2. Introduction
• Design of systems is a major field in engineering profession, which
includes analysis, design, fabrication, sales, R & D of systems, ….
• Existence of many complex systems is the testimonial of this
process. Ex: Airplanes, space vehicles, bridges, etc.
• Evaluations of these systems, whether it was the best one, has
been slow.
• Improved systems were designed only after a substantial
investment.
• Analyzing & design all possibilities can be time consuming & costly
affair.
• It requires computer technology and numerical computations to get
better system (less cost, easy to maintain & operate, and more
capability) in a short time.
• Design of systems can be formulated as problems of optimization in
which a measure of performance is to be optimized while satisfying
all constraints.
3. The Design Process
• Design is an iterative process which
implies analyzing several trial designs one
after another until an acceptable design is
obtained.
• Steps in the design process:
Need Recognition
Problem Definition
Synthesis
Analysis & Optimization
Evaluation
Presentation
• Many possibilities and factors must be considered during the problem
formulation phase.
Economic considerations
Interdisciplinary environment (Ex: Design
of a passenger car requires cooperation
among structural, mechanical,
automotive, electrical, human factors,
chemical, and hydraulics design
engineers.)
4. Continued …
• A system evaluation model:
Note:
• Re-examination may be required at any step, i.e. the feed back loops.
• Optimization concepts and methods can help at every stage in the
process.
5. Conventional Versus Optimum Design Process
• Conventional design process depends on the designer’s intuition,
experience, and skill.
• Conventional design process:
Advantages:
• Designer's experience and
intuition can be used for
conceptual changes in the
system to make additional
specifications in the procedure.
Disadvantage:
• Difficulties in treating complex
constraints and inputs.
• Can lead to uneconomical
designs
• Objective function that
measures the performance of
the system is not identified.
• Less formal.
6. Optimum Design Process
• It forces the designer to identify explicitly a set of design variables,
an objective function to be optimized, and constraint function for the
system.
Advantages:
• Designer gain a better
understanding of the problem.
• Formal.
• Substantial benefit from
designer’s experience &
intuition.
7. Optimal Design Versus Optimal Control
• The optimal control problem consists of finding feedback controllers
for a system to produce the desired output.
• The system has active elements that sense fluctuations in the output.
• System controls are automatically adjusted to correct the situation
and optimize a measure of performance, thus, control problems are
usually dynamic in nature.
• In optimum design, on the other hand, we design the system and its
elements to optimize an objective function and then the system
remains fixed for its entire life.
• Example: consider the cruise control mechanism in passenger cars.
The idea behind this feedback system is to control fuel injection to
maintain a constant speed; thus the system’s output is known, i.e.,
the cruising speed of the car. The job of the control mechanism is to
sense fluctuations in the speed depending upon road conditions and
to adjust fuel injection accordingly.
• Note: optimal control problems can be transformed into optimum
design problems, applications includes, Robotics and Aerospace
Structures.
8. Basic Terminology and Notation
• To understand and be comfortable with the methods of optimum
design, familiarity with linear algebra (vector and matrix operations)
and basic calculus is essential.
1. Sets and Points:
• Vectors & points can be used interchangeably.
• Bold lowercase letters are used to denote vectors & points.
• Bold uppercase letters are used to denote matrices.
Ex: is a point consisting of the two numbers.
is a point consisting of the n - numbers, often called
an n-tuple.
• The point with n - components can be collected into a column vector
as:
• where the superscript T denotes the transpose of a vector or a matrix.
9. Continued …
• A point or vector can also be represented in n-dimensional real space
denoted as Rn:
• Vector representation of a point P in three-dimensional space:
• For example, we may consider a set S of all points having three
components with the last component being zero, which is written as:
10. Continued …
• Members of a set are sometimes called elements.
• If a point x is an element of the set S, then we write x ∈ S.
• The expression “x ∈ S” is read, “x is an element of (belongs to) S.”
• Conversely, the expression “y ∉ S” is read, “y is not an element of
(does not belong to) S.”
• If all the elements of a set S are also elements of another set T, then S
is said to be a “subset of T.”
• Symbolically, we write S ⊂ T which is read as, “S is a subset of T,” or
“S is contained in T.”
• Alternatively, we say T is a superset of S, written as T ⊃S.
• As an example of a set S, consider a domain of the xl–x2 plane
enclosed by a circle of radius 3 with the center at the point (4, 4), as
shown in Figure below.
• Mathematically, all points within and
on the circle can be expressed as:
11. Continued …
2. Notation for Constraints
• Constraints arise naturally in optimum design problems.
• For example, material of the system must not fail, the demand
must be met, resources must not be exceeded, and so on.
• A constraint of this form will be called a less than or equal to type.
• It shall be abbreviated as “≤ type.”
• Similarly, there are greater than or equal to type constraints,
abbreviated as “≥ type.”
• Both types are called inequality constraints.
• Considering the above example:
• The set S is defined by the following constraint:
12. Continued …
3. Superscripts/Subscripts and Summation Notation
• Superscripts are used to represent different vectors and matrices.
• For example, x(i) represents the ith vector of a set, and A(k)
represents the kth matrix.
• Subscripts are used to represent components of vectors and
matrices.
• For example, xj is the jth component of x and aij is the i–j element of
matrix A.
• Double subscripts are used to denote elements of a matrix.
• To indicate the range of a subscript or superscript we use the
notation xi; i = 1 to n.
• Similarly, a set of k vectors, each having n-components, will be
represented as x( j ); j = 1 to k
• Summation notation: For example,
will be written as:
13. Continued …
• Example: multiplication of an n-dimensional vector x by an m x n
matrix A to obtain an m-dimensional vector y, is written as:
• If we let x and y be two n-dimensional vectors, then their dot
product is defined as:
• Or, in summation notation, the ith component of y is:
4. Norm/Length of a Vector
• Thus, the dot product is a sum of the product of corresponding
elements of the vectors x and y.
• Two vectors are said to be orthogonal (normal) if their dot product is
zero, i.e., x and y are orthogonal if x · y = 0.
• If the vectors are not orthogonal, the angle between them can be
calculated from the definition of the dot product:
• where ɵ is the angle between vectors x and y, and ||x|| represents
the length of the vector x, or the norm of the vector x.
14. Continued …
5. Functions
• The length of a vector x is defined as the square root of the sum of
squares of the components, i.e.,
• The double sum notation, for scalar quantity c, can be written in the
matrix form as follows:
• Since Ax represents a vector, the triple product of the above
equation will be also written as a dot product:
• Functions of n – independent variables:
• The ith function:
• If there are m – functions:
• A function f(x) of n – variables is called continuous at point x* if
for any ϵ>0, there is a ẟ>0 such that:
and
• Thus, for all points x in a small neighborhood of the point x*, a change in
the function value from x* to x is small when the function is continuous.
15. Continued …
• Ex: Continuous and differentiable functions:
• Ex: Continuous but not differentiable function:
• Ex: By graph: Continuous and discontinuous functions. (A) Continuous
function. (B) Continuous function. (C) Not a function (because it has
infinite values at x1). (D) Discontinuous function.
