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ME 233 Advanced Control II

Continuous time results 2

Kalman filters

(ME233 Class Notes pp.KF7-KF10)

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Outline

• Continuous time Kalman Filter

• LQ-KF duality

• KF return difference equality

– symmetric root locus

• ARMAX models

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Stochastic state model

Consider the following nth order LTI system with

stochastic input and measurement noise:

Where:

• deterministic (known) input

• Gaussian, white noise, zero mean, input noise

• Gaussian, white noise, zero mean, meas. noise

• Gaussian

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Assumptions

• Initial conditions:

• Noise properties (in addition to Gaussian),:

Conditional estimation

• Conditional state estimate

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• Conditional state estimation error covariance

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CT Kalman Filter

Kalman filter:

Where:

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Steady State KF Theorem:

1) If the pair (C, A) is observable (or detectable):

The solution of the Riccati differential equation

Converges to a stationary solution, which satisfies the

Algebraic Riccati Equation (ARE):

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Steady State KF Theorem:

2) If in addition to 1) the pair (A,B’w) is controllable (stabilizable), where

The solution of the Algebraic Riccati Equation (ARE):

is unique, positive definite (semi-definite) , and the close loop observer

matrix

is Hurwitz.

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Steady State Kalman FilterTheorem:

3) Under stationary noise and the conditions in 1) and 2),

The observer residual

of the KF:

becomes white

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KF as a innovations (whitening) filter

White noise

input Colored noise

output

White noise

output

plant

Kalman filter

W(s) Y(s)+

C Bw (s)

-

+

L

Y(s)

Y(s)^

~

C (s)

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LQR dualityCost:

Where:

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Kalman Filter & LQR DualityComparing ARE’s and feedback gains, we obtain the following

duality

LQR KF

P M

A AT

B CT

R V

CQT B’w = BwW 1/2

K LT

(A-BK) (A-LC)T

duality

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W(s) Y(s)+

C Bw (s)

-

+

L

Y(s)

Y(s)^

~

C (s)

KF return difference equality

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KF root locus for SISO Systems

W(s) Y(s)+

C Bw (s)

-

+

L

Y(s)

Y(s)^

~

C (s)

roots are plant poles

roots are Kalman filter poles

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KF symmetric root locus for SISO Systems

roots are plant poles

roots are Kalman filter poles

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KF Loop gain and phase margins (SISO)

Consider the closed loop sensitive transfer function

-

+

L

Y(z) Y (z)o

Y (z)o

~

C (z)

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KF Loop phase margins (SISO)

Since,

The phase margin of is greater than or equal

to 60 degrees.

-

+

L

Y(z) Y (z)o

Y (z)o

~

C (z)

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KF Loop gain margins (SISO)

Estimator was designed for

Estimator is guaranteed to remain asymptotically stable for

-

+

L

Y(z) Y (z)o

Y (z)o

~

C (z)

SISO ARMAX model:

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SISO ARMAX stochastic models

(Hurwitz)

Kalman filter innovations (residual)

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Y(s)

U(s)

+

Y(s)~

SISO ARMAX stochastic models

one random

white noise

input

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Outline

– Continuous time Kalman Filter

– LQ-KF duality

– KF return difference equality

• symmetric root locus

– ARMAX models

Additional material:

• Derivation of continuous time Kalman Filter

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Derivation of the CT Kalman Filter

1. Approximate the CT state estimation problem by a

DT state estimation problem .

2. Obtain the DT Kalman filter for the DT state

estimation problem.

3. Obtain the CT Kalman filter from the DT Kalman

filter by taking the limit as the sampling time

approaches to zero.

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CT Kalman Filter

Consider the following nth order LTI system with

stochastic input and measurement noise:

Where:

• deterministic input

• Gaussian, white noise, zero mean, input noise

• Gaussian, white noise, zero mean, meas. noise

• Gaussian

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Derivation of the CT Kalman Filter

1. Approximate the CT state estimation problem by a

DT state estimation problem .

2. Obtain the DT Kalman filter for the DT state

estimation problem.

3. Obtain the CT Kalman filter from the DT Kalman

filter by taking the limit as the sampling time

approaches to zero.

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Derivation of the CT Kalman Filter

1. Approximate the CT state estimation problem by a

DT state estimation problem :

• State and output equations:

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Derivation of the CT Kalman Filter

• Covariances (from pages 48-52 in random process

lecture 8):

Notice that:

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Derivation of the CT Kalman Filter

• Covariances:

Notice that:

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Derivation of the CT Kalman Filter

2. Obtain the DT Kalman filter for the DT state

estimation problem.

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Derivation of the CT Kalman Filter3) Obtain the CT Kalman filter from the DT Kalman filter.

• State Equation:

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Derivation of the CT Kalman Filter

Taking limit as

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Derivation of the CT Kalman Filter

Kalman filter gain

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Derivation of the CT Kalman Filter

Riccati equation

Subtracting from both sides and dividing by

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Derivation of the CT Kalman Filter

Taking

we obtain