ECSE 4962 Control Systems Design

Instructor: Professor John T. Wen

TA: Ben Potsaid

February 10, 2004

http://www.cat.rpi.edu/~wen/ECSE4962S04/

Parameter Identification

Today

• Friction identification

• Parameter identification

• Frequency domain identification

• Input excitation selection

Project proposal and presentation next Wed!

Single Joint Model

θL denotes pan or tilt angle (with coupling between the two axes ignored).

Goal of identification: find a1 , a2 , a3 , a4 by using the experimental data (choose V(t) and measure θL(t)).

{ {1 2 3 4

inputconstant

sgn( ) sin

Note that the equation has already been divided by the inertia.

L L L L

viscous Coulomb gravityfriction friction load

a a a V aθ θ θ θ+ + = +&& & &14243 14243

Coulomb Friction (dry)

Viscous Friction (oil)

force

mg

N

NFcoulomb µ=

ks µµµ == Assume

Combined Frictionviscouscoulombfriction FFF +=

Friction is constant while the object is moving

velocityBvFviscous =

Friction is proportionalvelocity

Friction Overview

Friction comes from:

•Motor Bearings

•Motor commutator

•Gearhead gears and bearings

•Belts and pulleys

•Joint Bearings

We will lump all of the friction sources together and represent their behavior with a combination of Coulomb and viscous friction models.

To identify friction, we will command a constant torque and then monitor the steady state joint velocity.

Sources of Friction

Model of Friction

1-DOF model (mount system to avoid gravity):

ID strategy: If V = constant = A, then à steady state value:Lθ&

1 2 3sgnssa a A a Aθ + =&

1

3

slope = aa

A

ssθ&

- cF

Stiction

Stribeck friction

1 2 3sgn( )L L La a a Vθ θ θ+ + =&& & &

1 2

3 3

sgnssa a

A Aa a

θ + =&or

2

3

aa

2

3

aa

−

1

3

slope = aa

Input = Constant Torque

Each line shows the velocity response for that torque.

System reaches a steady state velocity

No movement for given torque (T?0)

Friction Identification

Coulomb_pos = 0.075 NmCoulomb_neg = -0.07 NmViscous_pos = 1.46e-3 NmS/radViscous_neg = 1.56e-3 NmS/rad

Coulomb friction Slope of line is viscous friction

These are the results for the demo pantiltunit for joint 1

Friction Identification Result

Why identify friction?

You can cancel it!

1 21

3 3

( sgn( ))L L

a aV V

a aθ θ= − −& &

3 1L a Vθ =&&

If the friction model and id were exact, you can design controller V1 for a simple double integrator system!

Friction Cancellation

Caution:

• Cancellation is subject to zero-order hold

• Friction model is not exact (position dependence, load dependence, more complex function shape)

• Velocity is not perfect (velocity estimation is very important!)

Whatever leftover is considered as disturbance and must be dealt with by the controller.

Implementation

• You can construct a simple Simulink diagram with constant voltage output (you set the voltage) and measures the encoder signal.

• Set voltage to a constant, record the encoder signal, estimate the joint velocity, repeat.

• You may want to do several identical runs and average the result (record mean and std dev).

• You can use a MATLAB script to automate the whole thing too.

Identification of the Full Model

( 1) ( 1) ( 1) ( 1) ( 1)2

1 2 3 4

Multiply by and integrate over 1 sampling period:

sins s s s s

s s s s s

k t k t k t k t k t

kt kt kt kt kt

dt a dt a dt a Vdt a dt

θ

θθ θ θ θ θ θ+ + + + +

+ + = +∫ ∫ ∫ ∫ ∫

&

&&& & & & &

Do this for k=0,1,…,N-1, the length of the recorded data, and estimate a1 , a2 , a3 , a4 .

1 2 3 4sgn( ) sina a a V aθ θ θ θ+ + = +&& & &

2 2 2 21 21 1 1

3 1 4 1

1( ) ( ) ( )

2 2 2( ) (cos cos )

k k k k s k k s

k k k k k

a at t

a V a

θ θ θ θ θ θ

θ θ θ θ

+ + +

+ +

− + + + +

= − − −

& & & & & &

Identification of the Full Model (Cont.)

0 1 0 1 0 1 0 1 0

1 2 1 2 1 2 1 2 1

1 1 1 1 1

( ) ( ) 2 ( ) 2(cos cos )

( ) ( ) 2 ( ) 2(cos cos )

( ) ( ) 2 ( ) 2(cos cos )N N N N N N N N N

V

VA

V

θ θ θ θ θ θ θ θ

θ θ θ θ θ θ θ θ

θ θ θ θ θ θ θ θ− − − − −

− + − + − − − − + − + − − −

= − + − + − − −

& & & && & & &M M

& & & &

Put all the equations together and write it as A x = b

The minimum norm solution (recall IEA!) is x = (ATA)-1ATb

Note that ATA is a 4 x 4 matrix. MATLAB : x=pinv(A)*b;

1

2

3

4

aa

xaa

=

2 21 02 22 1

2 21N N

b

θ θθ θ

θ θ −

− − =

−

& && &

M& &

Model Validation

• Evaluate ||Ax-b||

• Apply a different V(t) and compare simulated response vs. actual response

Choice of Input

Choice of V(t) is very important – it has to be “sufficiently exciting”, i.e., A should be a well conditioned matrix (cond(A) should not be too large).

Try multiple sinusoids with different frequencies – randomly generate phase to avoid input peaking.

1 1 1 2 2 2( ) sin( ) sin( ) ...V t A t A tω φ ω φ= + + + +

Implementation

• You can construct a simple Simulink diagram with a multiple-sinusoid voltage output (adjust the amplitude to avoid saturation) and measures the encoder signal.

• You may want to do several identical runs and average the result (record mean and std dev).

• Write a MATLAB script to post-process data (with velocity estimation!) to generate the parameter estimate. Compare to friction ID.

• You can use a MATLAB script to automate the whole thing too.

Next Week

• Controller design and tuning

• Project proposal + presentation

Tomorrow at 6pm

• Bring lab notebook

• Groups 1-4: 6:00-7:00

• Groups 5-7: 7:00-8:00

Proposal

• Objective– Quantitative performance spec

• Design strategy– Model development– Performance spec vs. available components– Model identification– Simulation– Controller design and tuning– Design alternatives– Additional subsystem development (e.g.,

sensors)

Proposal

• Plan of action– Division of design into tasks– Assignment of responsibility– Partition into phases

• Verification procedure– Model identification and validation (std dev in

parameter estimation)– Test procedure (to check if spec if met!)– Robustness (tolerance of error, noise,

disturbance)• Bibliography

– Number references and use them in text!