Event-based State Estimation of Linear Dynamical Systems: Communication Rate Analysis

Transcription

1 2014 American Control Conference Estimation of Linear Dynamical Systems: Dawei Shi, Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Kowloon, Hong Kong

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

3 Discrete-time LTI process driven by white noise: x k+1 = Ax k + w k, (1) where w k is zero-mean Gaussian with covariance Q 0. The initial state x 0 is Gaussian with E(x 0 ) = µ 0 and covariance P 0 0. Smart sensor: y k = Cx k + v k, (2) where v k R m is zero-mean Gaussian with covariance R > 0. Assume (A, Q) is stabilizable, and (C, A) is detectable.

4 Data Scheduler At each time instant k, the estimator provides a prediction ˆx k k 1 of x k and sends it to the scheduler. The scheduler computes γ k according to: { 0, if yk C ˆx γ k = k k 1 δ 1, otherwise (3) Only when γ k = 1, the sensor transmits y k to the estimator.

5 Estimator For this type of scenario, several estimates have been proposed, e.g., [1]-[4]. We consider a simple estimator of the form proposed in [5]: ˆx k k 1 = Aˆx k 1 k 1, (4) ˆx k k = ˆx k k 1 + γ k P k C (R + CP k C ) 1 (y k C ˆx k k 1 ), (5) where P k evolves according to P k = AP k 1 A + Q γ k AP k 1 C (CP k 1 C + R) 1 CP k 1 A. [1] J. Wu, Q. Jia, K. Johansson, and L. Shi, sensor data scheduling: Trade-off between communication rate and estimation quality, IEEE Transactions on Automatic Control, vol. 58, no. 4, pp , [2] J. Sijs and M. Lazar, Event based state estimation with time synchronous updates, IEEE Transactions on Automatic Control, vol. 57, no. 10, pp , [3] D. Shi, T. Chen and L. Shi. Event-triggered maximum likelihood state estimation, Automatica, 50(1), pp , [4] D. Shi, T. Chen, and L. Shi, An event-triggered approach to state estimation with multiple point-and set-valued measurements, Automatica, 50(6), pp , [5] S. Trimpe and R. D Andrea, An experimental demonstration of a distributed and event-based state estimation algorithm, in Proceedings of the 18th IFAC World Congress, Milano, Italy, 2011.

6 Conditioned on the received information I k 1, the prediction error ê k k 1 := x k ˆx k k 1 is zero-mean Gaussian with Cov(ê k k 1 I k 1 ) = P k. Define z k := y k C ˆx k k 1. We have E(z k I k 1 ) = 0 and E(z k z k I k 1) := Φ k = CP k k 1 C + R. Define Ω := {z R m z δ}. We have E(γ k I k 1 ) = 1 f zk (z)dz, (6) Ω where f zk (z) = (2π) m/2 (detφ k ) 1/2 exp ( 1 2 z Φ 1 k z). Objective: To provide lower and upper bounds for E(γ k I k 1 ).

7 Define Ω 0 := {z z Φ 1 k z r2 } and Ω 0 := {z z Φ 1 k z > r2 }. Since Ω 0 Ω 0 = R m, Ω 0 f zk (z)dz = 1 f Ω zk (z)dz. 0 1 Ω 0 f zk (z)dz = Γ(m/2, r 2 /2)/Γ(m/2). Γ(m/2, r 2 /2) and Γ(m/2) can be iteratively calculated according to Γ(z + 1) = zγ(z), Γ(1/2) = π and Γ(a, b) = (a 1)Γ(a 1, b) + b a 1 exp( b), Γ(1/2, b) = 2 π[1 Q( 2b)], Q(z) = z 1 2π exp ( t2 2 )dt.

8 The tightest inner and outer ellipsoidal approximations of Ω Define Ω k,1 as the largest ellipsoid that is contained in Ω and satisfies Ω k,1 := {z R m z Φ 1 k z δ2 k,1 }. (7) Define Ω k,1 as the smallest ellipsoid that contains Ω and satisfies Ω k,1 := {z R m z Φ 1 k z δ k,1}. 2 (8) Figure 2 : Relationship of Ω k,1, Ω k,1 and Ω ( denotes the boundary of a set) for the case of m = 2.

9 Calculation of Ω k,1 and Ω k,1 The value of δ k,1 can be calculated as δ k,1 = max z Φ 1 k z, (9) z i {δ, δ}, i {1,2,...,m} where z = [z 1, z 2,..., z m ]. To calculate δ k,1, the following bi-level optimization problem needs to be solved: max i zi s.t. zi = max z z i (10) s.t. z (Φ 1 k )z = 1.

10 Calculation of Ω k,1 and Ω k,1 cont d Lower level problem: 2 max z s.t. z i z (Φ 1 k )z = 1. (11) The optimal solution to problem (11) equals z i = m j=1 α2 k,i,j, where α k,i,j = u k,i,j, u k,i,j is the element in the ith row and jth λk,j column of U k, U k Φ 1 k U k = Λ k and Λ k := diag{λ k,1, λ k,2,..., λ k,m }. The optimal solution to problem (10) can be written as max i m i=1 α2 k,i,j.

11 Theorem 1 For the state estimation scheme in Fig. 1 and the event-based scheduler in (3), the expected sensor to estimator communication rate E(γ k I k 1 ) is bounded by Γ(m/2, δ 2 k,1 /2) Γ(m/2) with δ k,1 = max zi {δ, δ}, i {1,2,...,m} δ δ k,1 =. max i {1,2,...,m} m j=1 α2 k,i,j E(γ k I k 1 ) Γ(m/2, δ2 k,1/2), (12) Γ(m/2) z Φ 1 k z and

12 Low complexity inner and outer ellipsoidal approximations of Ω Define S R m as the largest sphere contained in Ω: S := {z R m z z δ 2 }, (13) Define S R m as the smallest sphere that contains Ω: S := {z R m z z δ 2 m}. (14) Based on S and S, define Ω k,2 S as the largest ellipsoid that is contained in S and satisfies Ω k,2 := {z R m z Φ 1 k z δ2 k,2 }, (15) and define Ω k,2 as the smallest ellipsoid that contains S and satisfies: Ω k,2 := {z R m z Φ 1 k z δ 2 k,2}. (16)

13 Low complexity inner and outer ellipsoidal approximations of Ω cont d Figure 3 : Relationship of S, S, Ω k,2, Ω k,2 and Ω ( denotes the boundary of a set) for the case of m = 2.

14 Calculation of Ω k,2 and Ω k,2 3 For all z R m satisfying z Φ 1 k z = 1, 1/ λ k z z 1/λ k holds, where λ k and λ k are the smallest and largest eigenvalues of Φ 1 respectively. For z {z R m z Φ 1 k z r2 }, r 2 / λ k z z r 2 /λ k holds. Therefore we have δ k,2 = λ k δ and δ k,2 = λk mδ. k,

15 Theorem 2 For the state estimation scheme in Fig. 1 and the event-based scheduler in (3), the expected sensor to estimator communication rate E(γ k I k 1 ) is bounded by Γ(m/2, δ 2 k,2 /2) Γ(m/2) with δ k,2 = m λ k δ and δ k,2 = λ k δ. E(γ k I k 1 ) Γ(m/2, δ2 k,2/2), (17) Γ(m/2)

16 cont d Corollary 1 If the system in (1) is stable, the communication rate is bounded by Γ(m/2, δ 2 /2) Γ(m/2) E(γ k I k 1 ) Γ(m/2, δ2 /2), (18) Γ(m/2) as k, where δ = mλ 1 δ, δ = λ 2 δ, λ 1 = max{eig[(cp C + R) 1 ]}, P being the stabilizing solution to the Riccati equation P = AP A AP C [CP C + R] 1 CP A + Q, and λ 2 = min{eig[(c P C + R) 1 ]}, P being the stabilizing solution to the Lyapunov equation P = AP A + Q.

17 A Numerical Consider a second-order process of the form in (1) measured by a sensor with scalar-valued measurements (m = 1): [ ] [ ] A =, Q =, C = [ ], R = and δ = UB E(γk Ik) LB time, k Figure 4 : Plot of E(γ k I k 1 ) (UB and LB respectively denote the upper and lower bounds derived in Corollary 1).

18 1 can be applied to recover the communication rate analysis results in [1]. The proposed results can be extended to analyze the communication rate of general event-based estimation schemes { 0, if yk Y γ k = k 1, otherwise as well. Inner and outer ellipsoidal approximations of Y k need to be considered. [1] J. Wu, Q. Jia, K. Johansson, and L. Shi, sensor data scheduling: Trade-off between communication rate and estimation quality, IEEE Transactions on Automatic Control, vol. 58, no. 4, pp , 2013.

19 ment Natural Sciences and Engineering Research Council (NSERC) of Canada Research Grants Council (RGC) of Hong Kong FGSR Travel Award, University of Alberta Thank you!