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ME 233 Advanced Control II

Continuous time results 2
Kalman filters(ME233 Class Notes pp.KF7-KF10)

2Outline• 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 withstochastic input and measurement noise:

Where:•deterministic (known) input•

Gaussian, white noise, zero mean, input noise•

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

Gaussian

4Assumptions•Initial conditions:•Noise properties (in addition to Gaussian),:5

Conditional estimation• Conditional state estimate• Conditional state estimation error covariance6CT 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 theAlgebraic 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
matrixis Hurwitz.

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Steady State Kalman Filter
Theorem:3) Under stationary noise and the conditions in 1) and 2),
The observer residualof the KF:

becomes white10

KF as a innovations (whitening) filter
plant
Kalman filter
W(s)+C  (s) Bw

White noise
input

~Y(s)

Y(s) +-^
Y(s)Colored noise
outputC  (s) L

White noise
output11

LQR duality
Cost:

Where:

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Kalman Filter & LQR Duality
Comparing ARE’s and feedback gains, we obtain the following
dualityduality
LQR

KF

PM

AAT

B

CT

R

V

CQTB’w = BwW 1/2

KLT(A-BK)(A-LC)T13

KF return difference equality

W(s)+C  (s) Bw~Y(s)Y(s) +-^Y(s)C  (s) L

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KF root locus for SISO Systems
W(s)+C  (s) Bw~Y(s)Y(s) +-^Y(s)C  (s) L

roots are Kalman filter poles
roots are plant poles

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KF symmetric root locus for SISO Systems

roots are Kalman filter poles
roots are plant poles

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KF Loop gain and phase margins (SISO)
Consider the closed loop sensitive transfer function~oY (z)Y(z) +-o^Y (z)C  (z) L17

KF Loop phase margins (SISO)~oY (z)Y(z) +-o^Y (z)C  (z) LSince,
The phase margin ofto 60 degrees.

is greater than or equal

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KF Loop gain margins (SISO)~oY (z)Y(z) +-o^Y (z)C  (z) L

Estimator was designed for
Estimator is guaranteed to remain asymptotically stable for

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SISO ARMAX stochastic models
SISO ARMAX model:(Hurwitz)

Kalman filter innovations (residual)

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SISO ARMAX stochastic models

one random
white noise
input~Y(s)

Y(s)
+U(s)

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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 aDT state estimation problem .
2. Obtain the DT Kalman filter for the DT stateestimation problem.
3. Obtain the CT Kalman filter from the DT Kalmanfilter 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 withstochastic 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 aDT state estimation problem .
2. Obtain the DT Kalman filter for the DT state
estimation problem.
3. Obtain the CT Kalman filter from the DT Kalmanfilter 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 aDT state estimation problem :•State and output equations:26

Derivation of the CT Kalman Filter•

Covariances (from pages 48-52 in random processlecture 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 stateestimation 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31

Derivation of the CT Kalman Filter
Kalman filter gain

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Derivation of the CT Kalman Filter
Riccati equation

Subtractingfrom both sides and dividing by

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Derivation of the CT Kalman Filter
Takingwe obtain