A Stochastic Online Sensor Scheduler for Remote State Estimation with Time-out Condition

Transcription

1 A Stochastic Online Sensor Scheduler for Remote State Estimation with Time-out Condition Junfeng Wu, Karl Henrik Johansson and Ling Shi Stockholm, 9th, January / 19

2 Outline Outline 1. Introduction 2. Problem Setup 3. Event-Based State Estimation with Time-Out Condition 2 / 19

3 Introduction Figure: A typical networked control system from Internet 3 / 19

4 Introduction Remote state estimation exists in many applications of NCSs. Sensors send packets to a remote state estimator over a network. 4 / 19

5 Introduction Limited Recourses vs. Estimation Quality Sensors: limited energy budget Network: shared bandwidth 5 / 19 Goal: use sensor scheduling schemes to obtain a desired tradeoff between limited communication resources and remote estimation quality

6 Problem Setup System Model x k+1 = Ax k + w k, y k = Cx k + v k, x k =process state; y k =observation; w k =state noise, w k N (0, Q) (Q 0); v k =observation noise, v k N (0, R) (R > 0), independent of w k ; x 0 = initial state x 0 N (0, Π 0 ) (Π 0 0), uncorrelated with w k and v k 6 / 19

7 Problem Setup System Model x k Process y k ˆx s k Sensor and Preprocessor γ k Estimator Event-detector 7 / 19 Sensor runs a local Kalman filter to compute ˆx k s = E[x k y 0,..., y k ]. The estimation error ek s and the error covariance matrix Ps k are: ek s x k ˆx k s, Pk s E[(ek s )(es k ) y 0,..., y k ].

8 Problem Setup Kalman Filtering Preliminaries Definition h (X) AXA + Q, g λ (X) X λxc [CXC + R] 1 CX g(x) X XC [CXC + R] 1 CX Assume the local Kalman filter has entered steady state to simplify the discussion, = P, k 1, P s k where P is the steady-state error covariance such that g h(p) = P. 8 / 19

9 Problem Setup System Model At each time k, the sensor(smart or normal) decide whether or not it sends packet to the remote estimator: { 1, if ˆx s γ k = k is sent, 0, otherwise. A sensor data schedule θ is defined as θ {γ 1,..., γ k,...} 9 / 19

10 Problem Setup Consider the remote estimation problem where the communication resource is severe. We are interested in finding a schedule θ which solves the following problem: min θ s.t. lim sup T lim sup T + 1 T 1 T T 1 Tr (E [P k ]) k=0 T 1 γ k (θ) = Ψ, (1) k=0 where Ψ [0, 1]. 10 / 19

11 Incremental Innovative Information Let us define ε k as the incremental innovative information in ˆx s k compared with ˆx s k 1 : ε k ˆx s k Aˆx s k 1. (2) Lemma: The following statements on ε k hold: 1. ε k is zero-mean Gaussian, and for any d Z, ε k d and e s k are independent and E[e s k ε k d ] = ε j and ε k are independent for any j k. 3. E[ε k ε k ] = h(p) P. 11 / 19

12 A Stochastic Online Schedule Define the holding time τ(k) as follows: τ(k) k max 1 t k { t : γ t = 1 }. (3) We propose the following stochastic schedule θ e : { 0, if ɛk γ k = < δ τ(k 1)+1, 1, otherwise, (4) 12 / 19 where δ i [0, + ], i Z + are parameters to be designed. θ e can be equivalently defined as θ e = {δ 1, δ 2,...}. τ k s are the states of a Markov chain with S = {0, 1,...}.

13 Theorem: The MMSE estimate of x k is ˆx k = { ˆx s k, if γ k = 1, A τ ˆx k τ s, if γ k = 0. (5) Theorem: Under θ e, when γ k = 1, P k = P. And when γ k = 0, P k is given by τ 1 [ ] P k = P + [1 β(δ τ i )] h i+1 (P) h i (P), (6) i=0 where β(δ) = 2 2π δe δ2 2 by Q(δ) + δ 1 2π e x2 2 dx. [1 2Q(δ)] 1, and Q-function is defined 13 / 19

14 Theorem: The MMSE estimate of x k is ˆx k = { ˆx s k, if γ k = 1, A τ ˆx k τ s, if γ k = 0. (5) Theorem: Under θ e, when γ k = 1, P k = P. And when γ k = 0, P k is given by τ 1 [ ] P k = P + [1 β(δ τ i )] h i+1 (P) h i (P), (6) i=0 where β(δ) = 2 2π δe δ2 2 by Q(δ) + δ 1 2π e x2 2 dx. [1 2Q(δ)] 1, and Q-function is defined 13 / 19

15 Time-out Condition Note that for a θ e, S has infinite number of states, hence it is difficult to optimize θ e. However, the following proposition allows us to construct a θ e with a time-out condition. Proposition: Define θ e {δ 1, δ 2,...} as the optimal θ e. We have lim inf τ δ τ = 0. Remark: It allows us to approximate τ k = N, N + 1,... from recurrent states to transient ones if δ N µ c. We propose θ e (N) with a time-out condition: { 0, if τ(k 1) N 1 and ɛk γ k = < δ τ(k 1)+1, 1, otherwise. (7) 14 / 19

16 Time-out Condition Note that for a θ e, S has infinite number of states, hence it is difficult to optimize θ e. However, the following proposition allows us to construct a θ e with a time-out condition. Proposition: Define θ e {δ 1, δ 2,...} as the optimal θ e. We have lim inf τ δ τ = 0. Remark: It allows us to approximate τ k = N, N + 1,... from recurrent states to transient ones if δ N µ c. We propose θ e (N) with a time-out condition: { 0, if τ(k 1) N 1 and ɛk γ k = < δ τ(k 1)+1, 1, otherwise. (7) 14 / 19

17 15 / 19 Figure: A Markov chain modeling the evolution of τ k = {0,..., N}. Problem min θ e(n) Tr P + N j=1 j 1 π j k=0 [ 1 β(δj k ) ] h k+1 (P) h k (P) j i=1 s.t. π j = p i 1 + N l l=1 i=1 p, p j = [ 1 2Q(δ j ) ] r, j = 0,..., N, i π 0 Ψ.

18 15 / 19 Figure: A Markov chain modeling the evolution of τ k = {0,..., N}. Problem min θ e(n) Tr P + N j=1 j 1 π j k=0 [ 1 β(δj k ) ] h k+1 (P) h k (P) j i=1 s.t. π j = p i 1 + N l l=1 i=1 p, p j = [ 1 2Q(δ j ) ] r, j = 0,..., N, i π 0 Ψ.

19 An Upper Bound for J(θ e (N)) Lemma: 1 β(δ) [1 2Q(δ)] 2. Problem min Tr P + π 0 {p 1,...,p N } subject to π 0 = Ψ, N 1 Ψ π 0 N j j=1 i=1 j=1 i=1 j 1 p i p 2 r k=0 j p i 0, p j [0, 1], j = 1,..., N. It is a generalized geometric programming. j k h k+1 (P) h k (P) 16 / 19

20 Examples J (θoff ) J (ˆθ e ) J (θe ) J (θ r ) J (θ s ) Ψ Figure: Performance comparison 17 / 19

21 Examples e 2 k under θ off k e 2 k under ˆθ e k Figure: Realization of θoff and ˆθ e for Ψ = / 19

22 The End Thank You 19 / 19