Sherman’s Theorem

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Sherman’s TheoremFundamental Technology for ODTK

Jim Wright


Why?

Satisfaction of Sherman's Theorem guarantees that the mean-squared state estimate error on each state estimate component is minimized


Sherman Probability Density

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ShermanProbability Density Function sPx


Sherman Probability Distribution

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ShermanProbability Distribution FunctionSPx


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)


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)


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)}


Aurora Response to CME


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)


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}


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:


The conditional distribution function need not be Sherman; i.e., not symmetric nor convex


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:



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


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


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)


Measurement System Calibration

• Definition from Gaussian probability density function

• Radar range spacecraft tracking system example


Gaussian Probability Density N(μ,R2) = N(0,1/4)

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GaussianN0,1/4 Probability Density Function


Gaussian Probability Distribution N(μ,R2) = N(0,1/4)

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GaussianN0,1/4 Probability Distribution Function


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


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


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


After Calibration

Percentage Found Normal Distribution

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Range Residual Ratios DistributionRange Residual Ratios Distribution

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Ground Station: BOSS-A Sample Size: 3988 Satellite: V3333

GaussianN0,1 Histogram of Real Range Residual Ratios After Calibration


Nonlinear Real-Time Multidimensional Estimation

• Requirements

- Validation

• Conclusions

- Operations


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


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


Necessary & Sufficient ValidationRequirements

• Satisfy rigorous necessary conditions for real data validation

• Satisfy rigorous sufficient conditions for realistic simulated data validation


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


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


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)

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Sherman’s Theorem

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