Optimal Control: Finding Solutions in Engineering - Prof. E. Cliff, Study notes of Aerospace Engineering

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Optimal Control - what is a

solution?

We seek an

open-loop

function

t

∈

[

, T

]

→ u ( t ) ∈ Ω ⊂

IR

m

allowed ?What types of functions are

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continuous controls.We consider piecewise

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Optimize then Approximiate

➤ C of V - Four Classical Conditions

Euler - Lagrange

Legendre

Weierstrass

Jacobi

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Transcribing the Problem

POST Approach

TrajectoriesProgram to Optimize Simulated

control functionFinite parameterization of the

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Finite-dimensional Problem

Minimize

f (^) ( z )

while satisfying

g ( z ) = 0

problemNonlinear programming (NLP)

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Gill - Murray - Wright Chap 2

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Independent variables

z =

z 1

z ... 2

z n

 

∈ R

n

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Cost Functional

f

R

n

→ R

Constraints

g i (z)

i

= 1

,... , m

e

g j (^) (z)

≥ 0 j = m e

,... , m

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quadratic function

functionssum of squares of nonlinear

smooth nonlinear function

sparse nonlinear function

non-smooth nonlinear function

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Special Cases for Constraints

none

linear function

simple bounds

sparse linear functions

smooth nonlinear functions

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Comments

consideredDiscrete optimization is not

Review some linear algebra

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Real (Complex) Vector Space

elements (vectors) XA vector space is a set of

vector addition

x, y

∈ X → ( x + y ) ∈ X

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Subspaces

A

subspace

is subset S

X that

is a vector space itself,

i.e.

x , (^) y

S

(x + y)

S

x

S

, α

∈ I R → ( α

x)

S

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through the origindimensional subspaces are linesIn two dimensions the one

S =

is the zero-dimensional

subspace