1 Introduction to Optimum Design.pdf

Introduction to Optimum Design

Cheng-Liang Chen

PSE

LABORATORY

Department of Chemical Engineering National TAIWAN University

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Introduction to Design 

Several systems can usually accomplish the same task, and some are better than others



Any problem in which certain parameters need to be determined to satisfy constraints can be formulated as an   optimum design problem



The design of systems can be formulated as problems of optimization where a measure of performance  is to be optimized while satisfying all the   constraints 

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Iterative Design Process

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Conventional Design Process

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Optimum Design Process

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Conventional Design Process vs  Optimum Design Process 

  Conventional Design:

An objective function measuring system performance is not identified   Trend information is not calculated to make design decisions for improving systems  Most decisions are made based on designer’s experience and intuition 



  Optimum Design:

Optimum design process forces designer to identify explicitly a set of design variables, a cost function to be minimized (a performance function to be optimized), and the constraint functions for the system (  proper mathematical formulation of design problem)   Using trend information to make decisions   Aided by the designer’s interaction 

 ⇒

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Basic Terminology and Notation US-British SI Units

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Basic Terminology and Notation Sets and Points 

A point or vector in n-dimensional space

  x x   x  =   ..  = x 1 2

1

xn

x2

···

xn



T 

≡ (x1, x2, ··· , x

n

)

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

10

Geometrical representation for the set 2

 S  = x = (x , x ) | (x − 4) 1

Note: (0, 0)

∈ S,

2

1

(3, 3)

∈ S 

+ (x2

2

 − 4) ≤ 9

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Basic Terminology and Notation Notation for Constraints 

The set S   defines   points   within and on the circle of  radius 3 centered at (4, 4) 2

 S  = x = (x , x ) | (x − 4) 1

2

1

+ (x2

2

 − 4) ≤ 9

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Basic Terminology and Notation Norm/Length of A Vector

n



(x y) = xT y =

·

xiyi

i=1

||x||

=

x y =

·

√ x · x

n

√ 

xT x =

=

      x

2 i

i=1

||x||||y|| cos θ n

y = Ax =



n

 y = a x a x        = = x a x a (i)

i

i

ij

i=1

j =1

n

xT Ax = (x Ax)

·

j

n

i

i=1

n

ij

j =1

n

ij xixj

j

i=1 j =1

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Basic Terminology and Notation Functions

f (x) = f (x1, x2,

n

gi(x) =

n

g (x) =

··· , x ) g (x1, x2, ··· , x )  g1(x) g2(x) ··· i

T 



gm(x)

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Basic Terminology and Notation Continuous and Discontinuous Functions A function f (x) of  n  variables is called   continuous at a point x if  ∗

for any  >  0, there is a δ > 0 such that f (x) whenever x

 || − x || < δ  ∗

 |

− f (x )| <  ∗