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Implementation of adaptive control algorithm based on SPOC form

Winter 2011Supervisor: Dr. Ilan RusnakSubmitted by: Ofer Rosenberg

Roy Mainer

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Project Background

•Why do we need adaptive control?•What is an SPOC form?•What was accomplished in previous

projects?▫Implementations of linear systems

identification algorithms – in simulations and based on recorded data – not real time

▫Different algorithms returned different results when applied on the same system. Which is correct?

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Project Goals

•Implementation of SPOC form adaptive control algorithm – in REAL TIME

•Observation and measurement•Conclusions

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System Hierarchy

Matlab + Simulink

Dspace interfac

e

Linear motion system

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Project Development Steps:

•Introduction to the linear motion system and the Dspace interface.

•Understanding and repeating the results of previous related projects.

•Implementation of the SPOC algorithm in Simulink.

•Test the algorithm on both simulation and real time.

•Gathering results and conclusions.

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The SPOC Simulink blocks implementation

Ke

y

Ke

P0

Aed

1 Out1

z

1

X0

1.2e-9

Re

-C-

Qe

MatrixMultiply

Product9Matrix

Multiply

Product8

MatrixMultiply

Product7Matrix

Multiply

Product5

MatrixMultiply

Product4

MatrixMultiply

Product3

MatrixMultiply

Product2

MatrixMultiply

Product10

Product1

Product

z

1

P0

1

MatrixConcatenation3

1

MatrixConcatenation2

1

MatrixConcatenation1

2

MatrixConcatenation

uT

MathFunction1

uT

MathFunction

Inv

Ke

-K-

Gain2

-K-

Gain1

-K-

Gain

eye(9)

Constant5

eye(3)

Constant4

eye(3)

Constant3

zeros(3)

Constant2

diag(ones(1,2),1)

Constant1

eye(3)

Constant

[1x9]

Ce

Add6Add5

Add4

Add3

Add2

Add1

2

In2

1 In1

X0

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The SPOC Simulink blocks implementation - continued• Using matrix building blocks and algebra

we implement the SPOC algorithm.• Process noise estimation vector (Qe block)

is constant.

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Simulation and Real Time• Once the SPOC block was complete, our work

was divided to two parts – simulation and real time.

• Simulation:▫Mainly based on simulated inputs and transfer

functions. • Real Time:

▫Linear systems transfer function estimated in real time and recorded.

• Both methods estimate the transfer function during run time!

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Main System Simulink Diagram (simulation)

KcWhen openloop

gain=0

states

numerator

denominator

YY

To Workspace4

UU

To Workspace3

denum

To Workspace2

num

To Workspace1

In1Out1

State SpacePlant

Sine Wave2

Sine Wave1

Sine Wave

Scope

In1

In2

Out1

SPOC Subsystem

Kdc SW

In1

In2

In3

Out1

KDCSubsystem

Divide4

1Constant8

Add

0AckerGain In1

In2

In3

Out1

Out2

ACKERSubsystem

YY

UU

UU

X states

denom

denom

denom

KX

num

num

X0

KcKdc

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Main System Simulink Diagram (simulation) - continued• A 3rd order linear system requires an input

comprised of at least 3 non dependent signals.• Input is fed to both the transfer function (also 3rd

order) and the SPOC block.• Transfer function output is recorded and fed to the

SPOC block.• The SPOC block outputs are the state space vector,

the numerators and denominators vectors.• Other blocks: Acker and KDC are used to gain

stability by moving the poles to pre determined locations and normalizing the system’s gain respectively.

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Simulation ResultsNumerator Coefficients Denominator Coefficient

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5Numerator Coefficients

time [sec]

b1

b2

b3

0 10 20 30 40 50 60 70 80 90 100-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0Denominator Coefficients

time [sec]

a1

a2a3

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Simulation Results – continued

•Simulated transfer function:

•Estimated transfer function:

•Success!

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Main System Simulink Diagram (real time)

X0

shut downat simulation

end

start simulationwith 0 inputsfor 10 sec.

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Kc_out

9

inv_gain_out

8

states_out

7

denum_out

6

num_out

5

UU_norm

4

UU

3

error

2

YY

1

Kcx_out

DAC

to motor

states

numerator

limit

1

input gain

0.012

enc_scale_factor

denominator

0

Zero

Switch2Switch1

Switch

Sine Wave3

Sine Wave2

Sine Wave1

Sine Wave

In1

In2

Out1

SPOC Subsystem

RTI Data

PulseGenerator

1

One

MultiportSwitch

Mult

20

Kp

0

Kcx_gain0/0.1/0.5/1

In1

In2

In3

Out1

KDCSubsystem

Inv_GainManual Switch

Divide4

ENCODERMASTER SETUP

DS1104ENC_SETUP

Enc position

Enc delta position

DS1104ENC_POS_C1

1

Constant2

0

Constant1

1

Constant

0

Clock1

0

Clock

Chirp Signal

Add5

In1

In2

In3

Out1

Out2

ACKERSubsystem

v el_error

pos_error

YY

YY

YY

UU

error

error

xo

states

num

num

denum

denum

Kcx

kc

kc

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Main System Simulink Diagram (real time) - continued• Based on the simulation schematic.• Input switching to allow multiple choices

and stop the motor from reaching the rail end.

• Gain blocks based on previous empiric results.

• Dspace designated blocks:• Inputs are fed to motor through Dspace

D/A.• Outputs (motor position on rail) is fed

through position feedback.

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Process noise estimation vector

• The ratio Qe/Re affects the estimated coefficients convergence speed.

• Re – Measurement noise• Qe – Process noise

• Results show that high ratio improves estimation speed while low ratio reduces noise.

• In simulation we have no measurement noise so no need to switch.

• Convergence time is approximately 20 [sec] in real time.

• After 20 [sec] the Qe vector is switched to lower the ratio and reduce the noise.

Switch1e-6*diag([1 1 1 0 0 0 0 0 0 ]) + 1e-4*diag([ 0 0 0 1 1 0 0 1 1])

Qe1

1e-6*diag([1 1 1 0 0 0 0 0 0 ]) + 1e-8*diag([ 0 0 0 1 1 0 0 1 1])

New Qe1

0

Clock

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Real Time ResultsNumerator Coefficients Denominator Coefficient

0 10 20 30 40 50 60-150

-100

-50

0

50

100

150

200

250Numerator Coefficients

time [sec]

b1

b2b3

0 10 20 30 40 50 60-300

-250

-200

-150

-100

-50

0

50

100

150Denominator Coefficients

time [sec]

a1

a2

a3

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Numerator Coefficients convergence – zoomed

5 10 15 20 25 30-150

-100

-50

0

50

100

150

200

250Numerator Coefficients

time [sec]

b1

b2b3

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Real Time Results – continued

•Estimated transfer function:

•Success?▫We have no reference to compare with…

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ConclusionsResults table Conclusions summary

Comparison Parameter

Simulation Real Time

Accuracy Perfect Transfer function converged

successfully .system might be of higher order.

Convergence time Very fast, 23 [sec] without Qe switching.

Also fast, 30 [sec] with proper Qe.

Noise Noiseless Dramatic decrease in noise after Qe switching.

Influence of closed loop

None Influences the estimation result

• Linear system was expected to be of 3rd order. Results show it is probably of higher order.

• High ratio of Qe/Re improves convergence speed, while lower ratio reduces noise.

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Articles Links • The links will be opened from technion computers or any

computer who is registered to IEEE Xplore Digital Library.

• Real-Time Simultaneous State Estimation and Parameters Identification of Linear Drive System with the SPOC based Algorithm

• http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=00532255

• http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=721052• http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4793065

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Tubes

SPOC OPEN ZOOM• http://www.youtube.com/watch?v=aFKcut

mu7Jg&feature=g-upl SPOC OPEN LOOP

• http://www.youtube.com/watch?v=KeURm9FxpgA&feature=g-upl

SPOC ACKER GAIN 0.5 ZOOM• http://www.youtube.com/watch?v=nv7Uw

oHDDZI&feature=g-upl SPOC ACKER GAIN 0.5

• http://www.youtube.com/watch?v=He0nM6c7g14&feature=g-upl

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Bibliography• “Real-Time Simultaneous State Estimation and Parameters Identification of Linear Drive System with the SPOC based Algorithm” \ Dr. Ilan Rusnak (article)• "Feedback Control of Dynamic Systems 6th Ed." \ Gene F. Franklin, J

David Powell, Abbas Emami-Naeini (681.516)• "Linear Control System Analysis and Design with Matlab 5th. Ed" \

John j. D'azzo, Stuart N. Sheldon (681.511)• “Control for Unstable Nonminimum Phase Uncertain Dynamic

Vehicle” \ Dr. Ilan Rusnak (article)• “State Observability and Parameters Identifiability of Stochastic

Linear Systems” \ Dr. Ilan Rusnak (article)• “Simultaneous State Observability and Parameters Identifiability of

Discrete Stochastic Linear Systems” \ Dr. Ilan Rusnak • Internet and especially Wikipedia• Project book by Gil Kanashty:

•." מעבדתית" מערכת על ליניאריות מערכות של זיהוי אלגוריתמי יישום

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Special thanksDr. Ilan RusnakKoby KochaiOrly VigderzonGil Kanashty

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Thanks for watchingOfer RosnebergRoy Mainer

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Adaptive Control (Wikipedia)

•“Adaptive Control is the control method used by a controller which must adapt to a controlled system with parameters which vary, or are initially uncertain”.

•For example, as an aircraft flies, its mass will slowly decrease as a result of fuel consumption; a control law is needed that adapts itself to such changing conditions.

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SPOC Algorithm

•SPOC – States and Parameters Observability Canonical Form.

•SPOC algorithm is another method of estimating the transfer function of a system.

•By representing the observer canonical form of a 3rd order linear system, we can isolate the condition for stability.

•That condition can be solved using Kalman filter.

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Linear Motion System

• Our linear motion system is based on a DC motor, traveling back and forth across the rail.

• The DC system has speed and even acceleration feedback.

• The motor is controlled by the computer via the Simulink implemented controller.

• This system was believed to be of 3rd order.

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Acker Block

We changed the coefficientsof the polynom.

The algorithem didn'twork for fast poles!

2 Out2

1Out1

[ 1 1.2 0.47 0.06]

pvv -required closedloop polynomial

plv

2

controlability matrix

(0 1 0)

col 2

(1 0 0)

col 1

kx

To Workspace2

kxx

To Workspace1

Terminator

Scope1

Product6

MatrixMultiply

Product18

MatrixMultiply

Product17

MatrixMultiply

Product16

MatrixMultiply

Product15

Product14

MatrixMultiply

Product13

MatrixMultiply

Product12

Product11

2

MatrixConcatenation4

Inv

Divide1

eye(3)

Constant6

[0 0 1]

Ce1

3

In3

2

In2

1

In1

KX

KXnum

num

num

num

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Real-Time Simultaneous State Estimation and Parameters Identification of Linear Drive System with the SPOC based Algorithm

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Real-Time Simultaneous State Estimation and Parameters Identification of Linear Drive System with the SPOC based Algorithm - continued

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Real-Time Simultaneous State Estimation and Parameters Identification of Linear Drive System with the SPOC based Algorithm - continued