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Report

For

Project 5

Kalman Estimation of the State of a DC Motor

ASIF AL – RASHEED

EE 5521

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The state of a large dc motor, the shaft angle and the angular velocity of the shaft, is estimated from noisy

measurements of the shaft angle. The input to this motor is the applied voltage u(t) plus noise on this

voltage signal w(t). The output y(t) is the motor shaft angle θ(t) plus additive measurement noise v(t).

The transfer function for the motor is:

( ) 2

( ) ( 0.02)

s

U s s s.

Estimates of both states are required to monitor the status of the motor.

Now,

2( 0.02 ) ( ) 2 ( )

( ) 0.02 ( ) 2 ( )

s s s U s

t t u t

Let,

1 1 2

2 2 20.02 ( ) 2 ( ) 0.02 2 ( )

x x x

x x t u t x u t

The state model

1 1

2 2

0 1 0( )

0 0.02 2

x xu t

x x

1

2

( ) ( ) ( ) ( ) ( )

( ) 10 10 ( )

y t t v t x t v t

xy t v t

x

Initial condition for all of the simulations,

0 deg(0)

deg(0) 0

sec

.

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Case a:

22

4

deg1

secw

; 2 220 degv

;

0 degˆ(0)degˆ 0(0)sec

;

22

2 2

2

deg0 deg 0

sec(0)

deg deg0 0

sec sec

e .

Simulation results:

Fig 1: Plot of Kalman filter gain.

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Fig 2: Plot of estimation error standard deviation.

Fig 3: State and Estimation for shaft angle

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Fig 4: Estimation error for shaft angle

Fig 5: State, Estimation and Error for shaft angle

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Fig 6: State and Estimation for angular velocity

Fig 7: Estimation error for angular velocity

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Fig 8: State, Estimation and Error for angular velocity

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Case b: The same as case a, except

22

4

deg10

secw

Simulation results:

Fig 9: Plot of Kalman filter gain.

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Fig 10: Plot of estimation error standard deviation.

Fig 11: State and Estimation for shaft angle

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Fig 12: Estimation error for shaft angle

Fig 13: State, Estimation and Error for shaft angle

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Fig 14: State and Estimation for angular velocity

Fig 15: Estimation error for angular velocity

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Fig 16: State, Estimation and Error for angular velocity

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Case c: The same as case a, except 2 2200 degv

.

Simulation results:

Fig 17: Plot of Kalman filter gain

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Fig 18: Plot of estimation error standard deviation

Fig 19: State and Estimation for shaft angle

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Fig 20: Estimation error for shaft angle

Fig 21: State, Estimation and Error for shaft angle

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Fig 22: State and Estimation for angular velocity

Fig 23: Estimation error for angular velocity

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Fig 24: State, Estimation and Error for angular velocity

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Case d: The same as case a, except

5 degˆ(0)degˆ 1(0)sec

;

22

2 2

2

deg100 deg 0

sec(0)

deg deg0 10

sec sec

eS .

Simulation results:

Fig 25: Plot of Kalman filter gain

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Fig 26: Plot of estimation error standard deviation

Fig 27: State and Estimation for shaft angle

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Fig 28: Estimation error for shaft angle

Fig 29: State, Estimation and Error for shaft angle

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Fig 30: State and Estimation for angular velocity

Fig 31: Estimation error for angular velocity

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Fig 32: State, Estimation and Error for angular velocity

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Case e: The same as case a, except use the steady-state Kalman gain.

Simulation Results:

Fig 33: Plot of Kalman filter gain

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Fig 34: Plot of estimation error standard deviation

Fig 35: State and Estimation for shaft angle

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Fig 36: Estimation error for shaft angle

Fig 37: State, Estimation and Error for shaft angle

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Fig 38: State and Estimation for angular velocity

Fig 39: Estimation error for angular velocity

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Fig 40: State, Estimation and Error for shaft angle

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Comparison of the five cases:

Cases Kalman Filter

Gains

Estimation

error standard

deviations

Mean of

Estimation Error

Rate of

convergence

(s)

Standard Deviation

of Estimation

G1 G2 σe1 σe2 Shaft

angle

Angular

velocity

G1 G2 Shaft

angle

Angular

velocity

a 0.4806 0.1515 3.1 1.718 0.4861 1.6044 15 13 311.3252 4.3741

b 0.6992 0.3738 4.25 3.74 -0.114 0.2775 9 9 776.0154 9.1066

c 0.3007 0.05312 7.755 2.219 -0.436 0.3904 26 24 286.7357 3.0982

d 0.4806 0.1515 3.1 1.718 -0.288 0.0773 17 9 163.0103 3.6302

e 0.6004 0.2707 1.735 1.457 0.0625 0.1256 1 1 116.5068 4.8030

Comments:

A common trend that we observe among all cases is that the shaft angle estimate more accurately tracks

than the angular velocity estimate. From the results we see that using the steady-state Kalman gain yields

the best filter evident by the fastest rate of convergence, the lowest estimation error standard deviations,

and the lower values of mean and standard deviation of estimation and estimation errors. Also cases a and

d give almost identical results in terms of gains, convergence rates and estimation error standard

deviations but approach the steady state from opposite directions. Increasing the plant noise in case b

increases the output resulting in increase in gain. But increasing the output noise in case c reduces the

output, decreasing the gain. So we can conclude that the gain is proportional to the plant noise and

inversely proportional to the output noise.