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Classical Numerical, Slides of Aeronautical Engineering

Some concept of Advanced Control System Design for Aerospace Vehicles are Advanced Control System, Basic Principles, Calculus of Variations, Classical Control Three, Gain Scheduling and Dynamic Inversion. Main points of this lecture are: Classical Numerical Methods, State Equation, Optimality In Optimal Control, Necessary Conditions, Costate Equation, Boundary Condition, State and Costate Equations, Dynamic Equations, Formulation, Problems

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Uploaded on 04/27/2013

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Lecture – 26
Classical Numerical Methods to Solve
Optimal Control Problems
Dr. Radhakant Padhi
Asst. Professor
Dept. of Aerospace Engineering
Indian Institute of Science - Bangalore
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Lecture – 26

Classical Numerical Methods to Solve

Optimal Control Problems

Dr. Radhakant Padhi

Asst. Professor

Dept. of Aerospace Engineering

Indian Institute of Science - Bangalore

ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore

Necessary Conditions ofOptimality in Optimal Control

z

State Equation

z

Costate Equation

z

Optimal ControlEquation

z

Boundary Condition

,^

,

H

X

f

t X U

λ ∂

=

=



,^

,

H

g t X U

X

λ

∂ ⎛

= −

=

∂ ⎝



f

f ϕ X

λ

=

(

) 0

0

:Fixed

X t

X

0

,

H

U

X

U

ψ

λ

∂ ⎛

=

=

∂ ⎝

ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore

Classical Methods to SolveTPBVPs z

Gradient Method

z

Shooting Method

z

Quasi-Linearization Method

Gradient Method

Dr. Radhakant Padhi

Asst. Professor

Dept. of Aerospace Engineering

Indian Institute of Science - Bangalore

ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore

Gradient Method

(

)

(

)

(

)

f 0 f 0 f 0

T

f^

f

f

t

T

t t

T

t t

T

t

J
X
X
H
X

dt

X H
U

dt

U
H
X

dt

∫ ∫ ∫

ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore

Gradient Method z

After satisfying the state & costate equationsand boundary conditions, we have

(

)

t^ f^0

T

t

H
J
U

dt

U

z

Select

( )

,^

0

H

U

t

U

δ

τ

τ

∂ ⎡

= −

∂ ⎣

t^ f^0

T

t

H
H
J

dt

U
U

z

This leads to

ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore

Gradient Method: Procedure z

Assume a control history (not a trivial task)

z

Integrate the state equation forward

z

Integrate the costate equation backward

z

Update the control solution

This can either be done at each step while integratingthe costate equation backward or after the integrationof the costate equation is complete

z

Repeat the procedure until convergence

0

(a pre-selected constant)

t^ f

T

t

H

H

dt

U

U

γ

ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore

Gradient Method: Selection of z

Select

so that it leads to a certain

percentage reduction of

z

Let the percentage be

z

Then

z

This leads to

J

α

0

100

t^ f

T

t

H

H

dt

J

U

U

α

τ

=

0

100

t^ f

T

t

J

H

H

dt

U

U

α

τ =

ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore

13

MATHEMATICAL PERSPECTIVE: • Minimum time optimization problem• Fixed initial conditions and free final time problem SYSTEM DYNAMICS: Equations of motion for a missile in vertical plane. The non-dimensional equationsof motion (point mass) in a vertical plane are:

2

2

'^

sin( )

cos(

)

1

'^

[

sin(

)

cos( )]

where prime denotes differentiation with respect to the non-dimensional time

w

D

w

w

L^

w

M

S M C

T

S M C

T

M

γ

α

γ

α

γ

τ

= −

=

A Real-Life Challenging Problem

ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore

2

The non-dimensional parameters are defined as follows:

;^

;^

;

2

where

flight Mach numberflight path angle

thrust

mass of the

w

w

g

T

a S

V

T

S

M

at

mg

mg

a

M

T

m

ρ

τ

γ

=

=

=

=

= =

=

=

missile

reference aerodynamic area

speed of the missile

lift coefficient

drag coefficient

the acceleration due to gr

L

D

S

V

C

C

g

=

=

=

=

=

avity

the local speed of sound

the atmospheric density

flight time after launch

NOTE:

,^

are usually functions of

&

(tabulated data)

L

D

a t

C

C

M

ρ α

=

=

=

A Real-Life Challenging Problem

ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore

(^

2

2 2

Choosing

as the independent variable the equations are reformulated as follows:

sin( )

cos(

)

cos( )

sin(

)

cos

w

D

w

w

L

w

w

L

S M C

T

M

dMd

S M C

T

dt

a M

d

g S M C

γ

γ

α

γ

γ

α

γ

=

=

(^

2

0

0

0

( )

sin(

)

and the transformed cost function is

cos( )

sin(

)

A difficult minimum-time problem has been converted to a relativelyeasier fix

f

w

t

w

L^

w

T

dt

a M

J

dt

d

d

d

g S M C

T

π

π

γ

α

γ

γ

γ

γ

α

=

=

=

ed final-time problem (with hard constraint:

(

)

0.8)!

f

M

γ

=

A Real-Life Challenging Problem

ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore

Task

Solve the problem using gradient method. Assume

(0)

0.5 and

engagement height as 5 km. Next, generate the trajectories andtabulate the values of

for various

values.

Use the following system par

f

M

M

q

=

ameters

(typical for an air-to-air missile):

(^240) 0.

2

24,

0.53.

Use standard atmosphere chart for the atmospheric data.

D L m

kg

S

m

T

N

C C

= = =

= =

ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore

Necessary Conditions ofOptimality (TPBVP): A Summary

z

State Equation

z

Costate Equation

z

Optimal ControlEquation

z

Boundary Condition

,^

,

H

X

f

t X U

λ ∂

=

=



,^

,^

,

H

g t X U

X

λ

λ

∂ ⎛

= −

=

∂ ⎝



0

H U

=

f

f ϕ X

λ

=

(

) 0

0

:Fixed

X t

X

ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore

Shooting Method