TRANSFER FUNCTION FORMULATION OF ANALYSIS AND DESIGN PROBLEMS FOR

DISCRETE/CONTINUOUS CONTROL THEORY

by

Bardhyl Prishtina

Thesis Advisor: Dr. C.D. Johnson

Committee Member: Dr. Peter Gibson

Committee Member: Dr. Laurie L. Joiner

Date: December 05, 2003

1

Analog-Type ControlSignal

ZOH-Type ControlSignal

u(t)

t0

t

u(t)

T 2T 3T 4T 5T 6T 7T0

0 t

u(t)

T 2T 3T 4T 5T 6T 7T

Discrete/ContinuousType Control Signal

2

STATE SPACE REPRESENTATION

)()(

)()()(

tCxty

tButAxtx

)()(

)(~

)(~

])1[(

kTCxkTy

kTuBkTxATkx

)()(

)()(~

])1[(

kTCxkTy

kTBHkTxATkx

Linear and TimeInvariant

Continuous-TimePlant Model

Equivalent ZOH-Type DiscreteTime Plant Model

Equivalent D/C-Type DiscreteTime Plant Model

3

IDEA OF DISCRETE/CONTINUOUS-TYPE CONTROL

The control signal is allowed to vary continuously and strategically in time between two consecutive sample times. The D/C control can be viewed as generalization of a classical ZOH-Type discrete-time control. The control-variation rule is calculated and decided at the beginning at each intersample period. These control-variations can not be changed until the beginning of next sample period

4

D/C TYPE CONTROL SIGNAL

0 t

u(t)

T 2T 3T 4T 5T 6T 7T

Figure 1.2: A Typical “Smart” D/C Type Discrete-Time Control Signal Variations

[8]

5

LINEAR-IN-TIME TYPE D/C CONTROL

))(()()( 21 kTtkTCkTCtuLiT

t

u(t)

0 T 2T

C1(0)

C1(T)

C1(2T)

3T 4T 5T 6T

C1(3T)

C1(4T)

C1(5T)

C2(0)

C2(T)

C2(2T)

C2(3T)

C2(4T)

C2(5T)

Figure 1.3: Typical Linear in Time Type D/C Control Variations [8]

(1-2)

Polynomial-Spline Type Wave Form Model

6

QUADRATIC-IN-TIME TYPE OF D/C CONTROL

2321 ))(())(()()( kTtkTCkTtkTCkTCtuQiT

t

u(t)

0 T 2T

C1(0)

C1(T)

C1(2T)

3T 4T 5T 6T

C1(3T)

C1(5T)

C1(4T)

Figure 1.4: Typical Quadratic in Time Type of D/C Control Variations [8]

(1-3)

7

CUBIC-IN-TIME TYPE OF D/C CONTROL

34

2321 ))(())(())(()()( kTtkTCkTtkTCkTtkTCkTCtuCiT

Figure 1.5: Typical Cubic-in-Time Type of D/C Control Variations

(1-4)

t

u(t)

0 T 2T

C1(0)

C1(T)

C1(2T)

3T 4T 5T 6T

C1(3T)

C1(5T)

C1(4T)

8

EXPONENTIAL-IN-TIME TYPE OF D/C CONTROL

,

)(21 )()()( kTt

EiT ekTCkTCtu 0

Figure 1.6: Typical Exponential-in-Time Type D/C Control Variations [8]

(1-5)

t

u(t)

0 T 2T

C1(0)+C2(0)

C1(T)+C2(T)

C1(2T)+C2(2T)

3T 4T 5T 6T

C1(3T)+C2(3T)

C1(5T)+C2(5T)

C1(4T)+C2(4T)

9

EXACT DISCRETIZATION OF CONTINUOUS-TIME-STATE-SPACE EQUATION WITH D/C CONTROL

)()(

)()()(

tCxty

tButAxtx

)()(

)()(~

])1[(

kTCxkTy

kTBHkTxATkx

ATeA ~

dkTHBTkBH u

Tk

kT

),(],)1[()1(

(1-9)

(1-10)

(1-11) (1-13)

(1-12)

)()(

)()(

tDt

tHtu

TktkT )1( for

Du e )(

10

BLOCK-DIAGRAM OF D/C CONTROLLED SYSTEM USING REAL-TIME STATE OBSERVER

))(ˆ)(()()(ˆ~

])1[(ˆ 0 kTxCkTyKkTBHkTxATkx (1-14)

Plant StateObserver

Plant Model

D/C ControlAlgorithm

u (t)

+-

u (t)D/C

y(t)

y(kT)

u(kT)(kT) x(kT)

D/C

11

REVIEW OF THE TRANSFER-FUNCTION METHOD IN CLASSICAL ZOH-TYPE SYSTEMS ANALYSIS AND

CONTROL DESIGN

t

u(t)

T 2T 3T 4T 5T 6T 7T0

)()(

)(~

)(~

])1[(

kTCxkTy

kTuBkTxATkx

(2-1)

12

ATeA ~

T

TA BdeB0

)(~

BAzICzGP

~)

~()( 1

State Transition Matrix

Control Distribution Matrix

ZOH-Type Plant Transfer-Function Matrix

(2-7)

(2-8)

(2-12)

13

Gp(z)y(z)u(z)

ZOH

DOUBLE INTEGRATOR PLANT-MODEL WITH ZOH-TYPE CONTROL

)()( tuty

)(1

0

)(

)(

00

10

)(

)(

2

1

2

1 tutx

tx

tx

tx

)(

)(01)(

2

1

tx

txty

00

10A

1

0B 01C

(2-13)

(2-14)

(2-15)

(2-16)

yx 1 yx 2By selecting

14

2

0

1det)det(

AI

01 02

1110 )0()0()( ZfZfAf

)(f

110)( ZAf

00

1011 AZ

(2-17)

(2-18)

(2-19)

(2-20)

(2-21)

(2-22)

Thus, the characteristic polynomial of A and minimal polynomial of A are the same and hence

2)(

15

1)( f

10)( ZAf

10

0110 IZ

10

1

00

10

10

0111

TTTZIe TA

10

1~ TeA TA

(2-23)

(2-24)

(2-25)

(2-26)

(2-28)

16

T

TB 2~

2

T

Td

Td

TBdeB

TTTTA

211

0

10

1~2

000

)(

BAzICzGP

~)

~()( 1 (2-12)

2

22

22

2

)1(2

)1(

1

)1(2

)1(

012

1

10

)1(1

1

01)(

z

zT

z

Tz

zT

T

T

z

z

T

zzGP

2

2

)1(2

)1()(

z

zTzGP

(2-32)

17

Gp(z)y(z)u(z)

ZOH

INITIAL CONDITION RESPONSE AND ZOH-TYPE CONTROL SIGNAL FOR DOUBLE INTEGRATOR

8)0(1 x

8)0(2 x

Initial Conditions:

18

THE HARMONIC OSCILLATOR PLANT-MODEL WITH ZOH-TYPE CONTROL

)()()()( 2 tutyty n

(2-42)

)()(1 tytx )()(2 tytx If we select:

Then the matrix form of the state and output equations are:

)(1

0

)(

)(

0

10

)(

)(

2

12

2

1 tutx

tx

tx

tx

n

(2-43)

)(

)(01)(

2

1

tx

txty (2-44)

19

)cos()sin(

)sin()cos(~

TT

TT

A

nnn

n

nn

(2-45)

n

n

n

n

T

T

B

)sin(

)cos(1

~ 2

(2-46)

n

n

n

n

n

n

n

nN

nn

n

n

n

P T

T

Tzz

Tz

Tzz

TTzz

T

Tzz

Tz

zG

)sin(

)cos(1

1)cos(2

)cos(

1)cos(2

)sin()1)cos(2(

)sin(

1)cos(2

)cos(

01)(2

22

22

20Harmonic Oscillator (continued)

)sin(2

))cos(21(

)1)(cos(2

)1)cos(2(2

T

T

T

TK

n

nn

n

nnC

)sin()(cos2)cos(2

2

TTT

K

n

nnn

n

o

(2-49)

(2-50)

(2-51)4

23232

)1)((cos2

1)cos(2)(cos4))cos(6)(cos8()1)cos(4)(cos4)(cos8()(

zT

TTTTzTTTzzG

n

nnnnnnnCL

]~

[ 0 CKA

]~~

[ CKBACK

0K

Design such that system performs “deadbeat response”. Hence,

all eigenvalues of should be zero.

Also is designed for deadbeat response. All eigenvalues of

should be zero.

21

Therefore:

Must design value of T such that 1)cos( Tn ; ...,5,3, Tn

INITIAL CONDITION RESPONSE AND ZOH-TYPE CONTROL SIGNAL FOR HARMONIC OSCILLATOR

8)0(1 x

8)0(2 x

Initial Conditions:

22

TRANSFER-FUNCTION ANALYSIS OF A LINEAR PLANT WITH D/C CONTROL

)()(~

])1[( kTBHkTxATkx

)()( kTCxkTy

BHAzICzGP )]~

([)( 1

)]~

([)( 01

00 KCKKBHAzIzG DC

DCDC KzG )(

(3-1)

(3-2)

(3-8)

(3-15)

(3-16)

23

D/C “controlstate

Gp(z)Kdc(z)Go(z)+-y(z) x(z) (z) y(z)

-

D/C Compensator Gc(z)

FIRST ORDER PLANT WITH LIT-TYPE D/C CONTROL

)()(

)()()(

txty

tbutaxtx

))(()()( 21 kTtkTCkTCtu

01H

00

10D

10

1~ TeD TD

)1(),1(

20

)(

0

)( aTaTT

TaT

Ta eaTa

be

a

bdebdebBH

(3-17)

(3-18)

(3-20)(3-19)

(3-22)

24

LiT-Type D/C control

)(

)1(,

)(

)1()(

2 aT

aT

aT

aT

P eza

eaTb

eza

ebzG

ABHBHBHKTT

udbDC

~1

/

The Control gain matrix for a=3, b=5 and T=1 is computed to be:

015.0

021.0/ udbDCK

)(021.0)(1 kTxkT

)(015.0)(2 kTxkT

(3-24)

(3-26)

(3-27)(3-28)

(3-29)

25

Assuming: nBHrank )( where sxnRBH

INITIAL CONDITION RESPONSE AND LIT D/C CONTROL SIGNAL FOR FIRST ORDER PLANT

5)0(1 x

Initial Condition:

26

D/C DISCRETE-TIME TRANSFER-FUNCTION OF A DOUBLE INTEGRATOR PLANT WITH D/C-TYPE

CONTROL

d

TBH

T

10

101

1

0

10

1

0

2

622

32

TT

TT

BH

2

3

2

2

)1(6

)2(,

)1(2

)1()(

z

zT

z

zTzGP

(3-33)

(3-35)

(3-39)

27

ABHKDC

~1

23

2

612

46

TT

TTKDC

)(ˆ4

)(ˆ6

)( 2121 kTxT

kTxT

kT

)(ˆ6

)(ˆ12

)( 22132 kTxT

kTxT

kT

(3-41)

(3-45)

(3-47)

(3-48)

28Since s=n and nBHrank )( we have

zzTzzz

z

zGo

)2(

)2(2

)(

zzTzzzT

z

zGc

)2(30

)2(16

)(3

2

4

)23()(

z

zzzGCL

(3-50)

(3-52)

(3-53)

29

INITIAL CONDITION RESPONSE AND LIT D/C CONTROL SIGNAL FOR DOUBLE INTEGRATOR

8)0(1 x

5)0(2 x

Initial Conditions:

30

DOUBLE INTEGRATOR WITH REDUCED ORDER STATE OBSERVER BLOCK (C. D. JOHNSON DESIGN

RECIPE) [Ref #13]

)()()(ˆ 121211 kTTkTyTTkTx

)()()(])1[( kTkTykTHDTk

012 TC TT TTTT 12

1

121212

1

11

TT CCCT

1212~TATD 12

~TACH

][~

][ 1112 HDTACT

BHCT ][ 12

(3-54)

(3-55)

31

Choose T12 and T11 as follows:

3

)12()(

z

zzzGCL

(3-67)

(3-68)

(3-69)

32

Tz

zzG1

)(0

zT

zzT

z

zGC

3

2

18

10

)(

INITIAL CONDITION RESPONSE AND D/C CONTROL SIGNAL FOR DOUBLE INTEGRATOR PLANT AND

REDUCED ORDER STATE OBSERVER

8)0(1 x

5)0(2 x

Initial Conditions:

33

HARMONIC OSCILLATOR (D/C) CONTROL

)cos()sin(

)sin()cos(~ 1

TT

TTA

nn

nnn

))cos(1()sin(

))sin(())cos(1(21

32

TT

TTTBH

nnnn

nnnnn

)sin()(cos2)cos(2

~ 2

TTT

K

n

nn

n

o

(3-70)

(3-71)

(3-72)

34

zTz

TT

zzTz

Tz

zG

n

nn

n

n

n

o

))cos(2(

)1)(cos2()sin(

))cos(2(

)cos(2

)( 2

zTTTTTTzTTTT

zTTzTTTTTzTTTT

zTTTTTTTT

zG

nnnnnnnnnn

nnn

nnnnnnnnn

nnnnnnnnn

c

)]cos(2)cos()sin(4)(cos2))sin(2)cos([(

)]1)cos(2)(cos4([)](sin)cos(2))cos(1)(2sin(2))(sin)2sin()sin(2[(

)]sin()2sin()(cos4)cos(3)cos()2sin(2([

)(

2

23

22

32

4

2 )]cos(2)1)(cos4[()(

z

zTzTzG nn

CL

(3-73)

(3-76)

35Harmonic Oscillator D/C control continues:

INITIAL CONDITION RESPONSE AND LIT D/C CONTROL SIGNAL FOR HARMONIC OSCILLATOR

PLANT AND FULL ORDER STATE OBSERVER

8)0(1 x

8)0(2 x

Initial Conditions:

36

TRANSFER-FUNCTION ANALYSIS OF CLOSED-LOOP DISCRETE/CONTINUOUS CONTROLLED SYSTEM

)()()( 0 zGzGzG DCC

)()()()()( 1 zGzGzGzGIzG CPCPCL

)()(1

)()()(

zGzG

zGzGzG

CP

CPCL

MIMO:

SISO:

(4-1)

(4-2)

(4-3)

37

Gp(z)Gdc(z)Go(z)+-y(z) x(z) (z) y(z)

-

yc=0 +

GCL(z)

D/C CONTROLLED SYSTEM WITH FULL ORDER OBSERVER

0

1

0

~)( KCKKBHAzIKzG DCDCC

0

1

0

11

0

1

0

1 ~~~~)( KCKKBHAzIKBHAzICKCKKBHAzIKBHAzICIzG DCDCDCDCCL

0

1

0

10

1

0

1

~~1

~~)(

KCKKBHAzIKBHAzIC

KCKKBHAzIKBHAzICzG

DCDC

DCDCCL

(4-4)

(4-5)

(4-6)

38

Detailed Expressions:

MIMO:

SISO:

TRANSFER-FUNCTION ANALYSIS OF A D/C CONTROLLED SYSTEM WITH A REDUCED ORDER

PLANT-STATE OBSERVER (USING C. D. JOHNSON’S OBSERVER DESIGN RECIPE) [Ref #13]

1121211

11120 ][][)( HDzITTTKHDzITIzG DC

1121211

1112 ][][)( HDzITTTKHDzITIKzG DCDCC

1121211

1112

1

1121211

1112

1

][][~

1

][][~

)(HDzITTTKHDzITIKBHAzIC

HDzITTTKHDzITIKBHAzICzG

DCDC

DCDCCL

(4-16)

(4-21)

(4-22)

39

CLOSED LOOP SYSTEM WITH A STATE OBSERVER

Gp(z)Kdc(z)Go(z)+-y(z) x(z) (z) y(z)

-

D/C Compensator Gc(z)

40

DESIGN OF D/C DISCRETE TIME “COMPENSATOR” BY TRANSFER-FUNCTION METHOD

CC KzG )(bz

KzG C

C )(

az

bzKzG C

C

)(

)( ))(()(

bzaz

KzG C

C

))((

)()(

czaz

bzKzG C

C

))((

))(()(

czaz

dzbzKzG C

C

(5-1) (5-2)

(5-3)(5-4)

(5-5) (5-6)

41

Candidate Choices for compensator design process.

CLOSED-LOOP SYSTEM WITH DISCRETE-TIME “SERIES COMPENSATOR”

PlantD/C Series

Compensator+-

-y(kT) y(kT)(kT)

42

D/C COMPENSATOR DESIGN FOR FIRST-ORDER PLANT

)()(

)()()(

txty

tbutaxtx

(3-17)

aTeA ~

)1(),1(

2aTaT eaT

a

be

a

bBH

)(

)1(,

)(

)1()(

2 aT

aT

aT

aT

P eza

eaTb

eza

ebzG (3-24)

)()(1

)()()(

zGzG

zGzGzG

CP

CPCL

(4-3)

43

2

1)(

k

kzGC

aTaTaT

aTaT

CL

ebkTka

bk

a

bek

a

bk

a

bez

ekaTkkeakaka

b

zG

22211

)22211()(

2

2

aTaT eeka

b )1(1 0)(22

aTbeTa

b

a

bk

)1(1

aT

aT

eb

aek

02 k

0)1()( aT

aT

C eb

aezG z

ezG

aT

CL

)(

(5-8)

(5-10)

(5-9)

(5-11)

(5-12) (5-13)

(5-14)(5-15)

44

Trial Candidate:

Design of (k1, k2) for all roots at zero (deadbeat response)

DISCRETE-TIME INITIAL-CONDITION RESPONSE FOR FIRST ORDER PLANT

z

ezG

aT

CL

)( (5-15)

8)0( y

Initial Condition:

45

PURE GAIN D/C COMPENSATOR DESIGN FOR THE DOUBLE INTEGRATOR PLANT WITH LIT D/C

CONTROL

)()( tuty

yx 1 yx 2

2

3

2

2

)1(6

)2(,

)1(2

)1()(

z

zT

z

zTzGP (3-39)

2

1)(

k

kzGC (5-8)

46

Trial Candidate Design

k1 and k2 to be designed.

12333.015.0)22167.015.0(

)2213(167.0)213(167.0)(

32322

22

kTkTzkTkTz

TkkTzTkkTzGCL

022167.015.0 32 kTkT 012333.015.0 32 kTkT

2

101

Tk 3

182

Tk

3

2

18

10

)(

T

TzGC

(5-16)

(5-17) (5-18)

(5-19) (5-20)

(5-21)

47

Design of k1 and k2 for deadbeat response

D/C compensator design

DISCRETE-TIME INITIAL CONDITION RESPONSE FOR DOUBLE INTEGRATOR PLANT AND PURE GAIN D/C

COMPENSATOR

2

12)(

z

zzGCL

(5-22)

8)0( y

Initial Condition:

48

FIRST ORDER COMPENSATOR DESIGN FOR THE DOUBLE INTEGRATOR PLANT MODEL WITH LIT D/C

CONTROL

)()( tuty

yx 1 yx 2

2

3

2

2

)1(6

)2(,

)1(2

)1()(

z

zT

z

zTzGP

(3-39)

(5-23)

az

Kaz

K

zGC

C

C2

1

)(

49

Trial Candidate Design

)32

()262

1()2(

)2(6

)1(2)(

2

3

1

2

2

3

1

223

2

3

1

2

CCCC

CC

CL

KT

KT

zaKT

KT

zaz

KzT

KzT

zG

02 a 0262

1 2

3

1

2

aKT

KT

CC0

32 2

3

1

2

CC KT

KT

21

16

TKC

32

30

TKC

)2(30

)2(16

)(3

2

zT

zTzGc (5-35)

(5-33) (5-34)

(5-28)

50

Design of a, kc1, kc2 for deadbeat response

D/C compensator design

DISCRETE-TIME INITIAL CONDITION RESPONSE FOR DOUBLE INTEGRATOR PLANT AND FIRST ORDER

D/C COMPENSATOR

3

)23()(

z

zzGCL

(5-36)

8)0( y

Initial Condition:

51

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER WORK

In this Thesis it has been demonstrated that by using conventional discrete-time transfer-function block-diagram methods a D/C Type Discrete-Time Controller for time invariant MIMO systems can be formulated more directly than by using the conventional state-variable technique. The discrete-time block-diagram transfer-function formulation is simple and more direct compared to state-variable methods and revels very accurate results. In addition, we designed transfer-functions type D/C Controllers for some very common problems such as the: First Order, Double Integrator and Harmonic Oscillator plant model.

52

Summary

The conventional transfer-function technique used in this thesis to analyze and design D/C Controllers is a more direct method and should be useful in many applications. By using this technique, we have avoided the introduction of the state-variable, state-equations, state-observers, state “control-law” etc. in the design of D/C Control systems. Recommendations for Further Work

Further work in this are should address application of the transfer-function employed here to advanced-forms of D/C controllers, including “bumpless” and “smooth bumpless” D/C Controllers [16]. In addition, the practical application of D/C Control, using available “field-programmable analog-array chips”, would be an important step in demonstrating the practical utility and effectiveness of D/C Control.

53Conclusions