Intelligent controller design based on gain and phase margin specifications

Daniel Czarkowski and Tom O’Mahony*

Advanced Control Group,

Department of Electronics Engineering,

Cork Institute of Technology,

e-mails: dczarkowski@cit.ie * tomahony@cit.ie

ISSC 2004, Belfast 2

Overview

• Do advanced control structures significantly outperform PID for SISO systems?

• Compare– PID– 2-DOF PID– GPC

ISSC 2004, Belfast 3

Contents List

• Types of controllers

• Tuning– Gain and phase margin criteria– Non convex problem to be solved– Genetic Algorithms

• Models used in the evaluation

• Results

• Conclusions

ISSC 2004, Belfast 4

PID controller

• Controller structure

• Control law

• 3 Variables to tune

R(s)

D(s)

G(s)C(s)Y(s)E(s) U(s)

( ) ( ) ip d

KU s E s K K s

s

, ,p i dK K K

ISSC 2004, Belfast 5

2-DOF PID controller

• Controller structure

• Control law

• 6 variables to tune

D(s)

G(s)F(s)Y(s)

H(s)

U(s)R(s)

( ) ( )( ) ( ) ( ) ( ) ( )

1

dip

d

p

sK c R s Y sKU s K b R s Y s R s Y s

sKsK N

, , , , ,p i dK K K b c N

ISSC 2004, Belfast 6

GPC controller

• Introduced by Clarke et al., 1987

• Two degree of freedom structure

• Digital controller was used

• Unconstrained control algorithm

• 7 tuning parameters1

1 2, , , , ( )uN N N T z

ISSC 2004, Belfast 7

GPC properties

• Advantages– Two degree of freedom– Optimal controller– Can handle more complex systems– More flexible structure

• Disadvantages– No well developed tuning rules– More difficult to tune– Very few industrial implementations

ISSC 2004, Belfast 8

Design strategy

• Performance & robustness

• Performance– IAE servo + regulator

• Robustness– Gain and phase margin

1 2

1

1

0

( ) ( )t t

k k t

IAE e k e k

6 , 45

14 , 45

m m

m m

A dB

A dB

ISSC 2004, Belfast 9

Non-convex problem

• Inverse unstable system

00.2

0.40.6

0.81

0

0.1

0.2

0.3

0.410.5

11

11.5

12

KpKi

IAE

local minimum

global minimum

Avoid local minima!

3

1 2( )

( 1)

sG s

s

ISSC 2004, Belfast 10

Genetic Algorithms

• Stochatistic optimisation method– Gray Coding– Stochatistic Universal Sampling, SUS– Single point crossover– Maximum number of generations, 300– Population size, 100– Constraints on the controller parameters

ISSC 2004, Belfast 11

Direct the GA

• GA optimisation problem

• Penality factors on gain and phase margins

min . . 6 , 45Am m m mJ IAE s t A dB

0 2 4 6 80

2

4

6

8

10

Am (dB)

Am

0 10 20 30 40 50 600

2

4

6

8

10

m (deg)

m

ISSC 2004, Belfast 12

Models

• Benchmark test– Inverse unstable system– Integrating systems– Underdamped system– Conditionally stable system– 3 models with time delay– First order model

• 12 models were evaluated

(K. J. Åström 1998, 2000)

ISSC 2004, Belfast 13

Results

• Comparison of PID, 2-DOF PID and GPC, 6 , 45m mIAE A dB , 14 , 45m mIAE A dB

GPC outperforms the other two counterparts

1 2 3 4 5 6 7 8 9 10 11 120

2

4

6

8

10

12

IAE

Model number1 2 3 4 5 6 7 8 9 10 11 12

0

2

4

6

8

10

12

14

16

IAE

Model number

ISSC 2004, Belfast 14

Results

• Design based on minimum Am=6dB– GPC vs PID, average IAE decreased by 43%– GPC vs 2-DOF PID, average IAE decreased by 25%– 2-DOF PID vs PID, average IAE decreased by 24%

• Design based on minimum Am=14dB– GPC vs PID, average IAE decreased by 37%– GPC vs 2-DOF PID, average IAE decreased by 22%– 2-DOF PID vs PID, average IAE decreased by 15%

ISSC 2004, Belfast 15

Set-point following

• Model

• Design

• Results

0 5 10 15 200

0.5

1

1.5

y(t)

0 5 10 15 20-1

0

1

2

3

4

5

u(t)

Time (sec)

GPC

2-DOF PIDPID

, 6 , 45m mIAE A dB

3

1( )

( 1)G s

s

2.08 1.62 1.54

14.64

2

6.83

GPCPI

IAE

A

F

m

D DO

Better robustness achieved by PID controllers!

ISSC 2004, Belfast 16

• Model

• Design

• Results

Set-point following

0 10 20 30 40 50 60 70 800

0.5

1

1.5

2

y(t)

0 10 20 30 40 50 60 70 80-0.5

0

0.5

1

1.5

u(t)

Time (sec)

GPC

PID

2-DOF PID

, 6 , 45m mIAE A dB

13

51( )

( 1)sG

ses

42.9 41 33.52

6.27 6.02 6.

2

00

IAE

A

GP POF

m

CID D

GPC performs 25% better than the PID controllers!

ISSC 2004, Belfast 17

Set-point following

• Model

• Design

• Results

153

1( )

( 1)G s e

s

4, 451 ,m mdIAE A B

53.9 53.7 55.3

14.00 14.01 1

2

4.00

IAE

Am

PID GPCDOF

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

y(t)

0 20 40 60 80 100 120 140 160 180 200-0.5

0

0.5

1

1.5

u(t)

Time (sec)

GPC

GPC does not perform better than the PID controllers!

ISSC 2004, Belfast 18

Summary of work

• A GA approach to tuning controllers based on gain and phase margin was applied

• Novel optimisation function was proposed

• Twelve models were tuned

• Three controllers were evaluated

• The controllers were subsequently employed on a real time system

ISSC 2004, Belfast 19

Conclusions

• GPC performance depends on the sampling period

• Tuning strategy works well, but...

• GPC performed better in simulation, but...

• Do advanced control algorithms perform better?

Questions?