1. On Optimal Distributed Kalman Filtering in Non-ideal Situations
Marc Reinhardt, Benjamin Noack
Intelligent Sensor-Actuator-Systems Laboratory (ISAS),
Institute for Anthropomatics and Robotics,
Karlsruhe Institute of Technology (KIT),
Karlsruhe, Germany

2. http://www.faz.net/-gup-7kc5e, Wolfgang Eilmes
Motivation
Wolfgang Eilmes
EMPHASIZE
No simple task, isas drafted proposal with company Hirschmann
Benjamin Valeo

3. Calculate optimal estimate 𝑥 𝑘 of state 𝒙 𝑘
Monitoring
Proposal: Stability of Cranes
Monitoring of risk that crane tips
Different sensor types
Battery-powered sensors
Wireless communication
 Preprocessing at sensors
Problem Setup
Unknown state
Interconnected sensors
monitor stability of crane with multiple sensors
load sensor, windsensor, inclination sensor, etc.
Cranes are dynamic, prevent cable breaks, no electricity cables
Idea: battery + wireless communication
Achieve reliable communication, minimize battery costs/number of transmissions
Combine observations at sensors, communicate estimates if necessary
Uncertain measurements
Calculate optimal estimate 𝑥 𝑘 of state 𝒙 𝑘
Benjamin Valeo

4. Estimation Framework System Linear(-ized) evolution model
𝒙 𝑘+1 = 𝐴 𝑘 𝒙 𝑘 + 𝒘 𝑘
Linear(-ized) meas. models
𝒛 𝑘 = 𝐻 𝑘 𝒙 𝑘 + 𝒗 𝑘
Independent noise terms 𝒘 𝑘 , 𝒗 𝑘
Prediction
Filtering
Central optimal: Kalman filter
Problem Setup
Unknown state
Calculate optimal estimate 𝑥 𝑘 of state 𝒙 𝑘
Interconnected sensors
Uncertain measurements
LMMSE Estimator
Derive (recursive) linear estimator that minimizes argmin 𝑥 𝑘 𝑡𝑟 𝐶 𝑘 𝑤𝑖𝑡ℎ 𝐶 𝑘 = 𝑥 𝑘 − 𝒙 𝑘 2
Estimation
Estimator 𝑥 𝑘 is function of measurements
Error 𝑥 𝑘 − 𝒙 𝑘 is to be minimized
Abstract/general -> consider tractable mathematical problem
Assume: linear models can be inferred from real world problem: describe evolution of the state
Design variables: reduce complexity: focus on linear functions, stochastic error function: quadratic error
Mean squared error criterion
Only (recursive) linear functions
Benjamin Valeo

5. Sensor Network Sensor Network
Multiple nodes are connected to one network
Sensor Network
The same phenomenum is observed by the sensors
One Phenomenum
Relation to example in background
The processing is done at synchronized discrete time-steps
Discrete-time
Benjamin Valeo

6. Centralized Processing
Reference for sensor network algorithms
Optimal Estimator
Central Processing
Not scalable
Bad reliability
High data traffic
As a reference
Why bad solution
Benjamin Valeo

7. Communication in a Sensor Network
Links are not stable
Communication
Measurements must be processed locally
Local Processing
Benjamin Valeo

8. Distributed Sensor Network
Optimize average estimation quality
Decentralized Estimation
Optimize estimation at sink only
Distributed Estimation
Benjamin Valeo

9. Illustrating Example Sensor Network 100 sensor nodes Limited Range
No communication
Reconstruction of a 2D path of an object
Estimates are stored every 10th time step
Collection of data after 100 time steps
Sensor Network
Benjamin Valeo

10. Optimal Distributed Kalman Filter
Number of nodes as well as global measurement model is known to all nodes
Application to Example Scenario
Optimal
Results are equivalent to centralized processing
Kalman filter
The initial estimates 𝑥 𝑘 𝑖 , 𝐶 𝑘 𝑖 are globalized for 𝑖∈{1,…,𝑛}
Transformation of Local Estimates (Globalization)
𝑥 𝑘 𝑖 = 𝐶 𝑘 𝐶 𝑘 𝑖 −1 x k i
𝐶 𝑘 −1 = 𝑖=1 𝑛 𝐶 𝑘 𝑖 −1
Slightly modified Kalman filter
Presented by Koch
Benjamin Valeo

11. Optimal Distributed Kalman Filter (2)
Similar to standard prediction
Local Relaxed Prediction
With modified gains
Local Filtering
𝑥 𝑘|𝑘 𝑖 = 𝐶 𝑘|𝑘 𝐶 𝑘 𝑖 −1 x k i + 𝐻 𝑘 𝑖 𝑇 𝐶 𝑘 𝑣 𝑖 −1 z k i
𝐶 𝑘|𝑘 −1 = 𝐶 𝑘 𝑖 −1 + 1 𝑛 𝑖=1 𝑛 𝐻 𝑘 𝑖 𝑇 𝐶 𝑘 𝑣 𝑖 −1 𝐻 𝑘 𝑖
𝑥 𝑘+1 𝑖 =𝐴 𝑥 𝑘 𝑖
𝐶 𝑘+1 =𝐴 𝐶 𝑘 𝐴 𝑇 +𝑛 𝐶 𝑤
Fusion at Data Sink
𝑥 𝑘 = 1 𝑛 𝑖=1 𝑛 𝑥 𝑘 𝑖
𝐶 𝑘 = 1 𝑛 𝐶 𝑘
C bar is the same at each node
KEINE GÜLTIGE SCHÄTZUNG!
Benjamin Valeo

12. Illustrating Example Sensor Network 100 sensor nodes Defect / working
No communication
Reconstruct 2D path of an object
Estimate is stored every 10th time step
Collection of data after 100 time steps
Sensor Network
Benjamin Valeo

13. Missing Estimates / Failed Measurements
The number of defect nodes is only approximately known
Instead of 20, only 15 sensors are defect
85 are functioning
Example 1
Average RMSE
BLUE
DKF x
0.123
3.833 y
0.138
2.687
Result
The fused estimates are biased and significantly worse than the optimal ones
Benjamin Valeo

14. Distributed Kalman Filter
Similar to standard prediction
Local Relaxed Prediction
With modified gains
Local Filtering
𝑥 𝑘|𝑘 𝑖 = 𝐶 𝑘|𝑘 𝐶 𝑘 𝑖 −1 x k i + 𝐻 𝑘 𝑖 𝑇 𝐶 𝑘 𝑣 𝑖 −1 z k i
𝐶 𝑘|𝑘 −1 = 𝐶 𝑘 𝑖 −1 + 1 𝑛 𝑖=1 𝑛 𝐻 𝑘 𝑖 𝑇 𝐶 𝑘 𝑣 𝑖 −1 𝐻 𝑘 𝑖
𝑥 𝑘+1 𝑖 =𝐴 𝑥 𝑘 𝑖
𝐶 𝑘+1 =𝐴 𝐶 𝑘 𝐴 𝑇 +𝑛 𝐶 𝑤
Fusion at Data Sink
𝑥 𝑘 = 1 𝑛 𝑖=1 𝑛 𝑥 𝑘 𝑖
𝐶 𝑘 = 1 𝑛 𝐶 𝑘
C bar is the same at each node
Benjamin Valeo

15. Weak Points of the DKF Works well when all requirements are fulfilled
Model Knowledge
Global measurement model is not precisely known
Defect Nodes
The number of functioning nodes is not known exactly
Uncertainties
Measurement noise is variable or state dependent
Model Knowledge
Global measurement model is not precisely known
Defect Nodes
The number of functioning nodes is not known exactly
Uncertainties
Measurement noise is variable or state dependent
When the assumptions about models, number of nodes etc. are not met
DKF in Non-ideal Situations
Benjamin Valeo

16. DKF in Non-ideal Situation
Derive a multiplicative bias-correction matrix Δ 𝑘 with
𝐸{𝒙}= E{Δ 𝑘 𝑥 𝑘 𝑓 }
Basic Idea
Initialization
Prediction
Filtering
Correction Matrix Calculation
Standard DKF Equations
Initialization of DKF
Prediction of DKF
Filtering of DKF
Benjamin Valeo

17. DKF in Non-ideal Situation
We assume the globalized estimates to be combined according to
Correct the Fused Estimate
𝑥 𝑘 𝑓 = 1 𝑚 𝑖=1 𝑚 𝑥 𝑘 𝑖
𝐸 𝒙 =𝐸{ Δ 𝑘 1 𝑚 𝑖=1 𝑚 𝑥 𝑘 𝑖 }
with
Δ 𝑘 =𝐼
Initialization
Δ 𝑘+1 =𝐴 Δ 𝑘 𝐴 −1
Prediction
𝑥 𝑘+1 𝑖 =𝐴 𝑥 𝑘 𝑖
Filtering
With
𝐶 𝑘 −1 = Δ 𝑘|𝑘−1 𝐶 𝑘|𝑘−1 −1 + 1 𝑛 𝑖=1 𝑚 𝐻 𝑘 𝑖 𝑇 𝐶 𝑘 𝑣 𝑖 −1 𝐻 𝑘 𝑖
Δ 𝑘 = 𝐶 𝑘 𝐶 𝑘 −1
𝑥 𝑘|𝑘 𝑖 = 𝐶 𝑘|𝑘 𝐶 𝑘 𝑖 −1 x k i + 𝐻 𝑘 𝑖 𝑇 𝐶 𝑘 𝑣 𝑖 −1 z k i
Actually utilized models must be known
Benjamin Valeo

18. DKF in Non-ideal Situation
Globally optimal results when assumptions are correct (Δ=𝐼)
Unbiased estimates anyhow
Globally Optimal
Actual expected error can also be derived in closed-form recursively
Actual MSE covariance matrix
Regular state transition matrix
Regular correction matrix
Utilized models must be known at fusion node
Requirements
Relaxations
Reformulation
Hypothesizing KF
Benjamin Valeo

19. Missing Estimates
The number of defect nodes is only approximately known
Instead of 20, only 15 sensors are defect
85 are functioning
Example 1
Average RMSE
BLUE
DKF
Extended DKF x
0.096
3.814 y
0.115
2.645
Result
The fused estimates are unbiased and almost optimal
Benjamin Valeo

20. Full Example Example 1 Average RMSE BLUE HKF x 1.323 1.363 (3%) y
The number of defect nodes is only approximately known (30 vs. 24)
The measurement range is limited (15) and the quality decreases linearly with the distance to the object according to 𝐶 𝑘 𝑣 𝑖 =ceil 𝑝 𝑖 − 𝑝 𝑘 𝑥 𝐶 𝑘 𝑣
Example 1
Average RMSE
BLUE
HKF x
1.323
(3%) y
0.932
(0.7%)
Result
The fused estimates are unbiased and almost optimal
Benjamin Valeo

21. Example 2: Hypothesizing Filtering
Use hypothesis as substitute for unknown measurement capacity
Centralized KF with
𝐶 𝑘 𝑧 𝑠 = 𝐻 𝑘 𝑠 𝑇 𝐶 𝑘 𝑣 𝑠 −1 𝐻 𝑘 𝑠
bad optimization
𝐶 𝑘 𝑧 = 𝑠=1 𝑆 𝐻 𝑘 𝑠 𝑇 𝐶 𝑘 𝑣 𝑠 −1 𝐻 𝑘 𝑠
40 sensors, distance dependent measurement uncertainty
Benjamin Valeo

22. Evaluation Hypothesizing Filtering
Use hypothesis as substitute for unknown measurement capacity
good optimization
100 runs
Benjamin Valeo

23. Applications of Hypothesizing Filtering
Global measurement model is not precisely known
Model knowledge
The number of functioning nodes is not known exactly
Defect Nodes
Measurement noise is variable or state dependent
Uncertainties
Assumption
Only an assumption about the model is necessary
Correction
The number of nodes does not effect the estimation quality
Average
Only the global uncertainty must be estimated
Benjamin Valeo

24. Summary Bias Calculation
The globalized estimates of the DKF are biased
We have derived this bias in closed-form
Bias Calculation
The bias leads to bad estimates when assumptions are not met
We derived a correction matrix that allows to derive an unbiased estimate
Even when assumptions are not met, the result is almost optimal
DKF in Non-ideal Situations
The derivation of the correction matrix requires the utilized models to be available at the fusion center
Restrictions
Benjamin Valeo

25. Benjamin Valeo