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  • 1

    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

  • 2

    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

  • 3

    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

  • 5

    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

  • 6

    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

  • 7

    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

    10

    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

  • 11

    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

  • 12

    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

  • 13

    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

  • 14

    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