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Goal Programming and Goal Programming and

Isoperformance Isoperformance

March 29, 2004

Lecture 15

Olivier de Weck

© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

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Why not performanceWhy not performance--optimal ? optimal ?

“The experience of the 1960’s has shown that for military aircraft the cost of the final increment of performance usually is excessive in terms of other characteristics and that the overall system must be optimized, not just performance”

Ref: Current State of the Art of Multidisciplinary Design Optimization

(MDO TC) - AIAA White Paper, Jan 15, 1991

TRW Experience

Industry designs not for optimal performance, but according to targets specified by a requirements document or contract - thus, optimize design for a set of GOALS.

© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

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Lecture Outline Lecture Outline

• Motivation - why goal programming ? • Example: Goal Seeking in Excel • Case 1: Target vector T in Range = Isoperformance

• Case 2: Target vector T out of Range = Goal Programming

• Application to Spacecraft Design • Stochastic Example: Baseball

Forward Perspective

Choose x What is J ?

Reverse Perspective

Choose J What x satisfy this?

© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

A

Domain Range

B

T

fa

Jx

Target Vector

Many-To-One

Goal Seeking Goal Seeking

max(J)

T=Jreq

J

x *

min(J)

*

,i iso x

x i LB x ,, xmin

xi UB imax 4 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox

Engineering Systems Division and Dept. of Aeronautics and Astronautics

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Excel: ToolsExcel: Tools –– Goal Seek Goal Seek

Excel - example J=sin(x)/x

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-6 -5

.2 -4

.4 -3

.6

-2 .8 -2

-1 .2

-0 .4

0.

4 1.

2 2 2.

8 3.

6 4.

4 5.

2 6

x

J

sin(x)/x - example

• single variable x • no solution if T is

out of range

For information about 'Goal Seek', consult Microsoft Excel help files.

© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

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Goal Seeking and Equality ConstraintsGoal Seeking and Equality Constraints

• Goal Seeking – is essentially the same as finding the set of points x that will satisfy the following “soft” equality constraint on the objective:

xJ ( ) − Jreq Find all x such that ≤ ε

Jreq

Target massExample m 1000kgsat Target data ratexTarget J ( ) = Rdata ≡ 1.5Mbpsreq

Vector: Target Cost15M $Csc

© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

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Goal Programming vs. IsoperformanceGoal Programming vs. Isoperformance

Criterion Space Decision Space

(Objective Space) (Design Space) J2

is not in Z - don’t get a solution - find closest

x2

J1

S Zc 2

x1

x4

x3 x2

J1

J3

J2

J2

The target (goal) vector

= Isoperformance

T2

T1

The target (goal) vector

Case 1: is in Z - usually get non-unique solutions

Case 2:

= Goal Programming © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

Isoperformance Analogy Isoperformance Analogy

Non-Uniqueness of Design if n > z Analogy: Sea Level Pressure [mbar] Chart: 1600 Z, Tue 9 May 2000

2Performance: Buckling Load c EIπ =P Isobars = Contours of Equal Pressure Constants: l=15 [m], c=2.05 E l2 Parameters = Longitude and Latitude

Variable Parameters: E, I(r)

Requirement: P REQE = tonsmetric1000 ,

Solution 1: V2A steel, r=10 cm , E=19.1e+10 Solution 2: Al(99.9%), r=12.8 cm, E=7.1e+10

LL

LL LL

HH 1008

1008

1012

1008

1008

1008

1012

1016

1012

1012

1012

1016

1012

1004

1016

1012

1012

l2r

PE

E,I

c

Bridge-Column

Isoperformance Contours = Locus of

constant system performance

Parameters = e.g. Wheel Imbalance Us,

Support Beam Ixx, Control Bandwidth ω c 8 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox

Engineering Systems Division and Dept. of Aeronautics and Astronautics

Isoperformance and LP Isoperformance and LP T

• In LP the isoperformance surfaces are hyperplanes min c x

• Let cTx be performance objective and kTx a cost objective s t x ≤ ≤ x. . xLB UB

1. Optimize for performance cTx*

2. Decide on acceptable performance penalty ε

3. Search for solution on isoperformance hyperplane that minimizes cost kTx*

cTx*

= cT

k

c

Efficient SolutionIso

pe rfo

rm an

ce hy

pe rp

lan e x**

B (primal feasibility)

Performance Optimal SolutioncTx*+εεεε xiso9 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox

Engineering Systems Division and Dept. of Aeronautics and Astronautics

Isoperformance Algorithms

Empirical System Model

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Isoperformance ApproachesIsoperformance Approaches

(a) deterministic I soperformance Approach

Jz,req Deterministic

System Model

Isoperformance Algorithms

Design A

Design B

Design C

Jz,req

Design Space (b) stocha stic I soperformance Approach

Ind x y Jz 1 0.75 9.21 17.34 2 0.91 3.11 8.343 3 ...... ...... ......

Statistical Data

Design A

Design B

50%

80%

90%

Jz,req

Empirical System Model

Isoperformance Algorithms

Jz,req P(Jz)

© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

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Nonlinear Problem Setting Nonlinear Problem Setting

“Science Target Observation Mode”White Noise Input

Appended LTI System Dynamics

d J

Disturbances

Opto-Structural Plant

Control

(ACS, FSM)

(RWA, Cryo)

w

u y

z

ΣΣΣΣ ΣΣΣΣActuator Noise Sensor

Noise

[Ad,Bd,Cd,Dd]

[Ap,Bp,Cp,Dp]

[Ac,Bc,Cc,Dc]

[Azd, Bzd, Czd, Dzd]

Variables: xj J = RSS LOSz,2

© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

Performances

Phasing

Pointing

z,1=RMMS WFE

z=Czd qzd

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Problem Statement Problem Statement

Given

x q + Bzd ( ) x r LTI System Dynamicsq = Azd ( ) x d + Bzr ( )j j j z = Czd ( ) x d + D ( x r , where j = 1, 2,..., npx q + Dzd ( ) zr j )j j

And Performance Objectives T

T 1/ 2

1/ 2

J = F ( ) , e.g. J =z z z t= E z z = T

1 ( )2 dt RMS,z z i 2

0

Find Solutions x such thatiso

J ( x ) ≡ J ∀ i = 1, 2,..., nzz i iso z req i, , , − ≥1 and x j LB ≤ x j ≤ x j ,UB ∀ j = 1, 2,..., nAssuming n z ,

J x( ) − J τ,z z req Subject to a numerical tolerance τ : ≤ = ε

J 100,z req © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

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Bivariate Exhaustive Search (2D) Bivariate Exhaustive Search (2D)

“Simple” Start: Bivariate Isoperformance Problem First Algorithm: Exhaustive Search

Performance J (x x2 ) : z = 1z 1, coupled with bilinear interpolation

Variables x j = 1, 2 :j , n = 2 Number of points along j-th axis:

x − xj ,UB j ,LB

x1

x2 n = j

∆x

Can also use standard contouring code like MATLAB contourc.m

© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics

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Contour Following (2D) Contour Following (2D)

k-th isoperformance point:

Taylor series expansion

1 TTJ x ( k( ) = J x ) + ∇J x∆ + ∆x H x∆ + H.O.T.kkz z z xx 2 first order term second order term

T x∇J ∆ ≡ 0 τ Jz kp Tα (

∂Jz ∂x

J 1∇ = z ∂Jz

∂x2

1/ 2 −1

= 2 z req, t H