Extension of Set-Based Inference Mechanisms for
Predicate Logic Design Constraints with an
Application to Automotive Power Window Design
by
Amit Vishwanath Seshan
B.Tech, Mechanical Engineering (1998)
Indian Institute of Technology
Bombay
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2000
@ Massachusetts Institute of Technology 2000. All rights reserved.
A u th o r.......................................................
...............
Departmer'of Mechanical Engineering
May 8, 2000
C e rtifie d by .............................................................................................
William W. Finch
Research Scientist
Thesis SLIpprvisor
A cce pte d by ...............................................................
. .....................
Ain A. Sonin
Chairman, Department Committee on Graduate Students
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
SEP 2 0 2000
LIBRARIES
Extension of Set-Based Inference Mechanisms for Predicate
Logic Design Constraints with an Application to Automotive
Power Window Design
by
Amit Vishwanath Seshan
Submitted to the Department of Mechanical Engineering
on May 8, 2000, in partial fulfillment of the
requirements for the degree of
Master of Science
Abstract
In this thesis, Set Theory and Predicate Logic are applied to design Automotive Power
Window systems. This work addresses an electro-mechanical design problem, whose
solution is complicated by interactions between multiple sources of uncertainty. The
design problem is currently solved by engineers at the Ford Motor Company, using
time-consuming and error-prone iterative parametric design techniques.
Set-based tools use a formal mathematical notation to represent both, how and
when different sources of variation affect the a system. Such representations denote
design constraints using first order predicate logic expressions, called Quantified Relations. A set-based inference mechanism operates on quantified relations, drawing
information that allows designers to eliminate provably infeasible sets of parameter
values from a system's design space. This reduces the search space for parametric
design, reduces thrashing, and assists faster convergence towards a final design.
This thesis presents a set-based constraint model of a power window system, and
documents an attempt to apply current set-based tools to this model. With this
effort, it demonstrates that the existing set-based calculation tool, the Interval Propagation Theorem, is inadequate in addressing a class of design problems. This is due
to the restrictive symbolic form of constraint (with a single equation algebraic description) prescribed by the tool. This motivates exploration of alternative techniques to
overcome this limitation, using symbolic algebra and numerical methods.
This research culminates in the proof of the Extended Interval Propagation Theorem, a new tool enabling application of set-based design theory to a larger class of
engineering problems in practice (with multi-equation algebraic descriptions). The
thesis also explores additional technology and research issues with an aim to extensively employ the set-based paradigm in CAD tools for design automation.
Thesis Supervisor: William W. Finch
Title: Research Scientist
Acknowledgments
My heartfelt thanks to William Finch.
Foremost, for his constant support as my research advisor, and his wisdom in guiding
this project past many hurdles. Then for his patience and sincerity as a mentor;
particularly for his consistent and painstaking effort to help me think and write
clearly. Most importantly, for being a cheerful friend. His humor, optimism and trust
have constantly inspired me.
I would like to thank my friend, Arvind Sankar, for carefully listening to my
ramblings, clarifying my thoughts, and answering all my questions patiently. The
theoretical extension in this thesis owes much to insights derived from Arvind's counsel, dispensed on summer evenings in MIT's Muddy Charles Pub.
Thanks to Darrel Kleinke and the folks at the Ford Motor Company, whose interest
and financial support made this thesis possible. This research is supported in part
by the MIT Center for Innovation in Product Development under NSF Cooperative
Agreement Number EEC-9529140.
Thanks to the Center for supporting me and
providing an excellent environment for work and study.
Thanks to the MIT-Pakodas, Tzeho Lee and my CIPD colleagues for their friendship.
Finally, thanks to my friends Mihir Wagle, Rajashree Bhaskaran, Banga-
lore Pradeep, Vinod Suresh, Shyam Raghunandan, and Anand Ganti for showing
an interest in my work. Their encouragement (and criticism) helped my enthusiasm
rebound whenever it mattered the most.
Contents
Nomenclature
1 Introduction
1.1 Glazing System Design at Ford . . . . . . . . . . . . . . . . .
1.2 The Set-Based Paradigm . . . . . . . . . . . . . . . . . . . . .
1.2.1 Set-Based Inferences and Solutions . . . . . . . . . . .
1.2.2 Mathematical Tools to support the Set-Based Paradigm
1.3 Research Motivation . . . . . . . . . . . . . . . . . . . . . . .
1.4 Organization of the Thesis . . . . . . . . . . . . . . . . . . . .
2 Power Window Design Problem
2.1 Power Window System Components ......
2.1.1 Seals and Belt .................
2.1.2 Mechanisms ...............
2.1.3 Electrical System ..............
2.2 Design Requirements ..............
2.2.1 Stall Force ........
. ......
.
2.2.2 Glass Velocity . . . . . . . . . . . . . .
2.3 Problem Definition . . . . . . . . . . . . . . .
2.3.1 Proposed Methodology . . . . . . . . .
3
Set-Based Mathematics
3.1 Parametric Models . . . . . . . . . . . . . . .
3.1.1 Parametric Variables and Constraints .
3.1.2
3.2
. . . .
3.1.3 Pneumatic Actuator Example . . . . .
3.1.4 Motivation for a Set-Based Extension .
Set-based Models . . . . . . . . . . . . . . . .
3.2.1
3.3
Parametric Constraint Network
Set Variables - Closed Intervals
. . . .
3.2.2 Set Constraints - Quantified Relations
Causality in Engineering Systems . . . . . . .
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3.4
3.5
3.6
3.7
3.3.1 Causal Influences and Controllability . . . . . . .
3.3.2 Dependence and Temporality . . . . . . . . . . .
Causal Table Construction . . . . . . . . . . . . . . . . .
3.4.1 Causality, Design Intent and Quantifier Semantics
3.4.2 Formulating a Quantified Relation . . . . . . . .
3.4.3 Expressive Power of Quantified Relations . . . . .
Causal Constraint Network . . . . . . . . . . . . . . . . .
Inference mechanism for Set-Based Design . . . . . . ..
3.6.1 Bounding Sets . . . . . . . . . . . . . . . . . . . .
3.6.2 Bounding Sets Theorem . . . . . . . . . . . . . .
3.6.3 Interval Propagation Theorem . . . . . . . . . . .
An Example BST-IPT Design Inference . . . . . . . . . .
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4
Set-Based Model of the Cable Drum Power Window System
4.1 Electro-Mechanical System Model ....................
4.1.1 Complexity and Simplifying Assumptions . . . . . . . . .
4.1.2 Parametric Model Relations . . . . . . . . . . . . . . . .
4.2 Constraint Network and Causal Table . . . . . . . . . . . . . . .
4.3 Formulation of Quantified Relations . . . . . . . . . . . . . . . .
4.3.1 Choice of Quantifiers . . . . . . . . . . . . . . . . . . . .
4.3.2 Glass Velocity Quantified Relation . . . . . . . . . . . .
4.3.3 Stall Force Quantified Relation . . . . . . . . ... . . . .
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5
Set-Based Inferences for Power Window Design
5.1 Limitations of the Existing Mechanism . . . . . . . .
5.2 Methods to Enhance Applicability . . . . . . . . . . .
5.2.1 Symbolic Elimination of Intermediate Variables
5.2.2 Numerical Solution of a System of Equations .
5.2.3 Comparison of the two methods . . . . . . . .
5.3 Inference Results and Interpretations . . . . . . . . .
5.3.1 Drum Diameter Calculation . . . . . . . . . .
5.3.2 Torque Constant Calculation . . . . . . . . . .
5.4 Observations . . . . . . . . . . . . . . . . . . . . . . .
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Extending the Interval Propagation Theorem
6.1 Statement of the Extended-IPT . . . . . . . . . . . . . . . .
6.2 Proof of Extended-IPT . . . . . . . . . . . . . . . . . . . . .
6.3 Examples Using the Extended-IPT . . . . . . . . . . . . . .
6.3.1 Inference from Stall Force Quantified Relation . . . .
6.3.2 Inference from the Glass Velocity Quantified Relation
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104
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Conclusion
7.1
Research Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2
Limitations and Future Work . . . . . . . . . . . . . . . . . .
7.2.1 On Capturing Causality . . . . . . . . . . . . . . . . .
7.2.2 On Better use of Temporality . . . . . . . . . . . . . .
7.2.3 On Sensitivity Analysis . . . . . . . . . . . . . . . . . .
7.2.4 On Symbolic Elimination versus Numerical Methods .
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7.2.5 On Building Monotonicity Tables . . . . .
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7.2.6 On Quantifying more than k + 1 variables
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7.2.7 On Forms of Constraint . . . . . . . . . .
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Contributions . . . . . . . . . . . . . . . . . . . .
7.3
A DC Motor Model
A.1 DC Motor Theory . . . . . . . . . . . . . . . . . . .
A.1.1 Free Running Condition (No Load) . . . . .
A.1.2 Stall Condition (Maximum Load) . . . . . .
.
A.2 Experimental Determination of motor constants
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124
B Drag Force and Load Torque Equations
124
B.1 Engagement Lengths . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 125
B.1.1 Force Balance
127
C Performance Metrics
D Lemmas to Prove Extended-IPT
D.1 Implicit Function Theorem ......................
.....................
D.2 Supporting Lemmas ....
D.2.1 Existence of a Partitioning ..............
D.2.2 Existence and Uniqueness of the Explicit Function .
D.2.3 Perturbing a variable in 1'(x) . . . . . . . . . . . .
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E Monotonicity Tables
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Table
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Specific
Monotonicity
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E.1 Partition
E.2 A Filling Algorithm for M . . . . . . . . . . . . . . . . .. . . . . . . 135
9
List of Figures
2-1
Power Window Systems
. . . . . . . . . . . . . . . . . . . . . . . . .
31
3-1
Simple Parametric Model. . . . . . . . . . . . . . . . . . . . . . . . .
38
3-2
Simple Set-based Model for the Pneumatic Actuator
. . . . . . . . .
52
4-1
Parametric Constraint Network of Cable Drum Power Window System
66
A-i DC M otor Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
. . . . . . . . . . . . . . . . . . . .
A-2 DC Motor Linear Characteristics
A-3 DC Motor Characterisation Experiment
. . . . . . . . . . . . . . . . 120
B-i Glass Seal Configuration . . . . . . . . . . . . . . . . . . . . . . . . .
10
119
126
List of Tables
1.1
Table of Set Domains for the Quantified Relation 1.2 . . . . . . . . .
19
3.1
Examples of Causal Selectors. . . . . . . . . . . . . . . . . . . . . . .
43
3.2
Causal Table for Pneumatic Actuator, if p is assumed uncontrollable.
47
3.3
Causal Table for Pneumatic Actuator, if p is assumed controllable
. .
47
4.1
Relations that model the Electro-mechanical System
. . . . . . . . .
64
4.2
Causal Table for Electro-mechanical Model of Power Window System
67
5.1
Table of Interval endpoints for Diameter Inference . . . . . . . . . . .
83
5.2
Table of Relaxed Interval endpoints for Diameter Inference . . . . . .
85
5.3
Table of Interval endpoints for Torque Constant Inference from Stall
Force Q R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4
88
Table of Interval endpoints for Torque Constant Inference from Glass
Velocity Q R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
6.1
QR Specific Monotonicity Table for the Stall Force Quantified Relation 103
6.2
QR Specific Monotonicity Table for the Glass Velocity Quantified Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1
107
Relations to calculate Motor Constants from Experimental Data . . . 122
A.2 Experimental Data for Motor Characteristics Computations
. . . . . 122
A.3 Table of Parameters in the Electro-mechanical System . . . . . . . . . 123
11
Model Nomenclature
Variable
Associated
Description
Variation set
x
X
Normalised window position
X = [0, 1]; x = 1 => fully closed
position
1
L
Vector of Seal, pillar and belt lengths,
window geometry angles and lengths
1 = (l11 12,
A
Fdrag
lbelt.0, z,p)
Vector of Seal and Belt drag coefficients
J=
fdrag
-16,
(J1, 62,
... 64, 6 belt)
Total frictional drag force on the glass
(from all seal engagements and belt contact)
fshaft
Fshaft
Tangential force on the motor shaft
(arising from seal drag and glass weight,
corrected for frictional and transmission
losses by a linear model)
floss
Foss
Frictional Losses in window guides
7idrive
Hos8
Regulator Mechanical Efficiency
w
W
Glass Weight (in N)
d
D
Cable Drum Diameter
Tshaft
Tshaft
Torque on the motor shaft
W
Q
Angular velocity of the motor shaft
12
Vglass
Vlass
Cable speed or Glass Velocity
Vbat
Vbat
Battery Voltage terminal supply voltage
Vtest
Viest
Battery Voltage used for DC motor test
rbat
Rbat
Battery internal resistance
rine
Rune
Line resistance of wiring from terminal
to motor
rtest
Rtest
Line resistance used in DC motor test
rmotor
Rmotor
Motor internal resistance
if
If
Motor free running current
kt
Kt
Motor torque constant
kb
Kb
Motor back-EMF constant
i
I
Current in power-window circuit
Tstaul
Tstall
Motor stall torque
fStaii
Fstall
Window stall force
rImotor
Hmotor
Motor efficiency
Notation
x
Italicised lowercase letters denote value-variables.
Eg. x = 0.51, y =cow
X
Italicised uppercase letters denote set-variables.
Eg. X = [2, 6], Y = {cow,horse}
x
Vector of variables, x
X
Vector of sets, X = (X 1 , X 2 , ... Xn)
X
Collection (set) of sets
X
Collection of vectors of sets
a
Floor of a closed real interval associated with variable a
11
Ceiling of a closed real interval associated with variable a
=
(x1 , x 2 , .
.
xn)
(The set associated with a variable a is A = [a, ?])
13
x= a
Vector of ceiling assignments
(Each element of x is assigned the ceiling of a corresponding interval
from a specific set-vector A; If A = (A1 , A 2 ... A,)
then (X = a) =: xi = -di, i = 1, 2,. ...
n)
x
a
=
Vector of floor assignments
(Each element of x is assigned the floor of a corresponding interval
from a specific vector A; If A = (A1 , A 2 ... An),
then (x = a) =:> xi = aj, i = 1, 2,....n)
G(x)
A relation among values. (among elements of a variable vector x).
g(x)
An expression containing members of the variable vector x.
g(x) = 0
An algebraic equation in the variables x)
g(x -)
Expression g is positively monotonic in variable xi, or N > 0.
g(x)
XIb
Expression g is negatively monotonic in variable xi, or N < 0.
Lower bound on the domain X of a set variable X.
X DXI- VX C
Xub
.
Upper bound on the domain X of a set variable X.
X C B VX CXl
14
Chapter 1
Introduction
"Where shall I begin, please your Majesty?" he asked.
"Begin at the beginning", the King said, gravely,
"and go on till you come to the end: then stop."
-Lewis Carrol (1832-1898)
Uncertainty in the precise value of system parameters complicates the design,
manufacture and operation of many engineered systems.
Uncertainty arises from
various sources, including manufacturing variations, operator adjustments, and environmental changes.
Multiple sources of uncertainty interact with each other in
complex ways, making it difficult to predict and control the eventual performance of
an engineered system.
Product designers attempt to meet design goals by engineering their systems to
be robust against influences of uncertainty. They stand to benefit from automated
design tools that account for uncertainty while analyzing the behavior of engineered
systems. Such tools can appropriately guide design decisions in the product development process.
Previous attempts at this kind of automated reasoning for design, have used methods like interval mathematics, fuzzy logic, and probability distributions to capture uncertainty. Recently developed Set-Based Methods combine Set Theory and Predicate
Logic to represent and reason with uncertainty. They also overcome some shortcomings of earlier methods, constituting a reliable new approach to design automation.
15
This thesis demonstrates, validates and extends Set-Based Design methods by
application to a real-life design problem that is complicated by uncertainty. The
problem pertains to design of Automotive Power Window Systems at the Ford Motor
Company.
This chapter first introduces the design problem, and then proposes the set-based
problem-solving paradigm to address it. It surveys existing set-based tools, and explains why a set-based approach is appropriate for this particular design problem. It
finally lays out the organization of the thesis.
1.1
Glazing System Design at Ford
An automotive glazing system consists of the various mechanical elements that move
automobile glass windows along seal-guides in a vehicle's door pillars. This system
protects passengers from wind noise and water ingress. If the glass is actuated by a
DC motor, the system is called a power window system.
Power window system design is an important, frequently recurring problem at the
Ford Motor Co. The problem is solved afresh for each new vehicle model introduced.
Further into the vehicle life-cycle, body-shell styling changes repeatedly alter window
geometry. Such changes often warrant redesign of the power-window system, to ensure adequate performance. Engineers also prefer to source subsystem components
(e.g. DC motors, seals) from several alternative vendors, changing their selections
over time. For this, they must preserve relevant engineering knowledge, component
options etc. from prior, proven designs. The company devotes much effort to developing satisfactory glazing systems and requires a fast, efficient design tool. Elaborate
spreadsheets currently serve as computational tools for power window design. Numerous iterations with these tools are needed to develop satisfactory designs. The
current product development process is time-consuming and error-prone.
The components of the power window system work together to fulfill performance
requirements. Since manufacturingand environmentalvariations influence component
characteristics, the system's ultimate behavior in the field is hard to control. System
16
performance is thus adversely affected by uncertainty. This thesis proposes to solve
the power window design problem by using a new, set-based, approach. This should
make the design process quicker, and more robust against uncertainty.
1.2
The Set-Based Paradigm
This section introduces important concepts in set-based design to explain how this
problem solving paradigm method works. It discusses the state of the art in set-based
design tools, showing how they are relevant the power window design problem.
The set-based paradigm for robust design counters uncertainty to support engineering design. Using this paradigm, engineers communicate and reason about sets
of alternative designs, rather than trying to refine a particular design. Thus an important difference between set-based and traditional design procedures is that the
set-based paradigm seeks a collection (or set) of satisfactory designs that solve a
design problem, rather than a single, precisely defined, final design.
1.2.1
Set-Based Inferences and Solutions
A design process operates on large collection (a set) of different possible designs, called
the design space. The designer states certain design constraints or requirements that
a system is engineered to eventually satisfy. Designs that satisfy these constraints,
are termed feasible. All other designs are inconsistent with the proposed design goals,
and are termed infeasible.
Set-based design proceeds by making a sequence of set-based inferences about
portions of the design space. Each inference divides the relevant portion of the design
space, into two distinct subsets of designs
o a subset of provably infeasible designs, which can be safely discarded, possibly
shrinking the design space.
o a subset of potentially feasible designs, that are retained and processed further,
to move towards a final set-based solution to the design problem.
17
Set-based inferences thus narrow down the design space by progressively eliminating portions of the space that are provably infeasible. Successive shrinkages of
the design space must halt when residual portions of the design space cannot be discarded any further without the risk of sacrificing feasible designs. At this stage, the
remaining part of the design space contains all possible feasible designs for the system.
This residual part of the initial design space is the set-based solution to the original
design problem. If uncertainty is included in the reasoning for set-based inferences,
the set-based solution contains designs robust against expected uncertainty.
Set-based inferences reduce thrashing (wasted search/effort) in design by helping
engineers to focus design effort on appropriate regions of the design space. These
inferences reduce optimization search space, providing a pre-processing tool for rapid
design optimization.
1.2.2
Mathematical Tools to support the Set-Based Paradigm
Recent research in the area of set-based design, [1], has produced mathematical tools
that support the set-based design paradigm. These tools denote design constraints
as predicate logic expressions, a formal way of encoding design goals in a mathematically precise, hence unambiguous manner. These expressions are called Quantified
Relations.
Quantified Relations
To illustrate the kinds of constraints that can be posed and satisfied using quantified relations, three examples are presented here. The first example is set in a
non-engineering context, demonstrating set-based inferences on discrete domains. It
is intended to familiarize the reader with the concept of set bounds and set bound
inferences. The second example is more mathematical, showing set-based inferences
on continuous domains, with suggested design applications. The third example is a
real-life engineering constraint, drawn from a later chapter of this thesis.
18
Sets explicitly listed below Sets explicitly listed below
and unions thereof
and unions thereof
constitute the domain X
constitute the domain Y
A = {gnu, cheetah, zebra}, S = {Australia},
B = {koala, lion}
T = {Tanzania, Kenya},
C = {elephant},
U = {y I y is in South America}.
D = {llama},
Table 1.1: Table of Set Domains for the Quantified Relation 1.2
Example 1. Terrestrial wildlife distribution can be expressed in terms of a set of
animal species (types), and specific countries. Consider the relation
G(x, y) : x is native toy
(1.1)
where the variable x is a type of animal, and y is a country. This relation is true,
for some ordered pairs of actual values for (x, y). Examples of such pairs include
(anaconda,Brazil), (platypus,Australia), etc. We group animal types into sets
of animals. Similarly, we group countries into sets of countries. Individual sets,
explicitly listed in the columns of Table 1.1 are examples of sets of animals and
countries1 . Predicate Logic establishes a constraint between sets of animals and
sets of countries, by embedding G(x, y), in the expression below, called a quantified
relation.
QR(X, Y) : Vx E X 3y E Y - x is native to y
(1.2)
In natural language, it reads,
For any animal x, from a set of animals X, there exists a country y in a
set of countries Y, such that the relation "x is native to y" is true.
We instantiate (assign actual sets to) the variables X and Y, from their respective
domains, collections of sets2 . If one variable in QR(X, Y) is instantiated, the quan'So are appropriate combinations of these individual sets.
2, = {S, T, U, (S u T), (S U U), (T U U), (S U T U U)}. Likewise, X
includes the sets A, B, C, and
D and all possible unions of them, pairwise, in threes, and in fours.
19
tified relation may be satisfied by finding feasible set values for the other variable, in
accordance with the relation G(x, y). One (naive) way to do this is by exhaustively
searching the domain space. Set-based inferences can drastically cut computational
effort expended in such a search. We present some examples of such inferences.
Inference 1.
Let X = B = {koala, lion}
Which sets Y E Y satisfy QR(B, Y)?
In this simple example, it is easy (by inspection) to determine, or infer a lower bound 3 ,
Y* E Y, on all feasible Y, satisfying QR(B, Y). Evidently one possible bound 4 is
Y* = {Tanzania, Kenya, Australia} = S U T
(1.3)
Any larger set, Y', wholly containing Y*, also satisfies QR(B, Y). Using this inferred
bound, we divide Y into two distinct subsets.
= Y1 U Y2
Y= {S, T, U, (S U U), (T U U)}
Y2= {(S U T), (S U T U U)}
Sets in Y, are either disjoint with Y*, or only partially overlap it, and can all be
safely discarded from further consideration, as being provably infeasible. Sets in Y2
are the only sets in Y that completely contain the lower bound Y*, and some of them
may be feasible, satisfying QR(B, Y).
Inference 2.
Let Y = T = {Tanzania, Kenya}
Which sets X E X satisfies QR(X, Y)?
3Set containing the minimum possible number of elements needed to satisfy G(x, y) for this
particular instantiation of X. As explained later in Chapter 3, such a set is denoted Y1-6-, so that
V Y'IQR(X, Y'), we can guarantee that Y' E {YJYI b C Y C Y}.
'Other valid lower bounds, not actually members of Y, are {Australia , Tanzania} and
{Australia, Kenya}.
20
Again, due to the simplicity of this example, and its relatively small domains, it is
easy to infer an upper bound' X* E X, on all feasible X, satisfying QR(X, T). It is'
X* = { gnu, cheetah, zebra, elephant} = A U C
(1.4)
Any smaller set X', wholly included within X*, also satisfies QR(X, T). Divide
Z
into 2 distinct, mutually exclusive, collectively exhaustive subsets, X1 and X 2.
X
= X 1 UX
2
B,(AU B),(AU D),(AU BU C),(AU BU D),(AuCU D),
D, (BUC), (BUD), (CU D), (BU CUD), (AUB UCUD)
X2=
{A, C, (A U C)}
We safely discard every set in X 1 as being provably infeasible because it either partially overlaps X*, is disjoint with X*, or exceeds X* in membership. The elements
of X 2 are the only sets in X that are completely contained within the upper bound
X*, and maybe considered as potentially feasible solutions.
Inference 3.
Let Y = U = {y I y is in South America}7 .
Which sets X E X satisfy QR(X, U)?
Inspection shows that the upper bound on membership of X is the set X* = {llama}.
This eliminates all other members of X from consideration, removing entirely, the
need to search any further for a solution.
The Effectiveness of Set-Based Inferences
Each of the above inferences demonstrates the idea of a lower/upper set bound. The
set bound guides us in dividing the domain of possible solutions by clearly identifying
5 Set containing the maximum possible number of
elements permitting satisfaction of G(x, y) for
this particular instantiation of Y. As explained later in Chapter 3, such a set is denoted Xu., so
that V X'IQR(X', Y), we can guarantee X' E {XIX C Xu. C Z}
'Like before, we can also think of upper bounds that are not themselves included in X, E.g.
{gnu, cheetah, zebra, elephant,lion}.
7This is an Implicit Descriptionof a set, as opposed
to an explicit listing.
21
all provably infeasible sets. Removing such infeasible sets can considerably simplify
the search for feasible sets to satisfy a quantified relation over a given domain.
The space of ordered pairs (X, Y), within which the QR(X, Y), establishes a
constraint between sets, is the Cartesian product of the domains, X x Y. In this
example, the space, (X x Y), is a discrete set, containing a finite number of elements.
Each of the 3 inferences described here eliminates a significant fraction 8 of candidate
pairs (X, Y) E X x Y, from consideration for feasibility.
The bounds shown here were found simply by intuition/inspection. Inferences
with far more complex quantified relations, with numerous variables, and larger domain spaces would arise commonly in engineering applications. Finding set bounds
within these more complex domains is difficult; searching for feasible sets in discrete
domains is computationally expensive, even for moderate domain sizes. Eliminating
large numbers of provably infeasible sets from their domains by set-based inferences
is effective in satisfying quantified relations.
The use of inferred set bounds to partition domains by eliminating provably infeasible sets, is especially useful with quantified relations that embed continuous algebraic functions. Within the continuous domains of such functions, exhaustive search
is impossible.
Example 2. Algebraic relations in engineering, are often defined over continuous
real domains. Consider an equation in 3 real variables
G(x, y, z) : x + y = z
(1.5)
Assuming that the variables x, y, and z, lie within associated domain (sets) intervals,
X E X, y E Y, and z E Z, respectively, we relate the domain interval variables by a
'No attempt is made here to quantitatively measure the success of set-based reasoning here. An
interesting explanation of the relative efficiency of set-based inferences is found in chapter 1 of [1],
where the common game of "20 Questions" is used as a parallel, to explain the effectiveness of the
set-based method.
22
quantified relation of the form
QR(X,Y,Z):xEX
VyEY 3zEZ - x+y=z
(1.6)
This relation imposes a a constraint on the memberships of the sets X, Y, and Z, by
requiring that
There exists a value of x within the set X, such that for any value of y
in the set Y, there must exist some value of z within Z to guarantee that
x+ y
-
z = 0 holds.
The quantified relation is satisfied by certain ordered triples, (X, Y, Z). Two examples of set-based inferences are now presented. Each example instantiates 2 of the set
variables in the quantified relation to compute a bounding, (upper/lower) interval for
the third variable. Knowledge of the bounding interval helps us eliminate infeasible
regions of the domains (sets of real intervals along the x, y and z axes) from consideration, while satisfying the quantified relation.
Inference 1.
Let X = [0, 1] and Y = [0, 2].
Which values of Z will satisfy QR(X, Y, Z)?
A set-based inference mechanism for continuous algebraic relations (explained later
in Chapter 3) computes an lower bound Z*, on all intervals, Z C (-o, oo), that
satisfy QR([0, 1], [0, 2], Z).
The lower bound is Z* = [1, 2].
From the previously explained properties of an lower bound, any feasible interval, Z,
must completely contain Z* within it.
Inference 2.
Let X = [-1, 2] and Z = [0, 8].
Which values of Y will satisfy QR(X, Y, Z)?
A set based inference similar to the one above, computes an upper bound Y*, on all
23
intervals, Y C (-oo, oo), that satisfy QR([-1, 2], Y, [0, 8]).
The upper bound is Y* = [-2,5].
Any feasible interval, Y, must be completely contained within Y*.
We start out
knowing nothing specific about the location, or extent of a feasible interval Y. Thus
we assume that it may extend anywhere on the real line, Y C (-o, oc).
The inferred upper bound drastically reduces the space of possible solutions. The
inference eliminates any interval which is not a subset of [-2, 5], as being provably
infeasible.
The inferences shown above are easily verified by inspection, but the exact mechanism that generates them is a systematic numerical calculation. The inference is
based on the Interval Propagation Theorem [1]. This set-based design theorem uses
the monotonicities and quantifiers of variables in QR(X, Y, Z), to guide the use of
the relation G(x, y, z) : x + y - z = 0, in determining interval bound endpoints.
Mathematical tools, currently available to implement the set-based design paradigm
were developed [1] primarily for this exact class of quantified relations, with closed real
intervals as set domains. Prior to this research, the scope of the inference mechanism
had been restricted to quantified relations (like Relation 1.6), where the embedded
relation is a single, continuous, monotonic, asymptote-free algebraic equation
The next example illustrates a "real-life" engineering design constraint, posed as
a quantified relation. The example is drawn from chapter 4 of this thesis.
Example 3. Consider a system of 5 simultaneous, mutually independent, nonlinear
algebraic equations in 16 variables 9 .
G7(fmotor, floss, fdrag) W, lidrive)
A
fmotor -
G8 (Tmotor, fmotor, d)
i - if -
Gl(vattery,
rmotor, nine, kbw)
A joVater7rmtoirlneki
,
9
v -
0
0
i(rmotor+rine)
kb
kb
9 ,w,d)
=
= 0
:-Vbattery
)
A G(v
77drive
Tmotor - fmotor 2 =
A G9(if,,kT)
A
fdrag±W f1O3'
-
0
= 0
The variables are listed in the nomenclature at the outset of the thesis, and detailed explanations
of the equations are included in Chapter 4 and Appendices A through C.
24
This system describes
(a) the force balance between mechanical components,
(b) the electro-motive phenomena in a DC motor, and
(c) geometric relationship between linear and angular measures of force and velocity,
in an automotive power window system. Conceptually, the relation P1 is not very different from the function G(x, y, z) used in the previous example, though its algebraic
complexity is significantly greater.
Like before, we associate set domains (closed real intervals) with each variable, and
establish a constraint between the set domains by the following quantified relation.
QR1(Vattery, Rline, Rmotor , Kb, KT, If, Hdrive, Foss, D,
lW, Fdrag,7Vg) :
VVbattery G Vattery Vriine E Rline Vrmotor G Rmotor Vkb E Kb VkT E KT Vif C If
Vr7drive E Hdrive Vf 1 oss E Floss Vd E D Vw E W Vfdrag E Fdrag
3Vg E Vg = [0.125, 0.175]ms-1 - IF,
where rl is the system of simultaneous equations (J1 : G7 A G8 A G9 A Gio A Gil).
In natural language, this reads
For all possible variations in values of any of the system parameters,
(bounded by their respective intervals), there exists a value of glass velocity Vg, within the set V, such that the system I1 is satisfied.
The rest of this thesis is devoted to setting up such quantified relations, and
applying set-based tools to draw inferences from them.
Tools for Set-Based Design
The term "tools", as used in this report, includes theorems and algorithms that
operate on the quantified relations to make set-based inferences.
These set-based
methods are deterministic, and do not rely on probabilistic descriptions of uncertainty
(as illustrated in examples 1 and 2).
Set-based inferences overcome an important inadequacy of preceding robust design
tools like interval mathematics. Previous methods do not simultaneously incorporate
25
information about both, how and when a particular source of uncertainty affects the
system". Predicate Logic constraint representations capture this information and
allow set-based inferences to use it effectively.
Set-based tools implement the set-based paradigm on specific design domains.
They have been successfully applied to the design of simple mechanical systems described by an appropriate class of parametric (algebraic) models [1]. Likewise they
have been applied to catalog based design, where part numbers associated with specific component characteristics, are picked from discrete domains (catalogs/tables) to
satisfy performance constraints on a parametrically modeled system [1].
1.3
Research Motivation
This section combines the discussions from preceding sections. It suggests why there
is a distinct advantage in using set-based tools to solve the the power window design
problem. The expected advantage is presented as a list motivating factors for this
research.
1. Automotive power window systems are affected by numerous sources of variation, making design and analysis complicated. The set-based tools used in this
project have been developed with an aim to solve such problems efficiently.
2. This thesis addresses a specific instance of the power window design problem,
namely a type called the Cable Drum Power Window System. The mathematical model for the cable drum system has a large number of variables and
constraints. It is cumbersome for manual analysis and design. Studying this
problem will identify issues involved in extending and automating set-based
reasoning for application to even larger problems.
3. Power window systems are representative of a larger class of electro-mechanical
systems. Thus, set-based design for power windows yields insight into how
10References
[1], [2], and [3] contain examples to show that that such information, called causal
information is essential in making correct inferences reliably. This is also explained in greater detail
in Chapter 3, where set-based tools are presented in a more formal, rigorous setting.
26
the technique should be exploited for a more general electro-mechanical CAD
system. This technique can thus be generalized to other automotive subsystems
with appropriate types of parametric models (E.g. suspension, or drive-train
subsystems).
4. Developing a set-based model for the power-window system reveals limitations
of the theory that impede its application to complex, real-life problems. This
study identifies areas where the existing theory has to be augmented. It also
attempts to propose suitable modifications and extensions wherever possible.
Modeling and theoretical work required for this application were completed over a 2
year project at MIT's Center for Innovation in Product Development, starting in the
fall of 1998.
1.4
Organization of the Thesis
The chapters are structured as follows.
Chapter 1 is a gentle introduction to the glazing system design problem, and the
set based paradigm. It also explains research goals and outlines thesis organization.
Chapter 2 provides a detailed explanation of the Cable Drum power window
design problem. It describes all the relevant components and subsystems, lays down
design requirements and explains how the problem will be addressed by set-based
tools.
Chapter 3 is an extensive introduction to set-based mathematics. It presents a
theoretical, fairly detailed overview of existing set-based tools. This section also uses
a simple illustrative example to acquaint the reader with methods and tools by which
set-based theory is applied to solve design problems.
Chapter 4 applies modeling methods from Chapter 3, to formally specify the
Cable Drum Power Window System. It describes this particular electro-mechanical
system in the language of set-based mathematics, developing a parametric model and
constraint network for the system. It goes on to formally denote design intent by
27
developing suitable quantified relations using the power window model.
Chapter 5 draws set-based inferences from the model developed in Chapter 4. It
notes features of the problem formulation that motivate a new approach to set-based
inferences. It suggests and explores two new methods that extend set-based theory,
and presents results of design inferences about the power window system. It also
analyses the limitations of these inferences.
Chapter 6 provides a theoretical extension to an existing set-based mechanism.
It contains a proof of the new Extended Interval PropagationTheorem. This theorem
extends the applicability of the existing set-based tool called the Interval Propagation
Theorem [1], to a larger class of design problems.
Practical examples using the
theorem are included.
Chapter 7 concludes the thesis by pointing at directions for future work in the
area of set-based design research.
28
Chapter 2
Power Window Design Problem
The purpose of models is not to fit the data
but to sharpen the questions.
- Samuel Karlin (1923-)
This chapter first explores the components of a Cable Drum Power Window System. It then explains the objectives of Ford engineers in designing this system. It
notes the sources of uncertainty in components, and how they affect system performance. It also describes metrics to characterize satisfactory design, thus establishing
the scope of the research by clearly stating the design problem in engineering terms.
The chapter finally outlines a set-based solution procedure that will be implemented,
later, in Chapters 4 and 5.
2.1
Power Window System Components
This section details form and function of various components in a glazing system,
documenting important sources of variation in component characteristics. The descriptions included here, should familiarize the reader with the workings of a particular type of power window system, the Cable Drum Power Window System. Detailed
analytical models of component behavior are included in appendices A through C1 .
'Tables A.3 and A.2 contain numerical data for nominal values and variations of system
parameters.
29
2.1.1
Seals and Belt
Belts and seals isolate the car's occupants from wind, noise and rain.
Seals are
rubber linings that run round the edges of the automobile window. They are made
of extruded rubber, and are installed by pressing them snugly into a matching groove
that runs the length of the door pillars. They are also secured by screwing them into
the body of the door, further down within the metalwork of the door. Where two strips
of seal rubber meet at a right/acute angle (as in the corner between the B-Pillar and
the roof) they are bonded using a molded L-shaped connector. In some vehicles, such
sharp corners in a window are eliminated by rounding the window corners smoothly
(e.g. Ford's F-150 truck). In such cases, the seal is directly installed by bending a
single rubber lining round the smoothly curving perimeter.
A seal is characterized by its cross section, and the properties of the rubber it
is made of. The seal cross section includes ribs or extensions that help stiffen the
seal, and control the frictional force between the seal and the glass. The designer
must ensure that the power window system has enough actuating force throughout
its traverse, to overcome this frictional force, or seal drag, and move the glass.
At the upper limit of the window traverse, the actuation should be strong enough
to drive the glass snugly into the rubber lining, to ensure complete closure and proper
sealing. Seal drag increases significantly at the end of the traverse, because glass
movement is additionally restricted by
(a) wrinkles in the seal if it curved to transition from the B-pillar to the roof, or
(b) constriction because of the extra thickness of the plastic L-shaped connector.
The seals also serve to guide the glass while it is raised or lowered.
A rubber belt is installed on the upper edge of the door, as shown in Figure 2-1.
The belt scrapes against the outer surface of the glass. It wipes the glass as it retracts
into the door. The belt thus keeps water from entering the space within the door
where the electro-mechanical subsystem is housed.
Both, the seals and the belt oppose glass motion by contact friction. The amount
of friction against the glass varies with the environmental conditions, temperature of
30
Rubber Seal
A Pillar
B Pillar
B Pillar
- -Belt
Cross arms
----
Cable
Door body
Motor + Worm
Wormwheel
Motor + Worm
Cross Arm Mechanism
Wormwheel + Cable drum
Cable Drum Mechanism
Figure 2-1: Power Window Systems
the rubber, presence of moisture, age and wear, stiffening of seal cross section, etc.
This friction is the principal variation introduced in the glazing system by the seals.
It is measured as variation in the seal-drag and belt-drag friction coefficients.
2.1.2
Mechanisms
A power window system is a mechanism with a single degree of freedom.
Seals
along the door pillars kinematically constrain the glass. Motive power from a geareddown DC motor, drives the mechanism. A suitable actuating mechanism converts a
rotational input into translation along the seals.
Various actuating mechanisms are available. In the cable drum system, a pulley
with cables pulls the glass along rails. In a cross-arm system, a modification of a
slider-crank mechanism is used to move a pair of cross-arms that raise and lower the
window in its guides. Figure 2-1 illustrates the two systems.
Depending on the choice of mechanism there are variations in mechanical efficiency
and losses of the system. In the cross arm system, the mechanism efficiency varies
with window position. The frictional losses depend on lubrication, age, wear, and
compliance of structural members (flexing cross arms will waste some input power by
storing it as elastic energy). In the cable drum system, there is no issue of mechanical
advantage, but losses arise from friction and stretching of tensioned cables.
31
2.1.3
Electrical System
The electrical system consists of the automobile battery, alternator, wiring, and DC
motors that actuate the glazing mechanism.
The battery is a lead-acid unit that nominally supplies 12.6V DC output. Battery
voltage varies with environmental factors, primarily temperature. The car alternator
is connected in parallel across the battery terminals. When the engine is running, the
supply provides approximately 14.4 volts to the electrical system. The battery and
alternator are aggregated as a single DC source for further modeling, and the voltage
of that source is termed battery voltage.
The motor is a permanent magnet DC unit, with an integral worm-wheel regulator,
chosen to match the torque requirements of the window system. The motor in each
door is connected to the terminals by a length of wire, with a certain line resistance.
The length of wiring used varies from door to door depending on its proximity to
the battery location. This causes line resistance variation. Thus, each motor sees a
different impedance to the battery terminals. Line resistance also varies with external
temperature, but this variation is small compared to the length variations.
DC Motor characteristics are subject to manufacturing variations. A randomly selected motor has characteristics within tolerance bands specified by its manufacturer.
These tolerance bands are determined through motor tests described in Appendix A.
Line resistance and battery voltage variations are the primary sources of electrical
uncertainty considered in this thesis.
2.2
Design Requirements
The previous section mentions the various functions of the glazing system, along
with component descriptions. Successfully fulfilling all these functional requirements
is a broad design objective, made concrete by identifying measurable performance
characteristics as indicators of satisfactory design. Variation of such indicators, or
performance metrics, is limited to establish design constraints. The performance
metrics used in this project are Stall Force and Glass Velocity.
32
Though the power window system has many sources of uncertainty, this thesis
concentrates primarily on handling the effects of the following variations:
(a) variations in battery voltage (with environment).
(b) wiring resistance (manufacturing and temperature induced variation).
(c) window seal drag and belt drag (environmental and manufacturing).
(d) motor characteristic variations (due to manufacturing).
(e) glass weight, drum diameter etc.
(dimensional and physical variations due to
manufacturing).
2.2.1
Stall Force
Stall force, at any intermediate position of the window glass, is the force that must be
applied on the upper edge of the window, to just arrest its upward motion. Designers
would like to keep the stall force within the range of [100,250]N, at all positions of
window traverse.
If the stall force is too small, the window will run slow, especially if the seals are
damp or cold. The system may not have enough force to push past the edge of the
door seal to close tightly against the roof. This imperfect closure leads to improper
sealing and increased wind noise levels. These considerations govern the lower-bound.
If the stall force is too large, the window will run fast, and and there is a risk of
damaging some of the internal components (e.g.. the motor, its gearing, regulator
parts) due to excessive load on the system at closure. These considerations provide
the upper-bound.
2.2.2
Glass Velocity
The time taken for the window to close or open is an important consideration for
customer acceptance. The velocity with which the glass moves up (down), measured
at all points of traverse, is a metric for this closure (opening) time. A general design
requirement is that "the window must complete its full (fully open to fully closed, or
vice-versa) motion within about 3.5 seconds". This is appropriately translated into a
33
bound on glass velocity, based on the height of the window traverse.
For the dimensions of the particular vehicle modeled in this thesis, the variation in
the glass velocity should be held within a specific interval of [13,17]cm s-,
assuming
a supply voltage of 12.6V. When the voltage increases due to temperature, or after
the alternator starts working, this requirement is relaxed, and the window is allowed
to run faster.
Though it also possible to include other metrics (E.g. current drawn at stall) the
thesis concentrates only on stall force and glass velocity for simplicity of modeling.
2.3
Problem Definition
This section summarizes the power window design problem, and explains broadly,
how set-based tools will help solve it.
The power window system has numerous electrical and mechanical components
that are subject to uncontrolled variations. Its component characteristics (system
parameters) are thus uncertain within bounds. These component characteristics drive
the system behavior, and are related to performance metrics by appropriate functions.
The design problem is solved by finding system components that satisfy performance
requirements, robust to uncontrolled variations.
Set-based power window design satisfies performance constraints, by determining suitable sets of values for the component characteristics (system parameters) to
bound the performance characteristics as desired. These sets of values must be found
in accordance with the inter-related physical and geometric relations that govern
electro-mechanical behavior of the system. The glazing system design problem is
thus an attempt to satisfy this set of inter-related constraints; a ConstraintSatisfaction Problem or CSP.
2.3.1
Proposed Methodology
Set-based tools solve the CSP by eliminating provably infeasible designs, i.e. designs
that are guaranteed to be inconsistent with one or more constraints. Thus, set-based
34
tools help enforce consistency with the CSP formulation. The limits on the performance metrics are used to work backwards and calculate the allowable limits on all
other component characteristics, that will permit a feasible design. This process for
solving the CSP is called constraint propagation. It is accomplished as follows:
Step 1. Build a ParametricConstraint Model of the system. This model
contains inter-related functions (constraints) representing physical and geometric truths about the system. It is a computational tool that calculates
precise real values certain system parameters (E.g. glass velocity), given
precise real values for others (E.g. battery voltage)
Step 2. Associate an initial set of possible values with each system parameter included in the parametric model. The collection of these sets of
possible values, is the initial design space for the design problem.
Step 3. Express the design constraints in set-based notation to obtain
a list of Set-Based Constraintsconnected with the underlying parametric
model. These set-based constraints constrain the sizes of the sets from
which the system parameters derive values. This step converts the parametric constraint model into a set-based constraint model. The set-based
constraint notation supports set based inferences.
Step 4. Use an algorithmic procedure to operate on the set-based constraint model, using set-based inferences to discard provably infeasible
values from the associated set for each variable.
When the algorithmic procedure terminates, it exits with sets that are consistent
with the CSP formulation. They constitute the residual portion of the original design
space, where a feasible design might (or might not) exist. The eliminated parts of
the original design space are, however, provably infeasible, since they were discarded
in accordance with the set-based paradigm. Thus, if any feasible design existed in
the initially assigned variation sets, it must be captured in the residual associated
sets when the algorithm terminates. Chapters 4, 5 and 6 perform the steps of the
set-based solution described above.
35
Chapter 3
Set-Based Mathematics
When you have eliminated the impossible, whatever
remains, however improbable, must be the truth.
- Sir Arthur Conan Doyle (1859-1930)
This chapter provides mathematical background required to understand the existing set-based inference mechanism. It explains the mechanism in detail, using a
simple, practical example to illustrate the ideas.
It first reviews parametric models, commonly used mathematical representations
of real systems. These models are used as computational tools for engineering analysis
and design, and provide a precise, analytical "point tools" for computing with real
numbers. However, they are of limited effectiveness in capturing variation data. This
chapter demonstrates how set-based design helps engineers overcome this limitation.
For a class of design problems, a set-based model of the engineered system serves
as powerful design tool, by extending an underlying parametric model of the system.
This chapter discusses the motivation and technique for such extensions. It explains
the use of predicate logic expressions, called Quantified Relations, to denote design
constraints in set-based modeling to engineer systems.
A set-based inference mechanism is applied to design problems posed using quantified relations. An illustrative example clarifies the use of this inference mechanism.
This chapter also lays down the foundation of notation and concepts required for
proofs presented later in the thesis.
36
The reader may refer to [1], [2] and [3] for more details on quantified relations and
set-based inferences.
The Bounding Sets Theorem (subsection 3.6.2) and Interval
Propagation Theorem (subsection 3.6.3), described briefly here are proved in [1].
3.1
Parametric Models
This section discusses parametric models. It introduces the notion of a constraint
network representation of a parametric model, with an example to illustrate these
concepts. Later in the chapter, the same example is adapted for a set-based description, to explain the features of set-based models by analogy with parametric models.
As used here, the term parametric model denotes a symbolic, mathematical abstraction of a real-world, physical system. It is a computational tool in engineering,
useful for both, design and analysis of the system. Such a model assumes a particular
configuration (specification of connectivity and interactions between system components), providing mathematical relations that (approximately) represent the complex
functions of the physical system. A detailed explanation of terminology used here is
found in [1] and [11].
3.1.1
Parametric Variables and Constraints
A parametric model of a physical system consists of an n-tuple, x = (x , X2, ... Xn)
1
of real variables that describe the system. It includes a set of constraint functions,
{ G,(x), G2 (x) . ..
m (x) }.
An assignment of precise values to all elements of x, sat-
isfying all constraints, defines a particular instance of the system, a specific case of
the design of the system. The set of all such instances is the parametric design space.
Parametric models are "point models", where each possible design is characterized
by a precise point in the design space. Generally, a (proper) subset of elements in the
design vector x, called the design parameters can be varied continuously, to parameterize the design space, spanning all possible designs. Traditional parametric design
processes try to tune (or vary) the values in this subset, moving from point-to-point
37
f
g(p,af)=f-pa=0
PneumaticActuator
ParametricConstraintNetwork
Figure 3-1: Simple Parametric Model
in the design space, iteratively searching for feasible designs 1
3.1.2
Parametric Constraint Network
A parametric model can be represented as a bipartite graph. The graph has a set of
round nodes representing the variables, and a set of rectangularnodes representing the
constraints. Undirected arcs connect round (parameter) nodes to those rectangular
(constraint) nodes, in which the relevant variable appears explicitly. The graph, called
the parametricconstraint network, captures dependencies between the constraints. It
is bipartite since every arc connects a unique parameter-constraint pair.
3.1.3
Pneumatic Actuator Example
A simple physical system is now modeled to illustrate the ideas described in the
preceding subsections.
Consider a pneumatic actuator, with a pressure regulator.
that delivers a pressure p to a piston of area a, to produce a force f.
Its parametric model has three variables, and a single constraint.
The design
vector is x = (p, a, f), a point in 3. The constraint is a single algebraic equality 2 I
G(x) : g(f, p, a) =
f-
pa = 0. The parametric constraint network representation of
this system is shown in Figure 3-1.
'Likewise, design optimization drives this kind of iterative search towards minimizing an objective
function over many feasible designs.
2
The constraint is just the engineering definition of pressure: (pressure= force/area).
38
3.1.4
Motivation for a Set-Based Extension
The "point" representation of designs, inherent in parametric modeling, is ineffective
in including variation data, because a precise value assignment carries no information
about uncertainty. In the previous example, specifying a precise real value for the
pressure, say p = 50 psi, can deliver no information about a possible variation in
pressure within the range [45, 55] psi, due to an unreliable pressure regulator.
Uncertainty information can be included in parametric models by using inequality representations 3 . However, when the number of uncertainty-influenced variables
in a system is large, appending numerous inequalities makes the parametric model
computationally cumbersome.
Set-based models simplify the representation of variation, by avoiding the use of
inequalities. Another advantage they present over parametric models, is in the kind
of information that can be inferred by using them as design tools. The following
sections explain construction and properties of a set-based model.
3.2
Set-based Models
A set is a collection of objects. Sets capture variation quite naturally. A single set
can encompass a range of possibilities. Building a model whose variables are assigned
set-values allows variation data to be incorporated directly into the variables, rather
than expressing it in constraints. Such design variables that take on set-values are
called set variables. Set-based models [1] use set variables, related by set constraints.
This section explains how such a set-based representation is developed, starting with
a parametric model.
3
By appending a pair of inequalities to bound each parameter (representing limits on its allowed
variation).
39
3.2.1
Set Variables - Closed Intervals
For the particular class of problems addressed in this thesis, where real variables are
related by continuous equations, a set-based extension of a given parametric model
is constructed by associating a closed real interval, X, with each system variable,
x,. This association restricts the real variable, xi, to take on values from within the
interval (xi E X2 ). This model is now "set-based", because closed intervals are sets
of real numbers. Each association implicitly imposes a pair of inequality constraints
on the relevant real variable, i.e. if Xi = [ji, Ti], then xi E Xi = j5
xi x T. Since
Xi is used to symbolically denote an interval whose endpoints are unknown, and may
vary, it is a set variable. A more detailed discussion on set variables (and their types)
is included in [1].
Example: In the case of the pneumatic actuator, the variables a, p and f are associated with corresponding closed real intervals A, P and F. Each variable takes on
a value from the range specified by its associated interval. Thus a E A, p E P and
f
E F must hold. The endpoints of the intervals A, P and F will be decided when
the design problem is formulated.
The set-based extension of a parametric model has set-valued variables which
are assigned set values (intervals) instead of precise real values. Just as algebraic
constraints in a parametric model relate real variables, the set-based model uses
constraints that relate sets. Quantified Relations are constraints in a set-based model.
3.2.2
Set Constraints - Quantified Relations
A quantified relation [1], QR(X), is a well formed formula in first order predicate
calculus. It makes an assertion about the n-tuple of set variables appearing as its
argument, X. As a predicate, it is a true statement if values assigned to the elements
of the argument satisfy the relation it prescribes among them. A quantified relation
has the form,
QR(X) = qixl E X 1 q2 x 2 E X 2
40
...
qnxn E Xn
.
G(x)
(3.1)
The expression G(x) is a relation among the parameters x. In general, G can take
on several forms, including discrete, continuous or mixed relations. It is also referred
to in this thesis, as an embedded relation,since it is an algebraic relation amongst real
variables, contained within a quantified relation among sets.
Continuous relations in a real space R" include equations, inequalities and systems
of simultaneous equations or/and inequalities. A simple example of a continuous
relation is the equation (G(f, p, a) :
f
- pa = 0) used in the pneumatic actuator
model. Discrete relations in R" are explicit lists of n-tuples, satisfying G. Such a
discrete collection of n-tuples is usually compiled as a table. Tabular relations are
common in engineering in the form of charts and catalogs.
Each qi is either an existential (3), or a universal (V) logic quantifier. Thus each
term, qixi
e
Xj, is an assertion about set membership of a particular real-valued
variable, xi with its associated set-valued variable, Xi. If qj = V, the relation G(x)
must be satisfied for all values of xi contained in Xj. If qj = 3, there must exist at
least one value (possibly more values) for xi within X, such that G(x) is satisfied.
The pattern of logic quantifiers captures important information about the engineered system. The following section explores the importance of such additional data,
called causal information, in set-based design.
3.3
Causality in Engineering Systems
Like parametric models, set-based models characterize a real physical system by using
variables and constraints. Set-based models additionally include information about
both, how and when each source of variation affects a system variable. This information is called CausalInformation. This section develops guidelines for identifying and
storing causal information.
Example: Let us look a possible time history of the pneumatic actuator system, to
understand causality informally. Assume that the system is an industrial actuator,
where a fixed value of pressure is applied as an input to the piston, by regulating an
inlet valve. Further assume that this pressure is kept constant during the stroke of
41
the piston 4 . The constant pressure drives the piston outwards, generating a constant
output force to move a load5 . Thus, the three variables that characterize the system,
have a distinct sequence in time, according to which they assume numerical values.
The actuator is first designed and manufactured, thus, its piston area, a, is decided by a manufacturing process. Following this, an operator selects a pressure
setting (say by using a regulator knob), deciding the value of pressure, p, applied.
f (force), also gets a precise value,
pa. The value of f thus constrained to depend on the
At the instant pressure is selected, the variable
consistent with the relation f =
assignments of values to both a and p. This dependence is characterized not just by
a mathematical relationship, but also by noting that the value of f is a consequence
of priorly selected values for a and p. The following subsections explain more formal
ways of characterizing causality.
3.3.1
Causal Influences and Controllability
The underlying reason that governs the specific value assigned to a system variable
is a"cause" for that value assignment. Because such reasons "select" precise values
for system variables, they are called causal influences or selectors [1]. In some cases,
one can determine the actual selector (see Example 1 below) governing a particular
value assignment. In other situations, a selector involves several complex, intermeshed
reasons. Such a complex, underlying chain of causal events is abstracted into a single
representative idea (see Example 2 below).
Example 1: Causal influences are are italicized in the following statements.
"The designer selects a part number from a piston catalog, setting d."
"The operatorselects a value of pressure, p, using the knob on the regulator."
"The output force,
f, from the piston results
as a consequence of the above selections."
4A reasonable assumption in practice, if the compressed air supply line is connected to a sufficiently "stiff" pressure source, which delivers constant line pressure despite flow of air due to opening
the valve.
5Note that this situation is different from, another type of cylinder-piston arrangement: a syringe.
In a medical syringe the force is applied as an input, to the end of the piston (plunger) and pressure
built up within the syringe tube as a consequence of the force.
42
Causal Selector
Stochastic variations in manufacturingprocess
Changing conditions of environment
OperatorAdjustment
Interval Bound established from
Measurable process capability 6- limits.
Observed ranges of temperature and pressure.
Limits of control/calibration settings on a device.
Table 3.1: Examples of Causal Selectors.
Example 2: "The manufacturing process selects a precise value, say r = 102.42Q, of
a 100Q resistor with a 5% tolerance band."
"External environment (temperature and light intensity primarily) selects terminal
voltage v, of a Photo-voltaic cell."
Whenever a causal influence can be identified (or abstracted), it is characterized
in a formal causal model, by specifying its selector.
Selectors are then arranged
in a specific order to analyze their collective influence on the system over its timehistory. Examples of selectors are shown in Table 3.1, along with suggested methods
to determine interval bounds their effects.
Depending on its nature, each selector identified in a system is labeled as being
a controllable selector or an uncontrollable selector. The extent of its effect is rigorously specified by determining domains from within it selects values for its associated
variable. In general set-based design theory the domain of a selector is a set. In this
thesis (for models with real variables, related by continuous algebraic relations), the
domains of selectors are closed real intervals.
Random or unpredictable selectors
assign include random manufacturing variations, and unpredictable environmental
conditions like temperature and humidity.
Controllable selectors govern adjustable parameters like control settings, calibration settings etc. The system designer can assume that an agency with a goal-seeking
behavior, like an intelligent human operator, control algorithm, etc. will adjust these
parameters, so as to achieve a certain desired output. The range of possible variation (domain) of a controllable selector is relevant in determining whether it can
successfully help compensate uncontrollable influences.
43
3.3.2
Dependence and Temporality
Temporal order is the sequence in which causal agents act on a given system. It is
thus an ordering in time, according to which system variables are assigned precise
values.
Besides being temporally ordered, system parameters also have temporal dependence within relations. Every variable in a given relation is either temporally independent of, or temporally dependent on the other variables appearing explicitly in that
relation.
A temporally independent variable in a relation does not depend on any other
parameters in that relation to get its value assignment. Its value assignment is governed directly by a selector, or determined from another relation in the system, that
constrained it in accordance with certain prior selections.
A temporally dependent variablegets its value assignment as a result of a constraint
that relates it to a group of other variables. Thus, a given temporally dependent
variable must appear later in the temporal sequence than all the other (temporally
dependent, or independent) parameters it depends on.
Note that the notion of temporal dependence in causal models is different from the
notion of dependence between variables based on the symbolic form of the function
that relates them. The traditional concept of dependence is specific to a particular
calculation made using a function y = f(x). We say that the LHS variable, y, is
dependent on the independent RHS variable, x, if we specify a value for x, say x = a,
and then use the function to compute y = f(a). If the same function were rewritten 6
as x = f--1 (y), the relative dependence can be reversed in a calculation. We can now
supply a value, y = b, to the independent variable, y, and use this to compute 7 the
(functionally) dependent variable x = f-(b).
Dependence based on the symbolic form, or the role in a calculation, of function,
'this can be done under some assumptions about the continuity and differentiability of the function, f.
7
1n practice, it is not even necessary to symbolically invert f. Methods like Golden Section,
Regula Falsi, and Newton Iteration [6], [7], will numerically converge upon a solution for x, given a
value for y, even if the form of f 1- is not known.
44
is thus based on the "direction" of the calculation. It states what values are supplied
(as independent inputs) and which values are calculated (as dependent outputs).
However, temporal dependence is not decided by the way in which equations are
symbolically written (or used) in the parametric model. It is decided by the order
in which the system relations are used to determine value assignments. Temporal
dependence is thus abstraction denoting underlying causal ordering, not only of individual selectors, but also of variables that are indirectly constrained by successive
selections.
3.4
Causal Table Construction
From the previous discussion, causality includes temporal dependence, and controllability information.
Having identified the elements of causality, this section now
presents a tabular method to symbolically represent causality in engineering systems.
This Causal Table representation is a useful tool in building set-based models. Causal
Tables are temporal histories engineering systems, tabulated with data about controllability.
A discussion of temporality and controllability in the pneumatic actuator example
illustrates the causal table construction process. The time history of the actuator is
the same as the one discussed informally in the example of section 3.3. However, we
identify different selectors for the same set of variables to illustrate how the choice of
selectors can produce alternative causal models for the same physical system.
8
When large numbers of parametric constraints are inter-related in a constraint network representation, the notion of "direction" is formalized by directing the (initially undirected) arcs of
the parametric constraint network. The directions of these arcs will now indicate feasible paths in
the network, along which successive (functionally) dependent variables get value assignments. Any
change in the choice (functionally) independent variables in the system, is reflected by appropriately
re-directing the network. Arc directions on the parametric network do not capture the notion of
temporal dependence in the causal sense. References [4] and [5] provide algorithms by which to direct parametric networks into directed acyclic graphs, for efficiently solving systems of simultaneous
algebraic equations.
45
Temporality in the Pneumatic Actuator
Area, a, and pressure, p, are temporally independent variables in the system. Force,
f, is temporally
relation
f
dependent on two other variables, a and p, being constrained by the
- pa = 0.
Area selection precedes pressure selection in the time history of the system, since
the actuator must actually be constructed, before any pressure is applied. Force,
f,
appears after both, a and p in temporal order, because it is constrained only after
they are selected.
Controllability in Pneumatic Actuator
Pressure, p, may be set by an uncontrollable or a controllable selector, as per our
understanding (or assumption) about the causal influence that determines it. We can
thus consider 2 distinct cases:
Case 1. In a system where the regulator has a fluttering (unstable) valve,
pressure is dictated by an uncontrollable selector, because the human
operator of the device cannot regulate it precisely. (Causal Table 3.2)
Case 2. In a system (say with a precise valve) where an experienced
operator (or automatic controller) can adjust the regulator to a desired
setting, p would be set by a controllable selector. (Causal Table 3.3)
In either case, exact value of piston area, a, is decided by stochastic variation in the
manufacturing process for pistons, which is always an uncontrollable selector.
Causal Tables for the Pneumatic Actuator
Based on the above observations, we develop two distinct causal models for the pneumatic actuator, depending on the controllability assumption for p. These models are
shown in the causal tables 3.2 and 3.3 respectively. The next subsection uses these
46
Selector
1
2
Manufacturing
Valve Flutter
Control?
Sets
Variables
No
A
a
No
P
p
Relation
J Constrains
f -pa=0
f
Table 3.2: Causal Table for Pneumatic Actuator, if p is assumed uncontrollable
Selector
Control?
Sets
1
Manufacturing
No
A
a
2
Operator
Yes
P
p
Variables
Relation
Constrains
f - pa = 0
f
Table 3.3: Causal Table for Pneumatic Actuator, if p is assumed controllable
models to formulate design constraints developing appropriate QR's to pose and solve
a pneumatic actuator design problem.
3.4.1
Causality, Design Intent and Quantifier Semantics
Once the causal effects in a system are understood, a design problem can be posed
formally (using appropriate QR's) to engineer the system. The success of the subsequent design process depends not only on how the design goals are expressed, but
also on how congruous they are, with the inherent nature of the system itself.
The parametric model represents the designer's mathematical understanding of
physical relations in the system. Similarly, its causal table represents the designer's
logical understanding of causal relations in the system. The causal table is a model of
causal behavior, in much the same way as a parametric network is a model of physical
behavior.
The semantics of the quantifier pattern in a QR, must thus obey the causal behavior of the system ' (stored in its causal table) , just as the relation G(x) is obeys
(even if only approximately) the physical behavior of the system.
The string of quantified set-membership assertions in a QR essentially represents
a nested sequence of assertions based on temporal order. Current set-based inference
9
Naturally, if the designer's perception of causality in the system is altered for some reason, a new
causal table must be prepared, and design intent must be re-examined for consistency with (perceived) causality, to re-write QR's. A stronger connection between causality and design constraint
representation is a topic for further research.
47
technology accounts controllability information, but does not use this temporality
information 10 This will be addressed by future research in this area.
3.4.2
Formulating a Quantified Relation
Thus, the use of predicate logic allows QR's to incorporate causal data, while posing design problems.
As an example consider a simple design problem using the
pneumatic actuator.
Assume that the pressure regulator is just an open-close valve, connected
directly to the compressed air tank of an electric reciprocating compressor, in a machine shop.
Assume that the compressor is monitored by an electronic pressure sensor that switches the compressor on if pressure drops below 50psi, and
shuts off electric power if tank pressure exceeds 100psi. The air tank of
such a compressor cannot always provide a constant pressure11 . Thus, a
designer is confronted by uncertainty in available pressure. The uncertainty is bounded however, and pressure is guaranteed to lie in the range
[50, 100]psi.
Given this above information, let us design the piston to actuate a
vice/clamp for a small machine shop. The clamp must close due to the
force from the piston, whenever the open-close valve is opened.
Design Requirement is that the force generated by the piston is always
between [100, 125]lb. This is necessitated, say, by floor of 1001b to ensure
proper clamping of machined objects, and a ceiling of 125lb to ensure that
the clamped items are not damaged, or marked by the vice/clamp.
Determine the largest variation that can be tolerated in parameter a
10
This is evident from subsection 3.6.3 where controllability information is indirectly used from
quantifiers, whereas temporality is ignored in the inferences. However, quantified relations incorporate temporal data, allowing future extensions to set-based design technology to utilize causal
information even more effectively, enabling stronger, more efficient design inferences.
"Such a tank slowly bleeds air into the supply line as and when any pneumatic device drawing
power from the tank is used. Pressure drops as the mass of air in the tank slowly reduces.
48
without violating the force limitation. This calculation constitutes a design decision for the vice/clamp, using the given assumptions and data.
For this problem, we shall assume that pressure, p, is set by an uncontrollable
selector, since there is no way of knowing beforehand what pressure is delivered by
the regulator at a random time.. The relevant causal model is shown in Table 3.2. We
now illustrate how to build the pattern of quantifiers to develop a quantified relation
for this design problem, guided by the causal table.
This design problem aims at finding a set of areas, that will satisfy the the design
constraint, for any variation of p within its range P = [50, 100]psi. The universal
(V) quantifier requires the constraint to be satisfied for all possible values of p E P.
The variable p, is thus quantified using a V. Likewise, a is also uncontrollable, and is
quantified by a V quantifier.
In this model,
f
is a variable that the designer wishes to control, or restrict. Here,
the designer is merely interested in ensuring that the choice of area and pressure limits
will allow the force to lie anywhere within a given range, it is not required to span
any whole range of values, but merely required to exist somewhere within a specified
interval. Force,
f,
is thus quantified by an existential (3) quantifier
The quantified relation for the given pneumatic actuator design problem will read
QR(P,A, F) : Vp E P = [50,100]
Va E A
3f E F = [100,125] - f -pa = 0 (3.2)
The quantified relation above is a statement that for all possible uncontrollable
variations in pressure supply (in the range [50, 100]psi), the area of the piston chosen
from anywhere within A, guarantees that the force output from the piston will remain
within prescribed limits (the range [100, 125]lb), in accordance with the engineering
definition of pressure. Set-based inferences are drawn from this QR, later in this
chapter.
49
3.4.3
Expressive Power of Quantified Relations
As this subsection illustrates, the quantified relation is ideally suited for formally
stating goals or constraints in engineering design. It has the following advantages:
1. Formal notation makes QR's unambiguous, and mathematically rigorous.
2. Precise syntax of QR's permits automated parsing. Thus a parser can be used
to interpret a list of QR's and construct the set based model by programmed
rules. This is important in the context of automated CAD tools.
3. A QR supports relations of many types, and provides a common framework to
integrate different constraints (forms of G(x)) under an umbrella of set-based
constraint models.
4. The use of appropriate quantifiers, and their ordering within the QR permits
inclusion of causal information. The use of causal data allows QR based inferences to succeed even where other competing techniques, like conventional
interval propagation falter.
3.5
Causal Constraint Network
After QR formulation, the set-based model for a given design problem is complete.
Analogous to the parametric constraint network, a set-based model has a Causal
Network representation associated with it. This network representation is a datastructure that enables automation of set-based design, using network-consistency
graph algorithms. The Causal Network is constructed as follows:
Step 1. With each real variable, xi, in the parametric model, associate
a closed real interval Xi. This extends the parametric model introducing
set-variables into the model. Denote each set-variable by a round node.
Step 2. Complete the set-constraint bipartite network. Represent each
QR formulated
as a rectangular node, and connect it to all the set variables
appearing in it, by undirected arcs. This step produces a set-constraint
50
network analogous to the parametric constraint network.
Step 3. Connect the set-constraint network and the parametric constraint
network by directed causal arcs. The direction of each causal arc follows
the entry for that variable in the causal table. First, all Temporally independent variable nodes get arrows directed from their associated sets.
The remaining (temporally dependent) variables then get arrows directed
towards their associated sets.
The completed set-based model is thus made up of two underlying networks. The
parametric constraint network, and the set-constraint (QR) network. The two networks are connected together by causal arcs to form a complete set-based model,
called the Causal network. Each parameter node is connected to its corresponding
set node by a directed arc (arrow), whose direction denotes the nature of the causality associated with that parameter. The causal network created for the pneumatic
actuator problem is illustrated in Figure 3-2. An appropriate network consistency
[13] algorithm based on a set-based inference mechanism can now operate on this
set-based model to draw useful information to solve the design problem.
This thesis does not explore network consistency any further detail, since it primarily concentrates on providing an inference mechanism to enable algorithms that
enforce network consistency. However, the notion of the causal network, and consistency are useful in understanding set-based design automation, and are included here
for completeness.
3.6
Inference mechanism for Set-Based Design
The previous subsections formulated the problem for set-based inference mechanism
discussed here. This inference mechanism operates on the set-based model to generate
useful new information for the design process.
The inference mechanism is based on two theorems, the Bounding Sets Theorem
(BST) and the Interval Propagation Theorem (IPT). The theorems are not stated
formally in this report, but their implications are briefly explained.
51
CausalInfluence
VariableNode
G(p, a) : f-pa = 0
Algebraic Constraint
QR(PA, F)=Vp EP Va eA 3f EF.G(p,ajf)
p-
QuantifiedRelation
ParametricConstraintNetwork
Set Node
Set (QR) ConstraintNetwork
Set-based Causal ConstraintNetwork
Figure 3-2: Simple Set-based Model for the Pneumatic Actuator
3.6.1
Bounding Sets
The BST and IPT are tools to calculate bounding sets.
Informally, A bounding set for a given set variable specifies the group of elements
a given set must at least contain, or the group of elements it is at most allowed
to contain. In these two cases, it will be called the lower bound and upper bound
respectively. The notion of bound is now explained more rigorously.
A collection (set) of real variables, A, can always" be ordered by magnitude, to
produce a permutation, of its elements, arranged in increasing order of magnitude.
Given such a collection, A, there exists a pair of real numbers, (a, a), such that:
Va c A,
a < a <i.
The elements of this pair, a and u, are called the lower bound
and the upper bound on A respectively. Naturally, these bounds are not unique, and
any pair of numbers (a', ') that satisfies a' < a and
' > z, will also serve to bound
the set A (assuming A is non-empty).
The above concepts of ordering and bounds extend from real numbers to sets.
Consider a collection (set) of set-variables, X (a set of sets).The elements of X are
ordered by inclusion [8, pp.54-58]. The sets, XI-b. and Xu-b are called the lower and
upper bounds for the set, X, if they satisfy VX E ±,
These sets, XlA
12
and Xu.
XI.b- C X C Xu.b.
are collectively referred to as bounding sets for the
By the Trichotomy Law for real numbers [9, pp. 2 0]
52
collection X. Any non-empty X has at least one, possibly infinitely many upper
bounds (because any Xub' D Xu., is also an upper bound on X.) If X contains
two or more disjoint elements, its unique lower bound is the empty set, XI.b
=
0.
However, if this is not the case, then X may have multiple, possibly infinitely many
lower bounds (because any XI b-' C XI b. is also a lower bound on X.)
The concept of bounding sets is especially useful in describing collections of sets,
k
= {X X1b < X < X"-}, where the members (X's) are not discrete entities.
For instance, when
Z
is a continuous collection of intervals, this is a convenient
representation.
To specify a bounding set for a continuous collection of intervals, it is necessary
to numerically specify limits on both, the ceiling, and the floor, for any element
(interval) in that set. Thus we must specify the largest possible ceiling, and the
smallest possible floor, to define the upper bound. Likewise we must specify the
largest possible floor and the smallest possible ceiling, to define the lower bound.
Thus, in dealing with closed intervals, each bounding set computation involves two
distinct point calculations.
The BST and IPT operate on a quantified relation QR(X), supporting an inference
about a variable Xp appearing in its argument X. The BST establishes sufficient
conditions for a candidate X* to be a bound on the set Xp containing feasible x,'s
which satisfy the QR. The IPT uses the BST conditions to actually calculate such a
bound, performing the two point calculations that evaluate endpoints of X*.
The BST is a general theorem, applicable to all types of sets and relations in QR's.
It works even for discrete sets, continuous sets with more than one dimension etc.
The IPT is a computational tool based on the BST, for
(a) a specific class of sets, namely closed real intervals, and
(b) a specific class of relations, namely continuous, asymptote-free algebraic equations,
that are strictly monotonic in every variable.
53
3.6.2
Bounding Sets Theorem
The BST tells us what kind of bounding sets can be inferred using a given QR. Given
a quantified relation QR(X), and a domain of interest X, from which the elements of
the set-vector X take on assignments,
1. BST explains what can be inferred about the membership of a set X, appearing
in the vector X if all the other elements in X have definite set bounds on them.
2. BST provides a sufficient condition for any suggested bound X* on the set Xjr
(Xp C Xp) to be a lower or upper bound on all elements in Xi, that satisfy the
QR.
The statement of the BST uses a partitioning of the set-vector QR argument X
X
(3.3)
= (XP, Xy, X3)
Xp is the set on which a bound is inferred.
Xy is the set-vector, whose elements are universally quantified in Q(X).
XE is the set-vector, whose elements are existentially quantified in Q(X).
The superscript 1.b. indicates lower bound and u.b. indicates upper bound.
Thus, if a variable Xi is universally quantified (i
$
p),the notation X
would
indicate that a lower bound on all possible instantiations for Xi is substituted in the
QR while
making an inference about X,.
Thus, if a variable Xi is existentially quantified (i 4 p), the notation X1 would
indicate that an upper bound on all possible instantiations for Xi is substituted in
the QR while making an inference about Xp.
The same notation used when upper bounds are substituted in the QR. The notation is extended to entire vectors by adding the superscripts onto the partitions Xv
and X3 of the vector X, indicating the set assignments that are to be substituted for
each argument. With this notation, the BST's implications are stated below:
54
"Universal Quantifier -> Infer an Upper bound"
If the quantifier on x, is universal (Vxp E X,), BST directs us to infer that:
a suggested upper bound X* is indeed an upper bound on all feasible X, E X if it is
an upper bound for all X, satisfying QR(Xp, XVb.,
"Existential Quantifier
=
X
.
Infer a Lower bound"
If the quantifier on x, is existential (3x, E Xp), BST directs us to infer that:
a suggested lower bound X* is indeed a lower bound on all feasible X, E X if it is a
lower bound for all X, satisfying QR(Xp, Xb., X.b)
This theorem tells us nothing about how to come up with an X* that can be tested
as a bounding set by BST conditions. It merely tells us what conditions are sufficient
to guarantee that the suggested bound is indeed a bound. The IPT computes useful
X* candidates (guided by the BST) in a special context explained below.
3.6.3
Interval Propagation Theorem
The BST provides a sufficient condition for an interval to be a bounding set. The
IPT proposes a method to calculate such a bounding set, for a specific class of problems. The IPT uses the BST, but restricts the inferences to QR's having continuous
equations of the form G(x) : g(x) = 0 and closed real intervals. The equations must
also be strictly monotonic and free of asymptotes over the region of interest.
Suppose an inference is needed about bounds on a variable x, in the design vector
x. The equation g(x) = 0 is solved for this variable to get the explicit form xP =
gp(xv,
X3),
where the arguments of the right hand side are elements of the vector x
partitioned by quantifier.
Partitioning the elements further based on their monotonicity in the equation
g(x), w.r.t. x, (which can be assumed to have positive monotonicity1 3 IPT calculates
the bounding set X* on xP by evaluating the formula gp at certain endpoints of the
'3 If x, does not have positive monotonicity in the relation g(x), then it can always be made
positively monotonic, by multiplying as -1 x g(x) = 0, leaving the relation unchanged
55
intervals Xi. If xi E Xj, the formula will substitute either the ceiling -if or the floor
xi of Xi = [xi, T] according to the rules prescribed in the following cases.
Case 1
qp = V. Let
X* = [gp(x
,I R
,
X
-,x), gp(R +, x
3,x
1
i)
34
If this computed interval X* is non-empty, then it is an upper bound for all X,
satisfying QR(X).
Case 2
qp = -. Let
X* = [)
g(x
R +
(3.5)
If this computed interval X* is non-empty, then it is an lower bound for all X,
satisfying QR(X).
The IPT uses both monotonicity data as well as quantifier symbols to assign the
interval endpoints, as can be seen from the formulae.
Traditional interval propa-
gation uses similar interval calculation, but incorporates no quantifiers, thus losing
whatever causal information set-based computations include. An example in the next
section illustrates how the inclusion of quantifier (causal) data is vital to make reliable
bounding set inferences.
The two theorems described so far, together constitute a BST-IPT inference mechanism that operates on set-based models to deduce useful information about bounds
on the system variables characterize feasible designs.
The theorems are used to-
gether as a core for the Set EliminationAlgorithm. Set-based inferences based on the
BST-IPT mechanism compute bounding intervals by eliminating parts of the design
space 4. that are guaranteed to violate a given QR. The algorithm uses these inferences to operate on the causal network, altering the set-variable domains repeatedly,
14The
design space is a collection of points as explained earlier in this chapter. Associating sets
with the design variables permits us to aggregate points, grouping portions of the design space into
sets. Thus, IPT operates on the design space, treating it as a set of sets, rather than merely a large
set of points.
56
to satisfy each QR in the system, cycling through the QR's until it terminates with a
set of consistent residual intervals, an upper bound on the feasible design space. This
algorithm, its procedures and variants are explained and in detail in [1].
3.7
An Example BST-IPT Design Inference
In the example presented, there is only a single QR, and the set elimination algorithm
is unnecessary, since we compute a bounding set for the variable A by simple calculation. For this computation, the relevant IPT formula is used, along with relative
monotonicities obtained by treating a as a positively monotonic variable. Consider
the QR developed earlier in subsection 3.4.1.
QR(A, P,F) : Va E A Vp E P=[50,100]
3f E F = [100, 125] - G(p, a, f)
G(a, p, f) : g(a, p, f) = f - pa = 0
Rewriting the expression G(a, p, f) g(a, p, f) =
f-
pa = 0 so as to make g increase
with increasing values of a, we note that g is positively monotonic in a and p, and
negatively monotonic in
f.
Thus we have
g (p+, a+, f-)
Solving g(a,p, f) = 0 for a, we get
a = ga(f,p) =
p
This formula will be used to compute the bounding set for A.
The values to be
substituted in it are decided by quantifiers in the QR.
The quantifier on a in the QR is a V. By Case 1 of the preceding IPT statement,
the IPT can infer an upper bound on interval A. The inferred upper bound will
exclude any infeasible sets in A, and satisfy the design intent in the QR.
Values of variables appearing in gp are assigned using the formula in Equation 3.4,
based on quantifiers in QR(A, P,F) and monotonicities in g(a+, p+, f-). The upper
57
bound is thus calculated as follows:
Aub.
[g(
-
- ),(ga (P
A
[9a
_V
f
-
-3100'5
125 100
1
50
[_/
=
[1.25, 2] sq.inch
(3.6)
This result is a non-empty interval. The inference implies that any piston area outside
the range [1.25,2] sq.inch, is provably infeasible, and must be rejected from further
consideration in the process of developing a feasible actuator.
The inference mechanism does not guarantee that every value in the computed
valid interval will work 15 . It merely guarantees that everything outside the computed
set will fail. This is why the system is called a set-elimination method.
Note that conventional interval propagation produces a wrong answer in this case:
A= [
f
--
ga(P
)]
[V
/p]
[
100 125
-I
] = [1, 2.5] sq.inch (3.7)
100' 50
The computed interval is wrong because a pressure of 100 psi on the computed maximum feasible area 2.5 sq.inch will produce a force of 2501b, which clearly violates the
125 lb limitation. Interval propagation fails because it ignores the causality implicit
in the statement of the design problem.
The example concludes the chapter on set-based mathematics. The remaining
chapters of the report extend these ideas and demonstrate their application to the
actual power window system itself.
15
In this simple problem it is easy to see that this stronger condition is also true, but it should
not be assumed for a general QR based inference
58
Chapter 4
Set-Based Model of the Cable
Drum Power Window System
It can be shown that a mathematical web of some kind can be
woven about any universe containing several objects. The fact
that our universe lends itself to a mathematical treatment
is not a fact of any great philosophical significance.
-Bertrand Russel (1872-1970)
This chapter applies set-based mathematics and modeling concepts from the previous chapter, to develop a relevant set-based model for the Cable Drum power Window
System. It first constructs the parametric network and causal table for the system.
Then it uses these representations to develop Quantified Relations for the design
goals explained in the Cable Drum Power Window System description (Chapter 2).
The problem formulation completed here, is used in the following chapters to draw
set-based inferences about the power window system.
4.1
Electro-Mechanical System Model
This section develops a parametric model of the cable drum power window system.
The model combines physical and geometric relations governing various components
of the power window system, representing them as a system of inter-related alge-
59
braic equations.
The Advanced Vehicle Technology Center at Ford provided an
MS ExcellTM spreadsheet, containing the design-analysis model, currently used by
Ford engineers. The parametric model detailed in this section was built and tested at
MIT, then compared with the spreadsheet to clarify various modeling issues. Some
simplifying assumptions assist the modeling effort are described briefly, before presenting major components of the parametric, causal and network models.
4.1.1
Complexity and Simplifying Assumptions
The nomenclature included at the outset of this report lists the model variables
relevant to the power window system being designed. The power window system has
35 variables. It's variables and relations are grouped into two divisions (i) Motor
Characterization Experiment, and (ii) Electro-mechanical System.
The first division of the system, i.e. Motor Characterization Experiment, involves
some additional variables (not listed in the nomenclature, but described separately
in the Appendix A), that have been excluded to simplify analysis.
The reason for this exclusion is that existing causal representation is inadequate
to denote the causality within the motor characterization experiment. Thus, a causal
network (such as the one described in Subsection 3.5) currently impossible to construct for the motor characterization experiment. The theory of causality capture
must further be extended before the experiment can be included in the analysis.
To get around this problem, the motor constants are assumed known, and the
causality in the motor experiment is omitted from the set-based model. This thesis
concentrates on directly applying set-based tools to the electro-mechanical system
relations.
Despite this simplification, the parametric model of the cable drum power window
system is still quite large in comparison to problems addressed successfully by setbased methods in the past, (in fact, the largest to date). It contains 31 variables
related by a simultaneous system of 14 coupled non-linear algebraic equations.
60
4.1.2
Parametric Model Relations
The parametric model of the electro-mechanical system is a system of simultaneous
algebraic equations. The spreadsheet from Ford decomposes this system of simultaneous equations to handle them sequentially. The same practice is adopted in this
subsection also. Values for many system variables are calculated as temporally dependent output variables by solving a set of equations, taken one at a time, to evaluate
one unknown variable from each calculation.
Parameterizing Glass Movement.
The simultaneous algebraic equations that
model the power window system are always satisfied at a particular value of a parameter x, which represents window position. This parameter is moved within the
interval [0,1] to simulate the motion of the glass. Appendix B documents the drag
force and load torque equations satisfied by the system at every position of the window
glass. The parametric model calculates all temporally dependent system parameters
at a fixed window position, and is stepped through various positions, repeating these
calculations, to simulate the system behavior as the glass is raised or lowered. The
following paragraphs walk the reader through the sequence of algebraic equations
solved at each window position. These equations constitute the power window parametric model.
Glass-Seal Engagement Lengths -+ Drag Force. At any given value of x, we
first determine the engagement lengths of the different glass edges along the door
pillars, and along the beltline, at that position, x. The equations used for this en1
In this method, the constraint set {G 1 (x), G2 (x),... Gm (x)} is ordered in a specific sequence,
with each Gi rewritten to bring a single temporally dependent variable to its R.H.S., leaving only
priorly computed variables in its L.H.S. Thus, each equation computes precisely one output, using
previously known data. The solution proceeds in stages, with values "flowing" from one equation to
the next, till the constraint network is fully satisfied.
A more formal graph theoretic treatment of decomposition is found in [4] and [5]), where an
algorithm assigns directions to the (initially undirected) arcs of the parametric network to produce a
directed acyclic graph indicating a possible sequence of computations that satisfies the simultaneous
system, without actually having to solve them all simultaneously. In this thesis, the sequential
decomposition is readily available, (from the causal table), where the system relations are already
ordered sequentially.
61
gagement length calculation are shown in Appendix B (B.1). Understanding this
calculation also requires a detailed diagram of the window geometry, included in
Appendix B. The numerical specification of window geometry, including all the pillar lengths, seal lengths, and relevant angles, is stored in a vector of parameters 1
(data in Table A.3 of this section). This data is used to determine the engagement
lengths of the glass above and below the beltline in the A and B pillars, and along
the belt itself. The engagement equations return a vector of engagement lengths
1Aa(1,
x), lBa(1i X) lAb (, x), lBb(1, x) and lbelt(1, x), determined as functions of the win-
dow geometry, and glass position. Using these engagement lengths, we determine the
total drag force fdrag(x) at a particular window position, x, by multiplying engagement lengths with appropriate drag coefficients P. The above computations satisfy
the relation labeled G 6 .
G6
:
fdrag
= lAa 6 Aa
+ 1Ab 6 Ab + lBa 6 Ba +
1
BbBb
+ ibelt 6 belt
Drag Force -+ Motor Shaft Load. The aggregated force
fdrag,
(4.1)
from the seals
and the belt, combined with glass weight and frictional losses, appears as tension in
the cable of the cable-drum mechanism. This tension is the load ', fmotor, seen by
the motor shaft. It causes a torque Tmotor on the shaft4 , computed by using drum
diameter, d.
G 7 : fmotor
-
fdrag ± W
+ foss
(4.2)
ridrive
G
: Tmotor -
fmotor.
d
2
(4.3)
Motor Shaft Load -+ Current. Once the motor shaft load is known, we determine
the current in the circuit, corresponding to the instantaneous load torque, by using
2
see Equation B.7 in Appendix B, for more detail on drag force.
Equation B.8 in Appendix B for a detailed explanation of the linear force model for fmotor-.
4
See Equation B.9 in Appendix B for explanation of motor torque
3See
62
this equation from the DC motor Model.
G9 : i
(4.4)
= if + Tmotor/AlT
G9 relates current and torque by a linear model, using the characteristic constants
if, and AIT of the DC motor5
Current -+ Motor Shaft Speed. The current in the circuit is used in the relations
below, to determine the motor shaft speed', w (using the same DC Motor Model, and
also information about the battery voltage, line resistance and motor characteristics)
and thence the cable/glass velocity 7 .
G1 0
= Vbattery -
i(rmotor
kb
Gl :
Motor Stall Torque -+
+ ruine)
Vglass = W.-
d
2
(4.5)
(4.6)
Window Stall Force. Stall torque is the maximum torque
that the given DC motor can develop, for a particular choice of battery voltage and
line resistance. It is determined by the relation'.
G 1 2 :Tstall =
kt.(
Vbattery
rmotor
+ rine
-
if)
(4.7)
Stall torque determines the force needed to stall the glass (see Equation C.4 in Ap5See Appendix A for a detailed discussion of the Linear DC Motor Model. This calculation
Vba",.
. A motor-battery combination that
assumes that current remains bounded as i <
violates the constraint will not be able to provide any positive torque shaft, and cannot raise any
load.
6
This step assumes that speed is always positive. This assumption is valid as long as the previous
assumption about boundedness of current holds.
7See Equation C.1, under Performance metric calculations explained in Appendix C. This is a
simple application of a no-slip condition between the cable and the drum.
'This calculation assumes that appropriate values of Vbattery, rmotor and rTie are used, whereby
torque will evaluate to a positive number. The assumption is actually equivalent to the prior
Vbait,
, to ensure
assumption that the motor-battery combination should always satisfy if <
that shaft torque is always positive.
63
Temporally
Dependent
Variable
Constrained by
the Relation
Priorly Computed
Dependent
Variables
fdrag
G 6 (6, 1, X, fdrag)
lAa(1, x), lAb(1, x),
lBa (1, x), lBb(1, x)
Refer
Appendices
B and C
eq B.7
X)
______________lbelt(li
fmotor
G7(ldrive, floss, W, fdrag, fmotor)
fdrag
eq B.8
Tmotor
G 8 (d, fmotor, Tmotor)
fmotor
eq B.9
i
G 9 (if, AIT, T, i)
Tmotor, AIT(kT)
W
G1o(Vbattery, rine, rmotor, kb, i, )
Vglass
G11(d,wvglass)
rmotor, kb, i
W
Tstaul
G12(Vbattery, rine, if, rmotor, ATT Tstai)
rmotor, AIT(kT)
fstali
G13(7?drive,
rImotor
G14(Vbattery i,iW, Tmotor,,7motor )
eq A.4
eq 4.6
eq C1
eq A.9
eq C4
eq C.5
d, foss, w,
fdrag, Tstall, fstali)
W, fdrag, TstaII
'i, W, Tmotor
Table 4.1: Relations that model the Electro-mechanical System
pendix B) at the instantaneous window position, x.
G1 3 : fstaii = Tidrive
2Td
)ll
fdrag
-
w
(4.8)
-
DC Motor Efficiency. Dividing the mechanical output of the motor
(T
x w), by
electrical input energy (i x v), we compute the instantaneous motor efficiency9,
rmotor-
Table 4.1 summarizes the relations, the flow of values in the sequential computations, and indicates where the relevant theory is found in the Appendices. Table A.3
contains sample data propagated through the parametric system in calculations for
this thesis.
The tables presented in this subsection list system equations, dependencies, variables and their associated sets, with sample data. The equation numbers in the tables
where they are referenced are identical to the numbering in the appendices and graph
representations.
Parameter values from Table A.2 are combined to produce the motor constants as
9
See Appendix C for the relevant Equation C.5. This variable is included in the parametric and
network models, but no actual design constraints are posed on it in this thesis. It is included here
for completeness.
64
per constraint relations given in Table A. 1. When the motor constants are determined
from an experiment, the behavior of the electro-mechanical system can be simulated,
using motor constants, and additional independent parameters shown in Table A.3,
to satisfy the relations in Table 4.1".
4.2
Constraint Network and Causal Table
The previous section briefly sketched the model preparation process. This section
represents the parametric model as a constraint network, and then develops a causal
table from the relations described in this chapter.
The parametric constraint network is constructed from the relations and their
dependencies documented in Tables A.2 through 4.1. This network is similar to the
one built for the simple example, the pneumatic actuator in Figure 3-1 in subsection
3.5.
Figure 4-1 shows the constraint network for the Cable Drum power window
system. 1
Using the Tables A.2, A.1, A.3 and 4.1, the temporal ordering of the variables, and
their dependencies are easily visualized. The causal table for the electro-mechanical
system is compiled by including controllability information with this temporal order
to produce Table 4.2.
This section has described the major elements of a set-based model for the power
window system. The causal table will guide the development of the remainder of the
set-based model, i.e. the quantified relations. The next section explains QR construction for the power window design problem, and completes the problem formulation.
10Tables A.1 and 4.1 list each relation in implicit
form, measured variables first, followed by
previously solved variables (derived from preceding relations) and finally the unknown determined
from each equation.
"The diagram also has a few comments with its relation nodes, to indicate the nature of the
constraint relation, and an appropriate equation wherever possible. Note that it includes the motor
characterization experiment relations. These are excluded from further analysis, but shown here to
illustrate how the motor characteristics are evaluated.
65
W
+'rpafIoss
+4a n dr7*l,
ftall:
'Wh~l,= (fdra+
x
G7
-1;.Png ;f""qzt: onm r
sbe'-i
d
G8
motOr ~ m14";
-1
=w 4/
motor
r
G 4
M Res;1s"'1na
a? sua!",
Netwrk Rpresntaton o
o
G io
14gis spovnglasede
G
if
kI|Gtlis
G3
Gi
mdo-,qesue
exp
-
T
=
2+
Stall
-
t2
(2
-
tI
G5
p, w - ()2)
F dJ;t nce
bi r est'
2
G2
Network Representation of
h.,
Soivf-: ;"
o l
0niwo -ira
foir u
+ r)
nm
e n,
(fj)' ,
the Moving Glass Model
(01/25/99)
G14
-
e.-pi -1
2),;i2 -
Ioo
<V
n'itor
=
Figure 4-1: Parametric Constraint Network of Cable Drum Power Window System
Iinc
T@v
Selector
Control?
Sets
Variables
1
2
Mfg.
Env.
N
N
L
1
{A}
{6}
3
Program
N
X
x
4
5
6
Mfg.
Mfg.
Mfg.
N
N
N
HDrive
Floss
W
7
Relation
G6 :
Jili
fdrag
7
Mfg.
N
D
Constrains
fdrag
ldrive
floss
w
d
8
9
Mfg.(Expt.)
Mfg.(Expt.)
N
N
if
if
KT
kT
10
Env.
Mfg.(Env.?)
N
N
Vattery
Vbattery
11
12
Mfg.(Expt.)
N
Rline
KB
rnine
kb
G7:
fmotor
-flos
fdrag
W
f m otor
T = fmotor .
Tmotor
G8:
G:
Gio:
_i(rmotor+rtine)
kb
G:
_V
13
Mfg.(Expt.)
N
Rmotor
rmotor
=W
Vglass
G1:
kT
rmotortr
Tstall
c
G1 3 :
fstau + fdrag
+w + floss
=
Idrive.(
2
d "'')
G14:
77motor =
ha
L""|T.t
fstall
motor
Table 4.2: Causal Table for Electro-mechanical Model of Power Window System
67
4.3
Formulation of Quantified Relations
Chapter 2 highlighted the major performance specifications to be fulfilled by the power
window design process. This section casts the specifications in formal notation, as
quantified relations. It completes the set-based cable drum power window model.
Two quantified relations developed below constrain stall force and glass velocity. The
construction of these QR's is guided by the causality noted in Table 4.2.
4.3.1
Choice of Quantifiers
We briefly state the rationale behind our choice of quantifiers, before presenting the
quantified relations as design constraints on the power window system.
Universal Quantification. All uncontrollable selectors listed in the causal table
4.2 have to be countered, in order to constrain the performance metrics within the
prescribed limits. Thus, temporally independent variables that are governed by uncontrollable selectors, have been universally quantified, since their selectors might
place them anywhere within their interval domains, and the robust design will have
to ensure that the system parametric relations are satisfied for all possible values
arising from uncontrolled variations. This approach makes the design robust against
manufacturing variations in the motor characteristics, line resistance, regulator losses,
glass weight, seal drag, and component dimensions. The use of the V quantifier also
specifies a design robust to environmental variations that alter battery voltage, line
resistance, and seal drag. The variable x is actually controlled by the program that
simulates the window motion, but it is universally quantified because each constraint
(one on stall force, one on window velocity) must hold at any window position.
Existential Quantification. In each quantified relation developed in this section,
the last parameter (invariably a performance metric) is quantified existentially. To
the designer, this quantification expresses the intention that the parameter should
be bounded within specified limits. It does not matter which exact value it assumes
within that bounding interval, as long as it lies somewhere between the bounds 2 .
12
There is also a mathematical explanation instead of an intuitive design reason for the existential
68
4.3.2
Glass Velocity Quantified Relation
A quantified relation that requires the glass velocity to be held within [0.125,0.175]ms
1
robust to manufacturing and environmental variations in the electro-mechanical system components is as follows:
QR 1 (Vattery,
Vbattery C
Rline,
Rmotor, Kb, KT, If, Hdrive, FSS, D, W, A, L, X, V1)
vbattery Vriine C Rine Vrmotor C Rmotor Vkb C Kb VkT C KT Vif E If
V7Jdrive C Hdrive Vfloss C Foss Vd E D Vw C W V
1
]v9 E V = [0.125,0.175]ms
6c A V c L Vx c X
- F,
where I 1 is the system of simultaneous equations (F 1 : G6 A G7 A G8 A G9 A Gio A Gil)
G6(fdrag,,l ,X)
A G7(fmotor, floss, fdrag, W, irive)
A G8 (7Tmotor , fmotor , d)
A
A
fdrag
:
fmotor -
:
Tmotor - fmotor 2 =
-
:i-if
G9(if, T, kT)
GlO(Vbattery,
rmotor, rine, kbw)
A i(Vatey7'oorrin~b7)
A Gi(vg,w,d)
E 6 ili
:
:
7dr ive
-
W
0
fdrag±W-
s =
0
0
=0
Vbattery
kb
: V
=
i(rmotor+riine)
kb
=
0
=0
In natural language,
For all uncontrollable variations in the electrical parameters, motor characteristics, mechanical parameters (losses, friction effects etc.), the glass
velocity vg, must exist within the interval V = [0.125, 0.175]ms
1
, and
satisfy the system of equations F1 , characterizing geometric and physical
behaviour of the cable-drum power window system.
The quantified relation is further simplified before actually attempting to use it
for design inferences. Set-based inferences included in this report replace the string
quantifier. Both the relations contain algebraic equalities as their predicates. It is the equality
relation that directs the last variable to be existentially quantified. To understand this, one must
remember that according to the temporal ordering indicated by the pattern of quantifiers, prior value
assignments to all other system variables appearing in the quantified relation will leave only a single
degree of freedom before the last variable is instantiated. If this variable is quantified universally,
then it will have to span a whole range of values (specified as its range of variation) to satisfy the
quantified relation. But since it is a real variable constrained by the embedded equality relation, it
can assume precisely one value, and cannot span an interval. The last variable in each quantified
relation is therefore quantified by a 3 symbol.
69
,
VJEA VIEL
VXEX
by a single expression:
Vfdrag E Fdrag
For a given window geometry (1) and drag coefficient vector (J), relations in Appendix B determine the endpoints of interval
Fdrag.
The simplification suggested here
bypasses the relation G6 , directly providing interval limits on fdrag. It also avoids the
need to use the complicated glass engagement equations in any symbolic manipulations or numerical computations with the systems 1
The drag force variation causes uncertainty in glass velocity and stall force. By
quantifying fdrag instead of all the variables in (x, 1, 6) we can now use set-based
tools to draw inferences about the interval Fdrag and then use the fdrag relations'
to
calculate appropriate drag coefficients for the seals and belt. Using this simplification,
the glass velocity quantified relation is rewritten as shown below.
QR1(Vattery, Rline, Rmotor, Kb, KT, If, Hdrive, Floss, D, Wi, Fdrag, V)
VVbattery E Vbattery Vriine E Rline Vrmotor E Rmotor Vkb E Kb
VkT c KT Vif C If
V7ldrive E Hdrive Vf 108 s E Floss Vd E D Vw E W Vfdrag E Fdrag
3v 9 E V = [0.125,0.175]ms- 1 - ri
where F1 is the system of simultaneous equations (F1 : G7 A G8 A G9 A G10 A Gil)
A
G7(fmotor, fiss, fdrag, W, 7/drive)
:
fmotor -
A G 8 (Tmotor, fmotor , d)
:
Tmotor - fmotor
A
A
:i-if
G 9 (ifT,,kT)
GO(Vattery,
rmotor, rine,
kb, w)
A i(batrjmoo~lnek))
:
A Gii(v 9 ,wd)
:
fdrag"W
77drive
-
w=
=
0
= 0
=0
"Vbattery
kb
V-
a
i(rmotor+rine)
kb
=
0
0
The above QR has an embedded system of 5 simultaneous algebraic equations
that constrain 16 variables. At least 11 variables must be instantiated to perform
"The limits on fdrag are computed (Appendix B) by multiplying glass engagements with drag
coefficients: fdrag = lAa05 1 + lBa3 2 + 1Ab 6 3 + 1Bb6 4 + lbelt 6 belt
70
any useful computation with this syetsm, and the QR quantifies exactly 12 variables,
making it theoretically possible to specify values for 11 quantified variables and satisfy
171 to numerically determine the 12th variable.
4.3.3
Stall Force Quantified Relation
A quantified relation that requires the stall force to be held within [100,250]N, robust to manufacturing and environmental variations in the electromechanical system
components is as follows:
QR2(Vbattery, Rine, Rmotor, KT, If, Hdrive, Foss, D, W, A, L, X, Fstaii)
Vbattery E Vbattery Vrizne
V77drive G Hdrive
E
Rline
Vrmotor E Rmotor VkT E
Vif E If
KT
Vf 1 ss E Floss Vd E D Vw c W V6 E A V E L Vx c X
3fstay
E Fstaii
=
[100, 250]N - 172
where 172 is the system of simultaneous equations (172 : G 6 A G 12 A G13)
6
G6(fdrag,,1,X)
:
fdrag
-
Vbattery, rmotor, rine, if)
:
Tstall
-
k.( rmotor~rline
batter.
A G13(fstal, fdrag, W, 7Tdrive, Tstall, fioss)
:
fstaii
+
fdrag + W
A
G 12 (Tstaii,
kt,
ili
=
0
-
-
floss
if) = 0
-
7drive.(2j"n)
_
In natural language,
For all uncontrollable variations in the electrical parameters, motor characteristics, mechanical parameters (losses, friction effects etc.), the stall
force,
fstaii,
should exist within the interval Fstaii = [100, 250]N, and sat-
isfy the system of equations,
172,
that characterises geometric and physical
behaviour of the cable-drum power window system.
Using the same simplification adopted to remove variables and equations from
the glass velocity QR, we quantify fdrag instead of all the variables in (x, 1, J). We
can now use set-based tools to draw inferences about the interval
Fdrag
and then use
the fdrag values from this inference to determine seal drag coefficients appropriately.
Using this simplification, the glass velocity quantified relation is rewritten as shown
below.
71
QR2(Vbattery, Rline, Rmotor, KT, If, Hdrive, FIOSS,
VVbattery E Vbattery Vrline C Rine Vrmotor E
D, W, Fdrag, Ftaii) :
Rmotor VkT C KT Vif
G
if
V'7drive C Hdrive Vffoss G Foss Vd E D Vw C W Vfdrag C Fdrag
3fsta EEFstai = [100, 250]N - F2
where 1 2 is the system of simultaneous equations (172 : G 1 2 A G 1 3 )
A
G12(Tstall, kt, Vbattery, rmotor, rine, if)
Tstall -
G13(fstall, fdragiW, rrive,
fstall
TstalIl, floss)
k
t.(mt"
+ fdrag +
_ -
W -
if)
floss -
=
0
'7drive.(2T
all)
=
The above QR has an embedded system of 2 simultaneous algebraic equations
that constrain 12 variables. At least 10 variables must be instantiated to perform
any useful computation with this system, and the QR quantifies exactly 11 variables,
making it theoretically possible to specify values for 10 quantified variables and satisfy
172
to numerically determine the 12th variable.
The next chapter draws set-based inferences by operating on the stall force and
glass velocity quantified relations that have been set up in the preceding sections.
72
0
Chapter 5
Set-Based Inferences for Power
Window Design
In symbols one observes an advantage in discovery which is greatest
when they express the exact nature of a thing briefly and, as it were,
picture it; then indeed the labor of thought is wonderfully diminished.
-Gottfried
Wilhelm Leibniz (1749-1827)
The set-based constraint model for the power window system, with appropriate
simplifications, stands completed in the previous chapter. This chapter applies the
BST-IPT inference mechanism (from Chapter 3), to this constraint model, generating
set-based design inferences.
The cable drum power window system is representative of a larger class of design
problems. Such problems are exemplified by the quantified relations (subsection 6.3.2)
and parametric model in Chapter 4. These problems possess set-based constraints
that cannot be satisfied by direct application of currently available set-based tools.
This chapter identifies important inadequacies of existing set-based tools, in an
effort to explain this apparent shortcoming. It demonstrates a symbolic manipulation
technique to overcome the limitation, and presents example calculations of design
inferences made using the suggested algebraic technique.
It finally advocates a theoretical extension to the existing BST-IPT mechanism
to enhance its applicability. The theoretical extension relies on a numerical approach
73
that is expected to be computationally less expensive than symbolic manipulation.
5.1
Limitations of the Existing Mechanism
This section explores the applicability of the existing BST-IPT inference mechanism
to the power window design problem. It first compares QR formulations from the
preceding chapter with the symbolic pattern assumed (supported) by the existing
IPT calculation technique. This comparison determines that the new formulations
are incompatible with the existing capability of IPT. Two alternative methods are
therefore suggested, to address the observed incompatibility. Implementation issues
relating to these two methods are then discussed briefly.
The BST-IPT Inference Mechanism (Chapter 3). can be applied to a quantified
relation, as described in [1] only when the quantified relation has a form that satisfies
these conditions:
1. The embedded relation, G(x), must only comprise a single algebraic equation,
G(x) : g(x) = 0, that is continuous, strictly monotonic in every one of its n
variables, and free of asymptotes [1].
2. The quantified relation QR(X) must quantify every one of the n variables constrained by G(x), so that n -1
of them can be instantiated to draw an inference
about the nth variable, through a computation procedure that satisfies G(x).
However, the Glass Velocity and Stall Force quantified relations of Chapter 4 are
not of the exact symbolic form as specified above. In the quantified relations listed
in subsection 6.3.2, the following features are noted.
1. Each quantified relation has 17(x), a system of simultaneous algebraic equations
as the embedded relation, instead of the single equation G(x) as prescribed by
the IPT.
2. Each quantified relation, quantifies only some (say q) variables of the n system
variables (q < n). In the Glass Velocity QR, 12 variables are quantified from
74
amongst the 16 variables in the relevant portion of the parametric model. In
the Stall Force QR, 11 variables are quantified from amongst the 12 variables
in the relevant portion of the parametric model.
Thus, the BST-IPT inference Mechanism from Chapter 4 is not directly applicable to
the quantified relations for the power window system, because they do not conform
to the symbolic pattern specified by the existing IPT.
5.2
Methods to Enhance Applicability
The previous section determines that the currently available IPT is inapplicable to
power window design. Two possible approaches to overcome the limitation are as
follows.
1. Symbolic Algebra can help us modify the quantified relations, altering their
symbolic form to make them compatible with IPT. This has to be done in a
manner that preserves the semantics (design intent) of the quantified relations.
2. Numerical Methods can help us modify the Interval Propagation Theorem,
extending its reasoning appropriately, to address these apparently intractable
quantified relations. This approach leaves the quantified relations unchanged,
and increases the scope of the BST-IPT inference mechanism.
The following subsections analyze each of these approaches in more detail, explaining briefly, how they will be implemented. They also present an illustrative example
to explain the issues.
5.2.1
Symbolic Elimination of Intermediate Variables
One way of applying the existing IPT to a multi-equation QR, is symbolic elimination of variables to reduce its embedded multi-equation constraint, F(x), into a
single equation, G(x) : y(x)
=
0.
To do this, all variables that do not appear in the quantifier pattern, must be
eliminated by algebraic manipulation. This reduces the system of equations, F, into
75
a single equation, -/. The elimination process must also ensure that the reduced
equation, -y(x), explicitly contains all of the quantified variables. If such an elimination
is indeed possible, the IPT can be used for inferences about feasible sets satisfying
QR(X), using the equation1 -y(x).
Thus, symbolic elimination enables use of the existing inference mechanism without any (potentially difficult) extensions or modifications to the BST-IPT theory
itself. This theoretical simplicity comes at the cost of computational complexity incurred in the elimination process.
Programs like MAPLE and MathematicaTM use a symbolic computation paradigm
to manipulate algebraic equations. A quantified relation parser program, interfaced
with such a symbolic math package, can reduce expressions with complex systems
of embedded equations into appropriate (single equation) quantified relations that
the current IPT can address. However, if the system of equations in a QR is large
and complex, this method is unlikely to be a computationally efficient solution to the
problem, and is even prone to fail.
5.2.2
Numerical Solution of a System of Equations
An alternative to symbolic elimination is a numerical approach to solving the system
of equations F(x), by appropriately assigning values to some 2 be of its variables. This
process of instantiating variables must be governed by underlying logic of the existing
BST-IPT mechanism.
IPT relies on monotonicities of variables within an algebraic expression.
This
concept of monotonicity is extended to the coupled system, F, using partial derivatives
of appropriate variables3 Such an analysis allows us to attach a notion of monotonicity
the to system F, similar to monotonicity in algebraic expressions ([1], [11]).
'IPT is applicable if the reduced equation, -y, is continuous, strictly monotonic asymptote free.
If the system F(x) contains n variables related by m simultaneous equations (m < n), then
exactly k = n - m variables are instantiated, to determine m unknowns through a numerical solution
procedure.
3
The theoretical details of how this monotonicity information is captured and used, is the topic
of the next chapter.
2
76
The interval endpoint calculation rule from the existing IPT is extended to direct
interval endpoint assignments utilizing this monotonicity information about F. If an
inference is needed about X, quantified in QR(X), the logic of IPT is used first,
to assign values to (instantiate) all variables in x, except x, (see subsection 3.6.3).
To actually compute the bound (interval endpoints of X*), a program that solves
systems of simultaneous equations is invoked with the instantiation, to satisfy the
relation F(x). This determines the endpoints of the bound on X,.
This computation generates the desired inference 4 . This numerical procedure will
not only compute a set-bound on Xp, but also evaluates the intermediate variables appearing in F(x) (those that were eliminated symbolically in the previous subsection),
but not quantified in QR(X). These non-quantified variables do not affect satisfaction
of the QR, and are irrelevant.
There are numerous commercially available and public domain software packages
that accomplish system solution by Newton-Rhaphson iteration, or Constrained Optimization methods. These programs are quite robust, even when the number of
equations involved is large. Adopting a numerical approach avoids the need for complex manipulations to symbolically reduce equations. The potential drawbacks of this
method are:
1. It necessitates a procedure to infer relative monotonicity data from the coupled
equations F.
2. It also requires a theoretical extension to the BST-IPT mechanism, incorporating IPT logic into a formal method for appropriately instantiating parameters
in F(x) in a set based inference.
4
Here, x, maybe temporally dependent, or temporally independent. If xP is temporally dependent, the parametric constraint network, is usually already directed in a sequence of simple,
straightforward calculations that determine a value for x,. If x, is temporally independent, the
same directed constraint network is used, but an iterative procedure like Golden Section or Newton
Iteration must be used to solve F(x).
77
5.2.3
Comparison of the two methods
To illustrate how symbolic elimination of intermediate variables, differs from solving
a system of simultaneous equations, we consider a simple variation on the problem
of the pneumatic actuator. Chapter 3 develops a quantified relation to relate area,
pressure and force. The embedded relation G(a, p, f) is a single algebraic equation
f
- pa = 0, which is monotonic, and is easily addressed by the existing IPT.
QR(A, P,F): Va E A Vp E P 3f E F - G(a,p, f)
(5.1)
Now suppose the design problem was formulated in terms of piston diameter,
instead of area. The QR is then rewritten to include diameter, d, instead of area,
a. The embedded relation would now consist of a system of simultaneous equations
instead of a single equation.
QR(D, P,F) : Vd E D Vp E P 3f E F - F(d, a, p, f)
(5.2)
Here F(d, a,p, f) is the system of simultaneous equations:
G 1(a, d) :gi(a, d) = a -
7rd2
=0
4
G2 (a, p, f) : 92 (a,p, f) = f - pa = 0
(5.3)
(5.4)
The QR constrains d, but area a area is an intermediate variable that appears only
in the relation F.
Symbolic Elimination
The variable a is not quantified, and appears in both equations. The symbolic elimination approach will thus eliminate a and reduce F to a single equation -y,
IF
(d, p, f) : f - pi
4
=0
(5.5)
consisting only of the variables that are explicitly quantified in the QR. The QR is
78
thus transformed by the elimination process, into
QR(D, P,F): Vd E D Vp E P 3f E F - y(d,p, f)
(5.6)
This form is amenable to inferences by the existing BST-IPT mechanism. The relation derived by elimination of the intermediate variables is monotonic, and a direct
application of the IPT formulae discussed in subsection 3.6.3 will yield useful design
inferences.
Simultaneous System Solution
The alternative approach to elimination is implemented as follows. Assume that an
inference is needed about the bounds on piston diameter, d. The applicability of BST
is unchanged by the form of the relation IF, thus the set-based tool will attempt to
infer Du.b since d is quantified by a V. In order to compute the (minimum) floor and
(maximum) ceiling of D' b, the IPT application is now modified.
First, we generate relative monotonicity data. This involves comparing partial
derivatives of G1 and G2 w.r.t the system variables d, a, p and
f,
to infer the rel-
ative monotonicities. In this example, we need relative monotonicities of
f, p
and
d. We observe the behavior of the expressions g1 and 92 , when their variables are
increased or decreased. The expression gj, in the form shown above, increases with
increasing values of a, and decreases if d is increased. In the notation explained at
the outset of the report, we denote this monotonicity as gi(a+, d-). Likewise, the
monotonicity of variables in 92 can be written as 92 (a, p-,
f+).
If the monotonicity
in relation G1 is used to compare d with the variables in G 2 , we see that the required
relative monotonicities between the relevant variables (omitting a) are of the form
(f+,p-, d-).
Once the monotonicity analysis is complete 5 , we assign relevant floors and ceilings
5As a caveat, we are not currently certain that algorithmic processing of IF will always be able
to extract the required monotonicity information. Further research is necessary to determine the
conditions under which monotonicity analysis can span across variables in a system (allowing the
tables in the preceding footnote to be actually constructed). The method has been demonstrated
on relatively simple problems.
79
in the equation solution procedure, as guided by IPT. If we use set-bounds on, say
f
and p, while calling a solver on the system G1 A G 2 , the inference mechanism will
compute an upper bound on the set D, as well as corresponding values of a at the
interval limits.
This simple problem lends itself easily to simultaneous solution. When there are
several QR's, each quantifying a separate groups of variables, a mechanism is necessary to ensure that no constraint is violated by the values generated by the solver.
This will be accounted for by the network consistency algorithm, that enforces consistency between the solutions of multiple QR's in a system. However, the next chapter
will develop this method of making inferences in greater detail, because it is apparently less expensive, computationally, than trying to perform symbolic elimination.
In the remaining sections, this chapter illustrates symbolic elimination to make
inferences using the power window constraint model, and presents numerical results
obtained from such computations.
5.3
Inference Results and Interpretations
This section uses the models constructed in the previous chapters, and makes some
illustrative inferences using the Ford design data Please refer to Appendix A for tables
of parameter values used in these calculations. The inferences have been made by
applying the elimination technique discussed in the preceding section.
Illustrative inferences are made about the cable drum diameterand the DC motor
torque constant. One of the practical goals of the project leading up to this thesis, was to analytically assisting "correct" choices of motors and seals within Ford's
window design process. This decision is aided by knowledge about the seal friction
limitations, and motor sizing restrictions. This section thus makes inferences about
two component parameters, drum diameter d, and motor torque constant kt.
The drum diameter example works with a single QR, and introduces the idea of
QR
relaxation to help in arriving at feasible designs. The torque constant example
uses 2 QR's and calculates bounds on kt, that will satisfy the design requirements.
80
In each of the examples, the systems of equations appearing in the system QR's are
reduced into a single algebraic equation that is solved for the relevant parameter,
analyzed for monotonicity and used to calculate the required bounding sets.
5.3.1
Drum Diameter Calculation
We start with sample calculations to estimate permissible variation in drum size and
window weight, based on known variations in other parameters, and specified bounds
on stall force. Such calculations would be especially useful, if we were carrying over
drag and motor data from an older, proven design, and are trying to get a preliminary
spec on a new design before going into a more detailed design phase.
Assume that nothing is known about the size of the drum to be used for the cable
drum system. The QR's formulated in subsection 6.3.2 can be used to assist a design
decision in this situation, through an appropriate set-based inference.
The set-based inference mechanism will calculate bounding sets within which feasible drum diameters may lie.
As an initial illustrative example, this subsection
demonstrates IPT application using only the stall force QR. This is the simpler of
the 2 QR's and symbolic elimination is quite easy.
The simplification discussed for each QR in Chapter 4 (use of fdrag instead of
(x, 1, 6)) is used, to bypass the engagement length and drag force calculations. Drag
force values are directly assigned from the spreadsheet.
The quantified relation for stall force limits the stall force to lie between lON and
250N, irrespective of uncontrollable variations in all the other system parameters. The
QR is repeated here, with the simplifications incorporated.
QR2(Vattery, R ine, Rmotor, KT,
If, Hdrive, Floss, D, W, Fdrag, Fstaii)
Vbattery E Vbattery Vriine E Rine Vrmotor E Rmotor
VkT E KT Vif E If
V?7drive E Hdrive Vf1 ss E Foss Vd E D Vw E W
Vfdrag E Fdrag
3fstall E Fstaii = [100, 250]N - F 2
where 1F2 is the system of simultaneous equations (172 : G 12 A G 13 )
81
kt,
G12(Tstaul,
A
Vbattery, rmotor, nrine, if)
Tstall -
G13(fstall, fdrag, W) ' 1 drive, Tstall, floss)
fstaiu
+
k.(,Vbattery
rmotor+rline-0
fdrag
+ W
-
i)
= 0
7ldrive. ( 2s"al) = 0
fioss -
Simplification and Symbolic Reduction
The only intermediate variable to be eliminated between the 2 equations is
Tstal.
Thus the system of equations can be reduced into a single equation
172
<
72
fstall + floss + fdrag + w
2kt
77drive
d
Vbattery
rline
+
0
-
(5.7)
rmotor
The monotonicities in this relation can be determined, noting that the expression on
the L.H.S is increasing with d. Thus the monotonicities are
r+
7(v
r+
k-
72(Vbattery, iine, rmotor, k,t I,
+
+i
7drive,
+
d+ W+
fos, d+, w,
fag,
f+
fsta)
The reduced equation can be solved for d. The simplified expression for d is found as
72d
:d-
2
kt7ldrive
fstaul + fioss + fdrag + w
Vbattery
rine + rmotor
0
(5.8)
BST-IPT Application
The quantifier on d in the above QR is V. Thus, the BST indicates that we can infer
an upper bound on drum diameter. The interval endpoints for all the sets in this
inference, other than D, are shown in Table 5.1. Using quantifiers and monotonicities
shown, we apply Case 1 of the IPT. The IPT thus automatically determines which
endpoints of the parameter sets should be substituted in 72d to evaluate the floor and
ceiling of Dub. The resulting formula for Dub, written using the notation described in
Chapter 3 (IPT explanation) is:
72d (battery,
Eline, motor, -t,
ij'
Trive,
f ioss,
,
f drag'
_stai)]
The parametric model equations are now used to evaluate the endpoints of the
bounding set. The computations are performed at 12.6V battery voltage, and repeated at 14.4V, in accordance with the design requirements stated in Section 1.
82
SET
Vbattery
Rline
Rmotor
KT
If
Hdrive
Foss
Floor
Ceiling
12.6
0.044
0.2867
0.4980
4.5
0.5554
72.41
12.6
0.198
0.2867
0.4980
4.5
0.5554
72.41
Dub
Units]
V
Q
Q
Nm/A
A
N
Comments
Repeat verification at 14.4V
Front/Back Door
(spreadsheet data)
Motor test sheet data6
(measured from motor test sheet)
(from spreadsheet)
(from spreadsheet)
mm
BST =- infer Dub; quantifier = V.
W
36.75
36.75
N
(from spreadsheet)
Fdrag
34.37
100
68.45
250
N
N
(extreme points of the Fdrag : X E [0, 1])
From Performance spec
Fstali
Table 5.1: Table of Interval endpoints for Diameter Inference
Thus, voltage is held fixed at a nominal value while variations are permitted in all
other uncontrollable variables. The inference is made at two distinct values of battery
voltage. The results are as shown:
When Vbattery = 12.6V, IPT infers that Du.b = [47.2, 42.8]mm
When Vbattery
=
14.4V, IPT infers that
Dub
= [54.9, 50.2]mm
Both the bounding sets shown above are invalid intervals, since each has a floor larger
than its ceiling. Under the given conditions, we have
Du.b-
= 0, the null set. Thus,
there exist no feasible values of drum diameter that can satisfy the stall force QR,
at either value of battery voltage, for the given variations in all the other system
parameters.
QR Constraint Relaxation
To overcome this problem and still generate an estimate of feasible diameter, we note
that the interval endpoints are fairly close, and the constraint may be partially relaxed
to determine a feasible diameter. The
Dub
thus found, will not satisfy the original
constraint completely, but it provides an initial guess at a working design. To do this,
the constraints are relaxed as follows:
83
"
Limit the range of window travel over which the stall force is compulsorily
satisfied. This has been contracted to X
=
[0.25, 0.75], from X = [0, 1].
" Relax the tight range specified on stall force. The interval Ftaii is stretched.
Lowest allowed stall force has been set at 90N, instead of 100N. Similarly, the
upper limit on stall force has been raised from 250N to 260N.
Note that the set-constraints in a QR maybe relaxed by enlarging, or contracting
interval assignments. This change in interval endpoints must be made taking into
account the controllability of the parameters involved. In such relaxation, the quantifiers and causal information are preserved in the QR. The causal table is consulted
to guide such constraint relaxation 7
With the relaxed interval endpoints illustrated in Table 5.2, the parametric model
equations are invoked again to calculate Du-.
the results are as follows
When
Vbattery
= 12.6V, IPT infers that Du.b
=
[44.8, 45.0]mm
When
Vbattery
= 14.4V, IPT infers that Du.b
=
[52.1, 52.8]mm
The intervals computed here are both valid.
Interpretation of the Du-b. Inference
Thus, we have a bounding set on the interval of diameter values at each of the battery
voltage conditions specified. We started out with no information about what range of
diameters might be feasible. The only condition for a realistic design was positivity
of diameter , d > 0. Thus, d could theoretically take on any value in the range [0, o0).
The BST-IPT mechanism eliminates all portions of the real line which will provably
lead to violation of the stall force QR. It narrows down the choice of drum diameters,
to lie within the intervals shown above. Note that we have two intervals, one for each
nominal value of battery voltage.
7
Quantified relations may also be relaxed by switching quantifiers and keeping the set-values
intact. For instance switching the quantifier on say, variable w, from V to 3 will make the constraint
easier to satisfy [1]. However, QR relaxation by altering quantifiers alters the semantics of the QR
statement. This will change design intent encoded in the QR. In this work, we maintain consistency
with the causal table 4.2, so the quantifiers are kept intact.
84
Floor
SET
Vbattery
Rine
Rmotor
KT
If
Hdrive
Foss
12.6
0.044
0.2867
0.4980
4.5
0.5554
72.41
Ceiling
12.6
0.198
0.2867
0.4980
4.5
0.5554
72.41
Dub
Units
V
Q
N
Comments
Repeat verification at 14.4V
Front/Back Door
(from spreadsheet)
Motor test sheet data8
(measured from motor test sheet)
(from spreadsheet)
(from spreadsheet)
mm
BST => infer D b; quantifier = V.
Q
Nm/A
A
W
36.75
36.75
N
(from spreadsheet)
Fdrag
45.32
90
66.35
260
N
N
(extreme points of the Fdrag : X E [0.25, 0.75])
Relaxed Performance spec
Fstali
Table 5.2: Table of Relaxed Interval endpoints for Diameter Inference
Thus, if the cable drums are manufactured in-house, this inference tells the designer what process capability is necessary. If the drums are outsourced, or picked
from a catalog, the inference restricts the designer's attention to a specific range of
drum diameters. All feasible drums that can satisfy the stall force QR at the normal battery voltage of 12.6V, must have a diameter within the range [44.8,45.0]mm.
Membership of the bounding set is a necessary condition satisfied by a set-value that
is "not infeasible".
Note that the range of feasible diameters at the 14.4V operating point is different,
and does not overlap with the feasible diameter bounding sets at 12.6V. Thus, the
feasible design computed for a 12.6V nominal voltage, will violate the stall force QR
when the alternator is switched on. The designer can use this preliminary diameter
calculation to size the cable drum.
5.3.2
Torque Constant Calculation
A DC motor's torque constant serves to size the device, and aid in selecting a suitable
candidate from a catalog. In this report, we present inferences about the bounding
sets for this characteristic, that will permit constraint satisfaction of both, the stall
force and the glass velocity QR's.
85
Unlike the cable drum which has a single dimension, the DC motor has several
parameters that characterize it. Assumptions will be made about all other characteristics based on some existing design, in order to draw inferences about kt.
The stall force QR has been simplified and presented in the preceding subsection. The Glass velocity QR constrains window velocity to lie within specific limits,
irrespective of uncontrollable parameter variations. It is repeated here:
QR 1 (Vbattery, Riine, Rmotor, Kb, KT, If, Hdrive, Foss, D, W
FdragV)
E Vbattery Vriine E Rine VTmotor E Rmotor Vkb E Kb VkT E KT Vif E If
Vbattery
V77drive C Hdrive Vf1 ss E Floss Vd E D Vw G W Vfdrag E Fdrag
3vg E Vg = [0.125, 0.175]ms- 1 - IF
where 1 is the system of simultaneous equations (P1 : G7 A G8 A G9 A Gio A Gil)
A
G7 (fmotor, floss, fcrag, W, ?ldrive)
:
fmotor -
A G8 (7Tmotor , fmotor, d)
:
Tmotor - fmotor 2
fdragWfloss
77drive
A G9,(ifT,kT)
:
w)
:
Vbattery
A Gi(v9 ,w,d)
:
v 9- w = 0
G1o(Vbattery, rmotor, riine, kb,
-
fkb
-
=
= 0
0
T
0
W
i(rmotor+rine)
kb
=
As before, the simplification from Chapter 4 (subsection 6.3.2 introduces
0
fdrag
into
the QR for simplicity) is used, to bypass the engagement length and drag force calculations. Drag force values are directly assigned from the spreadsheet.
Simplification and Symbolic Reduction
The five simultaneous equations in r1 have three intermediate variables between them.
The intermediate variables are
fmotor, Tmotor
and current, i. Eliminating the inter-
mediate variables, we derive a single equation 71, that relates glass velocity to the
independent parameters.
71 4
'71
:
Vylass -
dVbattery -
2kb
(?line +
Tmotor)-if
86
d ( fdrag + W +foss)}] = 0 (5.9)
2kt
77drive
The monotonicities in this relation are determined, noting that the expression on the
L.H.S is monotonically decreasing with kt. The expression, 71 , is quadratic in d. This
might present a problem, since IPT requires strictly monotonic expressions.
To verify strictly monotonic behavior, we run a test with numerical values, holding
all other parameters at their nominal values and plotting the variation of vg, with
d. Increasing d in steps of 0.01m, it is observed that the parabolic variation of glass
velocity with drum diameter is indeed monotonically increasing for d > 0 9.
In the vicinity of d = 50mm, which is the region of interest to us (as per the
previous inferences about drum diameter), the Vglass is strictly increasing with d.
Monotonicities are assigned accordingly in the expression below:
Y1 (V+
r
77
, r motor, k+, k+ ,-
dive, fiOss, d±,w fragv
,
ss)
The reduced equation can be solved for kt. The simplified expression for kt is in the
relation
71(t
: Ylkt-
0
d(fdrag + floss+W)
2
27drive
kbvg:ak(
vbattery-2k d a
(5.10)
i
rine +rmotor
This inference in this subsection uses both QR's. The monotonicities for the stall
force equation are repeated here. Torque constant is made positively monotonic.
7 - ot
7 ( V+
Y2(Vbattery, rline, rMotor,
k + 1-f
+
-
s
I -l -
-
-
1dss
fdragi fstall)
drive,
The inference will also require the stall force relation Y2, solved for kt.
'Y2kt :
-
d
72ktkt
-Vbattery
2drive
fstall + floss + fdrag + W
0
(5.11)
rline+rmotor
f
BST-IPT Application to Stall Force QR
The quantifier on kt in each QR is V. Thus, the BST indicates that we can infer an
upper bound on torque constant from each quantified relation. The two QR's will
9
This test was carried out using only nominal data shown in Appendix A, except for drag force
and line resistance values which are pegged at their average values, 60N and 0.125 respectively.
The ordered pairs (d, vglass) computed in this test are listed here: (0,0), (0.01,0.050), (0.02,0.098),
(0.03,0.143), (0.04,0.186), (0.05,0.226), (0.06,0.264), (0.07,0.300), (0.08,0.333), (0.09,0.364).
87
SET
Floor
Vbattery
Rline
12.6
0.044
Ceiling
12.6
0.198
Rmotor
0.2867
0.2867
Q
KT
?
4.5
0.5554
?
4.5
0.5554
Nm/A
A
Foss
72.41
72.41
N
D
W
50.8
36.75
45.32
90
50.8
36.75
64.82
260
mm
N
N
N
if
Hdrive
Fdrag
Fstaii
Units
V
Q
Comments
Repeat verification at 14.4V
Front/Back Door
(spreadsheet data)
BST = infer Dub; quantifier = V.
(measured from motor test sheet)
(from spreadsheet)
(no variation assumed)
(no variation assumed)
(no variation assumed)
(extreme points of the Fdrag : x E [0.25, 0.75])
From relaxed performance spec
Table 5.3: Table of Interval endpoints for Torque Constant Inference from Stall Force
QR
yield their respective bounding sets for Kt, and these bounds will be combined to
make a design decision about the value of kt to be used.
The inference is carried out in 2 stages. The BST-IPT mechanism is applied
individually on each of the QR's, with all other sets in each inference (other than
Kt) tabulated accordingly, before the inferences are combined. To arrive at feasible
interval bounds, the performance constraints have been relaxed. The stall force constraint just as in the previous subsection. The Glass velocity constraint is relaxed
from [11,17]cm s-' to [8,20]cm s-1.
The set assignments for the stall force QR inference, are shown in Table 5.3.
Using quantifiers and monotonicities shown, we apply Case 1 of the IPT to derive the
formula for the upper bound
Kyu-
= [^x2kt
K
..
(Piattery, line, TmotorI kt)
Y2kt (;U battery, Eline, Emotor, kkt, ifd
fio,,
,Trive,
f
rag, Ltaii),
rive, floss, MciragI
,
fstall)
Using the parametric model, we calculate the bounding set endpoints shown:
When Vbattery
=
12.6V, IPT infers that
When Vbattery
=
14.4V, IPT infers that Kt".b
Kt. b =
=
Both bounding sets shown above are valid intervals.
88
[0.5617, 0.5641]Nm/A
[0.4789, 0.4855]Nm/A
BST-IPT Application to Glass velocity QR
The set assignments for the stall force QR inference, are shown in Table 5.4. Using
quantifiers and monotonicities shown, we apply Case 1 of the IPT to derive the
formula for the upper bound
KT-b.
f
fdrag) k2gass)7
[7.b
1k, (Ebattery, 7 ine, r mot or,
kt I --b I
'f' fl-drive' Yossid Ui Y
K 1kt (Ubattery,
, k,
f, 7 ldrive7
T-line,
motor
4oss
d
Ldrag ,ivgiass)]
Using the parametric model, we compute the bound on Kt as shown:
When
Vbattery =
12.6V, IPT infers that Kt.b
=
When
Vbattery =
14.4V, IPT infers that
= [0.4188, 0.4333]Nm/A
Ktb
[0.5205, 0.6505]Nm/A
The intervals computed here are both valid.
Interpretation of the Kt & Inference
Consider the upper bounds on Kt at 12.6V. The intervals returned by the stall force
and glass velocity QR inferences overlap. The Set Elimination algorithm will enforce
consistency with both the QR's simultaneously, by taking the intersection of the two
upper bounds. Thus, in order to satisfy the design constraints at a nominal 12.6V
battery voltage condition, the designer should pick a motor that has a torque constant
in the interval
[0.5617, 0.5641] n [0.5205, 0.6505] = [0.5617, 0, 5641]Nm/A
The upper bounds on Kt, computed at 14.4V neither overlap each other, nor overlap the feasible Kb - at 12.6V. This means that for the given ranges of variationsin
the parameters, the stall force and glass velocity constraints cannot be simultaneously
satisfied. Even though we have a feasible design at 12.6V, the constraints will be
violated when the alternator switches on, or if the battery voltage rises to 14V.
Just as with drum diameter, we started out with no information about what
range of torque constants might be feasible, and kt which has to be positive (kt > 0)
could theoretically take on any value in the range [0, oo). The BST-IPT mechanism
89
SET
Vbattery
Ruine
Rmotor
KT
Kb
If
Hdrive
Foss
D
W
Fdrag
Vlgass
Floor
12.6
0.044
0.2867
?
0.10008
4.5
0.5554
72.41
50.8
36.75
45.32
8
Ceiling I Units I Comments
12.6
V
Repeat verification at 14.4V
0.198
Q
Front/Back Door
0.2867
Q
(spreadsheet data)
?
Nm/A BST ->.infer Dub; quantifier = V.
0.10008 V/rpm (spreadsheet data)
4.5
A
(measured from motor test sheet)
0.5554
(from spreadsheet)
72.41
N
(no variation assumed)
50.8
mm
(no variation assumed)
36.75
N
(no variation assumed)
64.82
N
(extreme points of the Fdrag : X E [0.25, 0.75])
From relaxed performance spec
cm/s
20
Table 5.4: Table of Interval endpoints for Torque Constant Inference from Glass
Velocity QR
eliminates the infeasible values of kt from the designer's consideration. DC motors
are picked from manufacturer's catalogs, and this inference tells the designer what
ranges of kt values are "good". So the designer can use this set-based calculation to
select appropriate DC motors.
Note again, that the set-elimination mechanism employed here provides a necessary condition that eliminates provably infeasible sets. The remaining part of the
design space is not guaranteed to be feasible. Thus, motors chosen from catalogs, and
lying within the calculated bounding sets are not theoretically guaranteed to satisfy
the design constraints. But all motors that lie outside the envelope of the bounding
set are guaranteedto violate both the QR's, and may be safely discarded from further
consideration. This saves computational and search effort. The use of the bounding
set data reduces thrashing in the search for feasible designs.
5.4
Observations
In both examples discussed in this chapter, we ran into the problem of inferring
empty set bounds, and having to relax the QR constraints to come up with any
90
useful design inferences. These relaxations were made by empirically trying various
possibilities until a suitable one was found. This approach can be very time consuming
and inefficient.
Examining the reduced expressions, y's, to determine which variations strongly
influence the value in the LHS, helps the designer in choosing appropriate relaxations
quickly. A formal method to generate this information would constitute a tool for
Sensitivity Analysis [11] in the context of set-based constraints.
Examining the symbolic partial derivatives of the reduced expression 7, w.r.t all
the parameters contained within it, would be one possible approach to perform such
a sensitivity analysis. However, this involves much algebraic manipulation, and may
prove computationally expensive.
An alternative approach would be to carry out a QR factorization, or Singular
Value Decomposition (SVD) of the system Jacobian matrix J, or 1(x), at a particular
solution xO, of P(x). This is a relatively inexpensive numerical computation. These
matrix decompositions (explained in [7]) employ a pivoting technique that ranks the
system variables, i.e. elements of vector x, in terms of their relative influence on the
location of the solution point xO of 1(x), within the space Rn. Thus, an ordering of
pivots obtained from such a matrix decomposition can guide the designer in selecting
appropriate variables for which to relax interval bounds. This idea has not been
tested, or developed any further, and is included here as a pointer towards areas of
future research.
This next chapter explores the alternative path, an extension of the IPT logic, to
make it applicable to the QR's of Chapter 4, without altering the symbolic forms of
the QR's themselves.
91
Chapter 6
Extending the Interval
Propagation Theorem
Each problem that I solved became a rule
which served afterwards to solve other problems.
-Rene Descartes (1596-1650)
The previous chapter determined a limitation in the applicability of the existing
BST-IPT inference mechanism. It drew set-based inferences about the cable drum
power window system by using algebraic elimination to make the power window
quantified relations consistent with the symbolic form prescribed by the IPT.
This chapter presents an alternative approach, introducing a theoretical extension to the IPT, to avoid computationally expensive symbolic manipulation of the
quantified relations. It proves the Extended Interval Propagation Theorem (or
Extended-IPT). This new theorem uses logic from the BST-IPT inference mechanism to operate on a quantified relations embedding a whole system of simultaneous
equations. Thus, it overcomes the limitation of the IPT, which allows only a single
equation within a QR.
The chapter concludes by applying Extended-IPT to the power window design
constraints. Inferences drawn from the glass velocity and stall force quantified relations, using Extended-IPT are compared with those from IPT. This provides practical
examples of the applicability of the new theorem.
92
6.1
Statement of the Extended-IPT
Let x = (X1 ,X
2
,...
,
Xn) be a vector of n real variables.
Each variable, xi E x, is
associated with a closed real interval, Xi. Such intervals are grouped into a vector of
intervals, X
=
(X 1 , X2 ,...
,
Xn). Let F(x) be a system of m simultaneous, algebraic
equations. Let each component equation in F(x) be continuous, once-differentiable
and asymptote-free.
Let the system of equations be independent over a region of
interest. Assuming that n > m, such a system has k = n - m degrees of freedom 1 .
Partition the elements of x into two vectors2,
1.
Xk
X = (Xk, Xm),
where
is a vector of k independent variables. Independent variables get value
assignments directly from their associated intervals Xk, independent of any constraint imposed by the relation 1(x) .
2.
xm
is a vector of m dependent variables. Dependent variables get value
assignments indirectly from the values of independent variables, according to
the constraintimposed by the relation r(x) 3 .
Let the relation 1F(x) appear as the predicate of a quantified relation, QR(X(k+l)),
having the form
QR(X(k+l))
: qixi E X 1 q2 x 2 E X
...
2
qkXk E Xk ]Xk+1 E Xk+1
.
1(x)
In QR(X(k+l)), each symbol qi stands for a universal (V), or existential (3) quantifier.
The relation, QR(X(k+l)), and the partitioning, x =
(Xk, Xm),
Condition 1. All the independent variables in
Xk
Condition 2. Exactly one dependent variable,
Xd E
are quantified in QR(X(k+l))xm, is quantified in QR(X(k+l)).
Condition 3. The single quantified dependent variable,
Xd,
tonic manner with each of the independent variables in
Xk.
1
are assumed to satisfy:
varies in a strictly mono-
A perfectly constrained (n = m), or over-constrained (m > n) system does not permit parametric design through continuous variation of parameters . Both systems possess discrete sets of
solutions (distinct disjoint precise points in R"), if there is any solution at all.
2
Such a partition is guaranteed to exist (Lemma 2 in Appendix D). It can be analytically determined in the neighborhood of any point x 0 that satisfies F(x), by virtue of continuity and independence assumptions about F(x) in the statement of this Theorem.
3
Thus, 1F(x) is formally treated as a many-to-one map, F : Rk _ Rm [10]. It utilizes a given
value of Xk, to determine a suitable value of xm, satisfying F(Xk, Xm).
93
Condition 3 holds if the dependent variable Xd has a strictly positive or strictly
negative partial derivative w.r.t. each member of
Xk.
Each partial derivative is cal-
culated' holding all other members of Xk constant, while members of
xm
are allowed
to vary to satisfy IP(x). Thus Extended-IPT requires that
6
5 0 Vxj
xd
axi
EXk
Let Mx, be a symmetric (k + 1) x (k + 1) matrix 5 with elements mxd(i, j) defined
+
=
mXd (i, j) = sgn (x)OXj/
--
if
if
8x
9xi
> 0
t<
aXj
0
Let X, be the set for which Extended-IPT infers a set bound. Partition the vectors
X(k+1),
and x, as X(k+1)
=
(Xp, X(k)), and x = (xp,
x(,-1))
respectively. Here, Xp,
and xP are the variables that are bounded by the inference, with the rest of the n real
variables grouped together as
x(n-1),
and the remaining k quantified set variables are
in X(k). QR(X(k+l)) is now written as QR(Xp, X(k)) , to identify the variable X,.
Partition X(k) by quantifier in QR(Xp, X(k)), into two vectors of closed intervals,
Xv and X 3 . Partition the corresponding real variables in
of variables,
xv
x(n-1)
likewise, into vectors
and x3. Group the remaining (non-quantified) variables from
x(n-1)
into a third vector x(m-l).
Further partition the set and variable vectors by the sign of each variable's partial derivative w.r.t x,. This sign is obtained by looking up the pth column of
MXd,
corresponding to the variable x,. Thus6 the symbols mxd(1,p) through mxd (k +1, p)
identify the partitioned set-vectors X+, X-, X+, and X-, and the correspondingly
partitioned variable-vectors x+, x, x,
and x-.
4These partial derivatives are guaranteed to exist (Lemma 3 of Appendix D), in the neighborhood
of a point x0 that satisfies 17(x), by assumptions of independence, continuity, and differentiability
of F(x), stated priorly in the Theorem. They can be determined symbolically (not advised - this
extension seeks to avoid symbolic manipulation) or numerically (by finite differences [7]).
5
The table MXd is called the QR Specific Monotonicity Table, and is filled out by an algorithm,
which uses the first k elements in the (k + 1)th column as input data to logically fill out the remaining
k2 + k + 1 table entries efficiently, in a particular order (detailed explanation in Appendix E.)
6
This notation re-creates the assignment of monotonicity signs in the current IPT. Note that in
the pth row, the entry mxd (p, p) is always a (Appendix E), and we never have to worry about sign
inversion by multiplying throughout by -1, like it is sometimes necessary in the existing IPT.
94
Then
Case 1. q, = V. Let X* = [x*, g], with
P* = x,
=
(XP,7x+
x, F(x,,
-, X11
X7,
4,
xx(M-1))(.)
i3, X(m-1)
)
(6.2)
Then, if it is non-empty, X* is an upper-bound for all X, satisfying QR(Xp, X(k)).
Case 2. qp = 3. Let X* = [x*, x], with
=
x,| I(x,,
=x
+,X ,
(x,x+,
-,
4,
$X
X,
X(m-1) )
(6.3)
X(m-1))
(6.4)
Then, if it is non-empty, X* is an lower-bound for all X, satisfying QR(Xp, X(k)).
6.2
Proof of Extended-IPT
If a particular variable xP is perturbed from a point x0 satisfying F(xo),
(a) positively, or negatively by an appropriately chosen perturbation 6 > 0,
(b) without altering any other values in x,
then the relation 7(x) will no longer be satisfied (Lemma 4 of Appendix E).
Observation 1. Given such a choice of 6 that violates the embedded relation F(x),
the quantified relation QR(X,, X(k)) can still be satisfied by suitably altering values
of variables
x(k)
E X(k), other than the perturbed one (which is held constant at its
perturbed value).
Observation 2. The alterations for re-satisfaction must however be consistent with
set membership bounds in the argument of QR(Xp, X(k)). Thus, only existentially
quantified independent variables (in
x3)
may be altered to re-satisfy 1(x), because
changes in the universally quantified elements (in xv), will violate the set membership
implied by QR(Xp, X(k)).
95
Observations 1 and 2, suggest alteration of one or more
(a) existentially quantified variables in
xB,
(b) and/or non-quantified variables in x(m-1),
so as to re-satisfy F(x) at a nearby point in the domain space, after it has been
perturbed. Using these observations, we develop a proof for Extended-IPT.
Proof:
Case 1. qp = V. X* = [x*, if,
with
X* = X,
X*= X,
x+~,R,
Egx_,xm1
x,
R ,
3 x3
x
(65)
75x(m-1) )(6.6)
Assume that the theorem's consequent is false: that there exists an interval
X*, satisfying QR(Xp, X(k)). Then, either x > F or x, < x.
X',
Assume the first, so that
F(±6,
4V,
X,
I,
=
, x m-1)
even though the vector x 0 = (
- +6 with 6>O
, such that the relation
)is not satisfied,
, 4, x,
x4,
X5,
x(m1)
)satisfies
7(xo).
Sub-Case L.A
Negation of the consequent implies that F (i
+ 6,4,
x
be re-satisfied by altering existentially quantified variables in
, X,
3,
X(k)
X(m-1)
) can
E X(k), without
violating QR(Xp, X(k)). From Observations 1 and 2, if x, is increased, then IF(x) can
be re-satisfied by appropriately
1. increasing one or more independent variables, xi E Xk, which have axi
-"< 0,
or mXd (p,i) =
-
2. decreasing one or more independent variables, xi E
or mXd(p,i) =
Xk,
which have
+
7 The value of 6 here must be chosen in accordance with Lemma 4 of Appendix D.
96
'
> 0,
In this effort to re-satisfy F(x), all dependent variables may be allowed to vary freely,
since they are not constrained by quantification in QR(X,, X(k)). The specific dependent variable x, is held constant at its perturbed value,
*+ 6. Thus, ]P(x) can be
re-satisfied by increasing members of x-, and decreasing members of x.
Sub-Case 1.B
Negation of the consequent implies that 1 (T + 6,
4V,
x,
xX,
X3,
X(m-1)
)
can
be re-satisfied by altering existentially quantified variables in X(k) E X(k), without
violating QR(Xp, X(k)). From Observations 1 and 2, if x, is increased, then, 1F(x) can
be re-satisfied by appropriately
1. increasing one or more independent variables, xi E
or mXd (p,i) =
Xk,
which have ax, < 0,
xk,
which have ax' > 0,
-
2. decreasing one or more independent variables, xi E
or mXd (p, i) = +
3. increasing the dependent variable,
Xd,
4. decreasing the dependent variable,
Xd,
if -a- < 0, or mxd(pi) =
if
-9L >
-
0, or mzd ,i) = +
In this effort to re-satisfy ]P(x), all dependent variables (other than
Xd)
may be al-
lowed to vary freely, since they are not constrained by quantification in QR(Xp, X(k))Variation in the specific dependent variable
Xd
is constrained by the value of the in-
terval Xd, specified to bound it in QR(X,, X(k)). The independent variable x, is held
constant at its perturbed value, F + 6. Thus, 1(x) can be re-satisfied by increasing
members of x-, and decreasing members of x.
Conclusion of Case 1.
Both Sub-Cases 1A and 1B conclude that under the assumed perturbation, 1(x) can
be re-satisfied by increasing the members of x-, and decreasing members of x.
from the ceiling computation in Case 1,
P* =
x,
(XP
,
x1
I7xg, X3+
97
,
x(m-1))
But,
the variables in
are already at their prescribed upper-bounds (R3), and can be
x3
increased no further. Likewise, the variables in x3 are already at their prescribed
lower-bounds (x),
and can be decreased no further.
Thus, we have
, X+
xI
x_
) E ( Xi, X_ ) jr( FP_+
6, Xv,
,
iX
xgx
)
x-1
and the quantified relation cannot be satisfied. Hence, there is no X'
(6.8)
X,*satisfying
QR(Xp, X(k)). This contradicts the assumption that X* is not an upper bound, and
proves the validity of the computed upper-bound ceiling in Case 1. (both Sub-Cases
IA and 1B). A similar argument for the case x' < x* yields the result
3(
, x-
)
E ( Xi, X3
)rF(x-6, x,
4
,
x
X
(6.9)
(m-)
This completes the proof of Case 1.
Case 2. qp = 3. X* = [x*, x], with
*
= x
F(xV,,PI
=
(X,,
, x,
xj,)
Xg,
X1,
I,X
54, x-,
(m-1)
)
(6.10)
x(m-1)
)
(6.11)
Assume that the theorem's consequent is false: that there exists an interval
X' ;
X*, satisfying QR(Xp, X(k)). Then, either x < x or x' > x*.
Assume the first, so that x
F (T* -
,J4,Xg,
4R,
x,
x(m-)
even though the vector xo = (Y,
= X - 6 with 6
>
the relation
0 8, such that
) is not satisfied,
x4, X;,
X,
x-,
x(m- 1 )
) satisfies 1(xo).
Sub-Case 2.A
Negation of the consequent implies that F ( - 6,x,
8
, x,3,
x(m)
The value of 6 here must be chosen in accordance with Lemma 4 of Appendix D.
98
) can
be re-satisfied by altering existentially quantified variables in
X(k)
E X(k), without
violating QR(X,, X(k)). From Observations 1 and 2, if x, is increased, then, F(x) can
be re-satisfied by appropriately
1. decreasing one or more independent variables, xi E
or mXd(p,
=
which have
Xk,
"
< 0,
-
2. increasing one or more independent variables, xi E
Xk,
which have
k > 0,
or mXd(p, i) = +
In this effort to re-satisfy 17(x), variation in all dependent variables may be ignored,
since they are not constrained by quantification in QR(Xp, X(k)). The specific dependent variable xP is held constant at its perturbed value, F+ 6. Thus, ]P(x) can be
re-satisfied by increasing members of x-, and decreasing members of x.
Sub-Case 2.B
Negation of the consequent implies that F (F - 6,
x
4i
, x-,
be re-satisfied by altering existentially quantified variables in
X(k)
x,
x(m 1 )
E X(k),
) can
without
violating QR(Xp, X(k)). From Observations 1 and 2, if x, is increased, then, F(x) can
be re-satisfied by appropriately
1. decreasing one or more independent variables, xi E
or mXd (p,i) =
which have
'
< 0,
-
2. increasing one or more independent variables, xi E
or mXd (pI0 =
Xk,
Xk,
which have a
> 0,
+
3. decreasing the dependent variable,
Xd,
if
4
< 0, or mx(p,i)=
4. increasing the dependent variable, Xd, if dxl > 0, or m2d(p,i) = +
In this effort to re-satisfy 1(x), variation in all dependent variables (other than Xd)
may be allowed to vary freely, since they are not constrained by quantification in
QR(Xp, X(k)). Variation in the specific dependent variable Xd is constrained by the
99
value of the interval Xd, specified to bound it in QR(X,, X(k)). The independent variable x, is held constant at its perturbed value, F - 6. Thus, F(x) can be re-satisfied
by increasing members of x-, and decreasing members of x.
Conclusion of Case 2.
Both Sub-Cases 2A and 2B conclude that under the assumed perturbation, 1(x) can
be re-satisfied by decreasing the members of x-, and increasing members of x.
But,
from the ceiling computation in Case 2,
X*= x| F(x,, xj,
the variables in
x3
Kxx(m-1))
-3,
are already at their prescribed lower-bounds (R3), and can be
decreased no further. Likewise, the variables in x+ are already at their prescribed
upper-bounds (x+), and can be increased no further.
Thus, we have
,
(
,
X3
)
( X3, X3 )|F( g - 6, x,
Rg,
4,
,
x Xm
-1)
(6.13)
and the quantified relation cannot be satisfied. Hence, there is no X' ; X,* satisfying
QR(Xp, X(k)). This contradicts the assumption that X* is not a lower-bound, and
proves the validity of the computed upper-bound ceiling in Case 2. (both Sub-Cases
2A and 2B). A similar argument for the case x' > x* yields the result
,
x,
) E ( X,
X- )IF(
+)
)
(6.14)
This completes the proof of Case 2.
Q.E.D
100
6.3
Examples Using the Extended-IPT
It is easy to see that the quantified relations presented in Chapter 4, do indeed confirm
to the symbolic form specified by the Extended-IPT. This section illustrates the use of
the theorem by demonstrating how it can be applied to the power window quantified
relations. The relevant tables and instantiations are developed after verifying that
these quantified relations satisfy the conditions prescribed by Extended-IPT.
6.3.1
Inference from Stall Force Quantified Relation
In this example we first verify that the parametric system of equations, embedded
within the stall force quantified relation (subsection ??), does indeed possess the
monotonicity property required by Extended-IPT. Using this property, we construct
a monotonicity table, and instantiate the variables in the parametric model, using
the rules laid down by Extended-IPT. The stall force quantified relation reads:
QR2(Vbattery, Rine, Rmotor, KT, If, Hdrive, FieSS, D, W
VVbattery E Vbattery Vrizne
V7ldrive E Hdrive
Fdrag, Fstal
E Rine Vrmotor E Rmotor VkT E KT Vif E If
Vf1 ss
E Floss
Vd E D Vw E W Vfdrag E Fdrag
3fstal E Ftaii = [100, 250]N - 12
where 12 is the system of simultaneous equations (1F2 : G 12 A G )
13
G12(Tstall, kt,Vbattery, rmotor, rline, i ) : TstI - kt.(
- if) = 0
-Vattr
A G13(fstall, fdragi W,07drive, Tstalli, floss)
:
fstali + fdrag + W
-
floss
-
rlrie.(
2,"
)
The embedded parametric relation, 172, contains m = 2 simultaneous equations
in n = 12 variables. It quantifies only 11 of its 12 system variables. The vectors of
variables are explicitly written below.
X
(Vbattery, rimne, rmotor, kt, if, T/drive,
X
(Vbattery, Rline, Rmotor, Kt, Ifdrive , Foss, D,
fioss, d, w, fdrag, Tstall, fstali ),
W, Fdrag, Fstali)-
Partitioning these vectors by dependence is easy in this problem. The equations
are already "cascaded". Thus
Tstall
is computed from G 12 , and substituted into G 13
to constrain another output, fstall.
101
0
We can pick Ttal and ftaii, as dependent variables, since they are constrained by
the equations G 1 2 and G13 respectively, once the other 10 variables are independently
instantiated'. The partitioning is now denoted as
X = (Xk, Xm)
Xk =
Xm =
(Vbattery, rine, rmotor, kt, if, ldrive, fioss, d, w, fdrag)
(Tstai, fstall)
It is trivial to verify the above partitioning satisfies
Condition 1 of Extended-IPT, because all the independent variables are indeed
quantified in the Stall Force Quantified Relation
Condition 2 of Extended-IPT, because exactly on dependent variable, ftall is quantified in the Stall Force Quantified Relation.
Condition 3 of Extended-IPT is much harder to check. Using relatively straightforward qualitative reasoning, we determine (by inspection here, but by an automated
procedure ideally) that both, stall force
fstaii,
and stall torque
increase monotonically with increasing battery voltage
decrease monotonically with increasing line resistance
TstalI
Vbattery,
Trne,
decrease monotonically with increasing motor resistance rmotor,
increase monotonically with increasing torque constant kt,
decrease monotonically with increasing free running current if,
increase monotonically with increasing drive efficiency
Tidrive,
decrease monotonically with increasing force loss ffo0 s,
decrease monotonically with increasing free drum diameter d,
decrease monotonically with increasing glass weight w,
decrease monotonically with increasing drag force fdrag.
The above information guarantees the satisfaction of Condition 3, because there is
no non-monotonic variation. This qualitative information is used in the QR Specific
Monotonicity Table Filling Algorithm (Appendix E) to construct Table 6.1, containing
9
The system has k = n - m = 10 degrees of freedom.
102
Vbat
Vbatteryj
(D
rine
-
rTin
-
G
rmot
kt
-
if
7
+
-
+
+
-
-
7d
flos
d
W
fdrg
fsti
+
+
-
0
-
+
+
+
-
-
+
-
0
-
-
-
-
+
-
+
+
-
+
-
©
+
-
+
+
-
+
-
±
e
±
+
+
-
+
-
w
-
+
+
-+
&
+
-
fdrag
-
+
+
-
+
-
+
+
+
+
0
-
-
-
+
-
+
-
-
-
-
rmotor
kt
Tdrive
+
+
floss
d
if
fatall
+
+
+
(
-
-
+
-+
+
+
+
+
+
+
+
+
+
-
+
-
Table 6.1: QR Specific Monotonicity Table for the Stall Force Quantified Relation
the monotonicities (signs of partial derivatives) for the stall force quantified relation.
Any number of inferences using this QR can be drawn, once this table is filled. Each
such inference instantiates and solves the system P2 according to the Extended-IPT
statement, using the partitioning of variables, both by quantifier and monotonicity
as shown below, (considering the 2 possible patterns of signs, in any column/row of
the table).
F2(VbatteryV, ritnev, rmotorv, k-v, ijfv, %7riveV,fl~oksv, dV, w+, f+agv, faiu,
Tstai)
or
F2(VbiatteryV, rilnev, r-otorV, ktV, Z v, %+riveV, fi;osv, d-,
W-
f-ragV,
fstaliB, Tatail)
For a concrete instance, consider the variable d. It is quantified universally. We can
infer an upper bound on it (BST), by solving P2 twice, using the instantiations guided
by the signs in the column of Table 6.1 corresponding to d. These computations must
be done with a numerical solver for a system of simultaneous equations to evaluate
the endpoints of the interval D.b = [d*,]
d* =|2 (d, Tine,) motor , fioss,' drag I,
=
,
Vbattery
k
1ldrive f stall,
Tstaii)
d| 1 2 (d, rfine, rmotor, fioss, f drag~
7'f ) W,-battery kt, fI rive, LstaiI Tstali)
Comparing the expressions derived from Extended-IPT, with the formulae derived
using symbolic elimination (Chapter 5) for a similar inference,
103
DuA
-
[72d
battery, T ine,7motor
, t,-i,
, if)
72d(Ibattery, Tine, rmotor,'
1
?drive, floss',
-drag' fstali),
f ios,
idrive,
f drag Lfstall)
we see that instantiations guided by Extended-IPT produce the same results as the
formula for Du.b derived from IPT in Chapter 5. The Extended Interval Propagation
theorem thus uses the logic of IPT to instantiate a system of simultaneous equations,
to draw a set-based inference.
6.3.2
Inference from the Glass Velocity Quantified Relation
This subsection applies Extended-IPT to draw an inference from the Glass Velocity
Quantified Relation. The quantified relation is repeated below.
QR1(Vbattery, Rline, Rmotor, Kb, KT, If, Hdrive, Filss, D, W, Fdrag, V
Vbattery
Vbattery
/rline E Ruine Vrmotor G Rmotor Vkb
Vl7drive E Hdrive Vf 1 oss E Floss Vd E D Vw
Kb VkT
c W
3Vg E V = [0.125,0.175]ms'
Vif c If
KT
Vfdrag E Fdrag
- F1
where F1 is the system of simultaneous equations (F1 : G7A G8 A G9 A Go A Gil)
A
G7(fmotor, flons, fdrag,
A
G8
fmotor
(Tmotor , fmotor , d)
A
A
W, 7ldrive)
G9(ifT,kT)
-
fdrag W
7)drive
Tmotor -
:
Go(Vbattery, rmotor, rine, kb, w)
i
Vbattery
kb
A Gii(vgwd)
fmotor 2 =
_
"foss
=
0
0
,
0
W
_
i(r-otor+rjine) -
Wkb
0
-
vg -O=O
It's parametric relation, F1 , contains m = 5 simultaneous equations in n = 16
variables. It quantifies only 12 of its 16 system variables. The vectors of variables are
explicitly written below.
X -
X
(Vbattery, Irine, rmotor, kb, kt if , 7rdrive, floss, d, w, fdrag, fmotor, Tmotor, i, W, Vgiass),
(Mattery, Rline, Rmotor, Kt, Kb, If,drive , Foss, D, WV, Fdrag, Vgass).
Partitioning these vectors by dependence is again, quite simple.
is analogous to the one followed in the previous problem.
The process
The equations are al-
ready arranged to provide one output from each equation, substituted in the next,
104
to propagate values through the parametric network.
fmotor, Tmotor,
i, w and
vglass,
Thus, we can easily pick
as dependent variables, since they are constrained by
the equations G 7 through G 11 respectively, once the other 11 variables are independently instantiated10 . The partitioning is now denoted as
X = (Xk, xm)
Tnine,
kt, kbif,
Xk -
(Vbattery,
Xm =
(fmotor, Tmotor, i , W, Vglass)
rmotor,
7drive,
floss, d, W,
fdrag)
It is trivial to verify the above partitioning satisfies
Condition 1 of Extended-IPT, because all the independent variables are indeed
quantified in the Stall Force Quantified Relation
Condition 2 of Extended-IPT, because exactly on dependent variable, vglas, is quantified in the Stall Force Quantified Relation.
Condition 3 of Extended-IPT is again much harder to check. Using qualitative
reasoning, we determine that
vglass
increases monotonically with increasing battery voltage
decreases monotonically with increasing line resistance
Vbattery,
'line,
decreases monotonically with increasing motor resistance rmotor,
increases monotonically with increasing torque constant kt,
increases monotonically with increasing back-EMF constant kb,
decreases monotonically with increasing free running current if,
increases monotonically with increasing drive efficiency rdrive,
decreases monotonically with increasing force loss
floss,
increases monotonically with increasing free drum diameter 1 d,
decreases monotonically with increasing glass weight w,
decreases monotonically with increasing drag force
fdrag.
10 The system has k = n - m = 11 degrees of freedom.
"This is not straightforward. The variation is actually quadratic, and was checked by a numerical
test, documented in footnote 10, Chapter 5, to verify the strict monotonicity in the domain of
interest.
105
The above information guarantees the satisfaction of Condition 3, because there is
no non-monotonic variation. This qualitative information is used in the QR Specific
Monotonicity Table Filling Algorithm (Appendix E) to construct Table 6.2, containing the monotonicities (signs of partial derivatives) for the glass velocity quantified
relation. Any number of inferences using this QR can be drawn, once this table is
computed. Each such inference instantiates and solves the system IF according to
the Extended-IPT statement, using the partitioning of variables, both by quantifier
and monotonicity as shown below, (considering the 2 possible patterns of signs, in
any column/row of the table).
F1 (VatteryVI, ritnev, rMotorv,
d, w, fd+ragv,
t
Vglass,,
k
if V,
f108 8v,
T~riveV,
fmotor, Tmotor, i, W, Vglass)
or
F1
b(Vatteryv, rlinev, r-otorv, k
de, W-, f-agV,
Vi Z v, %7riveV,fiOssv,
Vglass3, fmotor, Tmotor, i, W, Vglass)
For a concrete instance of instantiation by Extended-IPT, consider the variable kt.
It is quantified universally. We can infer an upper bound on it (BST), by solving F,
twice, using the instantiations guided by the signs in the column of Table 6.2 corresponding to kt. These computations must be done with a numerical solver for a system
of simultaneous equations to evaluate the endpoints of the interval K b.
k = kt|F,(kt,7line,
motor,
fioss, f
rag,
ririve,
kbattery, kb, d7 Eglass, fmotor, Tmotor,
=
k|F1 (kt, !Iine, moto,, Los, fLdrag N-drive' 1,
Vbatter y, kb,
d,
[k*,kfl.
f
drag),
,
=
w, )
,
-drag1,f)
i,
Uglass, fmotor , Tmotor,
)
Comparing the expressions derived from Extended-IPT, with the formulae derived
using symbolic elimination (Chapter 5) for a similar inference,
KA-
=
[71kt (fbattery,
71kt (Ubattery,
line,
TEline,
motor,
b, Sf,
N-rive' fioss, d
d,
Emotor, kb, if, ?drive, loss'I 7
106
,
f
drag, [glass ),
, Lrag, Uglass)
Vbat
Trin
rmot
kt
kb
if
Tid
flos
d
W
fdrg
Vbattery
E
-
-
+
+
-
+
-
+
-
-
+
rsine
-
+
-
-
+
-
(E
-
-
-
+
-
+
+
+
+
-
-
dt
kb
+
-
+
+
+
+
-
-
rmotor
+
+
-
+
-
-
-
+
G
+
+
if
-
+
+
-
-
@
+
+
+
±
+
+
.2rive
T
+
-
-
+
+
-
-
+
-
+
+
+
-
-
+
+
+
+
fos
d
wfdrag
Vglass
-
I+
I-
i-
-
±
-
+
-
+
-
i+
-
-
t -
-
I+
-
-
I+
-
-
-
+
-
+
+
+
+
+
Vgjs
+
+
-
)
+
+
ED
-
-
-
E
-
I+
-
Table 6.2: QR Specific Monotonicity Table for the Glass Velocity Quantified Relation
we see that instantiations guided by Extended-IPT produce the same results as the
formula for K".b derived from IPT in Chapter 5. Several issues need to be addressed
before the theorem can be used by an automated reasoning tool. The table of monotonicities is currently built by qualitative reasoning. While this is easy for relatively
small systems, it is desirable to automate the determination of monotonicities, in
order to make Extended-IPT applicable to a more complex systems.
The next chapter presents the conclusions of the thesis, laying out some important
issues that must be addressed by future research.
107
Chapter 7
Conclusion
What we know is not much. What we do not know is immense.
- Pierre-Simon de Laplace (1749 - 1827)
This thesis has formally proved and demonstrated an extension to the existing
set-based inference mechanism for predicate logic design constraints. This concluding
chapter summarizes observations and research contributions from chapters 4 through
6. It prepares a list of issues that might be explored by future research in set-based
design theory. Using insights derived from current research, it identifies potential
challenges and suggests possible ways of addressing identified problem areas.
7.1
Research Summary
Starting with a detailed engineering description of the power window system (supported by analytical models in the appendices), the thesis focuses on specific design
goals of Ford engineers i.e. bounding glass velocity and stall force in the cable drum
power window system. This is followed by a formal, mathematically precise representation of the design intent, using first order predicate logic to denote design
constraints.
Surveying the state of the art in mathematical tools for set-based design, this thesis
attempts to apply currently available tools to the power window design problem. This
effort determines a limitation in the currently available approach: the BST-IPT set108
based inference mechanism [1], makes inferences only from a quantified relation that
embeds a single algebraic equation in its predicate.
The cable drum power window system is representative of a larger class of engineering systems, characterized by multiple subsystems (each with its own descriptive
model) interacting because of physical and geometric connectivity. The individual
mathematical models of various subsystems/components are inter-related and combined to capture the system behavior. This type of structure, common in engineering
systems, naturally leads to constraint models with systems of simultaneous algebraic
equations.
The "real-life" quantified relations, developed in Chapter 4 of the thesis, embed
not just single algebraic equations, but entire multi-equation constraintswithin their
predicates. These constraints have a symbolic form too complex to permit direct application of the current BST-IPT mechanism. This inadequacy is predicted in Section
8.2 of [1], which suggests that the form of algebraic models currently prescribed by
the IPT maybe too restrictive in practice. This thesis actually encounters the predicted limitation in a practical instance of engineering design. So it explores possible
methods to remove the limitation.
Chapter 5 first attempts to overcome the inadequacy of the existing IPT design
tool. It algebraically manipulates the power window QR's to make them consistent
with the form prescribed by IPT. Further, recognizing that this algebraic approach
can be computationally expensive, it formulates an alternative numerical approach
by extending existing set-based theory.
The resulting extension to set-based theory takes the form of a theorem, a calculation tool based on the the IPT. The Extended Interval Propagation Theorem
(Chapter 6) is the culmination of the research effort in this thesis1 . It can be directly
applied to quantified relations like those in the power window system. Examples
are provided to substantiate this claim. The philosophy of the theoretical extension
preserves the tested and proven logic of the BST-IPT mechanism.
'Additional tools and theoretical concepts required to understand and apply this newly developed
technology, are included in Appendices D and E.
109
7.2
Limitations and Future Work
In the course of modeling and analysis, the thesis records many instances where
set-based theory would benefit from new tools or further research.
Some of these
instances are highlighted here, with a brief description of the problems anticipated,
and potential approaches to solve them.
7.2.1
On Capturing Causality
The set-based modeling effort in this project is restricted to the electro-mechanical
model of the cable drum power window system. A more comprehensive picture would
emerge by including the motor-characterization experiment (described in Appendix
A), which was dropped from further consideration in Chapter 4.
Our current understanding of causality in terms of the "selector" concept explained in Chapter 3, is apparently inadequate to model the motor characterization
experiment. An effort to identify a selector for a motor constant (kt, kb, etc.) runs
into the issue of circular reasoning. The motor constant is determined in the motorcharacterization experiment, by measuring the current, voltage and torque in a test
system, under a particular quiescent condition (Appendix A). Thus, an initial attempt was made, to assign the experiment as a selector for the motor constants.
However, the physical quantities (i, v,
T)
that are measured are themselves dependent
causally on the motor constants that we try to calculate. This leads to undesirable
closed loops in the causal network. Notions of causality and methods to capture and
represent causality in engineering systems maybe able to resolve this issue.
7.2.2
On Better use of Temporality
The causal table representation contains temporal ordering of the selectors that influence the system. It sequentially orders value assignments to variables in the parametric model, suggesting a sequential decomposition of the system's simultaneous
algebraic relationships. The causal table motivates an approach of instantiating a
subset of the system variables and propagating their values through the network in
110
stages. At each stage, only a subset of nodes within the parametric network is determined by calculation to satisfy the network relations. The next stage of computation
will likewise satisfy more relations, and make new nodes consistent, in an incremental fashion. If the computation is terminated prematurely, a portion of the network
remains satisfied.
This idea indicates that we can choose to embed only certain relevant portions
(induced subgraphs or "sub-networks") of a larger parametric network within different
quantified relations on a single system2 . Identifying a sequential decomposition at the
stage of causal modeling itself can save a lot of effort in numerical computations with
the parametric model.
Research aimed at developing formal methods to use causal models in decomposing and directing parametric computations might prove useful Network directing
algorithms ([4], [5]) are an alternative, currently proven method for system decomposition. However, they tend to be complex to implement and do not always remove
inherently simultaneous sub-components in a constraint graph.
The causal table
might help us do this better since it inherently encourages an efficient, sequential
approach to parametric computation. The connection between causality and system
decomposition is currently quite tenuous and needs to be researched in great detail
before it can prove useful.
Current set-based tools account for controllability information in drawing inferences. A quantified relation contains temporal information in the sequential ordering
of quantifiers 3 . Neither the BST-IPT inference mechanism, not the Extended-IPT
makes any use of this temporal order to draw inferences, thus wasting this information.
Future research to include temporal concepts and nesting (sequence) of
quantifiers in making inferences might be able to draw stronger conclusions based on
2
This is actually done in Chapter 4, where only 5 of the 14 relations are embedded in the glass
velocity QR and only 2 of 14 relations are embedded in the stall force QR.
3
This ordering codes semantic information implicitly in the "nesting of quantification". The
scope of the quantifier in qixi E Xi extends all the remaining quantified expressions to its right,
i.e. over every qjxj E X 3 , j = i + 1,i + 2,...,k + 1. Thus we should read the meaning of
a quantified relation QR(X(k+l)) by interpreting nested parentheses in an expression of the type
QR(x(k+l)) = qlxl E X, - (q 2 x2 E
(q3X3 E X 3 - (...- (qk+lxk+1 E Xk+1) ...)) - F(X)-
111
set-based reasoning.
7.2.3
On Sensitivity Analysis
The concluding section of Chapter 5 stresses the importance of being able to carry out
a sensitivity analysis. This requirement is motivated by observing set-based inferences
drawn on the power window system. Sensitivity analysis in traditional optimization
studies [11] determines how robust an optimal solution is, when its parameters are
varied in the neighborhood of the solution.
In the set-based context, sensitivity
analysis on a set-bound computed for a variable Xp, indicates
A systematic method to determine an optimal order/sequence in which
to (a) relax the set bounds on one or more of the k other intervals,
XZ
E X, (i 4 p) instantiated in a quantified relation4 (b) change the
quantifier on one or more variables 5 , qj
(i = p), so as to compute an
improved' set bound on a variable X,.
The notion of "improving the set bound" is defined only in the context of a particular design problem. Sensitivity analysis can thus prescribe the course of action
necessary to achieve a cheaper, more robust feasible design.
Example 1. Infer a smaller lower bound for a controllable variable. This is useful in
any control/automation context, since such a bound can help select a smaller/cheaper
control actuator for a given control application. For instance, by one may be able
to select a lower-order filter or a pneumatic actuator with a smaller range of action
through an appropriate sensitivity analysis.
Example 2. Infer a larger/wider upper bound on an uncontrollable variable. This
allows cheaper (less tightly toleranced) manufacturing processes to be used to generate a particular characteristic parameter (dimension, material property etc.) in a
4
By moving the numerical interval endpoints in the instantiation of X. E.g. One simple type of
relaxation used in Chapter 4 involves setting Y- and xi further apart or closer together while holding
the mean, or midpoint of Xi constant.
5
Quantifier qj is "flipped" from V to 3 or vice versa, altering restrictiveness of quantification.
'Sensitivity analysis is especially relevant if the initially computed set bound is empty.
112
system. For instance, widening the upper bound on diameter of a cylindrical component through sensitivity analysis may allow us to use a turned component instead of
a finely ground one.
A sensitivity analysis would determine the manner in which designers should approach constraint relaxation, when faced with the prospect of invalid set bounds.
A method to approach constraint relaxation through QR factorization is presented
in the conclusion of Chapter 5. This would be an interesting area of future research,
since it will determine the practical usefulness of set-based inferences in "real-life"
engineering situations where designs often emerge out of compromise, and tradeoff
between competing constraints.
7.2.4
On Symbolic Elimination versus Numerical Methods
Chapter 5 presents the 2 alternative approaches to overcome IPT inapplicability, symbolic algebra and a numerical solution. This thesis eventually settles on a theoretical
extension using numerical methods, to avoid the complexity of symbolic manipulation. Symbolic computations do however offer certain advantages, like the ability to
produce closed form solutions, and a perfect analytical representation of the model.
Numerical methods are fraught with errors if not implemented very carefully. They
are at best, an approximation to an analytic alternative. A study of the tradeoffs
between these two competing approaches could prove useful. For instance, empirically comparing computational effort in several carefully constructed design problems
would reveal the relative strengths and weaknesses of the approaches in more detail.
Another interesting research area is to identify issues involved in merging the two
techniques 7 . The crucial issues involved are
(a) deciding conditions under which such decompositions is possible,
7
For relatively small problems, symbolic elimination wins. For large and complex problems, numerical solution of the system of equations seems more tractable. A marriage of the two methods will
may decompose a large problem into small sub-problems individually solved by symbolic elimination,
and then combine the results of this step by a numerical approach. This is similar to using a linear
sort algorithm (efficient for small problems, especially because of low data-management overheads)
as a sub-procedure within merge sort algorithm (asymptotically the fastest sorting algorithm) [14].
113
(b) finding the optimal decomposed sub-problem size at which symbolic elimination
should be used as a sub-procedure within a larger numerical framework, to make the
marriage of the two methods more efficient than just using using numerical solvers8 .
7.2.5
On Building Monotonicity Tables
The Extended-IPT assumes that the designer supplies the monotonicity information
to enforce condition 3 specified in its statement. This assumption depends on the
ability of human designer, to qualitatively reason about the physical system, use
symbolic mathematics, run simulations or perform other calculations and empirical
studies to actually determine the first k elements in the last column of the QR specific
monotonicity table Mx, (see Chapter 6 and Appendix E).
Without this input, the Extended-IPT can draw no inferences.
However, this
limitation is not as restrictive as it initially appears. The existing IPT draws similar
monotonicity information, but from a single relation. Automated large scale multidisciplinary optimization studies [11] also rely on monotonicity analysis to check optimization models for well-constrainedness.
Extensive research is recommended to
determine if we can adapt tools/methodologies from Al and optimization domains to
enable automation in the process of determining monotonicity. Of course, as identified in the concluding chapter of [1], set-based inference mechanisms that operate on
non-monotonic relationships will remove this restriction entirely.
7.2.6
On Quantifying more than k + 1 variables
The Extended-IPT allows the designer to quantify at most one dependent variable.
The numerical computation that determines the set bound actually expends enough
computational effort to determine all dependent variable values following an instantiation of the independent variables'. It may be possible to quantify more than one
dependent variable within the quantified relation, using a single instantiation to infer
8
In theoretical CS, optimal sub-problem size is determined by parameter balancing [14]
This is true of most solvers, especially if model-decomposition, or sequential satisfaction of
relations is not attempted to compute more efficiently.
9
114
more than a single set bound. This thesis has not directed any effort in this direction,
and the problem remains open for further research.
7.2.7
On Forms of Constraint
This thesis develops technology based on BST-IPT logic, applicable to systems of
simultaneous equations. An class of engineering problems that promises to be even
largerthan those modeled by multi-equation systems, is the class of problems modeled
by multi-inequality relations. Multi-inequality problems can possibly be adapted
and cast as multi-equation problems through the use of appropriately quantified slack
variables. This thesis has not explored inequality representations. This can be a fertile
area for future research problems.
7.3
Contributions
The scholastic contribution of the thesis is in overcoming the inadequacy of the existing set-based inference tool, the Interval Propagation Theorem. The results in this
thesis identify the limitations of IPT, and modify the theorem, enhancing its usefulness in engineering design. The logic of the BST-IPT inference mechanism is now
made applicable to a larger class of quantified relations, that have entire systems of
algebraic equations contained within their predicates.
The practical contribution of this thesis lies in in demonstrating set-based inferences on a real-life engineering design problem. This is the first instance where the
existing set-based theory has been tried on such a complex industrial problem. The
application has identified several areas of concern that must be explored before the
set-based paradigm can be embodied in large scale design automation CAD tools.
115
Appendix A
DC Motor Model
This appendix explains the parametric model of a permanent magnet DC motor. The
relations that govern DC motor operation are derived with appropriate explanations
of the underlying physics. The appendix also discusses an experimental method to
characterize a DC motor.
A.1
DC Motor Theory
Figure A-1 shows a schematic diagram of the DC motor. The motor generates an
electro-magnetic torque, Trn, on its rotor shaft internally. There is an opposing drag
torque, Td, acting on the shaft at the bearings. The drag torque is a result of viscous
damping by the air resistance, and the frictional losses at the bearings. A motor will
run at constant speed if a suitable external torque, Tshaft, is impressed on the shaft
baater
motor
Figure A-1: DC Motor Circuit
116
Td
haft
(e.g. by the cable drum mechanism) to oppose the electro-magnetic torque. Such a
motor is in a state of dynamic equilibrium, and the torque balance condition under
such a conditions gives us the relation
Tem -
Td
Tshaf t =
-
(A. 1)
0
Electro-magnetic theory reveals that the torque developed at the motor shaft is directly proportional to the current established in its armature coils. This gives us the
relation
Ter
=
kti
(A.2)
where kt is a constant of proportionality, called the torque constant. When the armature rotates, the motor produces a "back-EMF" voltage across its terminals. The
back-EMF opposes the applied battery voltage, and has a magnitude proportional to
the rate of rotation of the shaft. The back-EMF can thus be evaluated as kbw, where
kb is a constant of proportionality (back-EMF constant) and w is the angular velocity
of the rotor. Figure A-1 relies on this relation to introduce a voltage source that
opposes the battery. Applying Kirchoff's Loop law to the schematic circuit shown in
Figure A-1, the total potential drop across the loop sums to zero.
Vbattery
-
irline
-
irmotor
-
kbW
=
0
(A-3)
Solving for current i from the above relation yields
Vbattery -
kbW
+
nline
rmotor
In this work, a given motor is completely characterized by its physical motor constants,
kt, kb, rmotor and rd.
From the limiting cases of motor behavior, some additional
parameters can be defined to characterize the system.
117
A.1.1
Free Running Condition (No Load)
If there is no external load on the motor shaft
(Tshaft
= 0), the motor operates in
the free running (no load) condition. In this state, the electrical energy supplied by
the battery exactly balances energy dissipation by drag torque. The motor draws a
small current called the free running current, if, and attains a maximal free running
speed, wf. Free running current is analogous to the drag torque,
Td.
It is easier to
directly measure, than the drag torque itself, and is defined by applying Equation
A.2 at "no-load", to get the relation
(A.5)
if =
Free running current, if, is a constant for a given motor. Free running speed is a
constant for a given combination of motor, battery and line-resistance. Substituting
the free running conditions into equation A.3 the sum of potential drops across the
loop is zero.
Vbattery - ifrine - if rmotor - kbwf =
(A.6)
0
This relation can be solved for wf.
Vbattery Wf=
A.1.2
if (motor
+ rine)
(A.7)
kb
Stall Condition (Maximum Load)
The shaft can be brought to a halt (P = 0), by exerting a sufficiently large external
(shaft) load called stall torque, Tstal. By the definition of stall torque, the maximum
possible torque that the motor can deliver before it halts is
(Tstaii
+
Td).
From equation
A.2, the maximum current current in the circuit will correspond to the maximum
torque produced. At stall, the motor sinks a large current called stall current, tstaI,
which is also the largest current that can be established by the battery through the
118
i)
tstall
X=J/kT
IT
Ct
free
battery
free
ST
T
stall
T
stall
T
Figure A-2: DC Motor Linear Characteristics
given the resistances. Thus we have the following relations
_=
Tstall + Td
Vbattery
_
kt
rmotor
+ rine
Using equation A.5 to rewrite rd in terms of if, the above equation can be solved for
Tstaul
to get
TstaI
=
kt(
(A.9)
Vbattery
+ rnine
rmotor
For a given system, Tstal and Wf increase linearly with applied
Vbattery,
but scaled by
different factors -i- and kb1 respectively. The w - T line thus translates laterally
rmatar +rine~
outwards, perpendicular to itself, when Vbattery is increased.
A.2
Experimental Determination of motor constants
This project uses the linear model of a DC motor (Figure A-2), where shaft-torque,
T,
serves to correlate the speed and current characteristics. A set of four constants must
be specified for a motor to characterize it completely ({rmotor, kb, kt (or AIT), if (or Td)}).
The linear characteristics are simply written in terms of Wf, if and the slopes AST and
AIT:
w
= Wf + ASTT
(A.10)
i = if + AITT
(A.11)
119
Free Running Speed
I
11
Measured
10
1
------- Stal 1 urrent
5
9
dynamomete
20
8
Z!
-
7
6
15
5
rtachometer
4 -
10
1
3 -Measured
2
__
1
_
2
3
4
5
5
FeCurrenL
6
7
8
9
10
IN
=213
1
15
Stall Torque
Torque (T) in Nm
Figure A-3: DC Motor Characterisation Experiment
where the slopes
defined
wher thareslpesare
efied ass AsT
AT
=
-
(riine+rmotor)
and AIT
ktkb
=
1/kg. These relations
are relevant in the experiment that is used to characterize the motors. They are
just alternate compact notations for the motor characteristics that have been derived
earlier in this subsection. The model developed for the DC motor operation will be
used in the parametric model of the complete glazing system.
This section explains how engineers determine the values of the motor constants
experimentally. The motor manufacturer provides the experimental values measured
from a standard test on a sample motor. The experiment on a sample motor uses
a voltage source vtest and a line resistance rine, which are very precisely controlled
(as compared to the
Vbattery
and rnine in a real car). The experiment is performed as
follows.
Experiment
Two measurements of {T, w} pairs are taken as shown in Figure A-3.
" In the first measurement, torque
'T1
is precisely known and speed w1 is measured
within some error (tachometer error).
" In the second measurement, speed w2 is precisely known and the torque
T2
is
measured within some error (dynamometer error).
Two measurements of current are made (both with the same precision), under stall
120
and free-running conditions. Using the above experimental data, the motor characteristics are found by the following sequence of calculations. All the calculations
(listed below) are simple manipulations of the two-point analytic form of a straight
line (-I
=
_--1
-
slope). The exact relation between the straight line equation
used, and the measured and calculated constants can be seen from Figure A-3.
1. Stall Torque
Texpt
Tstal
2
+ W2
2
-1
-
WI
-
W2
(A.12)
2. Free-running speed
W1 +
expt
Wf2
W2
1
(A.13)
~1
3. Speed-torque slope
Wf
AST
(A.14)
- Ta
-
Tstall
4. Current-torque slope
1
All
-
1
stall Tstall
if(A.15)
5. Motor's internal resistance
rmotor
Vtest- -
rtest
istall
(A.16)
6. Back-EMF constant
kb
=
Vtest - if(rmotor + rtest)
Wf
(A.17)
'This step is equivalent to finding kt since the slope is just the reciprocal of the torque constant.
This fact must be used wherever an equation requires an experimentally determined kt.
121
Temporally
Dependent
Constrained by
the Relation
Priorly Computed
Dependent
Variable
Variables
estal
w e______Z
AIT
rmotor
kb
Refer
Appendix A
_G
G(Twi,2 w2 Tfv1 )
2 (TI,wT 212eq
2 ,w2 ,wf)
Tstall
A.12
A.13
eq A.15
-
eq A.16
w , rmotor
eq A.17
-_eq
-1
G3(istall, if, Tstall, AlT)
G4(istall, vtest, rtest, rmotor)
G 5 (ifVtestrtestWfrmotorkb)
Table A. 1: Relations to calculate Motor Constants from Experimental Data
variable E
Vtest
E
SET
Viest
rtest
E
Rtest
T1
Wi
T2
W2
E Ti
E Q1
E T2
E Q2
Nominalvalue
12.6V
0.18Q
2Nm
9.032 rad/s
9.07Nm
2.094 rad/s
istall
E
Istall
27A
if
E
If
4.5A
Table A.2: Experimental Data for Motor Characteristics Computations
122
variable E
Vbattery
E
SET
Vbattery
JNominal values
12.6V Nominal
14.4V Alternator On
nrine
E
Rline
x
E X
I
E L
0.044Q Front doors
0.198Q Rear Doors
X =[0, 1]
12 = 584mm
11 = 300mm
14 = 650mm
13 = 820mm
16 = 0mm
6 = 36.70
p = 80mm
= Omm
1 = 713.9mm
z = 100
l4, 1,1, , z, p}T
Figure B-1 (Section B.2, Appendix B)
12,
1 = {li 12,
6
E
A
= 0.0612N/mm
belt = 0.0201N/mm
61_4
6
{
}T
={6,
w
E
W
3.727kg
ndrive
E
Hdrive
0.554
floss
d
c
E
Floss
D
72.41N
50.8 mm
62,
63,
64,
6
belt }
Table A.3: Table of Parameters in the Electro-mechanical System
123
Appendix B
Drag Force and Load Torque
Equations
This appendix presents the force balance relations that calculate the glass engagement
with the seals, compute drag force and propagate the force information to determine
torque on the motor shaft.
Figure B-1 shows the engagement of the glass within the seals, and the various
characteristic lengths in the system. A parameter x, which is allowed to vary in the
interval [0,1], indicates the level to which the glass is raised. When x = 1 the window
is fully closed, and when x = 0 it is fully open. A particular point of interest in
the range of glass motion is specified by an intermediate value of x. A force-balance
calculation on the glass window system at every such point, shows how the drag
force and load torque vary with window position. The first step in this analysis is to
determine the lengths of the different seals applying friction forces against the glass.
B.1
9
Engagement Lengths
The actual vertical drop of the window is denoted h. A vertical length, p, of
the glass remains below the window belt-line permanently. Glass dimensions
are measured along the sides of the window (illustrated in the geometric model
of the moving glass as l through
16).
124
* The following equations are satisfied at any window position, from purely geometric considerations. The engagement lengths are denoted as lxy, where X
denotes which pillar is being considered (A or B), and y = a if we are looking
at the edge of glass above the belt-line. Likewise, y = b for glass below the
belt-line.
h + p - 12 cos z
if
lAa{
if
_
lBa=
_
+ h(l-x) K
-Oh(l-x)
z
13
COS z
12
if
12 +
h(
0
if
14
-
h(1-x)
h(1-x)
otherwise
11
-
12
-
if
lAa
-
14-COS z
1
K
13
0
< 0
Aa <
15
Ba
16
-
lBa
1
Scos z(h(1-x)-p)-li
cos ztan0
(B.3)
(B.4)
otherwise
if
12
(B.2)
14
16
lBb
B.1.1
11
otherwise
lAb
lbelz{
(B.1)
0
h(1-x)
15
11
=
(B.5)
otherwise
if
-
COS z
- p>h(1-x)
-
(B.6)
otherwise
Force Balance
The total drag force on the glass is calculated by multiplying engaged lengths with
appropriate seal/belt drag coefficients (measured in drag force per unit length of
125
Glass shown fullyraised (x=1)
h
.
17
1
beltline
P
A Pillar Seal
B Pillar Seal
Figure B-1: Glass Seal Configuration
engagement).
fdrag -
6
6
1lAa +
21 Ba +
6
3lAb +
6
4lBb +
6
beltlbelt
(B.7)
The resultant force on the motor shaft (the tension in the cable, that applies
torque on the shaft), fmotor, is computed by including the effect of glass weight, w,
and the drive mechanical efficiency 1. The model provided in the spreadsheet uses an
additional force loss component, floss that is lost in the joints, friction etc.2 The force
balance yields:
fmotor
=
fdrag k w + fioss
ridrive
(B.8)
Shaft torque is determined from fmotor, using the drum diameter d.
d
Tshaft =
fmotor- d
(B.9)
mechanical efficiency as used here is the force transfer function from the tangential force generated at shaft radius (by the motor shaft into the drive drum) to the force available at the output
of the cable-drum drive (serving to lift glass weight and overcome glass drag)
2
The + is applicable when the glass is being raised, the - when it is being lowered
1
126
Appendix C
Performance Metrics
This appendix explains how the performance metrics used in the QR's are actually
computed in the parametric model. The value of
Tshaft
is computed from the force
balance. It is then combined with the linear motor characteristics (using motor constants from the experiment). This is the connecting step between the electrical and
mechanical parts of the system and yields a value for motor speed, W (in rad s-).
1. Glass velocity
The linear speed of the cable (in ms- 1 ) is the glass travel velocity, v9.
By
applying a no-slip condition between the cable and the drum,
V9 = W.-
2
(C. 1)
2. Stall force
Denoted fsata,
stall force is defined as the extra force to be applied on the glass
at an instant when it is moving upwards, to bring the system to a halt. Using
this relation, stall force is determined by balancing forces and stall torque. At
stall, the motor's stall torque exactly balances the total torque applied by the
127
glass weight, stall force, drag and losses. 1
fmotor
(at stall) =
+ fstaii + w + floss
fdrag
'7drive
}
(C.2)
Using equation B.9, the force on the left hand side can be written in terms of
stall torque of the motor.
2stall
_
da
{idrive
fdrag + fstall + W
+
floss
(C-3)
The above equation is solved for fstal to get
fstaii = ?ldrive(
dau
-
fdrag -
W -
floss
(C4)
3. Mechanical Efficiency
Efficiency of the motor is determined as the ratio of shaft power to electrical
input.
W.Tshaft
rimotor =
(C.5)
Z-Vbattery
'The definition of stall force in the spreadsheet is suspect. It has been modified for the Set-Based
Model.
128
Appendix D
Lemmas to Prove Extended-IPT
This appendix presents the statement of the Implicit Function Theorem, of IFT,
a standard result from real analysis, which is used in this work. This subsequent
sections develop 3 additional results that are used as lemmas in support of the proof
of the Extended Interval Propagation Theorem.
D.1
Implicit Function Theorem
The Implicit function Theorem is stated here, without proof. It is a standard result,
from a family of theorems on continuous mappings. A formal proof is found in [10].
Lemma 1. (without proof)
The Implicit Function Theorem or IFT.
Let n, m, and k be positive integers, satisfying n = m + k.
Let A be an open set' in Rn.
Let be F : A -+ R m be a C' function 2
Write F in the form F(x, y), where x c
Rk
and y
E Rm
Suppose that (a, b) is a point in A, such that
'A set is open is every point in the set has a neighborhood lying in the set. An open set of radius
e and center xO is the set of all points x such that lix - x) 1 < e. In i-space, the open set is an open
interval.
2
A function is Cr if it is differentiable r times, with a continuous
129
rth
derivative
1. F(a, b) = 0 and
2. the determinant of the m x m matrix, whose elements are the derivatives of the
m component functions of IF, w.r.t the m variables written as y, evaluated at
the point (a, b), is not equal to zero, i.e.
Det ( )(a,b)
$
0
(i.e. the matrix is full-rank, or non-singular)
Then, there exists
(1) a neighborhood' B of a in
Rk ,
and
(2) a unique Cr function H : B -+ R",
such that H(a) = b and F(x, H(x)) = 0 for all x C B.
D.2
Supporting Lemmas
This section contains 3 lemmas to support the proof of Extended-IPT. The proofs of
these lemmas use the IFT, and the assumptions made in the statement of ExtendedIPT. They are best read in sequential order, since conclusions in each lemma are
carried over in proving the following lemma. The vector x referenced in these following
lemmas is the vector of parameters, x =
(Xi,
x 2, ... Xn), mentioned in the statement of
Extended IPT. The relation, F(x), is the system of m simultaneous nonlinear algebraic
equations that are assumed to be independent, once-differentiable, and asymptotefree over a domain of interest 4 .
3
An e-neighborhood of a point x E Rn, is a set of points inside an n-ball, with center x and
radius e > 0.
4The once-differentiable assumption about 1'(x) makes it a C' function. This fact carries over
into the decomposition suggested by the IFT.
130
D.2.1
Existence of a Partitioning
Lemma 2. In the neighborhood of a point xo satisfying F(xo), there exists a partitioning, x =
(Xki, Xm),
dividing the elements of vector xinto dependent, and indepen-
dent variables'.
Proof: The relation F(x) appearing in the Extended-IPT, is a system of m simultaneous, once-differentiable algebraic equations in n unknowns. It is a mapping from
an open set in R", to the space Rm
Extended-IPT assumes that these equations are independent everywhere over a
domain of interest. This assumption guarantees that the m x m Jacobian Matrix6
J
= 0_r
ax
is non-singular, or full-rank (rank = m) at all points in that domain.
Consider any such point, xo, within the domain, satisfying 17(xo). Matrix J must be
full-rank at xO. Since J is full-rank, a square (m x m) matrix S, that is also full-rank
(with rank = m) must be embedded7 in J. Matrix S can be identified by Singular
Value Decomposition' (or from the QR Decomposition) of Jacobian [7], [6].
In an orthogonal decomposition, of J like the SVD result shown below,
Jmxn = Umxm[ZmxmO] mxn
nxn
the first m columns of the matrix U constitute an orthogonal basis for the rangespace of J. The computational procedure for the decomposition employs a sequence of
column pivoting operations, and re-arranges the column indices of J while computing
U. Such a procedure uses a sequence of pivoting operations, and returns a vector of
integers containing the order in which the columns were chosen for pivoting. The
ordering in this vector puts columns keyed to dependent variables ahead of those
5
where the notions of dependence and independence are explained in the statement of the
Extended-IPT.
6
J is a matrix of partial derivatives of each equation, w.r.t. all the system variables.
7
Columns of S are a subset of the columns of J. The columns of J not included in S are merely
linear combinations of the m linearly independent columns of S
8
Alternatively we may use any procedure that determines a basis for the range-space of J. Many
such procedures can be found in the literature, and simple methods can be developed based on just
Gaussian Elimination.
131
keyed to independent ones. Thus, we identify the vector
Xk,
of independent variables
("x" in Lemma 1), as variables keyed to the first m columns of U. The remaining
variables are dependent, grouped as ("y" in Lemma 1). Thus, we can partition x
accordingly to get x =
(Xk, Xm).
QED
Note on Partitioning. The partitioning determined by such a numerical procedure
is only locally valid. It may change as we move across the domain of interest, as
relative scaling between the linearized equations will change, altering the pivot order
in an orthogonal decomposition of J. The aim of Lemma 2 is merely to guarantee that
some partitioning exists. This makes it certain that Extended-IPT can be applied.
The partitioning need not be unique, and one may use any other partitioning
of x, that is able to satisfy Conditions 1,2, and 3 of the Extended-IPT statement.
Such alternative partitions will also support the application of IFT to the system of
equations 1F(x).
D.2.2
Existence and Uniqueness of the Explicit Function
Lemma 3. In the neighborhood of any point xO satisfying 17(xo), there exists a
unique, continuous, differentiable function H, that computes the dependent variables,
xm,
using the independent variables, Xk, for a given partitioning x = (Xk,
Xm).
Proof: Let the values of the independent and dependent variables respectively be
XkO
and
xmo
at the point xO, i.e.
xO
=
(XkO, XmO)
satisfies F((XkO,
XmO)).
For the mapping realized by the system of equations in r(x), the matrix S, identified
in Lemma 2, is identical with the matrix of partial derivatives 9 used in the statement
of the IFT.
axm
5y_
Matrix S is non-singular or full-rank (rank = m) at the point xo (see proof of Lemma
2). Under this condition, IFT guarantees that there exists a positive number e, and
a unique once-differentiable10 function H, satisfying
9
The same matrix tested for singularity (checked for full rank) in IFT.
10By IFT, the explicit function H is C', if the implicit function
F is C 1 .
132
I(Xk, H(xk)) = 0
for all
Xk
satisfying IIXk
-
xkoII
< c
Given values of the independent variables Xk, function H calculates values of dependent variables,
Xm,
to satisfy IF((xk,
xm)).
The functional relationship,
xm
= H(xk),
holds within an E-neighborhood of the solution xO. QED
D.2.3
Perturbing a variable in F(x)
Lemma 4. Consider a particular variable x, E x. Partition the vector x, to identify
x, explicitly, x = (Xv,
X(n-1)).
Given a point xo = (xpO,
xo(n-1))
satisfying r (xo),
there exists a positive real number, 3, such that neither 1((xPO + 6, xo(n-1)), nor
7((xPO - 6, xo(n-), are satisfied. for any 6 E (0, 0).
Proof: By Lemma 3, H is valid within an c-neighborhood of the value of the
independent partition
XkO
in xO. All further references to E in this proof are indicate
the value of E defined by Lemma 3.
Any positive number, 6, will alter the state of equations in H. Perturbing the single
variable xp, positively (or negatively), will alter the value of the expression in the LHS,
or the RHS, of one or more equations in the representation
XP can only belong to one of the vectors,
Xk
xm
= H(xk). However,
or xm, but not to both. Thus, in an
equation within H, that is affected by the perturbation, the value of either the LHS
or the RHS expression will change, but not both. Such equation(s) that are altered
by the perturbation will necessarily have an inequality between the LHS and RHS,
and will no longer be satisfied.
Case 1. If x, E Xk, it is sufficient to select
Case 2. If xP E
Xm,
then
entire space Wm, then any
E , since H is restricted in domain.
=
# is either unrestricted (if the range of function
#3> 0 will suffice), or determined as follows:
Maximize each expression in the RHS of H, subject to the constraint I1xk
H is the
- xkoI1
< E.
Set / to the maximum value among all the computed maxima.
Thus, we can always find a suitable positive perturbation 6, such that H is violated
and the relations I((xpO + 6, xo(n-1)), and ]((xpO
QED
133
-
6, xo(n-1)), are no longer satisfied.
Appendix E
Monotonicity Tables
Chapter 7 repeatedly uses a table of monotonicity information, to instantiate variables. This appendix explains how such a monotonicity table is constructed.
It
describes 2 monotonicity tables,
(a) The Partition Specific Monotonicity Table, T, and
(b) The QR Specific Monotonicity Table M,,
It also details the Filling Algorithm by which the QR specific Monotonicity Table is
constructed using monotonicity data.
E.1
Partition Specific Monotonicity Table
Using the notation of Extended-IPT, this table, T is a k x m matrix, defined for a
particular partitioning x =
(Xk, Xm).
It contains the elements
T = sgn
(,)
The table has one row, for each independent variable xi E
for each dependent variable xj E
Xk,
and contains a column
xm.
The partial derivatives that are considered while building the table are described
in Chapter 7.
They are all computed at (or in the vicinity of) a point xo that
satisfies 17(xo). Each row in T contains the signs of partial derivative taken w.r.t. an
independent variable xi E xk. It is found by perturbing that independent variable xi,
134
while all other independent variables are held constant. All dependent variables in
xm
are allowed to vary freely in this computation, so as to keep F(x) satisfied. The
relative change in every dependent variable, from its value prior to the perturbation
is recorded as sgn
).
We do not currently have an automatic procedure to fill out the entries in T.
This might be possible by processing the entries in a symbolic Jacobian, but further
research is necessary to determine the conditions under which T can be derived form
the Jacobian. In the interim, we assume that it is built by the numerical (perturbation, finite difference) method outlined above, or simply filled manually by qualitative
reasoning. If we use qualitative reasoning, the symbols in T can come from intuitive
observations about a system's behaviors, experimental results, simulations, precise
finite difference calculation, or examination of an analytic/symbolic Jacobian etc.
Exactly one column of T must be known completely, to check the validity of
Condition 3, before Extended-IPT can be applied to any given to a quantified relation
that is known to satisfy Conditions 1 and 2 prescribed by the theorem. In Chapter 7,
the table T is not constructed explicitly for any of the examples. A list of qualitatively
inferred monotonicity relations is used as a substitute for a column of T.
Given the k-vector of +/- symbols in a column of T relevant to the dependent
variable
appearing in a quantified relation, a Filler Algorithm constructs Mxd,
which is the (k +1) x (k +1) matrix, or the QR Specific Monotonicity Table appearing
Xd
in the statement of the Extended-IPT. The entries of Mx, are defined
+
if
mxd(ij) =
-if
E.2
9xi > 0
xi
9xif < 0
(E.1)
A Filling Algorithm for Mxd
The table
MXd
is built by the QR Specific Monotonicity Table Filling Algorithm This
algorithm uses the vector of k elements in the column of T relevant to the dependent
variable Xd, as input data. It then computes the remaining k2 + k + 1 elements of
Mxd
efficiently. The notation used here is consistent with [14].
135
Filling Algorithm
Step 1. Fill every entry on the diagonal with a + sign.
For i = 1 to k + 1
m(i, i) <- +
Step 2. Read Input vector from T'
For i = 1 to k
m(i, k + 1) <- t(i, [Xd]) E T
Step 3. Fill the upper triangle
For i = k to 0
For j = i + 1 to k + 1
m(i, j) +- [m(i + 1, j).m(i + 1, p).m(i, p)
Step 4. Copy from upper triangle to fill symmetric lower triangle
For i = 2 to k + 1
For
j=
1 to i - 1
m(i, j) = m(j, i)
The explanation of the filling algorithm is as follows. To determine each element
m(i, j) of the matrix MXd, it uses one of the following rules.
Rule 1
If i = j, we are on the diagonal, and the partial of an element w.r.t. itself is unity,
with a positive sign, entered e in visual representations to distinguish it from other
derivatives' signs. The algorithm fills these as part of Step 1.
Rule 2
If xi E
xm
and xj E
Xk,
then xi is
Xd,
and mXd(i, j) is just the sign of the partial
derivative of the single quantified dependent variable
Xd.
This partial derivative is
guaranteed to exist, and maintain a uniform sign over a region of interest, by the
assumption of Condition 3. It is obtained as an input from the column of T, keyed
to the relevant variable
Xd,
using a simple table lookup. (in Chapter 7, it is found by
'Fill the upper k cells in the last, or (k + 1)'h column of MXd. This is done by copying the column
of T keyed to the variable Xd into these cells.
136
qualitative reasoning). These entries are read into M, in Step 2.
Rule 3
If Xi E xk and xj E xm, then xj is
Xd,
and the table entry mXd(i, j) is identical to
symmetric element (reflected in the diagonal), i.e.
mXd(j,
i), which is already defined
and entered in M.d , by Rule 1. (implicitly implemented by step 4)
Rule 4
If xi E
Xk
and xj E
Xk
then,
(a) The super-diagonal element m(i, j) is determined by an applying the derivative
chain rule and multiplying 3 table entries that are determined before it. The signs
are multiplied, as if a sequence of signed 1's are multiplied. Thus
m(i, j) = m(i + 1, j).m(i + 1, p).m(i,p)
because
2! +x
=
axp x gxi
axj
axj
Oxi+1
axp
->sgn(-i) = sgn(aa 1).sgn(-P).sgn(aam1) = sgn(2E+1).sgn(2+1).sgn(a)
(This logic is embodied in Step 3)
(b) The sub-diagonal element m(i, j) is found by copying the entry m(i, j) (reflected
in the diagonal, implementation is implicit in Step 4.)
137
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