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458.308 Process Control & Design

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458.308 Process Control & Design

Jong Min Lee

School of Chemical and Biological Engineering

Lecture 3: Dynamic Simulation & Analysis (part 3)

0 0.5 1 1.5 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time

Tank Level (h)

Linear Approximation

Exact Nonlinear Solution

(2)

Laplace Transform

Pierre-Simon Laplace (1749–1827) wikipedia.com

Objective

L.T. is used to create transfer function models, which are the basis for many control system design techniques

Advantage

L.T. allows for the analysis of ODEs by using simple algebraic manipulations

Definition

L [f (t)] = F (s) =

Z 1

0

f (t) · e st dt

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L 1 [F (s)] = f (t)

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t: time

s: complex number (a + bj)

L [c 1 f (t) + c 2 g(t)] = c 1 L [f (t)] + c 2 L [g (t)]

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L.T. is a linear operator

(3)

Linear ODEs ) ( )

( t u t dt y

dy + =

τ

Solution

( )

( τ ) ζ ζ

ζ τ

d u

y t

y

t t

t

) ( exp

) 0 ( exp

) (

∫ 0 − −

+

=

Laplace Transform

) ( )

( ) 1

( τ s + Y s = U s

Integration

Inverse Laplace Transform

1 ) ) (

( = +

s s s U

Y τ

Algebraic Manipulation

(4)

L.T. of Common Functions

1 Exponential function f

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(t) = e at

(often the solution of linear odes)

a > 0

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a < 0

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f (t)

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f

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(t)

L [f (t)] = L [e at ] = F (s) =

Z 1

0

e at · e st dt

<latexit sha1_base64="+5wAwzKrYAKWuGFyDGzfK8fg854=">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</latexit><latexit sha1_base64="+5wAwzKrYAKWuGFyDGzfK8fg854=">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</latexit><latexit sha1_base64="+5wAwzKrYAKWuGFyDGzfK8fg854=">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</latexit><latexit sha1_base64="+5wAwzKrYAKWuGFyDGzfK8fg854=">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</latexit>

=

Z 1

0

e (s+a)t dt = 1 s + a

h e (s+a)t i 1

0 = 1

s + a [0 1] = 1 s + a

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(5)

2 Derivatives L

 df (t)

dt =

Z 1

0

df (t)

dt · e st dt = ⇥

e st f (t) ⇤ 1

0

Z 1

0

( s · e st )f (t)dt

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= [0 f (0)] + s ·

Z 1

0

e st f (t)dt = f (0) + s · F (s)

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L

 df (t)

dt = sF (s) f (0)

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f

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g

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g

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f

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g 0

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f

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(6)

3 Time Delays (Dead Time)

: Fluid transport through pipes, measurement sample delays f (t)

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a particular function of time f (t ✓)

<latexit sha1_base64="tc2THGY/rX72imbx0l4j71yVtIo=">AAACIHicdVDLSgNBEJz1bXxFPXoZDIIeXHaDoAcPghcPHiIYFZJFeie9ZnD24UyvEJZ8h1f1a7yJR/0YcTaJ4LNhmKKqu6upMFPSkOe9OmPjE5NT0zOzlbn5hcWl6vLKmUlzLbApUpXqixAMKplgkyQpvMg0QhwqPA+vD0v9/Ba1kWlySr0MgxiuEhlJAWSpINokvs3b1EWCrctqzXPrXln8N/Ddwe/V2Kgal9X3dicVeYwJCQXGtHwvo6AATVIo7FfaucEMxDVcYcvCBGI0QTE4us83LNPhUartS4gP2K8TBcTG9OLQdsZAXfNTK8m/tFZO0V5QyCTLCRMxNIpyxSnlZQK8IzUKUj0LQGhpb+WiCxoE2Zy+uZS7tYlMf4MryBQIdHl7YCh0cdyv2MA+U+H/g7O663uuf7JTO9gfRTfD1tg622Q+22UH7Ig1WJMJdsPu2D17cB6dJ+fZeRm2jjmjmVX2rZy3D0e9ouQ=</latexit><latexit sha1_base64="tc2THGY/rX72imbx0l4j71yVtIo=">AAACIHicdVDLSgNBEJz1bXxFPXoZDIIeXHaDoAcPghcPHiIYFZJFeie9ZnD24UyvEJZ8h1f1a7yJR/0YcTaJ4LNhmKKqu6upMFPSkOe9OmPjE5NT0zOzlbn5hcWl6vLKmUlzLbApUpXqixAMKplgkyQpvMg0QhwqPA+vD0v9/Ba1kWlySr0MgxiuEhlJAWSpINokvs3b1EWCrctqzXPrXln8N/Ddwe/V2Kgal9X3dicVeYwJCQXGtHwvo6AATVIo7FfaucEMxDVcYcvCBGI0QTE4us83LNPhUartS4gP2K8TBcTG9OLQdsZAXfNTK8m/tFZO0V5QyCTLCRMxNIpyxSnlZQK8IzUKUj0LQGhpb+WiCxoE2Zy+uZS7tYlMf4MryBQIdHl7YCh0cdyv2MA+U+H/g7O663uuf7JTO9gfRTfD1tg622Q+22UH7Ig1WJMJdsPu2D17cB6dJ+fZeRm2jjmjmVX2rZy3D0e9ouQ=</latexit><latexit sha1_base64="tc2THGY/rX72imbx0l4j71yVtIo=">AAACIHicdVDLSgNBEJz1bXxFPXoZDIIeXHaDoAcPghcPHiIYFZJFeie9ZnD24UyvEJZ8h1f1a7yJR/0YcTaJ4LNhmKKqu6upMFPSkOe9OmPjE5NT0zOzlbn5hcWl6vLKmUlzLbApUpXqixAMKplgkyQpvMg0QhwqPA+vD0v9/Ba1kWlySr0MgxiuEhlJAWSpINokvs3b1EWCrctqzXPrXln8N/Ddwe/V2Kgal9X3dicVeYwJCQXGtHwvo6AATVIo7FfaucEMxDVcYcvCBGI0QTE4us83LNPhUartS4gP2K8TBcTG9OLQdsZAXfNTK8m/tFZO0V5QyCTLCRMxNIpyxSnlZQK8IzUKUj0LQGhpb+WiCxoE2Zy+uZS7tYlMf4MryBQIdHl7YCh0cdyv2MA+U+H/g7O663uuf7JTO9gfRTfD1tg622Q+22UH7Ig1WJMJdsPu2D17cB6dJ+fZeRm2jjmjmVX2rZy3D0e9ouQ=</latexit><latexit sha1_base64="tc2THGY/rX72imbx0l4j71yVtIo=">AAACIHicdVDLSgNBEJz1bXxFPXoZDIIeXHaDoAcPghcPHiIYFZJFeie9ZnD24UyvEJZ8h1f1a7yJR/0YcTaJ4LNhmKKqu6upMFPSkOe9OmPjE5NT0zOzlbn5hcWl6vLKmUlzLbApUpXqixAMKplgkyQpvMg0QhwqPA+vD0v9/Ba1kWlySr0MgxiuEhlJAWSpINokvs3b1EWCrctqzXPrXln8N/Ddwe/V2Kgal9X3dicVeYwJCQXGtHwvo6AATVIo7FfaucEMxDVcYcvCBGI0QTE4us83LNPhUartS4gP2K8TBcTG9OLQdsZAXfNTK8m/tFZO0V5QyCTLCRMxNIpyxSnlZQK8IzUKUj0LQGhpb+WiCxoE2Zy+uZS7tYlMf4MryBQIdHl7YCh0cdyv2MA+U+H/g7O663uuf7JTO9gfRTfD1tg622Q+22UH7Ig1WJMJdsPu2D17cB6dJ+fZeRm2jjmjmVX2rZy3D0e9ouQ=</latexit>

the value of the function 𝜽 time units in the past L [f (t ✓)] =

Z 1

0

f (t ✓ ) · e s(t ✓) · e s✓ dt

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0

f (t ⇤ ) · e st ⇤ dt ⇤

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time should be positive

L [f (t ✓)] = e s✓ F (s)

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(7)

4 Step Functions

Unit step function

used to simulate the ( ) in an input variable (e.g., flow rate being rapidly changed from one value to another)

f

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(t) 1

time

f (t) =

⇢ 1 (t 0) 0 (t < 0)

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L [f (t)] =

Z 1

0

f (t) · e st dt =

Z 1

0

1 · e st dt =

 1

s e st

1 0

= 1 s

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L [1] = 1 s

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L [c] = c s

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Referensi

Dokumen terkait

To design and develop a GUI simulation for fluid flow and water level control system that is user-friendly and easy to be used by students, by using the Eclipse and Android