IV-1

IV. Advanced Design Techniques

� Nonlinear PID

� Model Independent Methodology

� Examples

� H2 and H∞

� Fuzzy Logic/Neural Networks/Genetic Algorithms

� Scaling, Parameterization and Optimization

� …

IV-2

Nonlinear PID

� The benefits of using nonlinear gains� Small error, large gain� Reduce the role of integrator

� Nonlinear integral control� Resolve the overshoot issue� Maintain zero s.s. error and good disturbance rejection

� Nonlinear differentiator� Approximate differentiator� Noise immunity

Gao et al, ISA 2001, CDC 2001

( ( ))( ) ( ) d

p p i i d

d g eu k g e k g e k

dt= + +∫

IV-3

Nonlinear Feedback

u = e

u = fal( e )

e

u

1 d

-d

1

| | ( ), | | ,( ) 0

/ , | | ,

a

a

e sign e e dfal e d

e d e d−

>= >

≤

IV-4

Nonlinear Feedback

26

10 ( ),a

u e sign e e r y= = −, 0,y w u w= + =&

IV-5

Nonlinear Feedback

27

, 2 sin(10 ), 10 ( )a

y w u w t u e sign e= + = + =&

IV-6

Differentiation

� Pure differentiation

� Linear approximation

� Tracking differentiator

� Robust Exact Differentiator

des

dt⇔

( 1)m

s

sτ +

+−−=

=

)2

||)(( 22

12

21

R

xxtvxRsignx

xx

&

&

1/ 2

1

1

| ( ) | ( ( ))

( ( ))

x y

y y k x v t sign x v t

y asign x v t

=

= − − −

= − −

&

&

IV-7

Differentiation

IV-8

Example: NPID Control of Power Converter

S

ou

rce

Digital Control

Loa

ds

S

S

S

Converter

Converter

Load

Leveling

Power

Distribution

Unit

IV-9

Digitally Controlled Power Converter

IV-10

Model Validation: Steady State

55 60 65 70 75 80 85 9020

22

24

26

28

30

32

34

Duty Ratio (% )

Ou

tpu

t V

olt

ag

e (

V)

Current Load = 20 A m ps

Converter

S A B E R

M A TLA B

TF /TF i

IV-11

Model Validation: Transient Response

0.02 0.022 0.024 0.026 0.028 0.03 0.032 0.034 0.036

26

26.5

27

27.5

28

28.5

Time (s)

Outp

ut

Voltage (

V)

Current Load = 20 Amps

TFi

SABER

MATLAB

Converter

PI/NPID Simulink Block

IV-13

NPID controller Simulink Block

IV-14

NPID Control Design

G-Function for proportional control

2 1 2

p p

1

( ) sgn( ) | |k G ( ) ( )

| |

p p p p p

p

p p

k e k k e ee G e

k e e

δ δ

δ

⋅ + − ⋅ ⋅ >⋅ =

⋅ ≤

IV-15

NPID Control Design

G-Function for integral control

i i 2 1 2i

1ii

k G ( ) ( ) sgn( ) | |G ( ) 0( )

| |G ( ) 0k 0

i i i i i

i

i i

e dt k e k k e ee eG e

k e ee e

δ δ

δ

⋅ ⋅ + − ⋅ ⋅ >⋅ ≥ =

⋅ ≤⋅ <⋅

∫

IV-16

2 1 2

d d

1

( ) ( ) sgn( ) | |k G ( , ) ( )

( ) | |

d d d d d

d d

d d

k y k k y yy G y

k y y

δ δδ

δ

⋅ − + − ⋅ ⋅ − >⋅ − − =

⋅ − ≤

& & && &

& &

NPID Control Design

G-Function for derivative control

IV-17

A Simplified NPID Implementation

Kd

Signal

Conditioning

Setpoint

Profile

Kp

KiPower

Converter

)1(2

+s

S

1

s

Pulse

Count

NPID CONTROLLER

IV-18

Model Independent Design

� Control design with the math model of the plant

� Estimating the plant dynamics in real time

� Actively compensate for the disturbance

� Ultimate robustness

IV-19

Active Disturbance Rejection Control

y ay by w bu= − − + +&& &

0 0 0( )y ay by w b b u b u f b u= − − + + − + = +&& &

0( )f ay by w b b u= − − + + −&

1 2

2 3 0

3

1

x x

x x b u

x h

y x

=

= +⇒

= =

&

&

&

x Ax Bu Eh

y Cx

= + +

=

&

0

0 1 0 0

0 0 1 ,

0 0 0 0

0

[1 0 0], 0

1

A B b

C E

= =

= =

1 2 3, , , x y x y x f h f= = = = &&

Plant:

State

Space

IV-20

Extended State Observer (ESO)

ˆ ˆ ˆ( )

ˆ ˆ

: observer gain

ˆ : estimated state

ˆ

x Ax Bu L y y

y Cx

L

x

x x

= + + −

=

→

&

IV-21

Control Law

0 3 0 3

1 2

2 0

1

0 1 2

ˆ ˆ( ) / ,

ˆ ˆ( )p d

u u x b x f

x x

x u

y x

u k r x k x

= − →

⇓

=

= =

= − −

&

&

IV-22

Schematics

Transient

Profile

+_

+_PD +_ Plant

Extended

State Observer

(ESO)

1/b0 b0

r(t) v2(t)

v1(t)

u0(t) u(t) y(t)

w(t)

3x̂2x̂

1x̂

IV-23

Separation Principle

0

0

0 1 0 0 0

0 0 1 , , [1 0 0] , 0

0 0 0 0 1

,

ˆ ˆ ˆ( )

ˆ ˆ

, / , (1/ 0)[- - -1]ˆˆ 0

A B b C E

x Ax Bu Eh

y Cx

x Ax Bu L y y

y Cx

B Ex x rA BFB B b F b kp kd

x hLC A LC BF Bx

e

= = = =

= + +

=

= + + −

=

= + = =

− +

&

&

&

&

0

A BF A BF BFig eig

LC A LC BF A LC

+ = − + −

IV-24

Application of ADRC in Motion Control

IV-25

Simulation Model

Y (Position)

Zero-Order Hold1

Zero-Order Hold

Yd Out1

Out2

Trapezoid Profile

Td

Sum6 Sum5

Sum4 Sum2

Sum1 16.5 0.71s +s 2

Servo Plant Scope4

Saturation1

In1

In2 Out1

Non-linear Combination 1/b0 Gain2

In1 In2 Out1

Extended State Observer

Demux Demux2

IV-26

0 0.5 1 1.5 20

0.5

1

1.5

0 0.5 1 1.5 2-0.02

0

0.02

0.04

0 0.5 1 1.5 2-2

0

2

Position (rev.)

E rror (rev.)

C ontrol (volt.)

AD RC

Time (sec.)

P ID - -

PID vs. ADRC: nominal case

IV-27

0 0.5 1 1.5 20

0.5

1

1.5

0 0.5 1 1.5 2-0.1

0

0.1

0 0.5 1 1.5 2-5

0

5

Position (rev.)

Error (rev.)

Control (volt.)

ADRC PID - -

PID vs. ADRC: 100% load change

IV-28

PID vs. ADRC: 20% torque disturbance

0 1 2 3 4 50

0.5

1

1.5

0 1 2 3 4 5-0.05

0

0.05

0 1 2 3 4 5-2

0

2

Position(rev.)

Error(rev.)

Control(volt.)

ADRC PID - -

IV-29

Performance of the disturbance observer: simulation

0 1 2 3 4 5-30

-20

-10

0

10

20

30

a(t)

z3(t)

Total disturbance and its estimation

Time (sec.)

IV-30

Hardware Test Setup

PC-based

ADRC

DACDC Singal

ConditioningDC drive

(2 channels)

Electro-

mechanical

Plant

Encoders

Quadrature

CountingBoard

DisturbanceSignal

IV-31

Hardware Test: torque disturbance

0 2 4 6 8 10 120

0.5

1

1.5

0 2 4 6 8 10 12-0.1

0

0.1

0 2 4 6 8 10 12-5

0

5

Torque Disturbance Rejection Rev.

Rev.

Volts

Position

Position error

Control Command

ADRC

ADRC

ADRC

PID

PID

PID

IV-32

Performance of the disturbance observer: Hardware Test

0 0.5 1 1.5 2 2.5-50

-40

-30

-20

-10

0

10

20

Z3(t)

a(t)

D isturbance Observation

Time (sec.)

IV-33

PID vs. ADRC: sensitivity function

10-2

10-1

100

101

102

103

104

-120

-100

-80

-60

-40

-20

0

20

Y/Yd (dB)

Frequency (rad/s)

Position disturbance

PID

ADRC

IV-34

PID vs. ADRC: Torque Disturbance to Output

10-2

10-1

100

101

102

103

104

-120

-100

-80

-60

-40

-20

0

Y/Td (dB)

Frequency (rad/s)

Torque disturbance rejection

PID

ADRC

IV-35

H2 and H∞

� Optimal Control Methods

� Based on Small Gain Theorem

IV-36

Fuzzy Logic/Neural Networks/GA

� Incorporating human intelligence into control schemes

� Self-learning capability

� Practicality

IV-37

Latest Research

� Scaling � One controller is scaled for different plants

� Parameterization:� Making all control parameters a function of bandwidth

� Practical Optimization� Objectives (cost function)� Physical Limitations (constraints)� A design procedure to find the optimal solution

IV-38

Summary

� Basic Concepts

� Existing Design Methods

� Problem Solving Skills

� Advanced Techniques

� Real World Examples

IV-39

To Practice Control Design

� Good grasp of fundamentals

� Clear understanding of the system and problem

� Think through problems via analytical tools� What happened and why?

� A problem understood is almost a problem solved