Luiz F. O. Chamon, George J. Pappas and Alejandro Ribeiro. CDC 2017 December 12 th, 2017

Transcription

1 The Mean Square Error in Kalman Filtering Sensor Selection is Approximately Supermodular Luiz F. O. Chamon, George J. Pappas and Alejandro Ribeiro CDC 2017 December 12 th, 2017 Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 1

2 Goal Why does greedy sensor selection for Kalman filtering works when it shouldn t? Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 2

3 KFSS Problem (KFSS) Select up to s system outputs to estimate its internal states. Why the MSE? KF minimize S O subject to MSE(S) S s NP-hard [Natarajan 95, Zhang 17, Ye 17] Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 3

4 Greedy KFSS Definition Select sensors/outputs one at a time by choosing the one that most improves estimation at each step. function Greedy(q) end G 0 = {} for j = 1,..., q u = argmin MSE (G j 1 {v}) v O\G j 1 G j = G j 1 {u} end Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 4

5 Greedy KFSS Definition Select sensors/outputs one at a time by choosing the one that most improves estimation at each step. function Greedy(q) end G 0 = {} for j = 1,..., q u = argmin MSE (G j 1 {v}) v O\G j 1 G j = G j 1 {u} end Low complexity Sequential Near-optimal for supermodular objectives Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 4

6 KFSS Problem (KFSS) Select up to s system outputs to estimate its internal states. Why the MSE? KF minimize S O subject to MSE(S) S s NP-hard [Natarajan 95, Zhang 17, Ye 17] Estimation MSE is not supermodular [Tzoumas 16, Olshevsky 16, Singh 17, Zhang 17] Use a supermodular surrogate (e.g., log det) [Joshi 09, Shamaiah 10, Tzoumas 16] Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 5

7 Greedy KFSS Count Smoothing Relative suboptimality Filtering Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular

8 Greedy KFSS Count Smoothing Relative suboptimality Greedy KFSS is near-optimal Filtering Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular

9 Kalman filtering sensor selection (Approximate) supermodularity Near-optimality of greedy KFSS Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 7

10 Kalman filtering x k+1 = F x k + w k y k = Hx k + v k w k N (0, σ 2 wi) v k N (0, σ 2 vi) x 0 N ( x 0, Π 0 ) Problem (Filtering) Estimate x k based on outputs up to time k, i.e., Solution (Kalman filter) ˆx k = E [x k {y j } j k ] ˆx k = F ˆx k 1 + K k [y k HF ˆx k 1 ] Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 8

11 Kalman filtering x k+1 = F x k + w k y k = Hx k + v k w k N (0, σ 2 wi) v k N (0, σ 2 vi) x 0 N ( x 0, Π 0 ) Problem (Filtering) Estimate x k based on outputs in S O up to time k, i.e., Solution (Kalman filter) ˆx k = E [x k {(y j ) S } j k ] ˆx k (S) = F ˆx k 1 (S) + K k [(y k ) S H S F ˆx k 1 (S)] Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 8

12 KFSS Problem (KF sensor selection) Find S O, S s, that minimizes the estimation MSE minimize S s m 1 θ j MSE l+j (S) j=0 Myopic sensor selection: m = 1 Final estimation MSE: θ j = 0 for j < m 1 and θ m 1 = 1 Exponentially weighted error: θ j = ρ m 1 j, ρ < 1 Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 9

13 KFSS Problem (KF sensor selection) Find S O, S s, that minimizes the estimation MSE where minimize S s m 1 θ j Tr [P l+j (S)] j=0 P k (S) = F P k 1 (S)F T + σwi 2 }{{} P k k 1 +σv 2 h i h T i }{{} i S i-th sensor Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 9 1

14 Kalman filtering sensor selection (Approximate) supermodularity Near-optimality of greedy KFSS Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 10

15 Supermodularity Definition (Supermodularity) For A B O and u O \ B f (A) f (A {u}) f (B) f (B {u}) diminishing returns Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 11

16 Greedy supermodular minimization Theorem ([NWF 78]) Let S be the optimal solution of the problem minimize S s f (S) and G be its greedy solution. If f is (i) monotone decreasing and (ii) supermodular, then f(g) f(s ) f( ) f(s ) e Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 12

17 Greedy supermodular minimization Theorem ([NWF 78]) If f is (i) monotone decreasing and (ii) supermodular, then f(g) f(s ) f( ) f(s ) e Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 13

18 α-supermodularity Definition (Supermodularity) For A B O and u O \ B f (A {u}) f (A) f (B {u}) f (B) Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 14

19 α-supermodularity Definition (α-supermodularity) For A B O, u O \ B, and α 0 [ ] f (A {u}) f (A) α f (B {u}) f (B) If α 1: f is supermodular If α < 1: f is approximately supermodular Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 14

20 Greedy α-supermodular minimization Theorem ([Chamon-Ribeiro 16]) Let S be the solution of the problem minimize S s f (S) and G q be the q-th iteration of a greedy solution. If f is (i) monotone decreasing and (ii) α-supermodular, then f(g q ) f(s ) f( ) f(s ) e αq/s. Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 15

21 Greedy α-supermodular minimization Theorem ([Chamon-Ribeiro 16]) If f is (i) monotone decreasing and (ii) α-supermodular, then f(g q ) f(s ) f( ) f(s ) e αq/s. For q = s and α = 1, we recover the classical e 1 result If α < 1, then e 1 is recovered for q = α 1 s Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 16

22 α for KFSS What is α for KFSS? Combinatorial problem α = min A B O u O\B MSE (A) MSE (A {u}) MSE (B) MSE (B {u}) Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 17

23 α for KFSS Theorem ([Chamon-Pappas-Ribeiro 17]) The objective of KFSS is α-supermodular with α min l k l+m 1 λ min [P k (O)] [ ] λ max Pk k 1 P k (O) = ( P k k 1 + σ 2 H T H ) 1 P k k 1 = F P k 1 F T + σ 2 wi σ 2 v σ 2 w and small κ(f ) α 1 Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 18 v

24 α for KFSS n = 100 states and H = I (db) Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 19

25 Kalman filtering sensor selection (Approximate) supermodularity Near-optimality of greedy KFSS Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 20

26 α for KFSS n = 100 states and H = I Suboptimality certificate (db) Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 21

27 Conclusion Why does greedy KFSS works so well? The MSE in KFSS is not supermodular, but almost Greedy KFSS is efficient and has a guaranteed near-optimal performance Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 22

28 The Mean Square Error in Kalman Filtering Sensor Selection is Approximately Supermodular Luiz F. O. Chamon, George J. Pappas and Alejandro Ribeiro More details: luizf Chamon, Pappas, Ribeiro The MSE in KF Sensor Selection is Approximately Supermodular 23