore, 1996; Dey & Moore, 1997; Elliott, Aggoun, & Moore, 1995; Huang et al., 2019; Shaiju & Petersen, 2009). (2) In consideration of realistic implementability, a one-step event-triggered minimax state estimation problem is formulated and solved. Its solution has a nice feature that resulting estimates can be evaluated recursively in a closed form. The estimation performance can be guaranteed by the corresponding one-step minimax error cost. (3) For the multi-sensor system scenario, a one-step eventtriggered minimax estimator is also derived in a sequential fusion way for a general SET condition on each sensor. The remainder of this paper is organized as follows. In Section 2, the event-triggered minimax state estimation problem for uncertain systems with a relative entropy constraint is formulated and the corresponding equivalent linear exponential quadratic Gaussian problem is built. In Section 3, an information state is defined and its explicit recursive expressions are obtained, based on which the event-triggered linear exponential quadratic Gaussian problem is solved via DP. Motivated by the implementation issue of the optimal solution, a one-step event-triggered minimax state estimation problem is formulated and solved in Section 4. The extension to the multi-sensor system scenario is presented in Section 5. Simulation examples are provided in Section 6, followed by some concluding remarks in Section 7. Notations. Rm×n denotes the set of m × n real-valued matrices and for brevity, denote Rn := Rn×1 . The transpose of a vector or a matrix is denoted by ‘‘′ ’’. For any vector x ∈ Rn and positive semidefinite matrix Z ∈ Rn×n , define ∥x∥2Z := x′ Zx. For brevity, denote ∥x∥2 := x′ x. For any X ∈ Rn×n , |X | denotes the determinant of X . x0k is a shorthand for any vector sequence {x0 , . . . , xk }. I is an identity matrix with a context-dependent dimension. Pr(·) ˜ ·)) and E[·] (or E˜ [·]) denote the probability of a random (or Pr( event and the expectation operator with respect to a probability ˜ respectively. measure P (or P), 2. Problem formulation 2.1. Uncertain systems Consider the following discrete-time linear time-varying stochastic uncertain system within a time horizon T : xk+1 = Ak xk + w ˜ k, yk = Ck xk + v˜ k , z k = Lk x k ,

(1) n

m

where xk ∈ R is the system state, yk ∈ R is the measurement, and zk ∈ Rp is the variable to be estimated. Ak , Ck and Lk are known, whereas the noises w ˜ k and v˜ k and the initial state x˜ 0 are stochastic with unknown distributions. To characterize the admissible perturbed noise processes, it is necessary to specify a nominal assumption of noise processes and the initial state, which are denoted by {wk }, {vk } and x0 . In this paper, we consider a commonly used Gaussian assumption: wk and

J. Xu, Y. Tang, W. Yang et al. / Automatica 110 (2019) 108592

3

Fig. 1. Event-triggered minimax state estimation for uncertain systems.

vk are mutually independent white Gaussian noises with densities ψk (wk ) := N (wk ; 0, Qk ) and φk (vk ) := N (vk ; 0, Rk ), respectively, and the initial condition x0 is Gaussian with the density µ(x0 ) = N (x0 ; π0 , Π0 ), and independent of wk and vk . Here Qk , Rk , Π0 are positive definite matrices. Since the dynamic evolution of system (1) actually depends on x˜ 0 and the noise sequences {w ˜ k } and {˜vk }, system (1) is considered to be defined on the unknown noise space {˜x0 , w ˜ 0T , v˜ 0T } with the perturbed probability measure ˜ x˜ 0 , w P( ˜ 0T , v˜ 0T ). Correspondingly, we have the nominal probability measure P(x0 , w0T , v0T ). According to independent properties of x0 , wk and vk , we have dP(x0 , w0T , v0T )

=dP(x0 )

T ∏

The relative entropy constraint given by Definition 1 can capture many useful uncertainty description. For example, for uncertain systems (1), the random noises can be x˜ 0 = x0 + ξ , w ˜ k := ∆1 (xk ) + wk and v˜ k := ∆2 (xk ) + vk . Here, ξ is an unknown constant and independent of {wk } and {vk }, and ∆1 (xk ) and ∆2 (xk ) are unknown but subject to some bound, respectively. Then by (3), h(P˜ ∥ P) in this case can be explicitly obtained: h(P˜ ∥ P) =

1 2

∥ξ ∥Π −1 +

T 1∑ [ E˜ ∥∆1 (xk )∥2 −1 Qk 2 k=0

] + ∥∆2 (xk )∥2R−1 .

(5)

k

A specific uncertain system model can be ξ = 0, ∆1 (xk ) = δ I · xk , δ ∈ [−1, 1], ∆2 (xk ) = 0, and Dk = I, which satisfies the relative entropy constraint (4).

dP(wk , vk |x0 , w0k−1 , v0k−1 )

k=0

=µ(x0 )dx0

T ∏

ψk (wk )dwk

k=0

T ∏

2.2. Event-triggered strategy

φk (vk )dvk .

(2)

k=0

Here, dP represents the differential of the probability measure P, which involves the Lebesgue integral. We further use relative entropy (Dupuis & Ellis, 1997) to measure the discrepancy between probability measures P˜ and P as in Petersen et al. (2000) and Yoon et al. (2004). The relative entropy h(P˜ ∥ P) is defined by h(P˜ ∥ P) :=

{

E˜ [log

+∞,

dP˜ dP

],

P˜ ≪ P and log otherwise,

dP˜ dP

˜ , ∈ L1 (P)

where P˜ ≪ P means that the perturbed measure P˜ is absolutely continuous with respect⏐ to P and L1 is the L1 space such that ⏐

∫ dP˜ ˜ denotes ⏐⏐log dP˜ ⏐⏐ dP˜ < ∞. The set of all measures log dP ∈ L1 (P) dP P˜ satisfying h(P˜ ∥ P) < ∞ is denoted by P . It follows the chain rule for relative entropy (Lemma 7.9 in Gray, 2011) that

˜ x˜ 0 , w h P( ˜ 0T , v˜ 0T ) ∥ P(x0 , w0T , v0T )

(

)

T ( ) ∑ ( ˜ x˜ 0 )∥P(x0 ) + ˜ w =h P( h P( ˜ k , v˜ k |˜x0 , w ˜ 0k−1 , v˜ 0k−1 )∥ k=0

P(wk , vk |x0 , w0k−1 , v0k−1 ) .

)

(3)

Along the line of stochastic uncertainty description introduced in Petersen (2006), Petersen et al. (2000) and Yoon et al. (2004), the relative entropy constraint for uncertainty systems (1) is defined as follows. Definition 1 (Yoon et al., 2004, Definition 1). Given a constant d > 0, a probability measure P˜ ∈ P defines an admissible uncertainty if h(P˜ ∥ P) ≤ E˜ [Vu ] + d, Vu :=

T 1∑

2

∥Dk xk ∥2 ,

(4)

k=0

where Dk is the so-called uncertainty matrix. The set of all admissible measures P˜ ∈ P is denoted by Ξ .

Considering limited energy and communication resources, we introduce an event-triggered strategy to decide whether yk is sent to the remote estimator (see Fig. 1). If γk = 1, yk is sent by the sensor; otherwise no signal will be sent. Let Sk := {i ∈ N0:k |γi = 0}, and define Ik := {γi |i ∈ Sk } ∪ {yi |i ∈ N0:k \Sk }

with I−1 := ∅ as the available information set of the estimator at time k. Here, we consider a general stochastic event-triggered scheme1 (Shi et al., 2016)

˜ γk = 0|yk , Ik−1 ) =Pr( ˜ γk = 0|y0k , x0k , γ0k−1 ) Pr( ) ( 1 = exp − ∥yk − βk ∥2Yk ,

(6)

2

where Yk ∈ Rm×m is a positive definite weighting matrix, and βk ∈ Rm is known to the estimator based on the available information set Ik−1 such that βk does not need to be transmitted to the estimator. The parameter Yk plays a role of tuning the communication rate and in general a smaller Yk suggests a relatively lower communication rate. In particular, the conventional timetriggered communication is recovered if Yk → ∞. The choice of βk depends on practical available resources at the sensor side; the classical choices are the previously transmitted measurement and the one-step predicted value of yk (Trimpe & Campi, 2015). The first equality of (6) means that the transmission probability at time k only depends on the current measurement yk and past information Ik−1 . For the nominal system, the event-triggered process (6) becomes Pr(γk = 0|yk , Ik−1 ) = Pr(γk = 0|y0k , x0k , γ0k−1 )

) ( 1 = exp − ∥yk − βk ∥2Yk .

(7)

2

1 At each time k, the probability Pr(γ = 0|y , I ) in (6) can be implek k k−1 mented by generating an independent and identically distributed (i.i.d.)) random ( variable ζk uniformly distributed on [0, 1]. If ζk < exp − 12 ∥yk − βk ∥2Y and otherwise γk = 1.

k

, γk = 0

4

J. Xu, Y. Tang, W. Yang et al. / Automatica 110 (2019) 108592

Remark 1. Note that rigorously, the random process {ζk } in the event trigger should be also defined on the probability measures P˜ and P. Since there does not exist uncertainty in {ζk } such that

˜ x˜ 0 , w h P( ˜ 0T , v˜ 0T , ζ0T ) ∥ P(x0 , w0T , v0T , ζ0T )

(

)

With such a duality relation, for the unconstrained problem (11) we have Jτ = τ

{(

where

this difference has no influence on the problem considered in this paper.

J L (τ ) = E exp(τ −1 Vτ )

In this subsection, we present the event-triggered minimax optimal estimation problem considered in this paper. We focus on a causal estimator {ˆz0 , . . . , zˆT } such that the estimates are based on available information of the estimator at time k, i.e., zˆk = zˆk (Ik ) = Lk xˆ k (Ik ) = Lk xˆ k .

(8)

To evaluate the estimator performance, the estimation error cost is defined by T

1∑

Ve :=

2

T

∥zk − zˆk ∥2 =

k=0

1∑ 2

∥xk − xˆ k ∥2Wk ,

(9)

inf sup E˜ [Ve ].

(10)

xˆ 0T P˜ ∈Ξ

Remark 2. Note that though the above minimax problem has the same form as that in time-triggered communication (Yoon et al., 2004), the cost functions Ve are different. The estimate xˆ k here depends on the event-triggered process {γk }, leading to a more complicated estimator design and analysis. By using a Lagrange multiplier technique, the constrained minimax problem (10) can be transformed into an unconstrained form, which is presented in the following lemma. Lemma 1 (Petersen et al., 2000, Theorem 3.1). Consider the unconstrained minimax problem Jτ := inf sup{E˜ [Ve ] − τ [h(P˜ ∥ P) − E˜ [Vu ] − d]} xˆ 0T P˜ ∈P

{

[

= τ inf sup E˜ [τ

−1

xˆ 0T P˜ ∈P

]

}

Vτ ] − h(P˜ ∥ P) + d ,

(11)

where Vτ = Ve + τ Vu

=

T ∑ ) 1( ∥xk − xˆ k ∥2Wk + τ ∥Dk xk ∥2 . k=0

]

inf τ

(P0′ ) :

τ ∈Γ

{(

τ ∈Γ

)

}

inf log J L (τ ) + d .

(15)

xˆ 0T

It follows from (P0′ ) that to solve the event-triggered minimax optimal estimation problem, we first need to solve the following event-triggered LEQG problem inf J L (τ ),

(16)

xˆ 0T

and then find the optimal τ by τ ∗ = arg infτ ∈Γ Jτ . Thus, the eventtriggered minimax optimal estimator is achieved by the derived LEQG estimator with τ = τ ∗ . 3. Event-triggered LEQG estimation The solution to the optimal LEQG estimation problem with time-triggered communication was presented in Speyer, Fan, and Banavar (1992), where the solution is a Kalman-like estimator. In this section, we investigate the event-triggered LEQG estimation problem, which is shown to be more complicated and not solvable analytically. Since the expectation in the LEQG cost (14) is taken with respect to the nominal measure P, the perturbed measure P˜ does not need to be considered in this section. In the sequel, we provide a solution framework to this problem: it is first shown that a newly defined information state has simple recursive expressions and then based on the information state the problem can be solved via DP. 3.1. Recursive information state In this subsection, we present closed-form recursive expressions for a newly defined information state. The principle of constructing an information state is that it should contain all available information about the state of the system (p. 79 in Kumar & Varaiya, 1986). Therefore, the commonly used conditional density of xk is certainly an information state. However, to deal with the exponential cost, it is better not to use the conditional density. We should use a more suitable information state which includes part of the LEQG cost (Boel, James, & Petersen, 2002; Collings et al., 1996). To do this, we decompose the LEQG cost (14) as

[

inf sup E˜ [Ve ] = inf Jτ .

(14)

is the event-triggered LEQG cost. Thus, the minimax problem (P0 ) is equivalent to

2

Define Γ := {τ : τ > 0 and Jτ < ∞}. Then the value on the constrained minimax problem (10) is finite if and only if Γ ̸ = ∅. Furthermore, if Γ ̸ = ∅, then xˆ 0T P˜ ∈Ξ

[

k=0

where Wk = L′k Lk . Then given the uncertainty constraint in Definition 1, the event-triggered minimax optimal estimation problem can be formulated as (P0 ) :

(13)

xˆ 0T

( ) ˜ x˜ 0 , w =h P( ˜ 0T , v˜ 0T ) ∥ P(x0 , w0T , v0T ) ,

2.3. Event-triggered minimax optimal estimation problem

}

)

inf log J L (τ ) + d ,

(12)

Due to the lack of an explicit expression for the entropy h(P˜ ∥ P), it is rather difficult to obtain the solution to the parameterized stochastic game problem (11) directly. A feasible way is to convert this problem into an equivalent event-triggered linear exponential quadratic Gaussian (LEQG) estimation problem by using the well-known duality relation between free energy and relative entropy (Dai Pra, Meneghini, & Runggaldier, 1996).

(

J (τ ) = E exp L

T ∑

)] Ψ¯ k (xk , xˆ k )

,

(17)

k=0

where

Ψ¯ k (xk , xˆ k ) =

1(

) τ −1 ∥xk − xˆ k ∥2Wk + ∥Dk xk ∥2 .

2 Then we construct an information state in the following. Definition 2.

αk (xk ) :=

An information state αk (xk ) is defined by

∫

∫ ··· Rn

(18)

Rn

exp(Ψ0,k−1 )f (x0k , Ik )dx0k−1 , k = 1, 2, . . . , T ,

(19)

J. Xu, Y. Tang, W. Yang et al. / Automatica 110 (2019) 108592

where f (x0k , Ik ) represents the joint distribution of x0k and Ik , and k−1 ∑

Ψ0,k−1 =

Ψ¯ i (xi , xˆ i ).

i=0

The initial value is given by α0 (x0 ) := f (x0 , I0 ).

Lemma 3. Given x ∈ Rn , a sequence of matrices {Gi } ∈ Rmi ×n , a sequence of symmetric matrices {Xi } ∈ Rmi ×mi , and of ∑Na sequence G′i Xi Gi and vectors {bi } ∈ Rmi for 1 ≤ i ≤ N, define G1N := i = 1 ∑N ′ Gb := i=1 Gi Xi bi . If G1N is nonsingular, then N ∑

Remark 3. The information state defined here is more mathematically direct compared with the classical information state approaches to solve the estimation and control problems that involve an exponential cost (e.g., Collings et al., 1996; Dey & Moore, 1997; Elliott et al., 1995; Huang et al., 2019; Shaiju & Petersen, 2009). In these papers, the information state is defined under a new probability measure by using the change of measure approach (Elliott et al., 1995), which is somewhat complicated. Specifically, this approach is to introduce a new measure P¯ by defining a Radon–Nikodym derivative (Cohen & Elliott, 2015; Elliott et al., 1995) that establishes a connection between the new measure P¯ and the original measure P. Then it can be proved that ¯ {yk } is a sequence of i.i.d. random variables with densities under P, φk and is independent of {xk }. The information state under P¯ can be defined as

¯ k exp(θ Ψ0,k−1 )I(xk ∈ dx)|Ik ], αk (x)dx = E¯ [Λ (20) ¯ ¯ where E∏ [·] is the expectation operator corresponding to P and k ¯k = Λ l=0 φl (yl − Cl xl )/φl (yl ) is the defined Radon–Nikodym ¯ The advantage of introducing the change of derivative dP/dP. measure approach is the independent properties of {yk }, which

5

∥Gi x + bi ∥2Xi

i=1

−1

=∥x + G1N Gb ∥2G1N +

N ∑

(25)

∥bi ∥2Gi − ∥Gb ∥2G−1 . 1N

i=1

Proof. By using the result of completing the square for N = 2 (e.g., see Lemma 15 in Shi et al., 2016), we can prove this lemma by induction. ■ Based on Lemmas 2 and 3, we further have the following explicit recursive result for αk (xk ). Theorem 1. For the nominal system with the event-triggered scheme (7), the information state αk (x) is an unnormalized Gaussian density given by

( ) 1 αk (xk ) = Sk exp − ∥xk − x¯ k ∥2Σ −1 , 2

(26)

k

Here, Σk−1 is given by the following recursion:

may simplify the derivation. However, this paper shows that a suitable information state can still be defined in the original measure P to handle the problems with an exponential cost such that the familiar Bayes inference can be used.

⎧ [ ] −1 −1 −1 ⎪ Ck + Qk−−11 ⎨Σk = Ck′ Rk + (1 − γk )Yk −1 −1 ′ ¯ − Qk−1 Ak−1 Σk Ak−1 Qk−1 , ⎪ ⎩Σ ¯ k−1 = A′k−1 Qk−−11 Ak−1 + Σk−−11 − τ −1 Wk−1 − D′k−1 Dk−1 .

In the sequel, we present the recursive expressions for αk (xk ) by Bayes inference.

The above recursion holds if and only if A′k Qk−1 Ak + Σk−1 − τ −1 Wk − D′k Dk > 0 for all k < T . And x¯ k and Sk are given by the following recursions:

Lemma 2. For the nominal system with the event-triggered scheme (7), the information state αk (xk ) defined by (19) has the following recursive form: (1) if γk = 0,

αk (xk ) ∫ ( ) = exp Ψ¯ k−1 (z , xˆ k−1 ) ψk−1 (xk − Ak−1 z)αk−1 (z)dz n R ∫ · Pr(γk = 0|yk , Ik−1 )φk (yk − Ck xk )dyk ;

(21)

Rm

k

(2) if γk = 1,

αk (xk ) =φk (yk − Ck xk )

∫

exp Ψ¯ k−1 (z , xˆ k−1 )

(

Rn

(22)

The recursion starts from

∫ Rm

Pr(γ0 = 0|y0 )φ0 (y0 − C0 x0 )dy0

(23)

if γ0 = 0, and

α0 (x0 ) = φ0 (y0 − C0 x0 )µ(x0 )

(28)

Σk

The above recursions start from

)

· ψk−1 (xk − Ak−1 z)αk−1 (z)dz .

α0 (x0 ) = µ(x0 )

( ) ⎧ −1 ¯ k Σk−−11 x¯ k−1 − τ −1 Wk−1 xˆ k−1 Σk x¯ k = Qk−−11 Ak−1 Σ ⎪ ⎪ ] [ ] [ ⎪ − 1 ⎪ ⎪ + Ck′ Rk + (1 − γk )Yk−1 γk yk + (1 − γk )βk , ⎪ ⎪ ⎪ γk −1 −mγk ⎪ 1 1 ⎪ − − ⎪ ) 2 |Rk | 2 |Rk 1 + Yk | 2 |Qk−1 |− 2 ⎪ ⎨Sk = Sk−1 (2π{ [ 1 1 2 2 ¯ 2 ⎪ · |Σk | exp 2 ∥ˆxk−1 ∥τ −1 Wk−1 − ∥¯xk−1 ∥Σk−−11 ⎪ ⎪ ⎪ ⎪ + − ∥γk yk + (1 − γk )βk ∥2 ⎪ ⎪ [Rk +(1−γk )Yk−1 ]−1 ⎪ ⎪ ]} ⎪ ⎪ ⎪ ⎩ ∥Σk−−11 x¯ k−1 − τ −1 Wk−1 xˆ k−1 ∥2¯ + ∥¯xk ∥2 −1 . Σ

(27)

(24)

if γ0 = 1. Proof. See Appendix A. From Lemma 2, one can observe that the achievement of an explicit result of αk (xk ) requires a computation of the multiplication of Gaussian probability density functions. To this end, we present a general technique of completing the square in the following.

⎧ [ ]−1 ⎪ Σ0−1 = C0′ R0 + (1 − γ0 )Y0−1 C0 + Π0−1 , ⎪ ⎪ [ ]−1 ⎪ ⎪ ⎪ Σ0−1 x¯ 0 = Π0−1 π0 + C0 R0 + (1 − γ0 )Y0−1 ⎪ [ ] ⎪ ⎪ ⎪ · γ0 y0 + (1 − γ0 )β0 , ⎪ ⎪ ⎨ (1−γ ) n+γ0 m 1 1 − 2k 1 S0 = (2π )− 2 |R0 |− 2 |R− + Y0 | |Π0 |− 2 0 { [ ⎪ ⎪ ⎪ ⎪ · exp 12 ∥¯x0 ∥2 −1 − ∥π0 ∥2 −1 ⎪ ⎪ Σ0 Π0 ⎪ ⎪ ⎪ ]} ⎪ ⎪ ⎪ . ⎩ − ∥γ0 y0 + (1 − γ0 )β0 ∥2 −1 −1

(29)

[R0 +(1−γ0 )Y0 ]

Proof. See Appendix B. 3.2. Dynamic programming By the law of iterated expectations (Lemma 2.4.8 in Cohen & Elliott, 2015) and the definition of αk (xk ), the LEQG cost in (17)

6

J. Xu, Y. Tang, W. Yang et al. / Automatica 110 (2019) 108592

Moreover, one has

can be rewritten as

[ [

(

∑

J L (τ ) = E E exp

)

T

]]

k=0 ] ( ) ¯ ˆ αT (xT ) exp ΨT (xT , xT ) dxT /f (IT ) ,

[∫ =E Rn

inf J L (τ ) = inf E[J¯] = inf E[E[J¯|I0 ]] = E[J0 (x¯ 0 )],

Ψ¯ k (xk , xˆ k ) |IT

xˆ 0T

(30)

which shows that the LEQG cost is a function of the information state αT (xT ). In the following, we show that based on the recursive result for αk (xk ) in Theorem 1, the event-triggered LEQG estimator can be obtained by solving a DP problem. Theorem 2. The optimal estimates xˆ 0|0 , . . . , xˆ T |T of x0T which minimize the event-triggered LEQG cost E[exp(τ −1 Vτ )] can be obtained by solving the following DP equations:

∫

( ) αT (xT ) · exp Ψ¯ T (xT , xˆ T ) dxT /f (IT ),

JT (x¯ T ) = inf xˆ T

Rn

Jk (x¯ k ) = inf E [Jk+1 (x¯ k+1 )|Ik ] , k = T − 1, . . . , 0,

(31) (32)

xˆ k

where Jk (x¯ k ) is the cost-to-go function at time k and the expectation is taken with respect to yk+1 and γk+1 . Moreover, the resulting LEQG cost is given by inf J L (τ ) = E[J0 (x¯ 0 )],

(33)

xˆ 0T

where the expectation is taken with respect to y0 and γ0 . Proof. Write

( ) αT (xT ) exp Ψ¯ T (xT , xˆ T ) dxT

∫

Rn

J¯ :=

f (IT )

.

(34)

xˆ 0T

where the first equality follows from (30) and (34), and the third equality is due to (37). ■ In DP equations (31) and (32), Σk−1 x¯ k and Σk−1 which evolve according to the forward recursion expressions in Theorem 1, are the state variables, xˆ k is the control variable, and γk+1 ∈ {0, 1} and yk+1 ∈ Rm are the disturbance variables. Unfortunately, the solution to the DP problem cannot be analytically expressed, even for some specific SET condition such as the simple case βk = 0 and Yk = aRk (0 < a < ∞ is a scaling factor). We have to resort to numerical execution of the DP algorithm to approximately solve the DP. This is computationally challenging. To be specific, we first need to discretize xˆ k and yk+1 and we can take dn and dm points for xˆ k and yk+1 , respectively. Then according to the state equations of the DP problem, i.e., (27) and (28), we know that the dimension of the state space of Σk−1 is 2k+1 , and the dimension of the state space of Σk−1 x¯ k , denoted by Dk , can be computed recursively:

Dk = (dm + 1)dn Dk−1 , D0 = dm + 1,

(41)

which scales exponentially as k increases. Note that we have let βk be a constant in computing Dk for convenience. Thus for the finite horizon T , the number of computational operations for the DP algorithm at the stage k is dn dm Dk = d(k+1)n (dm + 1)k+1 dm (Chapter 6 of Bertsekas, 2005), which shows that the computational complexity scales exponentially with the dimension of system model considered and the time horizon ( T . Hence, the computational complexity of the DP method is O

∗

(40)

xˆ 0T

∑T −1 k=0

d(k+1)n

Let Jk (x¯ k ) be the optimal cost for the subproblem of (16) that starts at time k and ends at time T ,

(d + 1)

Jk∗ (x¯ k ) := inf E J¯|Ik .

Remark 4. It is shown in Han et al. (2015) that for linear Gaussian systems without uncertainty, closed-form expressions for the MMSE estimator with SET conditions can be obtained. In the case of uncertain systems with a relative entropy constraint, however, optimal LEQG estimates xˆ 0|0 , . . . , xˆ T |T and the optimal LEQG cost E[J0 (x¯ 0 )] with SET conditions cannot be analytically expressed. This is due to the combination of DP, exponential quadratic cost and binary decision variables γk introduced by the event trigger. Specifically, the expectation in DP equations (32) is taken with respect to γk+1 and yk+1 for all k ∈ N0:T −1 , leading to the presence of multiple different exponential quadratic functions of the decision variable xˆ k at each time k. Thus, the optimal solution of xˆ k would not be analytically expressed as a function of x¯ k . Note that this is different from the time-triggered LEQG state estimation problem (Speyer et al., 1992; Yoon et al., 2004). In that case since only a single exponential quadratic function of xˆ k exists at each time k, so that the optimal solution of xˆ k is linear in x¯ k . It is also worth mentioning that since the change of measure approach used in Collings et al. (1996), Dey and Moore (1997), Elliott et al. (1995), Huang et al. (2019) and Shaiju and Petersen (2009) does not lead to different results with respect to the traditional Bayes inference (Di Masi & Runggaldier, 1982), one cannot expect to obtain an analytical result by the change of measure approach.

[

]

(35)

xˆ kT

We shall show that Jk∗ (x¯ k ) = Jk (x¯ k ), k = 0, 1, . . . , T

(36)

by induction, so that for k = 0, it is proved that inf E J¯|I0 = J0∗ (x¯ 0 ) = J0 (x¯ 0 ),

[

]

(37)

xˆ 0T

and xˆ 0|0 , . . . , xˆ T |T generated by (31) and (32) are indeed minimizing the LEQG cost. First, for k = T , according to (35) we have JT∗ (x¯ T ) = inf J¯,

(38)

xˆ T

as J¯ is a function of IT . Therefore, JT∗ (x¯ T ) = JT (x¯ T ). Assume that at time k + 1, we have Jk+1 (x¯ k+1 ) = Jk∗+1 (x¯ k+1 ). Then at time k, Jk∗ (x¯ k ) = inf E J¯|Ik

[

]

xˆ kT

[ = inf E xˆ k

inf E J¯|Ik+1 |Ik

[

]

]

xˆ k+1T

[ ] = inf E Jk∗+1 (x¯ k+1 )|Ik xˆ k

= inf E [Jk+1 (x¯ k+1 )|Ik ] = Jk (x¯ k ).

(39)

xˆ k

where the second equality is due to the law of iterated expectations and the principle of optimality (Bertsekas, 2005) and the third equality is according to the definition of Jk∗+1 (x¯ k+1 ).

m

k+1

d

) m

.

In consideration of realistic implementability, we further look into a simpler yet more interesting one-step event-triggered minimax estimation problem, As we will demonstrate, this formulation allows us to derive a one-step event-triggered minimax estimator in recursive closed-form expressions.

J. Xu, Y. Tang, W. Yang et al. / Automatica 110 (2019) 108592

4. One-step event-triggered minimax estimation

and Ψˆ 0,k−1 is given by

The one-step problem is considered as follows. At each time k, only xˆ k is considered as the decision variable and assume that the estimates have been determined for all previous time instants 0, . . . , k − 1. That is, the estimation error cost Ve defined by (9) is relaxed as

Ψˆ 0,k−1 =

Ve,k =

k−1 1∑

2

1

∥xi − xˆ i|i ∥2Wi + ∥xk − xˆ k ∥2Wk ,

where estimates xˆ i|i , i < k are known and xˆ k|k is to be determined. Certainly, we require that xˆ k|k is a function of Ik . The one-step technique is widely used for estimation problems in the literature, e.g., Boel et al. (2002), Dey and Moore (1997) and Shi, Chen, and Shi (2014b). In this case, the optimization problem becomes a forward recursive estimation problem such that the information set Ik at time k is available for the minimax estimator design. To do this, we define a relative entropy between the conditional per˜ ·|Ik ) and the conditional nominal measure turbed measure P( P(·|Ik ) given the current available information Ik as follows:

˜ ·|Ik ) ∥ P(·|Ik ) h0 P(

(

{ :=

2

i=0

for k ≥ 1 and Ψˆ 0,−1 = 0. Here, Γk := {τk : τk > 0 and Jτk < ∞}. Then by using the duality relation (Dai Pra et al., 1996), (P1 ) can be further converted as the following equivalent problem (P1′ ) :

inf τk

{(

τk ∈Γk

}

)

inf log J L (τk ) + d ,

(47)

xˆ k

where

[

(

)

J L (τk ) = E exp Ψˆ 0,k−1 + Ψ¯ k (xk , xˆ k ) |Ik

]

(48)

is the one-step event-triggered LEQG cost. Therefore, the onestep event-triggered minimax estimator and the corresponding minimax cost (47) can be obtained via the derived corresponding LEQG estimator with τk∗ = arg infτk ∈Γk τk (log infxˆ k J L (τk ) + d). 4.1. One-step event-triggered LEQG estimation

)

˜ ·|Ik ) dP( dP(·|Ik )

E˜ [log

|Ik ],

+∞,

˜ ·|Ik ) ≪ P(·|Ik ), P( otherwise.

(43)

(

)

˜ ·|Ik ) ∥ P(·|Ik ) ≤ E˜ [Vu,k |Ik ] + d, h0 P(

(

)

Vu,k :=

k 1∑

2

∥Di xi ∥2 .

(44)

i=0

˜ ·|Ik ) is denoted by Ξk . Also, the set of all admissible measures P( Then the optimal estimation problem (P0 ) is relaxed as (P1 ) :

inf

sup

xˆ k P( ˜ ·|Ik )∈Ξk

E˜ [Ve,k |Ik ].

(45)

Remark 5. Note that the original event-triggered minimax optimal estimation problem (P0′ ) is considered on a finite-horizon T , while the one-step form can be available on an infinite horizon. Hence, the time k for the one-step problem is not restricted to the finite-horizon T . As the formulation of original optimal problem in Section 2, by using the Lagrange multiplier technique (Lemma 1), (P1 ) is transformed into the following unconstrained problem

{

[

Jτk :=τk inf sup E˜ [Ψˆ 0,k−1 + Ψ¯ k (xk , xˆ k )|Ik ] xˆ k P˜ ∈P k

˜ ·|Ik ) ∥ P(·|Ik ) − h0 P(

(

sup

xˆ k P( ˜ ·|Ik )∈Ξk

)]

E˜ [Ve,k |Ik ] = inf Jτk ,

(

)

]

xˆ k|k = arg min E exp Ψˆ 0,k−1 + Ψ¯ k (xk , xˆ k ) |Ik .

For this problem, we need to redefine the information state αk (xk ) in the one-step case such that

αk (xk ) :=

∫

∫ ···

Rn

Rn

exp(Ψˆ 0,k−1 )f (x0k , Ik )dx0k−1

Theorem 3. For the one-step event-triggered LEQG estimation problem (49), the optimal estimate xˆ k|k is given by xˆ k|k = Kk x¯ k , Kk = I + (Σk−1 − D′k Dk )−1 D′k Dk ,

(51)

where x¯ k and Σk are given via the following recursive equations for 0 < t ≤ k:

⎧ −1 ) ( ¯ t Σt−−11 x¯ t −1 − τk−1 Wt −1 xˆ t −1|t −1 Σt x¯ t = Qt−−11 At −1 Σ ⎪ ⎪ [ ] [ ] ⎪ −1 ⎪ ⎪ + Ct′ Rt + (1 − γt )Yt−1 γt yt + (1 − γt )βt , ⎨ [ ]−1 Σt−1 = Ct′ Rt + (1 − γt )Yt−1 Ct + Qt−−11 ⎪ ⎪ ⎪ ¯ t A′t −1 Qt−−11 , − Qt−−11 At −1 Σ ⎪ ⎪ ⎩ ¯ −1 −1 ′ Σt = At −1 Qt −1 At −1 + Σt−−11 − τk−1 Wt −1 − D′t −1 Dt −1 ,

(52)

and the initial values x¯ 0 and Σ0 are the same as in (29). The optimal estimator (51) yields finite LEQG cost J L (τk ) in (48), if and only if the following inequalities

At Qt At + Σt ′

where the function Ψ¯ k is defined in (18) with a time-varying τ such that ) 1( Ψ¯ k (xk , xˆ k ) = τk−1 ∥xk − xˆ k ∥2Wk + ∥Dk xk ∥2 , 2

(50)

and the initial value is again given by α0 (x0 ) := f (x0 , I0 ). The recursions for αk (xk ) in (28) now need to be made some changes to coincide with the redefined αk (xk ). Based on the redefined information state, we present the following result.

Σt−1 − τk−1 Wt − D′t Dt > 0, t = k, (46)

(49)

xˆ k

}

+d ,

τk ∈Γk

According to the LEQG cost (48), we have the following onestep event-triggered LEQG estimation problem:

[

˜ ·|Ik ) ∥ P(·|Ik ) < ˜ ·|Ik ) satisfying h0 P( The set of all measures P( ( ) ˜ ·|Ik ) ∥ P(·|Ik ) can be regarded +∞ is denoted by Pk . h0 P( as an a posteriori relative entropy compared with the previous defined relative entropy h(P˜ ∥ P). Such an a posteriori relative entropy between finite-state hidden Markov models is characterized in Xie, Ugrinovskii, and Petersen (2007). In this context, the relative entropy constraint in Definition 1 should be changed to the corresponding a posteriori form:

inf

k−1 ∑ ) 1 ( −1 τk ∥xi − xˆ i|i ∥2Wk + ∥Di xi ∥2

(42)

2

i=0

7

−1

−1

− τk Wt − Dt Dt > 0, t < k −1

′

(53) (54)

are satisfied. Moreover, the corresponding minimum LEQG cost is n

inf J L (τk ) = xˆ k

(2π ) 2 Sk

1

|Σk−1 − τk−1 Wk − D′k Dk |− 2 f (Ik ) ( ) 1 2 · exp ∥¯xk ∥Σ −1 K −Σ −1 , 2

k

k

k

(55)

8

J. Xu, Y. Tang, W. Yang et al. / Automatica 110 (2019) 108592

where Sk is obtained via the following recursion for 0 < t ≤ k St =St −1 (2π )

−mγt 2

1

1 + Yt | |Rt |− 2 |R− t

{ [ 1 · exp ∥ˆxt −1|t −1 ∥2τ −1 W 2

k

γt −1 2

1

¯ t| |Qt −1 |− 2 |Σ

where x˜ k , Λk and Zk are given by the following recursions:

1 2

− ∥¯xt −1 ∥2Σ −1

t −1

t −1

− ∥γt yt + (1 − γt )βt ∥2[R +(1−γ )Y −1 ]−1 + t

∥Σt−−11 x¯ t −1

t

t

− τk Wt −1 xˆ t −1|t −1 ∥Σ¯ t + ∥¯xt ∥2Σ −1 −1

2

t

]}

.

(56)

Proof. The proof is mainly based on the recursive results for x¯ k ,

Σk and Sk in Lemma 2. For the cost function in (49), we have [ ( ) ] ˆ ¯ E exp Ψ0,k−1 + Ψk (xk , xˆ k ) |Ik ∫ ( ) = αk (xk ) exp Ψ¯ k (xk , xˆ k ) dxk /f (Ik ) Rn ( ∫ Sk 1( = exp − ∥xk − x¯ k ∥2 −1 Σk f (Ik ) Rn 2 ) ) 2 2 dxk − ∥xk − xˆ k ∥τ −1 W − ∥Dk xk ∥ k

k

− 12

n 2

=(2π ) |Σk−1 − τk−1 Wk − D′k Dk |

Sk /f (Ik ) exp(qk /2),

⎧ [ ]−1 x˜ k = Ak−1 x˜ k−1 + Λk Ck′ Rk + (1 − γk )Yk−1 ⎪ ⎪ ] [ ⎪ ⎪ ⎪ · γk yk + (1 − γk )βk − Ck Ak−1 x˜ k−1 , ⎪ ⎪ [ ] ⎪ −1 1 ′ ⎪ Λ− ⎪ k = Ak−1 Λk−1 Ak−1 + Qk−1 ⎪ ⎪ [ ] − 1 ⎪ − 1 ⎪ + Ck′ Rk + (1 − γk )Yk Ck , ⎪ ⎪ ⎪ γk −1 ⎪ −mγk 1 1 ⎪ 1 2 ⎪ |Qk−1 |− 2 ⎨Zk = Zk−1 (2π ) 2 |Rk |− 2 |R− k + Yk | 1 { [ ⎪ ⎪ ¯ ⎪ · |Λk | 2 exp 12 ∥˜xk−1 ∥2 −1 ¯ −1 −1 ⎪ ⎪ Λk−1 Λk Λk−1 −Λk−1 ⎪ ⎪ ⎪ 2 ⎪ − ∥γ y + (1 − γ ) β ∥ ⎪ k k k k ⎪ [Rk +(1−γk )Yk−1 ]−1 ⎪ ⎪ ]} ⎪ ⎪ ⎪ ⎪ + ∥˜xk ∥2 −1 , ⎪ Σk ⎪ ⎪ ⎩ −1 1 ¯ k = A′k−1 Qk−−11 Ak−1 + Λ− Λ k−1 .

The initial values of x˜ k , Λk and Zk are the same as those of x¯ k , Σk and Sk in (29). Proof. According to Definition 2, it can be realized that αk (xk ) reduces to ηk (xk ) if τ −1 → 0 and Dk = 0. Then the recursive result for ηk (xk ) can be obtained via Theorem 1 by letting τ −1 → 0 and Dk = 0. ■

(57)

From the definition of ηk (xk ) and Lemma 4, we have

where

∫

qk =∥ˆxk ∥2 −1

τk Wk

n

f (I k ) =

− ∥¯xk ∥2Σ −1

Rn

k

+∥Σk−1 x¯ k − τk−1 Wk xˆ k ∥2(Σ −1 −τ −1 W k

k

′

(63)

1

ηk (xk )dxk = (2π ) 2 |Λk | 2 Zk .

(64)

Then applying the results of Theorem 3, Lemma 4 and (64) to the minimax problem (P1′ ), we obtain the following proposition.

−1

k −Dk Dk )

and the above integral exists if and only if

Σk−1 − τk−1 Wk − D′k Dk > 0.

(58)

Note that the inequality (58) itself ensures that qk is positive in xˆ k . Then by solving ∂ qk /∂ xˆ k = 0, the optimal estimate xˆ k|k in the context of one-step decision can be obtained: xˆ k|k = Kk x¯ k , Kk = I + (Σk−1 − D′k Dk )−1 D′k Dk .

(59)

Here x¯ k and Σk are given via recursive equations (52) and according to Theorem 1 the following condition At Qt At + Σt ′

−1

−1

− τk Wt − Dt Dt > 0 −1

′

(60)

is required to be satisfied for all t < k. Substituting (59) into (57), we can obtain (55). The proof is then complete. ■ 4.2. Determination of the one-step event-triggered minimax estimator According to (P1′ ) and the representation of the minimal LEQG cost (55), it is known that to determine the optimal Lagrange multiplier τk∗ , we need an explicit expression of f (Ik ). In the sequel, we define an information state ηk (xk ) to help evaluate f (Ik ). Definition 3.

The information state ηk (xk ) is defined by

ηk (xk ) := f (xk , Ik ),

(61)

where f (xk , Ik ) represents the joint distribution of xk and Ik . Lemma 4. The information state ηk (x) is an unnormalized Gaussian density given by

) ( 1 ηk (xk ) = Zk exp − ∥xk − x˜ k ∥2Λ−1 , 2

k

(62)

Proposition 1. For the one-step event-triggered minimax estimation problem (P1′ ), the minimax estimator is given by the corresponding LEQG estimator (Theorem 3) with

τk∗ = arg inf τk τk ∈Γk

{(

}

)

inf log J L (τk ) + d ,

(65)

xˆ k

where inf log J L (τk ) = log Sk − log Zk + 0.5 ∥¯xk ∥2 −1

(

xˆ k

Σk Kk −Σk−1

) − log(|Σk−1 − τk−1 Wk − D′k Dk ∥ Λk |) . Moreover, the resulting one-step minimax cost is τk∗ (log infxˆ k J L (τk∗ )+ d). Remark 6. Note that τk (log infxˆ k J L (τk ) + d) < ∞ is guaranteed by the constraint τk ∈ Γk , which is equivalent to the conditions (53) and (54). An explicit expression for τk∗ is not easily obtainable from (65), but we can find τk∗ numerically over τk ∈ [τk,b , ∞), where τk,b is the critical value below which the conditions (53) and (54) are violated. The one-step minimax cost (47) at each time k is then achieved by τk∗ and the estimation performance is guaranteed by this minimax cost. Also, it follows from (53) and (54) that if the uncertainty matrix Dk is very large, τk,b will not exist. Thus, Dk has an upper limit, which is a limitation of the relative entropy approach to deal with estimation and control problems of uncertain systems (Petersen et al., 2000; Yoon et al., 2004). Remark 7. By writing θk = τk−1 , it is interesting to note that the proposed one-step minimax estimator has a similar form with the event-triggered risk-sensitive estimator proposed in Huang et al. (2019). The parameter τk−1 plays a role similar to that of a risk-sensitive parameter. This is due to the fact that the onestep minimax problem also involves an exponential quadratic

J. Xu, Y. Tang, W. Yang et al. / Automatica 110 (2019) 108592

cost, which coincides with the risk-sensitive cost. The difference is that the quadratic function in the exponential of (49) includes the term ∥Dk xk ∥2 , which is introduced by the relative entropy constraint (44). In addition, our estimation algorithm involves the optimization of τk , while it is difficult to find an optimal risksensitive parameter in Huang et al. (2019). The comparison with the risk-sensitive estimator will be provided in the simulation section and it is shown that the existence of Dk can lead to improved robustness of estimation performance. 5. Extension to the multi-sensor system scenario In this section, we extend the estimation results in the previous sections to the multi-sensor system scenario in a sequential fusion way. Consider a set of M sensors such that for the nominal system the measurement processes are yik = Cki xk + vki , i = 1, 2, . . . , M .

(66)

where ∈ R is the ith sensor measurement, v i ∈ N1:M are mutually independent Gaussian noises with densities φki ∼ N (0, Rik ) and independent of wk and x0 . Then the nominal prob1 ability measure P is defined on the noise space {x0 , w0T , v0T ,..., M v0T }. Now consider that each sensor i is equipped with an event trigger to decide whether yik is sent to the remote estimator. Likewise, let γki ∈ {0, 1} be the transmission decision variable of yik . Define Ilk := {i ∈ N1:l |γi = 0}, 1 ≤ l ≤ M. We further define yik

⋃ {

Ikl :=

i k,

mi

} j M {γij |j ∈ IM k } ∪ {yi |j ∈ N1:M \Ik }

Theorem 4. For the multi-senor system (66) where each sensor is equipped with the event-triggered scheme (67), the one-step event-triggered minimax estimates xˆ ik|k are given by xˆ ik|k = Kki x¯ ik , Kki = I + ((Σki )−1 − D′ k Dk )−1 D′ k Dk , i = 1, 2, . . . , M .

where ¯ and are evaluated via the following recursions for 0 < t ≤ k: (1) for 1 < i ≤ M,

⎧ [ ]−1 ⎪ (Σti )−1 x¯ it = (Cti )′ Rit + (1 − γti )(Yti )−1 ⎪ ⎪ ( ) ⎨ · γti yit + (1 − γti )βti + (Σti−1 )−1 x¯ it−1 , [ i ] i −1 i ′ i i −1 −1 i ⎪ Ct ⎪ ⎪(Σt ) = (Ct ) Ri−t 1+ (1 − γt )(Yt ) ⎩ −1 + ( Σt ) ; (2) for i = 1, [ ⎧ i −1 i ¯ t (ΣtM−1 )−1 x¯ M (Σt ) x¯ t = Qt−−11 At −1 Σ ⎪ ⎪ ] t[−1 i ⎪ ⎪ − τk−,i1 Wt −1 xˆ M Rt + ⎪ t −] 1|t −1( + ⎪ ) ⎪ ⎪ i i −1 −1 i i ⎪ (1 − γ )(Y ) γ y + (1 − γti )βti , ⎨ t t t t [ ]−1 i (Σti )−1 = (Cti )′ Rit + (1 − γti )(Yti )−1 Ct + Qt−−11 ⎪ ⎪ −1 −1 ′ ¯ ⎪ − Qt −1 At −1 Σt At −1 Qt −1 , ⎪ ⎪ ⎪ ⎪ ¯ t−1 = A′t −1 Qt−−11 At −1 + (ΣtM−1 )−1 − τk−,i1 Wt −1 Σ ⎪ ⎪ ⎩ − D′t −1 Dt −1 .

′

Pr(γ = 0| ,

i−1 Ik )

(

1

= exp − ∥ 2

yik

−β ∥

i 2 k Yi k

)

(67)

(P2′ ) :

inf τk,i

}

)

inf log J L,i (τk,i ) + d ,

τk,i ∈Γk,i

(68)

xˆ k

(

)

]

J L,i (τk,i ) = E exp Ψˆ 0,k−1 + Ψ¯ k (xk , xˆ k ) |Iki .

(69)

Functions Ψ¯ k and Ψˆ 0,k−1 are the same as those for the singlesensor case. We also have the following LEQG estimation problem: xik|k

ˆ

[

) ] ˆ ¯ = arg min E exp Ψ0,k−1 + Ψk (xk , xˆ k ) |Iki

(

(70)

xˆ k

and define the information state for the multi-sensor system scenario:

αki (xk ) :=

∫

∫ ··· Rn

Rn

(ΣtM )−1

−

τk−,i1 Wt

− Dt Dt > 0, t < k. ′

(75) (76)

Moreover, τk∗,i is given by

τk∗,i = arg inf τk,i

{(

τk,i ∈Γk,i

}

)

inf log J L,i (τk,i ) + d ,

(77)

xˆ k

exp(Ψˆ 0,k−1 )f (x0k , Iki )dx0k−1 .

inf log J L,i (τk,i ) xˆ k

( = log S˜ki − log Z˜ki + 0.5 ∥¯xik ∥2(Σ i )−1 K i −(Σ i )−1

k k k ) − log(|(Σki )−1 − τk−,i1 Wk − D′k Dk ∥ Λik |) .

Here for all 0 < t ≤ k, if 1 < i < M, S˜ki is evaluated via S˜ki = S˜ki−1 (2π )

(71)

The initial value is given by α0i (x0 ) := f (x0 , I0i ). Then we have the following result on the one-step event-triggered minimax estimation for multi-senor systems.

−mi γki 2

− 12

|Rik |

|(Rik )−1 + Yki |

γki −1 2

(

) 1 · exp − ∥γki yik + (1 − γki )βki ∥2[Ri +(1−γ i )(Y i )−1 ]−1 , 2

where

[

(74)

where

,

where Yki ∈ Rmi ×mi is positive definite, and βki ∈ Rmi is known to the estimator based on Iki−1 . As in the case of a single sensor, we use a relative entropy constraint to describe the system uncertainty. Likewise, for the multi-sensor case, the analytical solution for the event-triggered minimax optimal estimator cannot be obtained. Thus, we consider the one-step problem as (P1′ ):

{(

−1

At Qt At +

with I00 := ∅ as the available information set of the estimator before considering the information from sensor l + 1 at time k. Write Ik0 := IkM−1 . Then a general stochastic event-triggered scheme for each sensor can be designed as

(73)

The optimal estimator (72) exists if and only if (Σti )−1 − τk−,i1 Wt − D′t Dt > 0, t = k,

⋃ j {γk |j ∈ Ilk } ∪ {yjk |j ∈ N1:l \Ilk }

yik

(72)

Σki

xik

i∈N0:k−1

i k

9

k

k

k

and if i = 1, S˜ki has the same form as that of Sk in (56). S˜ki and Σki reduce to Z˜ki and Λik respectively if τk−,i1 → 0 and Dk = 0. Proof. The proof can be completed by following a similar procedure as that in the single-sensor case. Remark 8. In this paper, we focus on a general SET scheme that the transmission probability for sensor i at time k relates to the previous information set Iki−1 . This can happen when we choose βki = Cki xˆ ki−1 . Therefore, x¯ k|k and xˆ ik|k are calculated by sequentially updating the measurement information as the way in Shi et al. (2014a). However, if the event-triggering scheme is the form of Pr(γki = 0|yik ) (Shi et al., 2016; Weerakkody et al., 2016), i.e., it is independent of the previous information set, then x¯ k|k can be calculated in a lumped fashion by updating all the measurement information at time k simultaneously.

10

J. Xu, Y. Tang, W. Yang et al. / Automatica 110 (2019) 108592

6. Simulation examples In this section, we illustrate the previous proposed estimation results by two examples: the single-sensor and the multi-sensor system scenarios. (1) Single-sensor system. Consider the following second-order uncertain system 0.6 0

0.2 −0.6

1

1

]

0 1

]

[ xk+1 = yk =

[ [

zk =

1 0

]

xk + w ˜ k,

xk + v˜ k , xk .

(78)

The nominal noise signals wk and vk are assumed to be Gaussian with covariance matrices Qk = I and Rk = 1, respectively. Also, the initial condition x0 is assumed to be subject to the distribution N (0, I). We will apply our one-step event-triggered (ET) minimax estimation technique in Proposition 1 to the uncertain systems (78) with the corresponding relative entropy constraint (44). Here, the uncertainty matrix Dk and the constant d are given by Dk = 0.38I , d = 10−6 .

Fig. 2. Optimal Lagrange multiplier τk∗ when δ = 0.38 for a single-sensor system.

(79)

Considering the uncertain systems with a set of system parametric perturbations, let the perturbed process noise be

w ˜ k = δ I · xk + wk , δ ∈ [−0.38, 0.38].

(80)

The true measurement noise v˜ k and initial condition x˜ 0 are assumed to be the same as the nominal case. Then according to (5), the relative entropy between the conditional nominal measure P and the conditional perturbed measure P˜ is h0 P˜ k (·|Ik ) ∥ Pk (·|Ik ) =

(

)

k 1∑

2

E˜ [∥xi ∥22

δ Qi−1

i=0

|Ik ]

k

<

1∑ 2

E˜ [∥Di xi ∥2 |Ik ] + d,

i=0

which strictly satisfies the conditional relative entropy constraint (44). To show the robustness of the one-step ET minimax estimator, the ET Kalman filter (KF) designed for a nominal system (Han et al., 2015) and the ET risk-sensitive estimator proposed by Huang et al. (2019) are also considered. For the three estimators, we use the ‘‘send-on-delta’’ triggering strategy (Miskowicz, 2006) on the general SET scheme (7), i.e., βk in (7) is the previously transmitted measurement, 2 and choose the triggering parameter Yk = 0.5. Thus, the triggering time instants for the estimators are the same. We search the optimal Lagrange multiplier τk∗ numerically to obtain the one-step ET minimax estimator. Fig. 2 plots τk ∗ for k = 1, . . . , 300 when δ = 0.38 under a realized available information set. In this robustness comparison, the time horizon is T = 100 and we perform a Monte Carlo simulation with 10 000 trials] to compute the estimation [ error cost

∑T

k=1

Fig. 3. Estimation error costs under different values of the risk-sensitive parameter θ when δ = 0.38.

E˜ ∥xk − xˆ k|k ∥2L′ L /2 . Since for the risk-sensitive k k

for the estimators under entire range of δ . 3 From Fig. 4, we can observe that the one-step minimax ET estimator is less sensitive to perturbations compared with the ET risk-sensitive estimator and the ET KF. This verifies the analysis in Remark 7 that the existence of Dk in the relative entropy constraint can indeed lead to improved estimation robustness. (2) Multi-sensor system. In this example, we compare the onestep ET minimax estimator in Theorem 4 with the time-triggered optimal minimax estimator (Yoon et al., 2004) to illustrate the effectiveness of the proposed estimator when the communication resource is limited. Since the derived minimax estimator in Yoon et al. (2004) is aiming for one-step delayed measurement equation, we give the optimal minimax estimator for the current-state measurement equation case in Appendix C. And this estimator is used in this example. In this case, we consider that the system is measured by the following three sensors 1

]

xk + vk1 ,

0

]

xk + vk2 ,

1

]

xk + vk3 ,

estimator it is difficult to give an effective method to suggest how to select an optimal risk-sensitive parameter θ , we fix δ = 0.38 and use the Monte Carlo method to find a relatively better value of θ . Fig. 3 reveals that the θ = 0.37 is the best choice, which has the smallest cost. Then Fig. 4 plots the estimation error costs

y1k =

[

y2k =

[

y3k

[

2 Although in Han et al. (2015) and Huang et al. (2019) the form of the ‘‘send-on-delta’’ triggering strategy for the SET scheme is not considered, the corresponding ET estimators using ‘‘send-on-delta’’ triggering strategy can be derived via similar procedures in Han et al. (2015) and Huang et al. (2019).

3 Since different δ ’s lead to different true systems, the sensor-to-estimator communicate rates are not the same for different δ ’s even using the same Yk . For Fig. 4 the average communicate rates decrease from 0.0.7985 to 0.4792 on δ ∈ [−0.38, 0.38].

=

1 1 0

(81)

J. Xu, Y. Tang, W. Yang et al. / Automatica 110 (2019) 108592

Fig. 4. Estimation error comparison for perturbed systems.

with noise covariances R1k = 1, R2k = 0.1, R3k = 0.1. Likewise, the ‘‘send-on-delta’’ triggering strategy is used and let Yk1 = 1, Yk2 = 4, Yk1 = 3.

(82)

In this case, we consider the system perturbation parameter δ = 0.38. The optimal τ for the time-triggered minimax estimator is τ = 0.1681, which is computed via (C.6) in Appendix C. The estimation performance for the two estimators is shown in Fig. 5. The average communication rates for the sensors 1, 2, and 3 are 0.5688, 0.5938∑ and 0.6625, respectively and γk in Fig. 5 3 is computed by 1/3 i=1 γi . From Fig. 5, we can observe that despite the obviously decreased communication rates for each sensor, the one-step ET minimax estimation performance does not degrade much compared with the time-triggered optimal one. On the other hand, this comparable estimation performance also shows the proper approximation for adopting the one-step idea.

of uncertainty has been investigated. Since an analytical optimal solution is not able to be obtained, a one-step ET minimax estimation problem is further studied and its solution has a recursive closed form. It is shown that the one-step ET minimax estimator has a similar form with the ET risk-sensitive estimator proposed in Huang et al. (2019). The one-step estimation results for the multi-sensor system scenario are also obtained. Finally, by examples of uncertain systems with a relative entropy constraint, the simulation results show that the proposed one-step ET minimax estimator has a better robustness than the ET Kalman filter and the ET risk-sensitive estimator proposed by Huang et al. (2019) and has an accepted estimation performance at a reduced communication cost. One possible future research direction is to explore a tractable ET minimax estimator that has a better robustness than the onestep one. For the minimax estimation formulation considered, it would be of interest to analyze the worst-case dynamics, for which the related problem of minimax control was considered in Yoon, Ugrinovskii, and Petersen (2005). Future work also includes sensor scheduling (Shi, Cheng, & Chen, 2011; Xu, Wen, Ge, & Xu, 2017) for uncertain systems with a relative entropy constraint and consensus minimax estimation for a distributed sensor network with relative entropy constraints. Appendix A. Proof of Lemma 2 According to the definition of αk (xk ) in (19), if γk = 0, we have

α∫k (xk ) ∫ = ··· exp(Ψ0,k−1 )f (x0k , γk = 0, Ik−1 )dx0k−1 ∫Rn ∫R n = ··· exp(Ψ0,k−1 ) Pr(γk = 0|x0k , Ik−1 ) Rn

· f (xk |xk−1 )f (x0k−1 , Ik−1 )dx0k−1 ,

(A.1)

Pr(γk = 0|x0k , Ik−1 )

=

In this work, an event-triggered (ET) state estimation problem under minimax formulation with a relative entropy description

Rn

where the last equality follows from the Bayes’ law and the independent property of noises. Since

∫

7. Conclusion

11

∫R

m

= Rm

Pr(γk = 0|x0k , yk , Ik−1 )f (yk |x0k , Ik−1 )dyk Pr(γk = 0|yk , Ik−1 )φk (yk − Ck xk )dyk ,

Fig. 5. Estimation performance of one-step ET minimax estimator when system perturbation parameter δ = 0.38.

(A.2)

12

J. Xu, Y. Tang, W. Yang et al. / Automatica 110 (2019) 108592

we further have

Again by Lemma 3,

αk (xk ) ∫ ∫ = ···

s2 = ∥z ∥Σ¯ k − ∥ˆxk−1 ∥2τ −1 W

Rn

Rn

( ) exp Ψ¯ k−1 (xk−1 , xˆ k−1 ) exp(Ψ0,k−2 )

f (x0k , yk , Ik−1 ) =f (xk , yk |x0k−1 , Ik−1 )f (x0k−1 , Ik−1 ) =f (yk |xk )f (xk |xk−1 )f (x0k−1 , Ik−1 )

(A.3)

(A.4)

Appendix C. Time-triggered minimax optimal estimator for the current-state measurement model case (A.5)

■

Appendix B. Proof of Theorem 1 We prove this theorem by induction. As in Lemma 2, we consider two cases γk = 0 and γk = 1, respectively. Since the derivation for α0 (x0 ) can be easily completed following a similar argument as the general case αk (xk ), we omit the derivation for α0 (x0 ) here. Assume that at time k − 1,

αk−1 (xk−1 ) = Sk−1 exp − ∥xk−1 − x¯ k−1 ∥Σ −1 2

2

)

.

(B.1)

k−1

Then we prove that (26) holds for αk (xk ). For brevity, xk , yk are simplified as x and y in the following. According to Lemma 2, for γk = 0 we have

αk (x) − 12

|Rk | |∫Qk−1 | ) ( ) exp −s1 /2 dy exp −s2 /2 dz ,

(

(B.2)

Rn

s1 = ∥y − Ck x∥2−1 + ∥y − βk ∥2Yk Rk

(

)−1

x¯ k , k = 0, 1, . . . , T .

− ∥z − xˆ k−1 ∥τ −1 W

k−1

− ∥Dk−1 z ∥ + ∥Ak−1 z − x∥2Qk−1 . 2

Here, Σk and Pk are given by the following recursive equations: −1 −1 1 −1 ¯ ′ Σk−1 =Ck′ R− k Ck + Qk−1 − Qk−1 Ak−1 Σk Ak−1 Qk−1 , − 1 − 1 − 1 ′ − 1 ¯ k =Ak−1 Qk−1 Ak−1 + Σk−1 − τ Wk−1 − D′k−1 Dk−1 , Σ Pk =D′k Dk + A′k Pk+1 (I − Qk Pk+1 )−1 Ak ,

Σk−1 − τ −1 Wk − D′k Dk > 0, Σk−1 − Pk > 0

(C.3)

are required to be satisfied. Also, x¯ k is given by the following recursion:

s1 =∥y − βk − (Ck x − βk )∥2−1 + ∥y − βk ∥2Yk

where J L (τ ) is computed by

Rk

=∥y − βk − Rk + Yk + ∥Ck x − β ∥

2 k (Rk +Yk−1 )−1

(Ck x − βk )∥

.

(C.4)

(C.5)

In addition, the optimal Lagrange multiplier τ is given by

( ) τ ∗ = arg inf τ inf log J L (τ ) + d ,

) −1 −1

(C.2)

and the following conditions

By using the completing square technique given by Lemma 3 and the matrix inversion lemma, we have

(

(C.1)

1 −1 ′ Σ0−1 =C0′ R− 0 C0 + Π0 , PT = DT DT , 1 Σ0−1 x¯ 0 =Π0−1 π0 + C0 R− k y0 .

2

Σk−−11

xˆ k|k = I − Σk Pk

The above recursions start from

and s2 =∥z − x¯ k−1 ∥

This minimax estimator can be derived following the argument in Yoon et al. (2004) or a similar procedure in this paper. For the current-state measurement model (1) and the relative entropy constraint uncertainty description in Definition 1, the time-triggered minimax optimal estimator is given by

( ) ¯ k Σk−−11 x¯ k−1 − τ −1 Wk−1 xˆ k−1|k−1 Σk−1 x¯ k =Qk−−11 Ak−1 Σ 1 + Ck′ R− k yk .

where

2

(B.6)

k

Here, the expressions of Σk , x¯ k and Sk are given by (27) and (28) when γk = 0. Similarly, we can show that (B.6) holds for γk = 1. Thus this proof is complete. ■

Rn

1

Note that the integral with respect to z in (B.2) is finite if and only if A′k Qk−1 Ak + Σk−1 − τ −1 Wk − D′k Dk > 0. Using Lemma 3 again, we obtain 2

· ψk−1 (xk − Ak−1 z)αk−1 (z)dz .

Rm

(B.5)

( ) 1 αk (x) = Sk exp − ∥x − x¯ k ∥2Σ −1 .

Rn

·

1

1

− 1 ¯ k| 2 =Sk−1 |Rk |− 2 |R− + Yk | 2 |Qk−1 |− 2 |Σ (1( k )) · exp ∥ˆxk−1 ∥2τ −1 W − ∥¯xk−1 ∥2Σ −1 k−1 k−1 (2 1( · exp − ∥Ck x − βk ∥2(R +Y −1 )−1 + ∥x∥2Q −1 − k 2 k k−1 )) ∥A′k−1 Qk−−11 x + Σk−−11 x¯ k−1 − τ −1 Wk−1 xˆ k−1 ∥2Σ¯ . k

· f (yk |xk )f (xk |∫xk−1 )f (x0k−1 , Ik−1 )dx0k−1 ( ) =φk (yk − Ck xk ) exp Ψ¯ k−1 (z , xˆ k−1 )

=Sk∫ −1 (2π )

1

1

α∫k (xk ) ∫ = ··· exp(Ψ0,k−1 )f (x0k , yk , Ik−1 )dx0k−1 ∫R n ∫R n ( ) = ··· exp Ψ¯ k−1 (xk−1 , xˆ k−1 ) exp(Ψ0,k−2 )

− 12

(B.4)

αk (x)

and further obtain

n − m+ 2

k−1

¯ k is given by (27). Substituting (B.3) and (B.4) into (B.2) where Σ and using the fact that the integration of a Gaussian density function over the entire space equals one, we have

where the last equality follows from the definition of αk−1 (xk−1 ). For γk = 1, by the Bayes’ law, we similarly have

(

k−1

k

Rm

The proof is then complete.

+ ∥¯xk−1 ∥2Σ −1 + ∥x∥2Q −1

−∥A′k−1 Qk−−11 x + Σk−−11 x¯ k−1 − τ −1 Wk−1 xˆ k−1 ∥2Σ¯ ,

· f (xk |xk−1 )f (x0k−1 , Ik−1 )dx0k−1 ∫ · Pr(γk = 0|yk , Ik−1 )φk (yk − Ck xk )dyk ∫ Rm ( ) = exp Ψ¯ k−1 (z , xˆ k−1 ) ψk−1 (xk − Ak−1 z)αk−1 (z) Rn ∫ · dz Pr(γk = 0|yk , Ik−1 )φk (yk − Ck xk )dyk ,

Rn

k−1

2 Rk +Yk−1

τ ∈Γ

− 21

(B.3)

(C.6)

xˆ 0T

J L (τ ) =|ΣT−1 − τ −1 WT − D′T DT |

T ∏ k=1

1

1

Fk |Π |− 2 |R0 |− 2

J. Xu, Y. Tang, W. Yang et al. / Automatica 110 (2019) 108592 1 −1 − 2

−1 −1 ′ { · |R0 − R0 C0 K0 Σ0 C0 R0 | 1 · exp − ∥π∥ −1 −1 [ −1 ′ Π

2

1

+Π 1

}

C0 R0 C0 −(K0 Σ0 )−1

]−1

Π −1

,

1

¯ k | 2 |Rk |− 2 Fk =|Qk−1 |− 2 |Σ

− 12

1 −1 ′ −1 · | R− k − Rk Ck Kk Σk Ck Rk |

.

(C.7)

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Jiapeng Xu received the B.Eng. degree from the Department of Automation, Three Gorges University, Yichang, China, in 2014 and the M.Eng. degree in control engineering at Hangzhou Dianzi University, Hangzhou, China, in 2017. He is currently pursuing the Ph.D. degree in control science and engineering at the East China University of Science and Technology, Shanghai, China. His research interests include cyber– physical systems, event-triggered state estimation and distributed filtering.

Yang Tang received the B.S. and Ph.D. degrees in electrical engineering from Donghua University, Shanghai, China, in 2006 and 2010, respectively. From 2008 to 2010, he was a Research Associate with The Hong Kong Polytechnic University, Hong Kong. From 2011 to 2015, he was a Post-Doctoral Researcher with the Humboldt University of Berlin, Berlin, Germany, and with the Potsdam Institute for Climate Impact Research, Potsdam, Germany. Since 2015, he has been a Professor with the East China University of Science and Technology, Shanghai. His current research interests include distributed estimation/control/optimization, cyber–physical systems, hybrid dynamical systems, and artificial intelligence and their applications. Prof. Tang was a recipient of the Alexander von Humboldt Fellowship and the ISI Highly Cited Researchers Award by Clarivate Analytics in 2017 and 2018. He is a Senior Board Member of Scientific reports, an Associate Editor of the Journal of the Franklin Institute, Neurocomputing, the Proceedings of the Institution of Mechanical Engineers, Part I—The Journal of Systems and Control Engineering, and a Leading Guest Editor of the Journal of the Franklin Institute and CHAOS.

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J. Xu, Y. Tang, W. Yang et al. / Automatica 110 (2019) 108592 Wen Yang is a Professor at East China University of Science and Technology. She received her B.Sc. degree in Mineral Engineering in 2002 and M.Sc. degree in Control Theory and Control Engineering from Central South University in 2005, Hunan, China, and Ph.D. degree in Control Theory and Control Engineering from Shanghai Jiao Tong University, Shanghai, China, in 2009. She was a Visiting Student with the University of California, Los Angeles, from 2007 to 2008. Her research interests include Information fusion, state estimation, network security, coordinated control, complex networks and

reinforcement learning.

Fangfei Li received the Ph.D. degree in applied mathematics from Tongji University, Shanghai, China, in 2012. From Aug. 2016 to Aug. 2017, she was a Research Fellow with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. From Jan. 2018 to Feb. 2018, she was a Senior Research Assistant with the Department of Mathematics, City University of Hong Kong. She is currently an Associate Professor with the Department of Mathematics, East China University of Science and Technology, Shanghai. She has published more than 30 papers in international journals. Her research interests include logical systems, semi-tensor

product of matrices, secure control of cyber–physical systems, artificial intelligence, and their applications. She is an Associate Editor of IEEE ACCESS.

Ling Shi received the B.S. degree in electrical and electronic engineering from Hong Kong University of Science and Technology, Kowloon, Hong Kong, in 2002 and the Ph.D. degree in Control and Dynamical Systems from California Institute of Technology, Pasadena, CA, USA, in 2008. He is currently an associate professor at the Department of Electronic and Computer Engineering, and the associate director of the Robotics Institute, both at the Hong Kong University of Science and Technology. His research interests include cyber– physical systems security, networked control systems, sensor scheduling, event-based state estimation , and exoskeleton robots. He is a senior member of IEEE. He served as an editorial board member for The European Control Conference 2013–2016. He was a subject editor for International Journal of Robust and Nonlinear Control (2015–2017). He has been serving as an associate editor for IEEE Transactions on Control of Network Systems from July 2016, and an associate editor for IEEE Control Systems Letters from Feb 2017. He also served as an associate editor for a special issue on Secure Control of Cyber–physical Systems in the IEEE Transactions on Control of Network Systems in 2015–2017. He served as the General Chair of the 23rd International Symposium on Mathematical Theory of Networks and Systems (MTNS 2018).