Power Systems Control

Prof. Wonhee Kim

Ch.3. Controller Design in Time Domain

2

Stability in State Space Equation: State Feedback

x x u

y x u

t A t B t

t C t D t

u xt K t

x x x

x

t A t BK t

A BK t

K should be chosen such that (A-BK) is Hurwitz!

3

Observer DesignSystem

Observer design

- Aim:

- Estimation error define:

- Observer

- Estimation error dynamics

x Ax Bu

y Cx

x̂ x

ˆx x x

ˆ ˆ ˆ

ˆ ˆ

x Ax Bu L y y

y Cx

ˆ ˆ ˆx x x Ax Bu Ax Bu LC x x

A LC x

L should be chosen such that (A-LC) is Hurwitz!

System

Control input using estimated state

System with control input

Closed-loop System

4

Separation Principle

x Ax Bu

y Cx

ˆu Kx

ˆx Ax BKx

A BK x BKx

x A BK x BKx

x A LC x

K and L can be separatively designed

5

Controllability and Observability

6

Controllability and ObservabilityControllability: for any desired eigenvalues, K exists such that (A-BK) has the desired

eigenvalues

Controllability matrix has n rank.

Observability: for any desired eigenvalues, L exists such that (A-LC) has the desired

eigenvalues

Controllability matrix has n rank.

1n

conC B BA BA

1n

T T T T T T

obO C A C A C

1

ob

n

C

CAO

CA

7

State Feedback Controller Design for Reference Tracking

Regulation)

Using final value theorem, steady-state response with step reference should be 1.

where Kdc is a dc gain for C(sI-(A-BK))-1B

x x u

y x

t A t B t

t C t

u xt K t Jr t

x x x

x

t A t BK t BKr t

A BK t BJr t

x x u

y x

t A t B t

t C t

1Y sT s C sI A B

U s

1Y sT s C sI A BK B J

R s

0 0 0

lim lim lim 1dcs s s

y sT s R s T s JK

1

dc

JK

8

Tracking Controller Design1. System

2. Controller design

- Aim:

- Tracking error define:

- Tracking Error dynamics

- Controller

1 2

2 3

1 1n n n

x x

x x

x a x a x bu

1x r

1 1

1n

n n

e r x

e r x

1 1 2

1 1

n n

n n n n

e r x e

e r x r a x a x bu

1 2

1 1n n n

e e

e k e k e

1 1 2 2 1 1

1 n

n n n nu r a x a x a x k e k eb

9

Tracking Controller Design Example1. System

2. Controller design

- Aim:

- Tracking error define:

- Tracking Error dynamics

- Controller

Control gain k1, k2 should be designed such that Ac = [0 1; k1 k2] is Hurwitz

1 2

2 1 1 2 2

x x

x a x a x bu

1x r

1 1

2 2

e r x

e r x

1 1 2

2 2 1 1 2 2

e r x e

e r x r a x a x bu

1 2

2 1 1 2 2

e e

e k e k e

1 1 2 2 1 1 2 2

1u r a x a x k e k e

b

10

Tracking Controller II Design1. System

2. Desired state dynamics: ud is designed for the desired state

- Tracking error:

- Aim:

- Tracking error dynamics:

- Controller

x Ax Bu

y Cx

0 asde x x t

d

d d

d

e x x

Ax Bu Ax Bu

Ae Bu Bu

du u Ke

d d d

d d

x Ax Bu

y Cx

de x x

d d

d d

e Ae Bu B u Ke

Ae Bu Bu BKe

A BK e

11

Tracking Controller II Design: DC Motor1. System

2. Desired constant velocity ωd, id and vd:

3. Desired state dynamics: ud is designed for the desired state

- Tracking error:

- Aim:

- Tracking error dynamics:

- Controller:

1

1t

B KiJ

i K Ri vL

0 asde x x t

d

d d

d

e x x

Ax Bu Ax Bu

Ae Bu Bu

1 2

d

t d d d d

u u Ke

K Ri k k i i

d d d

d d

x Ax Bu

y Cx

de x x

d d

d d

e Ae Bu B u Ke

Ae Bu Bu BKe

A BK e

1 1

0 , 0d d t d d dB Ki K Ri vJ L

,d dd d t d d t d

B BRi v K Ri K

K K

12

Control Algorithm Implementation

DC Motor System

Velocity feedback (y=ω)

Input voltage

(v)

1

1t

B KiJ

i K Ri vL

y

1

2

1 ˆˆ ˆ ˆ

1ˆ ˆˆ ˆ

ˆˆ

t

B Ki l y yJ

i K Ri v l y yL

y

1 2ˆˆ

dd

t d d d d

Bi

K

v K Ri k k i i

Control Board

d

vˆˆ , i

13

Control Algorithm Implementation

DC Motor SystemControl Board

DC Motor

+ Encoder

Input voltage

(v)

Velocity feedback (y=ω)

14

Control Algorithm Implementation

DC Motor SystemControl Board

DC Motor

+ Encoder

Input voltage

(v)

Velocity feedback (y=ω)

Load

disturbance

Load disturbance may degrade control performance of DC motor!

15

Tracking Controller: Position Control of DC Motor

DC motor with unknown load disturbance, d for position control:

Not normal form!

Definition of Acceleration state variable

With α, DC motor model:

Now DC motor model becomes normal form, thus, tracking controller can be applied!

1

1t

B Ki dJ

i K Ri vL

2

2 2 2

1

1

t

B Ki dJ

B KiJ

KKB BK B KR Ki d i v

J J J JL JL JL

Note that the disturbance can include

the parameter uncertainties as well as

the external disturbance

16

Extended State Observer Design: DC MotorDC motor with unknown load disturbance, d for position control

Extended state d with assumption that d is constant

Extended state observer to estimate full state and disturbance using position feedback

1

1t

B Ki dJ

i K Ri vL

1

1

0

t

B Ki dJ

i K Ri vL

d

1

2

3

4

ˆ ˆˆ

1 ˆ ˆˆˆ ˆ

1 ˆˆ ˆˆ

ˆ ˆ

t

l

B Ki d lJ

i K Ri v lL

d l

17

Extended State Observer Design: DC MotorEstimation error

Estimation error dynamics

ˆˆ ˆˆ, , ,i i i d d d

1

2

3

4

1 ˆˆˆ

1 ˆˆt

l

B Ki d lJ

i K Ri lL

d l

1

2

3

4

1 0 0

1

0

0 0 0

t

A

l

B Kl

J J J

K iRi lL L d

d l

l1, l2, l3 and l4 should be chosen such that A is Hurwitz!

18

Experimental Results

Position control with normal observer

(without disturbance rejection)

Position control with ESO

(with disturbance rejection)