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Event-basedState

Estimation,CommunicationRate Analysis

Dawei Shi,Tongwen Chenand Ling Shi

PreliminariesProblemDescription

Main ResultsFundamentalLemma

Main Result 1

Main Result 2

Example

Discussions

Acknowledg-ment

2014 American Control Conference

Event-based State Estimation of LinearDynamical Systems: Communication

Rate Analysis

Dawei Shi†, Tongwen Chen† and Ling Shi‡

† Department of Electrical and Computer Engineering, University of Alberta,Edmonton, AB, Canada

‡ Department of Electronic and Computer Engineering, Hong Kong Universityof Science and Technology, Kowloon, Hong Kong

www.ece.ualberta.ca/~dshi

www.ece.ualberta.ca/~dshi

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Event-based State Estimation

Figure 1 : Block diagram of the event-based remote estimation scenario.

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Problem Description

• Discrete-time LTI process driven by white noise:

xk+1 = Axk + wk, (1)

where wk is zero-mean Gaussian with covariance Q ≥ 0.• The initial state x0 is Gaussian with E(x0) = µ0 and

covariance P0 ≥ 0.• Smart sensor:

yk = Cxk + vk, (2)

where vk ∈ Rm is zero-mean Gaussian with covarianceR > 0.

• Assume (A,Q) is stabilizable, and (C,A) is detectable.

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Event-based Data Scheduler

• At each time instant k, the estimator provides a predictionxk|k−1 of xk and sends it to the scheduler.

• The scheduler computes γk according to:

γk =

0, if ‖yk − Cxk|k−1‖∞ ≤ δ1, otherwise (3)

• Only when γk = 1, the sensor transmits yk to the estimator.

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Event-based State Estimator

• For this type of scenario, several estimates have beenproposed, e.g., [1]-[4].

• We consider a simple estimator of the form proposed in [5]:

xk|k−1 = Axk−1|k−1, (4)

xk|k = xk|k−1 + γkPkC>(R+ CPkC

>)−1(yk − Cxk|k−1),(5)

where Pk evolves according to

Pk = APk−1A> +Q− γkAPk−1C

>(CPk−1C> +R)−1CPk−1A

>.

[1] J. Wu, Q. Jia, K. Johansson, and L. Shi, Event-based sensor data scheduling: Trade-off between communication rate andestimation quality,” IEEE Transactions on Automatic Control, vol. 58, no. 4, pp. 1041-1045, 2013.

[2] J. Sijs and M. Lazar, “Event based state estimation with time synchronous updates,” IEEE Transactions on AutomaticControl, vol. 57, no. 10, pp. 2650-2655, 2012.

[3] D. Shi, T. Chen and L. Shi. “Event-triggered maximum likelihood state estimation,” Automatica, 50(1), pp. 247-254, 2014.

[4] D. Shi, T. Chen, and L. Shi, “An event-triggered approach to state estimation with multiple point-and set-valuedmeasurements,” Automatica, 50(6), pp. 1641–1648, 2014.

[5] S. Trimpe and R. D’Andrea, “An experimental demonstration of a distributed and event-based state estimation algorithm,” inProceedings of the 18th IFAC World Congress, Milano, Italy, 2011.

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Communication Rate Analysis Problem

• Conditioned on the received information Ik−1, the predictionerror ek|k−1 := xk − xk|k−1 is zero-mean Gaussian withCov(ek|k−1|Ik−1) = Pk.

• Define zk := yk − Cxk|k−1. We have E(zk|Ik−1) = 0 andE(zkz

>k |Ik−1) := Φk = CPk|k−1C

> +R.

• Define Ω := z ∈ Rm| ‖z‖∞ ≤ δ. We have

E(γk|Ik−1) = 1−∫

Ω

fzk(z)dz, (6)

where fzk(z) = (2π)−m/2(detΦk)−1/2 exp (− 12z>Φ−1

k z).

• Objective: To provide lower and upper bounds for E(γk|Ik−1).

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Fundamental Lemma

• Define Ω0 := z| z>Φ−1k z ≤ r2 and Ω⊥0 := z| z>Φ−1

k z > r2.Since Ω0 ∪ Ω⊥0 = Rm,

∫Ω0fzk(z)dz = 1−

∫Ω⊥0

fzk(z)dz.

Lemma 1 ∫Ω⊥0

fzk(z)dz = Γ(m/2, r2/2)/Γ(m/2).

• Γ(m/2, r2/2) and Γ(m/2) can be iteratively calculatedaccording to Γ(z + 1) = zΓ(z), Γ(1/2) =

√π and

Γ(a, b) = (a− 1)Γ(a− 1, b) + ba−1 exp(−b),Γ(1/2, b) = 2

√π[1−Q(

√2b)],

Q(z) =∫∞z

1√2π

exp (−t2

2 )dt.

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The tightest inner and outer ellipsoidalapproximations of Ω

• Define Ωk,1 as the largest ellipsoid that is contained in Ω andsatisfies

Ωk,1 := z ∈ Rm| z>Φ−1k z ≤ δ2

k,1. (7)

• Define Ωk,1 as the smallest ellipsoid that contains Ω andsatisfies

Ωk,1 := z ∈ Rm| z>Φ−1k z ≤ δ2

k,1. (8)

Figure 2 : Relationship of Ωk,1, Ωk,1 and Ω (∂ denotes the boundary of a set) for the case ofm = 2.

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Calculation of Ωk,1 and Ωk,1

• The value of δk,1 can be calculated as

δk,1 = maxzi∈δ,−δ, i∈1,2,...,m

√z>Φ−1

k z, (9)

where z = [z1, z2, ..., zm]>.

• To calculate δk,1, the following bi-level optimization problemneeds to be solved:

maxi z∗is.t. z∗i = maxz zi

s.t. z>(Φ−1k )z = 1.

(10)

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Calculation of Ωk,1 and Ωk,1cont’d

• Lower level problem:

maxz zis.t. z>(Φ−1

k )z = 1.(11)

Lemma 2

The optimal solution to problem (11) equals z∗i =√∑m

j=1 α2k,i,j ,

where αk,i,j =uk,i,j√λk,j

, uk,i,j is the element in the ith row and jth

column of Uk, U>k Φ−1k Uk = Λk and Λk := diagλk,1, λk,2, ..., λk,m.

• The optimal solution to problem (10) can be written asmaxi

√∑mi=1 α

2k,i,j .

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Main Result 1

Theorem 1For the state estimation scheme in Fig. 1 and the event-basedscheduler in (3), the expected sensor to estimator communicationrate E(γk|Ik−1) is bounded by

Γ(m/2, δ2k,1/2)

Γ(m/2)≤ E(γk|Ik−1) ≤

Γ(m/2, δ2k,1/2)

Γ(m/2), (12)

with δk,1 = maxzi∈δ,−δ, i∈1,2,...,m

√z>Φ−1

k z andδk,1 = δ

maxi∈1,2,...,m√∑m

j=1 α2k,i,j

.

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Low complexity inner and outer ellipsoidalapproximations of Ω

• Define S ⊂ Rm as the largest sphere contained in Ω:

S := z ∈ Rm| z>z ≤ δ2, (13)

• Define S ⊂ Rm as the smallest sphere that contains Ω:

S := z ∈ Rm| z>z ≤ δ2m. (14)

• Based on S and S, define Ωk,2 ⊂ S as the largest ellipsoidthat is contained in S and satisfies

Ωk,2 := z ∈ Rm| z>Φ−1k z ≤ δ2

k,2, (15)

and define Ωk,2 as the smallest ellipsoid that contains S andsatisfies:

Ωk,2 := z ∈ Rm| z>Φ−1k z ≤ δ2

k,2. (16)

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Low complexity inner and outer ellipsoidalapproximations of Ω cont’d

Figure 3 : Relationship of S, S, Ωk,2, Ωk,2 and Ω (∂ denotes the boundary of a set) for thecase of m = 2.

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Calculation of Ωk,2 and Ωk,2

Lemma 3

For all z ∈ Rm satisfying z>Φ−1k z = 1, 1/λk ≤ z>z ≤ 1/λk holds,

where λk and λk are the smallest and largest eigenvalues of Φ−1k ,

respectively.

• For z ∈ z ∈ Rm|z>Φ−1k z ≤ r2, r2/λk ≤ z>z ≤ r2/λk holds.

Therefore we have δk,2 =√λkδ and δk,2 =

√λkmδ.

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Main Result 2

Theorem 2For the state estimation scheme in Fig. 1 and the event-basedscheduler in (3), the expected sensor to estimator communicationrate E(γk|Ik−1) is bounded by

Γ(m/2, δ2k,2/2)

Γ(m/2)≤ E(γk|Ik−1) ≤

Γ(m/2, δ2k,2/2)

Γ(m/2), (17)

with δk,2 =√mλkδ and δk,2 =

√λkδ.

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Main Result 2 cont’d

Corollary 1

If the system in (1) is stable, the communication rate is boundedby

Γ(m/2, δ2/2)

Γ(m/2)≤ E(γk|Ik−1) ≤ Γ(m/2, δ2/2)

Γ(m/2), (18)

as k →∞, where δ =√mλ1δ, δ =

√λ2δ,

λ1 = maxeig[(CPC> +R)−1], P being the stabilizing solution tothe Riccati equation

P = APA> −APC>[CPC> +R]−1CPA> +Q,

and λ2 = mineig[(CPC> +R)−1], P being the stabilizingsolution to the Lyapunov equation

P = APA> +Q.

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A Numerical Example

Consider a second-order process of the form in (1) measured bya sensor with scalar-valued measurements (m = 1):

A =

[0.8 0.20.3 0.6

], Q =

[0.3618 0

0 0.3035

],

C = [0.218 1.041], R = 0.0910 and δ = 0.8.

0 50 100 150 200 250 3000.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

time, k

LB

E(γk|Ik)

UB

Figure 4 : Plot of E(γk|Ik−1) (UB and LB respectively denote the upper and lower boundsderived in Corollary 1).

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Discussions

• Lemma 1 can be applied to recover the communication rateanalysis results in [1].

• The proposed results can be extended to analyze thecommunication rate of general event-based estimationschemes

γk =

0, if yk ∈ Yk1, otherwise

as well.

• Inner and outer ellipsoidal approximations of Yk need to beconsidered.

[1] J. Wu, Q. Jia, K. Johansson, and L. Shi, Event-based sensor data scheduling: Trade-off between communication rate andestimation quality,” IEEE Transactions on Automatic Control, vol. 58, no. 4, pp. 1041-1045, 2013.

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Acknowledgment

• Natural Sciences and Engineering Research Council(NSERC) of Canada

• Research Grants Council (RGC) of Hong Kong

• FGSR Travel Award, University of Alberta

Thank you!