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CHE302 Process Dynamics and Control

Korea University ^6-1

CHE302 LECTURE VI

DYNAMIC BEHAVIORS OF REPRESENTATIVE PROCESSES

Professor Dae Ryook Yang

Fall 2001

Dept. of Chemical and Biological Engineering Korea University

REPRESENTATIVE TYPES OF RESPONSE

• For step inputs

^U(s) ^Process_G(s) ^Y(s)

?

Unstable, oscillation, complex RHP poles Unstable, no oscillation, real RHP poles Inverse response, negative (RHP) zeros Overdamped oscillation, 2^ndor higher order Underdamped oscillation, 2^ndor higher order 1^storder+Time delay

Nonzero initial slope, no overshoot or nor oscillation, 1^storder model

Type of Model, G(s) Y(t)

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1 ^ST ORDER SYSTEM

( ) ( ) ( ) ( 1) ( ) ( )

dy t y t Ku t s Y s KU s

τ dt = − + →

^L

τ + =

• First-order linear ODE (assume all deviation variables)

• Transfer function:

• Step response:

– – –

( )

( ) ( 1)

Y s K

U s = τ s

+

Time constant

Gain

/

With ( ) / ,

( ) ( ) (1 )

( 1)

t

U s A s

Y s KA y t KA e

s s

τ

−

=

= → = −

+

^L

( ) (1

/

) 0.632

y τ = KA − e

⁻^{τ τ}

≈ KA

t y(t)

KA

0.632KA

τ (1

^t/

) 0.99 4.6 (Settling time=4 5 ) KA − e

⁻ ^τ

≥ KA ⇒ ≈ t τ τ : τ

/

(0)

^t

/

0

/ 0 (Nonzero initial slope) y ′ = KAe

⁻ ^τ

τ

t₌

= KA τ ≠

• Impulse response

• Ramp response

• Sinusoidal response

/

With ( ) ,

( ) ( )

( 1)

t

U s A

KA KA

Y s y t e

s

τ

τ τ

−

=

= → =

+

^L ^t

y(t)

τ

K A τ

2

/ 2

With ( ) / ,

( ) ( ) ( )

( 1)

t

U s a s

Y s Ka y t Ka e Ka t

s s

τ

−

=

= → = + −

+

^L

[ ]

² ²

2 2

With ( ) sin /( ),

( ) ( 1)( )

U s A t s

Y s KA

s s

ω ω ω

ω

τ ω

= = +

= →

+ +

^L

L

t/

y(t)

τ

1 2 3 4 5

( ) K u t

t

y(t) 1

φ= −tan⁻ωτ

2 2

1 KA ω τ +

2 2 ( )

1 K u t ω τ + /

2 2

( ) ( cos sin )

1 KA

t

y t ωτ e

^τ

ωτ ω t ω t

ω τ

=

−

− +

+

(2)

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• Ultimate sinusoidal response

– The output has the same period of oscillation as the input.

– But the amplitude is attenuated and the phase is shifted.

– High frequency input will be attenuated more and phase is shifted more.

( t → ∞ )

/ 2 2

2 2

1 2 2

( ) lim ( cos sin )

1 ( cos sin )

1 sin( ) ( tan )

1

t t

y t KA e t t

KA t t

KA t

ωτ

τ

ωτ ω ω

ω τ

ωτ ω ω

ω τ

ω φ φ ωτ

ω τ

∞ →∞ −

−

= − +

+

= − +

+

= + = −

+

0

Amplitude

Phase angle

2 2

1 1

ω τ 1

= <

+

Normalized Amplitude Ratio

(AR_N)

Phase angle

= − tan

⁻¹

ωτ

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BODE PLOT FOR 1 ^ST ORDER SYSTEM

• AR plot asymptote

• Phase plot asymptote

• It is also called “low-pass filter”

2 2 0

( 0) lim 1 1

N

1 AR ω

ω

ω τ

→ =

→

=

+

2 2

1 1

( ) lim

N

1 AR ω

ω

ω τ ωτ

→ ∞ =

→∞

=

+

1 o

0

( 0) limtan 0

φ ω

ω ⁻

ωτ

→ = −

→

=

1 o

( ) limtan 90

φ ω → ∞ = −

ω_→∞ ⁻

ωτ = −

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1 ^ST ORDER PROCESSES

• Continuous Stirred Tank

– With constant heat capacity and density

A

Ai A

V dc qc qc dt = −

h V c_Ai, q_i

c_A, q

( ) 1

( ) ( / ) 1

A

Ai

C s q

C s = Vs q = V q s

+ +

h V T₀, q_i 0 T, q

( )

ref

p p ref

p ref

d T T

VC qC T T

dt

qC T T

ρ ρ

ρ

− = −

− −

0

( ) 1

( ) ( / ) 1

T s q

T s = Vs q = V q s

+ +

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INTEGRATING SYSTEM

• • Transfer Function:

• Step Response

– The output is an integration of input.

– Impulse response is a step function.

– Non self-regulating system

( ) ( ) ( ) ( )

dy t Ku t sY s KU s

dt = →

^L

=

( ) ( ) Y s K U s = s

t y(t)

Slope= K 2

With ( ) 1/ ,

( ) ( )

U s s

Y s K y t Kt

s

=

= →

^L

=

(3)

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INTEGRATING PROCESSES

• Storage tank with constant outlet flow

– Outlet flow is pumped out by a constant-speed, constant- volume pump

– Outlet flow is not a function of head.

h V q_i

q Area = A

i

A dh q q dt = − ( ) 1

i

( ) H s

Q s = As ( ) 1

( ) H s

Q s = − As

2 ^ND ORDER SYSTEM

• 2

^nd

order linear ODE

• Transfer Function:

• Step response

– Varies with the type of roots of denominator of the TF.

• Real part of roots should be negative for stability:

• Two distinct real roots ( ): overdamped (no oscillation)

• Double root ( ): critically damped (no oscillation)

2

2 2 2

2

( ) ( )

2 ( ) ( ) ( 2 1) ( ) ( )

d y t dy t

y t Ku t s s Y s KU s

dt dt

τ + ζτ + = →

^L

τ + ζτ + =

2 2

( )

( ) ( 2 1)

Y s K

U s = τ s ζτ s

+ +

Time constant

Gain

Damping Coefficient

ζ > 1 ζ = 1

ζ ≥ 0

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• Case I with U(s)=1/s

• Case II

• Case III ( ζ > 1)

1 2

/ /

1 2

2 2

1 2 1 2

( ) ( ) 1

( 2 1) ( 1)( 1) ( )

t t

K K e e

Y s y t K

s s s s s s

τ τ

τ ζτ τ τ τ τ

− −

 − 

= + + = + +  → =   − −  

L

( ζ = 1)

( )

^/

2 2 2

( ) ( ) 1 1 /

( 2 1) ( 1)

K K

t

Y s y t K t e

s s s s s

τ

τ τ τ



−



= + + = + →

^L

=  − + 

(0 ≤ < ζ 1)

2 /

2 2

( ) ( ) 1 cos sin ( 1 )

( 2 1)

K

t

Y s y t K e t t

s s s

ζ τ

α ζ α α ζ

τ τ

⁻

ατ τ

−

   

= + + → =   −   +     =

L

Natural frequency

• Ultimate sinusoidal response

– Other method to find ultimate sinusoidal response

[ ]

2 2 2 2

With ( ) sin ,

( )

( 1)( )

U s A t

Y s KA

s s s

ω ω

τ τ ω

=

= →

+ + +

L

1

2 2

2 2 2 2

( ) sin( ) ( tan 2 )

(1 ) ( 2 ) 1

y t KA ω φ t φ ζωτ

ω τ ζωτ ω τ

= + = −

−

− + −

2 2 2 2

( ) ( )

( 2 1) (1 ) 2

s j

K K

G s G j

s s j

ω

τ ζτ τ ω ζτω

= →

→

=

+ + − +

2 2 2 2 2 2

( )

(1 ) (1 ) ( 2 )

K K

AR G j ω j

τ ω τω ω τ ζωτ

= = =

− + − +

1

Im( ( ))

1

2 ( ) tan G j tan

G j ω ζωτ

φ = R ω =

⁻

= −

⁻

-( ) -

For ( s + + α j ω ), ( ) has and it becomes as ( y t e

^α⁺^j^ω^t

e

^{j t}^ω

t → ∞ α > 0).

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BODE PLOT FOR 2 ^ND ORDER SYSTEM

• AR plot

• Phase plot

• Resonance

( )

²

2 2 2 2

1 1

( ) lim

(1 ) (2 )

AR

N

ω

ω τ ζωτ ωτ

→ ∞ =

→∞

=

− +

1 1 o

2 2

( ) limtan limtan 180

ω

1

ω

ζωτ ζ

φ ω

^→∞ ⁻

ω τ

^{→ ∞} ⁻

ωτ

→ ∞ = − = − = −

− −

Resonance

(

_N

) / 0

d AR dω =

2 max

1 2 ζ

ω τ

= −

for 0 < < ζ 0.707

The amplitude of output oscillation is bigger than that of input when the resonance occurs .

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1ST ORDER VS. 2ND ORDER (OVERDAMPED)

• Initial slope of step response

• Shape of the curve (Convexity)

{

²

}

1st order: (0) lim ( ) lim 0

1

s s

KAs KA

y s Y s

τ s τ

→∞ →∞

′ = = = ≠

{

²

} +

²

2nd order: (0) lim ( ) lim 0

2 1

s s

y s Y s KAs

s s

τ ζτ

→∞ →∞

′ = = =

+ +

1st order: ( ) y t ′′ = − Ke

⁻^t/^τ

< 0 (For 0) No inflection K > ⇒

1 2

/ /

1 2 1 2

2nd order: ( ) ( )

( as t ) Inflection

t t

KA e e

y t

τ τ

τ τ τ τ

− −

′′ = − −

−

+ → − ↑ ⇒

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CHARACTERIZATION OF SECOND ORDER SYSTEM

• 2

^nd

order Underdamped response – Rise time (t

_r

)

– Time to 1

^st

peak (t

_p

)

– Settling time (t

_s

)

– Overshoot (OS)

– Decay ratio (DR): a function of damping coefficient only!

– Period of oscillation (P)

2

2 / 1

P = πτ − ζ

(

²

)

/ exp / 1

OS = a b = − πζ − ζ

( )

2 2

/ ( ) exp 2 / 1

DR = c a = OS = − πζ − ζ

1 2

( cos ) / 1 ( 1)

t

r

= τ π n −

⁻

ζ − ζ n =

/ 1

2

t

p

= τπ − ζ

/ ln(0.05) t

s

≈ − τ ζ

1 2

τπ ζ

− ²

2 1 τπ

ζ

− ²

3 1

τπ ζ

−

( )

1 exp+ ζ τt/

2

1 exp 3 1

ζπ ζ

 

 

+  − 

2

1 exp 1

ζπ ζ

 

 

+  − 

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2 ^ND ORDER PROCESSES

• Two tanks in series

– If v

₁

=v

₂

, critically damped.

– Or, overdamped(no oscillation)

• Spring-dashpot (shock absorber)

– By force balance

c_Ai, q_i

c_A, q V1

V₂

1 2

( ) 1

( ) (( / ) 1)(( / ) 1)

A Ai

C s

C s == V q s V q s

+ +

( mg + f t ( )) − ky cv − = ma

( ( ))

my ′′ = − − ky cy ′ + mg + f t

2 2

2 ( )

4 m c m

y y y f t

k mk k

  ′′ + ′ + =

 

 

%

τ ζ

(can be <1: underdamped )

(5)

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Underdamped Processes

• Many examples can be found in mechanical and electrical system.

• Among chemical processes, open-loop underdamped process is quite rare.

• However, when the processes are controlled, the responses are usually underdamped.

• Depending on the controller tuning, the shape of response will be decided.

• Slight overshoot results short rise time and often more desirable.

• Excessive overshoot may results long-lasting oscillation.

POLES AND ZEROS

• Poles (D(s)=0)

– Where a transfer function cannot be defined.

– Roots of the denominator of the transfer function – Modes of the response

– Decide the stability

• Zero (N(s)=0)

– Where a transfer function becomes zero.

– Roots of the numerator of the transfer function – Decide weightings for each mode of response – Decide the size of overshoot or inverse response

• They can be real or complex

1 1 1

1

1 1

( ) ( 1)

( ) ( ) ( 1)

m m

n n

N s K b s b s b s

G s D s a s a s a s

− −

+ + + +

= =

+ + + L +

L

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• Real pole from

– Mode:

– If the pole is at the origin, it becomes “integrating pole.”

– If the pole is in RHP, the response increases exponentially.

• Complex pole from ( τ s + 1)

/

e

⁻t ^τ ^Re

Im

1 /τ

−

τ↑ τ↓

t y(t)

τ↑

( τ

2 2

s + 2 ζτ s + 1) (-1 < < ζ 1) s 1

= − τ

1

2

s ζ j ζ α j β

τ τ

= − ± − = − ±

2 2

2

1 1

(function of only)

s ζ ζ τ

τ τ

= + − =

2

1

1 tan (function of only)

s ζ ζ

ζ

−

R = ±

Re Im

ζ τ/

−

1−ζ τ2/

1 ζ τ2/

− −

τ ↓

ζ↓

ζ = cos θ

θ

– Modes:

– Assume is positive.

– If , the exponential part will grow as t increases: unstable – If , the exponential part will shrink as t increases: stable – If , the roots are pure imaginary: sustained oscillation

• Effect of zero

– The effects on weighting factors are not obvious, but it is clear that the numerator (zeros) will change the weighting factors.

2 2

/

(cos sin )

1 1

(cos sin )

t j t t

t

e e t j t

e t j t

α β α

ζ τ

β β

ζ ζ

τ τ

− ± −

−

= ±

− −

= ±

ζ <

0 τ

ζ >

0

ζ=

0

1

1 1

( ) 1 1

( ) ( ) (

_n

) ( )

ⁿ

(

_n

)

G s N s w w

s p s p s p s p

= = + +

+ L + + L +

(6)

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EFFECTS OF ZEROS

• Lead-lag module

– Depending on the location of zero

(a)

The lead dominates the lag.

(b)

The lag dominates the lead.

(c)

Inverse response

1

( ) ( 1)

( ) ( ) ( 1)

N s K

a

s

G s D s s

τ τ

= = +

+

^Lag

Lead

1

1 1

( 1) 1

( ) ( 1) 1

a a

KM s

Y s KM

s s s s

τ τ τ

τ τ

+  − 

= + =   + +  

1

0 τ

a

> > τ

0 ≤ τ

a

< τ

1

0 > τ

^a

/1

1

( ) 1 1

^a ^t

y t KM τ e

^τ

τ

⁻

   

=   − −      

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• Overdamped 2

^nd

order+single zero system

(a)

The lead dominates the lags.

(b)

The lags dominate the lead.

(c)

Inverse response

1 2

( ) ( 1)

( ) ( ) ( 1)( 1)

N s K

a

s

G s D s s s

τ

τ τ

= = +

+ +

1 1 2 2

1 2 1 2 1 2 1 2

( 1) 1 ( ) 1 ( ) 1

( ) ( 1)( 1) 1 1

a a a

KM s

Y s KM

s s s s s s

τ τ τ τ τ τ τ

τ τ τ τ τ τ τ τ

+  − − 

= + + =   + − + + − +  

1 2

1 / 2 /

1 2 2 1

( ) 1

^a ^t ^a ^t

y t KM τ τ e

^τ

τ τ e

^τ

τ τ τ τ

− −

 

=   + − + −  

1

0 (assume

1 2

) τ

a

> > τ τ τ >

0 < ≤ τ

a

τ

1

0 > τ

^a

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• Other interpretation

– Since , 1 is slow dynamics and 2 is fast dynamics.

1 2

1 2 1 2

( 1)

( ) ( 1)( 1) ( 1) ( 1)

K

a

s K K

G s s s s s

τ

τ τ τ τ

= + = +

+ + + +

1

1 1

2 1 2

1/

( 1) ( )

a a

s

K s K

K s

_τ

τ τ τ

τ

₌₋

τ τ

+ −

= =

+ −

2

2 2

1 1 2

1/

( 1) ( )

a a

s

K s K

K s

_τ

τ τ τ

τ

₌₋

τ τ

+ −

= =

+ −

1

(

1

1)

K τs+

2

(

2

1)

K τ s+

U(s) Y(s)

Y₁(s)

Y₂(s)

1 2

τ τ >

t y₁(t)

y₂(t)

t

y(t) K₁> K₂

2

0

K <

1

0

K >

t y(t) K1<K2^t

y₂(t)

2

0

K >

y1(t)

t

1

0

K <

1

τ

a

> τ τ

a

< 0

t y(t)

t y₁(t)

1

0

K >

1 a

0 τ τ ≥ >

y₂(t)

t K2 <ε

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Re(s) Im( s )

Unstable Region

Faster response

Faster response More

oscill.

Less oscill.

Faste r resp

onse

Shorter period of

oscill.

EFFECTS OF POLE LOCATION

Shorter period of

oscill.

(7)

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EFFECTS OF ZERO LOCATION

Re(s) Im( s )

Dominant Pole

Inverse response region

Deeper valley

0

More overdamped

More overshoot Real LHP zero:

Bigger inverse response Real RHP zero:

Complex RHP zero:

Zero at origin:

Complex LHP zero:

valley

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