1

數位控制（五）

2

data hold

TTs

se

sG

TT

TkxkTxkTxkTh

se

sG

TkTxkTh

Ts

h

Ts

h

11)(

0,))1(()(

)()(

holdorder -first

1)(

0),()(

holdorder -zero

2

1

0

3

0)()(y(t)

summationnsconvolutio

)()(y(t)

)()(y(t)

:integralnconvolutio

kkTtgkTx

dgtxor

dtgx

x(t) G(s) y(t)

In continuous-time

In discrete-time

g(kT) is the system’s weighting sequence

g(t) is the system’s weighting sequence

x(t)G(s)

y(t)x*(t)

X(z)

4

pulse transfer function Transfer function

continuous-time system Laplace transform

Pulse transfer function discrete-time system z transform

)()(

)(

)()(

)()(

)()(

)()(

)()(

)()()(

0 0

0

)(

0

0 0

0

0

zXzY

zG

zXzG

zhTxmTg

zhTxmTg

zhTxhTkTg

zkTyzY

hTxhTkTgkTy

m

k

h

m

hm

h

k

k

h

k

k

h

G(z)

X(z) Y(z)

5

x(t) G(s) y(t)x*(t)

X(z)

G(z)X(z)Y(z)

(s)(s)XG(s)Y

sXsGsY

x

***

transform Laplacestarred thetaking

)()()(

signals sampled impulse theis *

*

x(t) G(s) y(t)

GX(z)Y(z)

sGXG(s)X(s)(s)Y

sXsGsY*

** )]([][

)()()(

6

Pulse transfer function of cascade elements

x(t)y(t)x*(t)

as 1

bs 1

u*(t)

x(t)y(t)x*(t)

as 1

bs 1

)1

1)(

1

1(

11)()(

)()(

11

zezebs

Zas

ZzHzGzXzY

bTaT

)1)(1(

)(111)]()([

)()(

11

1

zeze

zeeabbsas

ZzHzGZzXzY

bTaT

bTaT

7

Pulse transfer function of closed-loop system

R(s) G(s) C(s)E*(s)

+ -

H(s)

E(s)

)(1

)(

)(

)(

)(1

)()()(

)(1

)()()()()()( since

)()()()(

)()()()()(

*

******

****

*

zGH

zG

zR

zC

zGH

zRzGzC

sGH

sRsGsCsEsGsC

sEsGHsRsE

sEsGsHsRsE

8

Pulse transfer function of a digital controller

nn

nn

D

nn

nn

n

zaza

zbzbb

zE

zMzG

zEzbzEzbzEb

zMzazMzazMzaM(z)

nkebkebkeb

11

110

110

22

11

10

n21

1)(

)()(

is controller digital theoffunction transfer the

)()()(

)()()(

is transformz the

)()1()(

n)-m(ka2)-m(ka 1)-m(kam(k)

m(k), isequation difference by the generatedoutput the

and e(k), is controller digital a input to suppose

9

Closed-loop pulse transfer function

r(t) Plant c(t)

e(kT)

+ -

e(t)Digital Controller

Zero-order hold D/A

s

e Ts1R(s) Plant C(s)E*(s)

+ -

E(s)G*D(s)

)(

sG

Sampler A/D

m(kT) u(t)

)()(1

)()(

)(

)()()(

1

zGzG

zGzG

sR

sCsGsG

s

e

D

DP

Ts

M*(s)

10

PID Controller Proportional-Integral-Derivative (PID)

Controller Most traditional and has been use

successfully for over 50 years. P: action proportional to the actuating error

between the input and the feedback signal. I: action proportional to the integral of the

actuating error D: action proportional to the derivative of the

actuating error

11

Analog PI Controller

LY-498

LY-498D

FY/498Z

FY/498Y

OUT

(Level)

(Level set)

(STM Flow)

(FW Flow)

(Integral)

(Integral)

FlowMismatch

FlowMismatch

Set

(Gain)

Level ControllerLC-498A

NCB 5

FY-498ENSA2

Flow ControllerFC-498NCB5

(Gain)

12

Foxboro 控制器之 PIDA Block 規劃

FW Flow Set

FW Flow

OUT

LAG

13

PID Controller Ziegler-Nichols Stabi

lity Boundary Tuning方式，係於如右上圖之閉迴路下，逐漸調高 Proportional Gain ，直至 Ku 產生如右中圖所示之震盪為止，量測震盪週期 Pu ，再依所示公式估算最佳化 P與 I 值。

14

負載響應

在控制參數已調整至最佳化下，該波形之過衝量 Ω(overshoot) 以及衰減比 δ(decay ratio) 皆應趨近於 0.2 。

15

PB 值最佳化調整 比例帶 Proportional Band

(PB)= 100/P. PB 主要決定負載波形之減

幅 (damping) 以及對稱性， 左圖顯示在最佳化積分時間

下不同 PB 設定下之負載波形：

較小之 PB(Popt/1.5) 會造成衰減比 (decay ratio) 較大，亦即減幅太輕 (light damping) 波形，應調高其 PB 。

太大之 PB(Popt*1.5) 會造成第一個波峰呈現非對稱波形並需較長時間回到穩態，應調低其 PB 。

圖、不同比例帶 PB 之負載響應波形

16

I 值最佳化調整 積分時間 I 值主要決定

負載波形之過衝量 (overshoot) ，

左圖顯示在最佳化 PB下不同 I 值設定下之負載波形：

較小之 I(Iopt/1.5) 會造成較大之過衝 (overshoot) ，應調高其 I 值。

較大之 I(Iopt*1.5) 會造成沒有過衝，亦即 undershoot ，應調低其I 值。

圖、不同積分時間 I 之負載響應波形

17

Pulse transfer function of a digital PID controller

td

i dt

tdeTdtte

TteKtm

0

)()(

1)()(

s

e Ts1R(s) Plant C(s)E*(s)

+ -

E(s)G*D(s)

)(

sG

M*(s)

18

Discretize the m(t) (positional form)

gain derivative

gain integral

gain naliproportio2

)()1(1

)(

))1(()(2

)())1((2

)()0()()(

)()(

1)()(

11

0

T

KTK

TKT

K

KKKwhere

zEzKz

KKzM

TTkekTe

TkTeTkeTee

TT

kTeKkTm

dttde

TdtteT

teKtm

dI

iI

IP

DI

P

di

td

i

19

Transient response of PID Controller

s

e Ts1R(s) C(s)E*(s)

+ -

E(s)G*D(s)

)(

sG

M*(s)

)1(1ss

)1)(3679.01(

2642.03679.011

21

zz

zzR(z) C(z)+ -)1(

11

1

zKz

KK D

IP

m Response.Unit Ramp

m Response.StepUnit

0528.06642.05906.18528.11

0528.02963.01452.05151.0)()(1

)()()()(

4321

4321

zzzz

zzzzzGzG

zGzGzRzC

D

D

20

Realization of digital controllers and filters

mnzazaza

zbzbzbb

zXzY

G

zaza

zbzbb

z

zKzKKKKK

zKz

KK

zEzM

zMGcontroller

zEzKz

KKzM

nn

mm

D

DDPDIP

DI

PD

DI

P

,1)(

)( general in

1

1

)2()(

)1(1)(

)()(

)()1(1

)(

22

11

22

110

22

11

22

110

1

21

11

11

21

Direct programming

)()()()(

Y(z)Y(z)-Y(z) Y(z)2

21

10

22

11

zXzbzXzbzXzbzXb

zazazam

m

nn

22

Standard programming

nn

mm

zazazazbzbzbb

zYzH

zHzY

zXzY

22

11

22

110

1

1)(

)()(

)()(

)()(

23

Series programming

21

21

1

1

21

1

1

1

1

)()()()(

zdzc

zfze

za

zb

zGzGzGzG

ii

ii

i

i

p

24

Parallel programming

21

1

1

21

11

)()()()(

zdzc

zfe

za

bA

zGzGzGAzG

ii

ii

i

i

p

25

Infinite vs Finite

)0()())1(()()()0(

)()(

filter response impulse-Finite

1)()(

filter response impulse-Infinite

0

22

11

22

110

xkTgTkxTgkTxg

hTkTxkTgy(kT)

zazaza

zbzbzbb

zXzY

k

h

nn

mm