Chapter 5 Dynamics and Regulation of Low-order Systems

Chapter 5 Dynamics and Regulation of Low-order Systems

§ 5.1 General Effects of Feedback

§ 5.2 Dynamics and Regulation of 1st-order System

§ 5.3 Dynamics and Regulation of 2nd-order System

§ 5.4 Time-domain Specifications of System Performance

§ 5.1 General Effects of Feedback (1)• Closed-loop System:

CascadeController

G1

PlantG2

r(t) y(t)u(t)

d(t)

-

++

-

e(t)

SensorH y=x

Controller

Plant with disturbance

• Negative Feedback Control:

u

controller

x

plantInput OutputCommand Response

Disturbance

r(t) y(t)

§ 5.1 General Effects of Feedback (2)• Proportional Regulator:

error

setpoint

Response

t

error

setpoint

Response

t

Under Regulation

Out of Regulationt) , f(e(t) Minimize:Objective

)K Gain nal(Proportio )t(eKu(t) :ControllerFunction) Step (Unit Ur(t) :Command

Function) Response (Impulse g(t) G(s), :Plant

pp

s

KpPlantG(s)

r(t) y(t)

u

Load

-

++

-

R(s) Y(s)e(t)

setpoint

, D(s)

Kp

r

Regulator

c(t)

u(t)

e

y(t)

Load

Plantg

§ 5.1 General Effects of Feedback (3)

Mechanical and Hydraulic Controller

p3

4

1

2

i

o1 K

RR

RR

)s(E)s(E)s(G

Electronic Controller

Governing of mechanical plant is liberated by using OP to realize electronic controller.

pi

o1 K

ab

)s(X)s(X)s(G

1b)s(a

aK If

+

--

+

R1

R2

R3

R4

e0

ei OP OP

+b+a

b)s(Xi

)s(Xo

sK

b+aa

)s(Xe

eX

oX

iX

a

b

Density Oil - ρ

AreaPiston - A ,ρA

A=K:piston 1

0)≈gain (Pressurexq

=K gainflow :valve pilot 1

• Proportional Controller: pK


• Error Response1E(s) R'(s)

1 G(s)H(s)

• Effectiveness of Feedback:

a

Large Loop Gain for the minimization of error responseIf E 0 System is in inertia status when error is not detected by control system.

E(s)If 1 GH 1 R'(s)

E(s)or GH 1 R'(s)

G Y(s)R(s) +

H

aE (s)

(1) Set point regulation

Following the first law of Newtonian Mechanics:• At rest Zero error in position regulation• At constant velocity Zero error in speed regulation

H1 Y(s)R(s) + GHH1 E(s)R'(s)

§ 5.1 General Effects of Feedback (5)• Dynamic Response

The dynamic response is governed by new poles as D1(s)D2(s)+N1(s)N2(s)=0

For unity feedback system, feedback has no effect on system zeros.

)s(D)s(N

=H(s) , )s(D)s(N

=)s(G If2

2

1

1

0)s(N)s(N (s)(s)DD 2121

Poles of Closed-loop Transfer Function:

1 2

1 2 1 2

( ) ( )Y(s) R(s) ( ) ( ) ( ) ( )

N s D sD s D s N s N s

§ 5.1 General Effects of Feedback (6)(2) Parameter Variations

( ) G T( ) ( ) 1 GH

( ) 1 If GH 1 ( )

Y ssR s

Y sR s H

HH

HH ,G

GG

G variation LargeH

HH

H ,GG

GG variation Small

error Relative ,HH ,G

G H G,

error AbsoluteH, G, H ,G

≈≈

I/O transmission is dominated by sensor for high loop gain.

• System variations

The effect of large variation is investigated by using robust analysis.

•


TH

= (Complementary sensitivity function)1 1

GH 1 S 1

TH

T GH LTSH GH L

HIf

Variation of H

The Sensitivity of I/O variation is reduced by employing high loop gain.

TG

ln 1 1 (Sensitivity function)ln 1 1

GH 1 S

TG

T TTSG G GH L

GIf

• Sensitivity analysis (small variation) Variation of G

Feedback sensor directly and constantly affect the I/O transmission.

In general, is higher than . High loop gain reduces theI/O variation due to system parameters.

GG

HH

Loop gain

§ 5.1 General Effects of Feedback (8)(3) Disturbance Rejection

2

1 2

1 2

1

2 2 1

2

GY =D 1+G G H

If G G H >>1, High Loop Gain

Y 1 =D G H

YWith moderate G , G 1 , then G H >>1 0D

Disturbance transmission is rejected with high loop gain and moderate gain G

G1(s) G2(s)R Y

D+

++

-

H(s)

R=0aE -

• Disturbance Transmission

• Disturbance Error

D HGG1

HGE

21

2a

• Overall Effects of Feedback

Advantages : Command following

Disturbance rejection

Improve system robustness

Disadvantages : Reduce gain

Increase system complexity

Introduce instability

R Y

+Total error

D


Note: steady state = set point - |offset error| - |disturbance error| |parameter uncertain error|

§ 5.2 Dynamics and Regulation of 1st-order Systems (1)

• Definition – A system that can store energy in only one form and location. Physical Examples:

tt=0+

Input signal

C

R

e i eo

j

Exponentialgrowth

Exponentialdecay

Static gain

System pole-zero diagram

C

K

x

f Input signal

Output signal

M gt=0+, Loading

R C

t=0+

Measure T Output Signal

Input Signal Tb

Thermal Reservoir

Output signal

Input signal

Input signal Output

signal

t=0+ , ON

t=0+

Open gate

hiho(t)

§ 5.2 Dynamics and Regulation of 1st-order Systems (2)• System and Input Model Differential Eq. Model: Transfer Function Model:

Standard Form of Pure Dynamics

( ). . x(0)

cx kx f tI C

0<t , 0

0≥t ,F0)t(f:Input

k} {c,:parameters System x: state System

constant Time :

Time :Unit ,kc } {k,:parameters System

)s(F )s(X

kcs1

)s(X

s11

)s(F

k1

R(s) +

s1

Y(s))s(R ( )Y s

s11

gainstatic Pure gaindynamic Pure

sF)s(F:Input 0


• System Dynamics:(1) Step response from differential equation

constant Time ,kc

)e1(k

Fex

k

Fe)

k

Fx()t(x

/t0/t0

0/t00

Transient stateSteady

Response Natural Response Forced


Response of initial relaxed system

)e-(1kF x(t) 0x t/-0

0

2%x-x(t)

)kc( 4t :Transientk

F x:stateSteady

s.s.

0s.s.

1 2 3 4 /t

k Fx(t)

0

1.0

1e1%2.63

2%


(2) Step response from pole-zero diagram equation: 1 s 0

Pole-zero diagram, 0(Pure dynamics)

Characteristic

Steady state (Pure static gain)

Transient (Pure dynamics)

1

1, s

( )

st

t

t

e

x t ke

ωj

1

Gain=1

0F kF0

k1

0

kF0

t

t ,x .s.s

kFx , 0

.s.s

1 2 3 4 /t

0x63.2%

Pure Dynamics of Transfer Function1

1s

§ 5.2 Dynamics and Regulation of 1st-order Systems (6)Overall response

)e1(k

F x(t).e.i

k

FK 0x)0( x.C.I

k

FKex)t(x)t(x

/t0

000

0t

.s.st


• Closed-loop Regulation Ex: Liquid-level regulation

+

-

sh r

brK K

bK

1RCsR

huae

oq

-

sh h

oq

rK

r

shgain point set

bK

b

h)k(k gain sensor rb

K

u

ae

gain controlflow alproportion

K

)t(e a

bK

)t(hh o

)p()t(q q

o

oo

R

C

constant=pi

ipi)q(

)t(qq ii

)t(hhso

h(t) :variable Regulatedr , h , q , q

:point Operatingr:point Set

000i

§ 5.2 Dynamics and Regulation of 1st-order Systems (8)• Unity Feedback Regulator

• Set Point Regulation

Equivalent I/O transmission

Proportional regulator (Kp) changes the static amplification (K’) and response

speed ( ).

p

p p

p

KY(s) K ' , K' , 'R(s) 1 's 1 K 1 K

E(s) 1 sR(s) (1 K ) s

Kpr(t) y(t)

D(s)

-+

-

R(s) Y(s)E(s)e(t) s1

1

plant modelcontroller model

+

Y(s)s'1

1

)s(R'K

)s(X

gainstatic gaindynamic Pure

'

0r

r(t)

t


(1) Effect of Kp on Pure Dynamics

Pole-zero diagram

Response is faster when the proportional gain is increased.

X(s) Y(s)

s'11 '

1s :pole

j

1

11

)10K( p )1K( p )0K( p

Gain = 1

pK1'

plant of constant time

gain control alproportion

constant time loop-closed

pK ' pole

0

1

10

2

11

1

2

11

Steady state error (offset) is reduced when the proportional gain is increased.

(2) Effect of Kp on Steady State Response )s(R )s(X

'K

p

p

K1K

'K

gain control alProportion

pK 'K

1

10

5.0

909.0

p

t

K11

)t(e limoffset

1.00.909

0.5

0.s.s rX

t

Open-loop plant10Kp

1Kp

(offset) error stateteady s

Open-loop gain 1


• Set Point and Response Regulation

0 0.1 0.2 0.3 0.4 1.0

0.5

0.9091.0

0.50.0909

63.2%

63.2%

63.2%

0

y(t)r

t

0909.0 , 10K 'p

'open-loop response,

5.0 , 1K 'p

)e1(Kr)t(c'/t'

0


Response speed is a pure dynamic behavior, therefore the comparisons of response speeds are referred to the steady state responses at different Kp.

• Disturbance Rejection

Y(s)

s'11

d)s(D

'Kd

d dp p

d

1K ' , '1 K 1 K

E(s) Y(s)Steady state error: -K '

pK 'Kd

0

1

10

1

5.0

909.0

-1

-0.0909

-0.5

t

Open-loop plant

10Kp

1Kp

(offset) error state steady0

y(t)d

0d

d(t)

t

010Kp

1Kp

0Kp pK1

1

t

''0( ) (1 )d

t

y t d k e

0

y(t)d


For infinite high gain Kp, the unity feedback regulator approaches ideal static system.

Dynamics) eDisturbanc and point (Set 0'' gain)static ce(Disturban 0'K

gain)static point (Set 1K' ,K

d

d

p

1r(t) y(t)

0

d(t)

0r

y(t)

t

set point and response

0

r(t),

• High Gain Regulation


Finite gain regulation will improve the speed of dynamic response but it has offset error in steady state.

• Definition – A system having two separate energy-storage elements. Physical Examples:

C

R

eoO u t p u ts i g n a l

I n p u ts i g n a l

t = 0 + , O N

L

ei

j

Exponentialgrowth

Exponentialdecay

Over damping(Exponential decay)

Static gain

System pole-zero distribution

c

k

x(t)

Input signal

Output signal

M g t=0+

m

Loading

§ 5.3 Dynamics and Regulation of 2nd-order Systems (1)

j

Critical damping(Critical condition between

oscillatally and non-oscillatallyexponential decay)

Static gain

j

Under damping(Sinusoidal-modulatdexponential decay)

Static gain

R 2C 2

hO u t p u tS i g n a l

1h

I n p u tS i g n a l

t = 0 + o p e n g a t e

1C

• System and Input Model Differential Eq. Model: Transfer Function Model:

Standard Form of Pure Dynamics

x(0).C.I)t(fkxxcxm

0<t , 0

0≥t ,F0)t(f:Input

k} c, {m,:parameters Systemx x,:states System

Frequency Natural :Ratio Damping :

n

mk ,

mk2c

} , {k, :parameters System

n

n

)s(F )s(Xkcsms

12

)s(X2

nn2

2n

s2s )s(F

k1

R(s) + Y(s)

ss n

n

22

2)s(R ( )Y s2

nn2

2n

s2s

gainstatic Pure gaindynamic Pure

sF)s(F:Input 0

R(s) + Y(s)

sn

sn

2

+

parameterStatic

parameterDynamic


• Step Response From Differential Equation For underdamping and initial relaxation system

1-

n

n2

d

n

d2

t

0.s.s

cos

constant Time ,1frequency Damping ,1

frequency Natural ,mk

)tsin(1

e1)kF(

)t(xx

)t(x

1mk2c

n

0.7~0.55 system Instrument 0.6~0.4 design) (Well system Control

0.02,~0.01 system Structure : of value Typical

dn


2%x-x(t)

)2mc( ,4t :Transient

kF x:stateSteady

s.s.

nn

0s.s.

0

21

21

2

21

3

tn

)kF(

)t(x0

2

t

1

e1

2

t

1

e1

undershoot

steady state

overshoot 21

2

period

1.0

envelope

envelope

d

n

The plant oscillated at .The response decays due to the exponent of - t.The overshoot (undershoot) only depends on the value of .


• Step Response From Pole-zero distribution n

2n

2nn

2 1-s i.e. 0,s2s :equation sticcharacteri

Steady state

Transient

dst j1s ,e

j

1 n

d 1

0F kF0

1k

0

kF0

t

Pole-zero distribution Poles distribution

Response:

j0

0

1 1

10

121

1

mode natural from from pole-zero distribution


cos 707.0~45o

5.0~60o

Overall response

d d

n

1 1( j )t ( j )t0

1 2

t0

d2

Fx(t) K e K ek

I.C. x(0) 0 x(0) 0

F e x(t) (1 sin( t ))k 1


Proportional Controller Servo Motor

Load

aEInput Device

aV

Potentiometer

T

mGear Train

Amplifier

mT

dT

-)s(Va

aR1

iKmm BsJ

1

m

dT (s)

mT +m(s)1

s

1N

• Closed-loop Regulation Ex: DC Servomechanism

Servo motor

Motion and power transmission by reduction gear

Position sensor by potentiometer

)s(Va m(s)

dT (s)

)sJB(sRK

mma

i

1N

m(s)

N1 (s)

(Motor) (Load) NT (s)

(Motor) (Load)mT (s)

(Gear Ratio; N)

sKV(s)(s)


Command response

Disturbance responsemips

am

am

sipn JKKK

NR2

B' ,NRJ

KKK'

(s)

NKKKR

ips

)s(Td

2nn

2

2n

's''2s'

d(s)2

nn2

2n

's''2s'

(s)

-

a

i

RK

pKmm BsJ

1

dT (s)

+

N1 (s)

s1d(s) +

-

sK

sK

N1

s p i a

d da m m s p i a m m s p i

K K K R / N(s) [ ] (s) [ ]T

R Ns(J s B ) K K K R Ns(J s B ) K K K


• Unity Feedback Regulator

• Set Point Regulation

Equivalent I/O transmission

Proportional regulator (Kp) changes the static amplification (K’) and dynamic

response( ) through increasing the stiffness of a system.

2 2p n n

2 2 2 2n p n n n

pn p n n n

pp

K K' 'Y(s)R(s) s 2 s (1 K ) s 2 ' 's '

K' 1 K , ' , K' , ' '

1 K1 K

Kp

D(s)

-+

-

R(s) Y(s)E(s)2

nn2

2

n

s2s

+

Y(s))s(R'K

)s(U2

nn2

2n

's''2s

'

gainstatic gaindynamic Pure


' ,'n

(1) Regulation of Pure Dynamics

Pole-zero distribution

)s(V Y(s)2

nn2

2n

's''2s'

j' 1''s 1,

' 1''s :poles

n2

n

n2

n

j

)10K( p

)1K( p

)0K( p

1pK

0

1

10

2

11

' 'n

n

n2

n11


1)'0(

(2) Steady-state response The same result as that of 1st-order system with offset error

pK11

y(t)

t

10Kp

0.1 ,5.0 n

1Kp

Open loop response

• Disturbance Rejection

Y(s))s(D'Kd 2

nn2

2n

's''2s'

p

p

1'1 K

( ) ( )1- state error:

1 K

d

d

K

E s Y s

Steady

0d

d(t)

t


10Kp

1Kp 0Kp

t

pK11

- 0.0909

- 0.5

- 1

0.1 ,5.0 n

0

y(t)d

• Step Testing of Black-Box System

(1) Static test

(2) Dynamic test (Within linear range)

§ 5.4 Time-domain Specifications of System Performance (1)

Black BoxInput Output

)t(r y(t)

Two channel Recorder

Controlled Environment

Initial Relaxation System

Input

Output

Linear Range

Slope, a

0r

r(t)

t

Input:

Max. Overshoot

t

0

y(t)y (t)r a %

%

1.00.9

0.5

0.1

dT

rT

pT

sT

s.s.y

r

d

s

p

max s.s.

s.s.

T : Rise TimeT : Delay TimeT : Settling Time ( %, ex. 2%, 5%)T : Peak Time

P. O.: Percentage Overshooty y 100%

y

(1) Performance testing

• Step Testing and Identification of Low-order Systems

2%) (at 4T 69.0T

2.2T:system orderst1

s

d

r

2

21

r r,0~100%n d

dn

sn

p 2d n

1

2nd order system (0 0.7):

10.8 2.5 1T (T tan ( ))

1 0.7T

4T

T1

P.O. 100 e

.O.P

0

7.0

o90

o45

%100

%5


(2) System Identification1st-order system:


2nd-order system:

}L ,{a } A,r{:Measure

00

00

}a ,T ,r{:Measure 000

Time Residence Average

sT1rG(s)

0 )1(

ar

0

0

2

nn2

2

n0

20

n20

s2s

rG(s)

1T

2 ,)a /log(21

1

0

0ar r

AT

Time Dead Normalized

10 ,TL

0ar

00

Response

t

Input0r

a

0T

0aa

Response

t

Input0r

0A

0L

0a

0sL

0

0

0

esL

aG(s)

1 (2)

• Effects of Additional Poles an Zeros§ 5.4 Time-domain Specifications of System Performance (4)

Effect of real pole

Effect of real LHP zero

The effect of the real pole is to make the response more sluggish.

The effect of the real zero is to make the response more oscillatory.

)s(R Y(s))1/s2/s)(1s(

1

n2

n2

)s(R1/s2/s

1s

n2

n2

Y(s)

j

1 n

1st-orderpole

2nd-orderpole

)1(

1

j

1 n

1st-orderzero

2nd-orderpole

)1(

1

1

t

y(t)

1

21

0.5 ,0.1n

11

1

y(t)

t

1

21

0.5 ,0.1n

11

§ 5.4 Time-domain Specifications of System Performance (5)Effect of real RHP zeroRHP zero: Nonminimum-phase zero

The effect of nonminimum-phase zero is to cause initial reversal motion in step response.

j

1n

1)s(R1/s2/s

1s

n

2

n2

-1( )Y s

1

t

y(t)

1

21

0.5 ,0.1n

11

• Dynamic Model Simplification


Order reduction: High order system Low order model Ignore real pole (zero) in oscillatory system modes:

Pole-zero cancellation:

Ignorance of far away poles and zeros: Poles and zeros only have s.s. effects

n101 if

j

1 n

1

j

n

1

j

1

j

n

1

j

1

j1

Chapter 5 Dynamics and Regulation of Low-order Systems

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Transcript of Chapter 5 Dynamics and Regulation of Low-order Systems