Design of Disturbance Rejection Controllers for a Magnetic Suspension System

By: Jon DunlapAdvisor: Dr. Winfred K.N. Anakwa

Bradley UniversityApril 27, 2006

Outline Of Presentation: Goal System Information Previous Lab Work Preliminary Lab Work Internal Model Principle Design Process Results Conclusion

Goal Multiple Controllers for Multiple Disturbances

Digital Controllers Created In Simulink xPC Target Box Serving as “Controller

Container” Minimize Steady-State Error, Overshoot and

Setting Time

Act As A Stepping Stone From Previous Work Practical Use in Antenna Stabilization

Method Method of Choice:

Internal Model Principle B.A. Francis & W.M. Wonham

The Internal Model Principle of Control Theory, 1976

Chi-Tsong Chen Linear System Theory and Design, 3rd, 1999

Analogous to an Umbrella

Functional Description Host PC using Simulink and xPC software xPC Target Box with Controllers

Magnetic Suspension System Feedback Incorporated 33-210

+Reference

Set P

oint

Error Controller on xPC

Target BoxControl Signal ‘U’ +

Disturbance

Disturbance Model

Magnetic Suspension

System

Photo sensor

Ball Position

Position Signal -

Block Diagram

Controller and Plant

Ball Position

Set Point

Reference Signal

Disturbance

Previous Lab Work

Using Classical Controller Will It Reject Disturbances?

7.67

1/961s +-12

Transfer Fcn1

7.67

1/961s +-12

Transfer Fcn

ste

To Workspace2

sin

To Workspace1Step3

Step1

Step

Sine Wavenum(z)

z -z2

DiscreteTransfer Fcn1

num(z)

z -z2

DiscreteTransfer Fcn

U + DER U

U + DER U

Previous Lab Work Results of Classical Controller With

Disturbance

Rejected Step Disturbance0 0.5 1 1.5 2 2.5 3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

.25 Input Reference

input w/.1sin(pi*t) disturbance

input w/ .1 unit-step disturbance

Preliminary Lab Work Laplace Transfer

Functions Found

Later Converted To Discrete Using Zero-order Hold

Disturbance Laplace Equation

k*Cos(a*T)

k*Sin(a*T)

k-unit step

k-unit ramp

22

*

as

sk

22

*

as

ak

s

k

2s

k

Internal Model Principle Uses a Model to Cancel Unstable Poles of

Reference and Disturbance Inputs to Provide Asymptotic Tracking and Disturbance Rejection

Model Is: Least Common Multiple of Unstable or Zero Continuous Denominator Poles Ramp Disturbance Input = 0,0 Step Reference Input = 0 Model = P = 0,0

2s

2s

2s

2ss

Design Approach

Disturbance Removed – Model Inserted Must Stabilize Plant at all Times Disturbance Should Never Affect Plant

Output 3 Known, 2 Unknown A(z)D(z)P(z) + B(z)N(z)

1

Reference

N(z)

D(z)

Plant

B(z)

A(z)*P(z)

Controller w/ Model

E

Diophantine Equation A(z)D(z)P(z) + B(z)N(z) = F(z) Want:

Choose Poles to Form F Polynomial Discrete, Close to 1

Need: Order of Controller

Order of D(z)P(z) – 1 = Order of Controller Order of F

2*(Order of D(z)P(z))-1 = Order of F

Keep In Mind - Account for Model when Implementing Controller

Controller Order Assumes Denominator without Model

Adding Model Increases Order Beyond Designed Value Ex. If D(z)P(z)=4, then Controller=3 But Model=2 so Controller Denominator really should

be 1

012

201

012

23

3 1*

pzpzpaza

bzbzbzb

Pole Placement Ideal Situation

Tsettle = 60ms %O.S. = 18 = .479 Wn=

Wn= 139.1788 Wn<

*Tsettle

4

21100

e%O.S.

001.,*22

TT

Pole Placement Problems and Solution

Complex Poles Give Oscillations Wn < Poles>.92

All Poles Close To 1 Is Too Slow Speed Up With Poles Closer To Origin

Iterative Design Approach Required Working Poles For Ramp Rejection At:

.9947, .9716, .9275, .9, .01 F(z) =

001.,*22

TT

0.0081-0.8408z3.4615z-5.4326z3.8038z-z 2345

Diophantine Solution A(z)D(z)P(z) + B(z)N(z) = F(z)

Combine D(z)P(z) to Equal D*(z)

For Each X(z)=

System of Equations To Be Solved Simultaneously A0D*0+B0N0=F0…AnD*n+BnNn=Fn

nxxx ...10

Diophantine Solution

Using Previous Example:01

012

23

3

aza

bzbzbzb

510

10

10

10

43210

10

43210

321100 ...

0000

0000

0000

*****0

0000

0*****

* fff

nn

nn

nn

ddddd

nn

ddddd

bbbaba

Actual Values For Ramp Controller

N(z) = D(z) = P(z) = F(z) =

B(z) = A(z) =A(z)P(z) =

0.0081-0.8408z3.4615z-5.4326z3.8038z-z 2345

4-6.634e4z-6.634e

12.001z-z2

0.091z

149.3966-458.5402z469.0901z-159.9474z 23

12z-z2

0.091-0.8179z1.909z-z 23

xPC Simulink Implementation

Ref_D_Num(z)

Ref_D_Den(z)

ReferenceTransfer Function

Dis_D_Num(z)

Dis_D_Den(z)

Disturbance Transfer Function

du/dt

Derivative1

du/dt

Derivative

MM-32Diamond

Analog Output1

D/A Output ToPlant

Numd(z)

Dend(z)

ControllerTransferFunction

1

Constant1

1

Constant

MM-32Diamond

Analog Input1

A/D Input FromBall Position

DR

E Uc U+

Results – Stability

1. 2V Step Disturbance at 2.00V Set Point 2. 5V/s Ramp Disturbance at 2.00V Set Point

1. 2.

2.042 2.042

Results – Tracking

1. 5V/s Ramp Disturbance

6Hz .5V Peak-PeakSine Wave Input

5.98Hz .7V Peak-PeakSine Wave Output

Conclusion Problems

Simulation Does Not Match Plant Pole Locations are Hard to Find <300mV Error at Start Up

Future Continue to Implement Sinusoidal and

Square Wave Fine Tune Ramp With Better Poles

Questions?

Magnetic Suspension System Control and Disturbance Signal Create

Current Current Induces Magnetic Field Field Suspends Ball Sensor Translates Location into Voltage

xPC Target Box and Host PC Using ±10 V ADC and DAC Host Uploads Controller and Commands Process Position Data and Passes Control

User Input: Plant, Reference,

Disturbance

Find Roots of Disturbance Denominator

Find Roots of Reference

Denominator

Do Both Have At Least 1 Stable Root?

Display ErrorEnd Program

NoYes

Find Least Common Multiple of

Stable Roots

Convolve Plant Denominator with

Model

Determine Number of Poles Required

Create Plant Matrix Based on Numerator and

Denominator after Convolution

Solve For Compensator Coefficients

Extract Discrete Compensator

Numerator and Denominator

Send To Simulink Block

Convert From Continuous To

Discrete Time with User Input

Sampling Time

Modeling Hybrid Systems“Simulink treats any model that has both continuous and discrete sample times as a hybrid model, presuming that the model has both continuous and discrete states. Solving such a model entails choosing a step size that satisfies both the precision constraint on the continuous state integration and the sample time hit constraint on the discrete states. Simulink meets this requirement by passing the next sample time hit, as determined by the discrete solver, as an additional constraint on the continuous solver. The continuous solver must choose a step size that advances the simulation up to but not beyond the time of the next sample time hit. The continuous solver can take a time step short of the next sample time hit to meet its accuracy constraint but it cannot take a step beyond the next sample time hit even if its accuracy constraint allows it to.”

http://www.mathworks.com/access/helpdesk/help/toolbox/simulink/ug/f7-23387.html