Sherman’s Theorem
description
Transcript of Sherman’s Theorem
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Sherman’s TheoremFundamental Technology for ODTK
Jim Wright
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Why?
Satisfaction of Sherman's Theorem guarantees that the mean-squared state estimate error on each state estimate component is minimized
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Sherman Probability Density
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ShermanProbability Density Function sPx
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Sherman Probability Distribution
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ShermanProbability Distribution FunctionSPx
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Notational Convention Here
• Bold symbols denote known quantities (e.g., denote the optimal state estimate by ΔXk+1|k+1, after processing measurement residual Δyk+1|k)
• Non-bold symbols denote true unknown quantities (e.g., the error ΔXk+1|k in propagated state estimate Xk+1|k)
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Admissible Loss Function L
• L = L(ΔXk+1|k) a scalar-valued function of state
• L(ΔXk+1|k) ≥ 0; L(0) = 0
• L(ΔXk+1|k) is a non-decreasing function of distance from the origin: limΔX → 0L(ΔX) = 0
• L(-ΔXk+1|k) = L(ΔXk+1|k)
Example of interest (squared state error):
L(ΔXk+1|k) = (ΔXk+1|k)T (ΔXk+1|k)
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Performance Function J(ΔXk+1|k)
J(ΔXk+1|k) = E{L(ΔXk+1|k)}
Goal: Minimize J(ΔXk+1|k), the mean value of loss on the unknown state error ΔXk+1|k in the propagated state estimate Xk+1|k.
Example (mean-squared state error):
J(ΔXk+1|k) = E{(ΔXk+1|k)T (ΔXk+1|k)}
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Aurora Response to CME
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Minimize Mean-Squared State Error
Smoother V3333 Atmospheric Density Simulator V3333 Atmospheric Density
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Atmospheric Density Overlay dp/pAtmospheric Density Overlay dp/p
Minutes after Midnight 01 Sep 2003 00:00:00.00
Satellite: V3333 Process: Smoother, Simulator
Time of First Data Point: 01 Sep 2003 00:00:00.00
OverlayAtmospheric Density / : Simulated and Smoothed (simDATA)
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Sherman’s Theorem
Given any admissible loss function L(ΔXk+1|k), and any Sherman conditional probability distribution function F(ξ|Δyk+1|k), then the optimal estimate ΔXk+1|k+1 of ΔXk+1|k is the conditional mean:
ΔXk+1|k+1 = E{ΔXk+1|k| Δyk+1|k}
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Doob’s First TheoremMean-Square State Error
If L(ΔXk+1|k) = (ΔXk+1|k)T (ΔXk+1|k)
Then the optimal estimate ΔXk+1|k+1 of ΔXk+1|k is the conditional mean:
ΔXk+1|k+1 = E{ΔXk+1|k| Δyk+1|k}
The conditional distribution function need not be Sherman; i.e., not symmetric nor convex
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Doob’s Second TheoremGaussian ΔXk+1|k and Δyk+1|k
If:
ΔXk+1|k and Δyk+1|k have Gaussian probability distribution functions
Then the optimal estimate ΔXk+1|k+1 of ΔXk+1|k is the conditional mean:
ΔXk+1|k+1 = E{ΔXk+1|k| Δyk+1|k}
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Sherman’s Papers
• Sherman proved Sherman’s Theorem in his 1955 paper.
• Sherman demonstrated the equivalence in optimal performance using the conditional mean in all three cases, in his 1958 paper
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Kalman
• Kalman’s filter measurement update algorithm is derived from the Gaussian probability distribution function
• Explicit filter measurement update algorithm not possible from Sherman probability distribution function
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Gaussian Hypothesis is Correct
• Don’t waste your time looking for a Sherman measurement update algorithm
• Post-filtered measurement residuals are zero mean Gaussian white noise
• Post-filtered state estimate errors are zero mean Gaussian white noise (due to Kalman’s linear map)
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Measurement System Calibration
• Definition from Gaussian probability density function
• Radar range spacecraft tracking system example
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Gaussian Probability Density N(μ,R2) = N(0,1/4)
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GaussianN0,1/4 Probability Density Function
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Gaussian Probability Distribution N(μ,R2) = N(0,1/4)
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GaussianN0,1/4 Probability Distribution Function
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Calibration (1)
N(μ,R2) = N(0,[σ/σinput]2)
N(μ,R2) = N(0,1) ↔ σinput = σ
σinput > σ• Histogram peaked relative to N(0,1)
• Filter gain too large
• Estimate correction too large
• Mean-squared state error not minimized
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Calibration (2)
σinput < σ• Histogram flattened relative to N(0,1)
• Filter gain too small
• Estimate correction too small
• Residual editor discards good measurements – information lost
• Mean-squared state error not minimized
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Before Calibration
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Range Residual Ratios DistributionRange Residual Ratios Distribution
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Ground Station: BOSS-A Sample Size: 4039 Satellite: V3333
Gaussiannon-N0,1 Peaked Histogram of Real Range Residual Ratios Before Calibration
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After Calibration
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Ground Station: BOSS-A Sample Size: 3988 Satellite: V3333
GaussianN0,1 Histogram of Real Range Residual Ratios After Calibration
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Nonlinear Real-Time Multidimensional Estimation
• Requirements
- Validation
• Conclusions
- Operations
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Requirements (1 of 2)
• Adopt Kalman’s linear map from measurement residuals to state estimate errors
• Measurement residuals must be calibrated: Identify and model constant mean biases and variances
• Estimate and remove time-varying measurement residual biases in real time
• Process measurements sequentially with time
• Apply Sherman's Theorem anew at each measurement time
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Requirements (2 of 2)
• Specify a complete state estimate structure
• Propagate the state estimate with a rigorous nonlinear propagator
• Apply all known physics appropriately to state estimate propagation and to associated forcing function modeling error covariance
• Apply all sensor dependent random stochastic measurement sequence components to the measurement covariance model
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Necessary & Sufficient ValidationRequirements
• Satisfy rigorous necessary conditions for real data validation
• Satisfy rigorous sufficient conditions for realistic simulated data validation
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Conclusions (1 of 2)
• Measurement residuals produced by optimal estimators are Gaussian white residuals with zero mean
• Gaussian white residuals with zero mean imply Gaussian white state estimate errors with zero mean (due to linear map)
• Sherman's Theorem is satisfied with unbiased Gaussian white residuals and Gaussian white state estimate errors
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Conclusions (2 of 2)
• Sherman's Theorem maps measurement residuals to optimal state estimate error corrections via Kalman's linear measurement update operation
• Sherman's Theorem guarantees that the mean-squared state estimate error on each state estimate component is minimized
• Sherman's Theorem applies to all real-time estimation problems that have nonlinear measurement representations and nonlinear state estimate propagations
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Operational Capabilities
• Calculate realistic state estimate error covariance functions (real-time filter and all smoothers)
• Calculate realistic state estimate accuracy performance assessment (real-time filter and all smoothers)
• Perform autonomous data editing (real-time filter, near-real-time fixed-lag smoother)