Modeling, Simulation, and Control of a Real System

Robert ThroneElectrical and Computer EngineeringRose-Hulman Institute of Technology

Introduction

• Models of physical systems are widely used in undergraduate science and engineering education.

• Students erroneously believe even simple models are exact.

Introduction

• Obtained ECP Model 210a rectilinear mass, spring, damper systems for use in both system dynamics and controls systems labs.

• Models for these systems are easy to develop and students have seen these types of models in a variety of courses.

Introduction(mass, springs, and encoder)

Introduction(motor, rack and pinion, damper, and spring

connecting to first cart)

Introduction

We developed four groups of labs for the ECE introductory controls class for a one degree of freedom system:

• Time domain system identification.• Frequency domain system identification.• Closed loop plant gain estimation.• Controller design based on the model.

Parameters to Identify

In the transfer function model

we need to determine

• the gain

• the damping ratio

• the natural frequency

( )K

( )

( )n

2

2

( )2

1n n

KH s

s s

Time Domain System Identification

• Log decrement analysis

• Fitting the step response of a second order system to the measured step response

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.5

0

0.5

1

1.5

Time (sec)D

ispl

acem

ent

(cm

)

1

2

34

5 6 7

0 0.2 0.4 0.6 0.8 1 1.2

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (sec)

Dis

plac

emen

t (c

m)

EstimatedMeasured

0 0.5 1 1.5 2

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (sec)

Dis

plac

emen

t (c

m)

EstimatedMeasured

Frequency Domain System Identification

• Determine steady state frequency response by exciting the system at different frequencies.

• Compare to predicted frequency response.

• Optimize transfer function model to best fit measured frequency response.

Model/Actual Frequency Response(from log-decrement)

101

48

50

52

54

56

58

60

62

64

66

68

Mag

nitu

de (

dB)

Frequency (rad/sec)

= 0.0877, n = 25.43

ModelActual

Model/Actual Frequency Response(from fitting step response)

101

50

52

54

56

58

60

62

64

66

68

Mag

nitu

de (

dB)

Frequency (rad/sec)

= 0.1, n = 26.7

ModelActual

Model From Frequency Response

101

54

56

58

60

62

64

66

Mag

nitu

de (

dB)

Frequency (rad/sec)

= 0.19081, n = 26.1252

ModelActual

Closed Loop Plant Gain Estimation

• We model the motor as a gain, , and assume it is part of the plant

• We use a proportional controller with gain • The closed loop system is

• The closed loop plant gain is then

motorK

pK

clpg motorK K K

Closed Loop Plant Gain Estimation

• Input step of amplitude A

• Steady state output

• The closed loop plant gain is given by

ssy

clpg

1ss

p ss

yK

K A y

Results with Controllers

After identifying the system, I, PI, PD, and PID controllers were designed using Matlab’s sisotool to control the position of the mass (the first cart).

Both predicted (model based) responses and actual (real system) responses are plotted on the same graph.

Integral (I) Controller

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (sec)

Dis

plac

emen

t (c

m)

ModelActual

``It doesn’t work!’’

Proportional+Derivative (PD) Controller

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (sec)

Dis

plac

emen

t (c

m)

ModelActual

PID Controller(complex conjugate zeros)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

Dis

plac

emen

t (c

m)

ModelActual

PID Controller(real zeros)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time (sec)

Dis

plac

emen

t (c

m)

ModelActual

State Variable Feedback

0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Time (sec)

Pos

ition

(cm

) k

1 = 0.3, k

2 = 0.02, f = 0.40764

ModelActual

Conclusions

Students learn:

• Simple, commonly used models are not exact, but still very useful.

• Simple models are a reasonable starting point for design.

• Motors have limitations which must be incorporated into designs.

Conclusions

We have extended these labs to include

Model matching• ITAE • quadratic optimal• polynomial equation (Diophantine)

2 and 3 DOF state variable models

Acknowledgement

This material is based upon work supported by the National Science Foundation under Grant No. 0310445