Duality and Equivalence in Estimation and Control

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Duality and Equivalence in Estimation and Control

• (chapter 15 in Linear Estimation)
• Katrin Strandemar and Märta Barenthin

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Contents

• Definition of dual basis
• Reasons for introducing dual bases
• Applications for dual basis
• Equivalence basic properties
• Equivalence applications
• Criticism

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Use of more general vectors This allows more
general concepts of inner product
Definition of dual basis
In this case Using a ring rather than a field
allows us to use matrix-valued inner products
GRAMIANS
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Dual basis vectors
Definition of dual basis
Set of linearly independent vectors
quantities we want to estimate
observations
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Dual basis gramian
Definition of dual basis
GRAMIAN
Since z,y linearly independent vectors
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Dual basis definition
Definition of dual basis
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Algebraic Specification
Definition of dual basis
Span the same linearspace
From definition
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Algebraic Specification
Definition of dual basis
With gramian
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Geometric Specification
Definition of dual basis
span same space
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Estimators via the dual basis
Reasons for introducing dual bases
Ex. 2. Project z onto Ly
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Why introduce dual bases?
Reasons for introducing dual bases
Ex. 1. Project x onto Lz,y
if z and y orthogonal
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Application to Linear Models
Applications for dual basis

• Suppose satisfy
• Then the dual basis also satisfy
  a linear model
• where
  
• 
  
  
• Remember Katrins slide interesting connection!
  

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Equivalent Stochastic and Deterministic Problems
Equivalence basic properties

• Equivalence between a stochastic and
  deterministic problems.
• Stochastic filtering problem consider zero mean
  random variables
• Now construct equivalent deterministic
  least-squares problem.

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Equivalence basic properties

• Equivalent deterministic least-squares problem
  given a vector y and a matrix H, solve
• So when we solve the stochastic problem we are
  also solving the deterministic problem.
• Equivalent problems gain matrix
  identical.

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Equivalence for Dual Problems
Equivalence basic properties

• Remember Katrins part about duality.
• This suggests that there are dual stochastic and
  deterministic problems that are equivalent.
• Stochastic
• Deterministic
• Same gain matrix
• dual stochastic and the dual deterministic
  problems are equivalent.

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Summary
Equivalence basic properties
Stochastic/deterministic
Type of base
Duality
Same solution (equivalence)
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Solve deterministic problem with smoothing
techniques
Equivalence applications

• Deterministic problem solve dual
  stochastic problem instead.
• Use formulas based on the Bryson-Frazier
  smoothing formulas.
• Application linear quadratic tracking
• Deterministic tracking problem

Keep close to track
Limited input
Condition on terminal state
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Equivalence applications

• Solve the dual stochastic problem instead.
• The solution to this problem can be found
  with Bryson-Frazier (BF)-like smoothing formulas

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Summary

• We have introduced a dual basis.
• Estimation problem can be solved in dual basis
  instead.
• Equivalence between stochastic and deterministic
  problems.
• Use duality/equivalence to solve for example
  linear tracking problem with smoothing formulas.

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Criticism

• High level of generality
• Difficult to get an intuitive feeling about dual
  bases.
• Applications more discussion about advantages
  with dual/equivalent approach compared to other
  methods (for example dynamic programming).

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Questions?
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Bonus Slide More Applications
Equivalence applications

• Application 2 problems with causality
  constraints.
• Application 3 measurement feedback control.
• Compute a feedback law when the feedback is
  restricted to be a function of noisy
  measurements.
• This is called the separation principle
• Feedback law is a function of the state
  estimator.
• Estimated state is calculated separately from
  control law.
• Application 3 problems in the frequency domain.