1ME 233 Advanced Control II

Lecture 6
Least Squares Estimation(ME233 Class Notes pp. LS1-LS5)

2Notation
Let X and Y be continuous random vectors withjoint PDF

Let x and y be respectively outcomes of X and Y
and3

Marginal Expectation (review)
Let X and Y be continuous random vectors withjoint PDF
Marginal Expectation (mean) of X

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Marginal Expectation (review)
Let X and Y be continuous random variables withjoint PDF
Marginal Expectation (mean) of X

new notation(following the ME233 class notes)

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Marginal Expectationis the minimum least squares
marginal estimator of X, i.e.
• For any deterministic vector z

Euclidean norm

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Marginal Expectation

Proof:

0

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Conditional Expectation (review)
Let X and Y be continuous random vectors withjoint PDF
Conditional Expectation (conditional mean)of X given and outcome Y = y8Conditional Expectation (review)
Conditional Expectation (conditional mean)of X given and outcome Y = ynew notation(following the ME233 class notes)

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Conditional Expectation
Notice that the conditional expectation

can be interpreted as a function of the random variable Y.

10Conditional Expectation
Lemma:For any functionof the random vector Y, withthe appropriate dimensions

we can replace X by its conditional expectation

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Marginal Expectation

Proof:First examine the left-hand side:12

Marginal Expectation

Proof:First examine the left-hand side:

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Marginal Expectation

Proof:
Now examine the right-hand side:

Not a function of x

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Marginal Expectation

Proof:
Therefore,

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Conditional Expectation
Theorem:is the least squares minimum estimator of X
given Y, i.e.

for all functionsof Y of appropriate dimensions

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Marginal Expectation

Proof:

17

Marginal Expectation

Proof:
Define0Sincefor all outcomes,

Conditional Expectation for Gaussians(review)
Whenwhere18

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Conditional Mean for Gaussians

When0

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Conditional Mean for Gaussians

When

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Conditional Mean for Gaussians

When00

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Least Squares Estimation: Property 1• The conditional estimation error
are uncorrelated•

andare orthogonalandand23

Least Squares Estimation: Property 1

Proof

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Least Squares Estimation: Property 1

Proof00

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Least Squares Estimation: Property 1

Proofscalar

Why does trace
commute withexpectation?

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Deterministic interpretation ofProperty 1

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Recursive LS Estimation
Let X , Y and Z be jointly Gaussian R.V.s

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Recursive LS Estimation
1. Assume that we already know of outcome Y = y
and we have obtained

inverse of an M × M matrix

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Recursive LS Estimation
1. Assume that we already know of outcome Y = y
and we have obtained

2. Now we also know the outcome Z = z

How do we efficiently compute?

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Non-Recursive LS Estimation
1)

Define the vector2)

Computeinverse of an (p+M) × (p+M) matrix

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Least Squares Estimation: Property 2
Assume thatThen,

where

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Deterministic interpretation of Property 2

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Least Squares Estimation: Property 2
Proof:

00
because Z and Y
are uncorrelated

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Least Squares Estimation: Property 2
Proof:

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Least Squares Estimation: Property 2
Proof:

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Least Squares Estimation : Property 3
What happens when

Z and Y are correlated?

Then,This warrants further explanation…

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Recursive LS Estimation
Using Y , we can estimate X and Z by theirconditional means:
The conditional mean of X

The conditional mean of Z

The corresponding conditional estimation errors are:Uncorrelated withY (by Least Squares Property 1)

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Recursive LS Estimation
We have:
The conditional mean of X

The conditional mean of Z

If we get the outcomes Y=y and Z=z
The corresponding conditional estimation errors become:This is still randomThis is now an outcome

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Deterministic interpretation of Property 3

40Computation ofwhere:

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Least Squares Estimation : Property 3a) Recursive estimate

where:

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Least Squares Estimation : Property 3b) Recursive estimation error

where:

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Derivation of Recursive LS Estimation1)

Define the vector2)

Computeinversion of an (p+M) × (p+M) matrix

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Solution: use Schur complement• Givenand• Compute the Schur complement of

which is the conditional covariance45

Solution: use Schur complement of• Given• Then46

Non-Recursive LS Estimation

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Use Schur complement

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Use Schur complement

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Use Schur complement

expected value of X given outcome y

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Use Schur complement
We will now show that

The expected value ofgiven the outcome

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Computation ofThe conditional mean of Z given Y = y :52Computation ofTherefore,

We will now computeto verify thatusing the LS result:

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Computation ofUsing Gaussian least squares results:

0

Estimation errors always have zero means

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Computation ofUsing Gaussian least squares results:

the conditional covariance55

Computation ofUsing Gaussian least squares results:

Notice that, from the Schur complements result,

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Computation ofUsing Gaussian least squares results:057

Computation ofUsing Gaussian least squares results:

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Computation ofTherefore,

and

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Non-Recursive LS Estimation Error

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Use Schur complement

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Use Schur complement

62Summary• The conditional mean is the least squares
estimator:• For Gaussians, the conditional mean is anaffine function

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Summary
The conditional mean can be computed
recursively:

1. If we first know of outcome Y = y64

Summary
The conditional mean can be computed
recursively:

2 If we afterwards know of outcome Z = zthen

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Course Outline• Unit 0: Probability• Unit 1: State-space control, estimation• Unit 2: Input/output control• Unit 3: Adaptive control

Finished