Optimal Control, Guidance and Estimation · 2017. 8. 4. · OPTIMAL CONTROL, GUIDANCE AND...

Lecture Lecture Lecture Lecture –––– 22222222

Transcription Method to Solve Optimal Control Transcription Method to Solve Optimal Control Transcription Method to Solve Optimal Control Transcription Method to Solve Optimal Control

ProblemsProblemsProblemsProblems

Prof. Radhakant PadhiProf. Radhakant PadhiProf. Radhakant PadhiProf. Radhakant Padhi

Dept. of Aerospace Engineering

Indian Institute of Science - Bangalore

Optimal Control, Guidance and Estimation

OPTIMAL CONTROL, GUIDANCE AND ESTIMATION

Prof. Radhakant Padhi, AE Dept., IISc-Bangalore2

Topics

� Motivation

� Philosophy of Transcription Method

� Pseudospectral Transcription

� A Toy Problem

� An Application Problem

MotivationMotivationMotivationMotivation






Optimal Control Formulation

Indirect Approach:-Variational Calculus

Direct Approach:-Dynamic Programming

-Transcription Method



Necessary Conditions of Optimality

through Variational Calculus

(Dualization)

� State Equation

� Costate Equation

� Optimal Control Equation

� Boundary Condition

( ), ,H

X f t X Uλ

∂= =

∂ɺ

( ), ,H

g t X UX

λ∂

= − = ∂

ɺ

f

fX

ϕλ

∂=

∂( )0 0

:FixedX t X=

( )0 ,H

U XU

ψ λ∂

= ⇒ = ∂



Shooting Method Philosophy

� Guess the initial condition for costate

� Compute the control at each grid point

� Propagate the state and costate

� Calculate the final boundary condition and error in the costate at the final time

� Correct the costate vector at the initial time based on this error at the final time

� Repeat the procedure

( )10λ

( )0 2λ

fλ∆

t

( )tλ

ft0t



Problems in Shooting Method

� Sensitivity of the procedure to the initial guess value of costate

� Costates do not have ‘physical meaning’: complicates the issue of selecting ‘good’initial values (it is usually done through guessing a control history)

� Costate equation is normally unstable for stable state dynamics: Long-duration prediction is not good!



Multiple Shooting Approach

� Strategy: “Divide-and-Rule”; i.e. divide the control application duration to multiple segments and solve the individual segments independently (possibly in a parallel setting to speed up the solution).

� This approach is called “Multiple Shooting”

� It brings in additional constraints of continuity and smoothness at the ‘joining points’.

� Extension of this philosophy leads to “Transcription Method”: A Direct Approach!

t

( )x t

ft0

ti

t

Transcription MethodTranscription MethodTranscription MethodTranscription Method






Philosophy of Transcription Method

� Convert the dynamic system variables into a finite set of static variables (or parameters)

� Pose an equivalent “static optimization” problem

� Solve this static optimization problem using static (paramter) optimization methods [e.g. using Nonlinear Programming (NLP)]

� Assess the accuracy

� Repeat the steps if necessary



Problem Objective & Philosophy

Philosophy: Select grid points, Discretize the states and control

variables, Convert the problem to a nonlinear programming (NLP) problem

and solve that problem, preferably in a computationally efficient manner.

0

0

0

( (.), (.)) ( ( ), ( )) ( ( ), ( ))

(

in t

) ( ( ), ( ))

( ( ), ( )) 0

( ( ), ( )) 0

in t

ft

f

t

f

J E t t L t t d t

t f t t

M in im ize

S u b je c t to

w i

e t t

h

th e n d p o c o n d itio n s

a n d p a th c o n stra s

t t

= +

=

=

≤

∫x u x x x u

x x u

x x

x u

ɺ

Objective:



( ) ( ) ( )0 0 1 1, , ,N Nx u x u x u

Free variables :

⋯

Ref: I. M. Ross, Lecture notes, Short course in AIAA GNC-2010, Toronto

Key Components of

Direct Transcription



( ) ( ) ( ) ( ){ }0 0 1 1 1 1, 2 , 2 , ,2

N N N N

hF x u F x u F x u F x u− −≈ + + + +⋯

Key Components of

Direct TranscriptionReference: I. M. Ross, Lecture notes, Short course in AIAA GNC-2010, Toronto



Direct Transcription

ϕ ϕ

= ⇒ =

= ⇒ =

≤ ≤ ≤ ⇒

≤

⋯

* *

0 0 0

N

0 0

1 1

N N

x x x x

x x

x u

x ux u

x u

0( )

( ( )) 0 ( ) 0

( , ) 0

( , ) 0( ( ), ( )

int

i

) 0

( , )

t

0

n

f

End po conditions

Path const

t

t

h

hh

h

ra s

t t



Direct Transcription:

Other Ideas

� Better Approximation of State Dynamics

• Higher-order Finite difference

• RK Methods

• Polynomial Approximations in Segments

� Better Approximation of Cost Function

• Higher-order Approximations

• Quadrature Approximations

� Finite Element Approach ……..etc.



Transcription Method

� Accuracy

• Higher number of grid points

• Indirect Transcription

� Computational Efficiency

• Sparse Algebra

• Mesh Refinement



References

� C. R. Margraves and S. W. Paris, “Direct Trajectory Optimization Using Nonlinear Programming and Collocation”, J. of Guidance, Vol.10, No.4, 1987, pp. 338-342.

� P. J. Enright and B. A. Conway, “Discrete Approximations to Optimal Trajectories Using Direct Transcription and Nonlinear Programming”, J. of Guidance, Control, and Dynamics, Vol. 15, No. 4, 1992, pp.994-1002.

� H. J. Pesch, “A Practical Guide to the Solution of Real-Life Optimal Control Problems”, Control and Cybernetics, Vol.23, No.1/2, 1994, pp.7-60.

� D. G. Hull, “Conversion of Optimal Control Problems into Parameter Optimization Problems”, Vol.20, No.1, 1997, pp.57-60

� J. T. Betts, “Survey of Numerical Methods for Trajectory Optimization”, J. of Guidance, Control, and Dynamics, Vol. 21, No. 2, 1998, pp. 193-207. (The author has written a Book as well)

Pseudospectral TranscriptionPseudospectral TranscriptionPseudospectral TranscriptionPseudospectral Transcription






Problem Objective

Philosophy: Discretize the states (and the control) using

Pseudospectral (PS) method, Convert the problem to a “lower-dimensional” nonlinear programming (NLP) problem and solve that problem in a computationally efficient manner.

0

0

0

( ( ), ( )) ( ( ), ( ))

( ) ( ( ), ( ))

( ( ), ( )) 0

( ( )

in

, ( ))

t

0

int

ft

f

t

f

Minimize

Subject to

wit

J E t t L t t d

h end po conditions

and path constra

t

t f t t

e

s

t t

h t t

= +

=

=

≤

∫x x x u

x x u

x x

x u

ɺ



Steps involved…

� 1. Approximate x(t) and/or u(t)?

� 2. Selection of grid points� How are these points selected?

� Uniform grid is not a very good choice!

� 3. Discretize the differential equation using PS method� Finite-difference Vs Spectral

� Sparse Vs Dense differentiation matrix

� 4. Approximate the integral equation� Quadrature rules

� 5. Apply an efficient finite optimization technique and

solve the lower dimensional NLP problem.

0

ˆ( ) ( )N

n n

n

x t a tφ=

= ∑ φ=

= ∑0

ˆ( ) ( )N

n n

n

u t b t



1. Approximation

ˆ: ?R f SMALL= −xɺ

It can be thought of as a function which satisfies the

boundary condition and makes the residualx̂

φ φ φ= …0 1: ( , , , )NNP- Define Trial functions:

Approximate φ φ= =

= =∑ ∑0 0

( ),ˆ ˆ( ) ( ) ( )N N

n nn n

n n

x t a u t bt t

- Define Test functions: χ χ χ…0 2( , , , )N

, 0 {0,1,..., }n

R n Nχ = ∀ ∈

( ) ( ( ), ( ))t f t t=x x u ɺ

( )n n

t tχ δ= −



2. Selection of grids

Ref: I. M. Ross, Lecture notes, Short course in AIAA GNC-2010, Toronto



2. Selection of grids

Grid of collocation points (or grid points) tn, n=0,…,N are points

such that it satisfies the state equation exactly at these points.



3. Approximating the

differential equation

( )

� � �

φ φ φ

δ

φ φ φ

= = =

= = =

=

=

=

−

= =

∑ ∑ ∑

∑ ∑ ∑

ɺ

ɺ

ɺ

ɺ …

A number A number A number

x x u

x x u

a

Multiply with on both sides:

a

0 0 0

0 0 0

( ( ), ( ))

ˆ ˆ ˆ( ( ), ( ))

( ) ( ), ( )

( ) ( ) , ( ) , 0,1, ,

N N N

n n n n n n

n n n

n

N N N

n n n n n n n

n n n

n n

tf

f

f a t b t

t t

f

t

t t

t

t a t b t n N

φ φ= =

= =∑ ∑0 0

ˆ ˆ( ) ( ), ( ) ( )N N

n n n n

n n

x t a t u t b t

State Equation Constraint:

Approximations:



� The Chebyshev polynomials

� Polynomial Approximation

� The differential yields

0

1

2

2

3 2

3

4 3 2

4

( ) 1

( ) 2 1

( ) 8 8 1

( ) 32 48 18 1

( ) 128 256 160 32 1

T t

T t t

T t t t

T t t t t

T t t t t t

=

= −

= − +

= − + −

= − + − +

0 0 1 1 2 2 3 3 4 4

0

0 1

0 1 1 2 2 3

2 3 4

3 4 4

ˆ( ) ( ) ( ) ( ) ( ) ( )

ˆ( ) ( ) ( ) ( ) ( ) ( )

Select Grid Points & Evalua ,te , ,: ,

x t a T t a T t a T t a T t a T t

x t a T t a T t a

t t t t t

T t a T t a T t

= + + + +

= + + + +ɺ ɺ ɺ ɺ ɺ ɺ

=

ɺɺ ɺ ɺ ɺ ɺ

ɺ ɺ ɺ ɺ ɺ ɺ

ɺ ɺ ɺ ɺ ɺ ɺ

ɺ ɺ ɺ ɺ ɺɺ

ɺɺ

00 0 1 0 2 0 3 0 4 0

1 0 1 1 1 2 1 3 1 4 1

0 2 1 2 2 2 3 2 4 22

0 3 1 3 2 3 3 3 4 33

0 44

ˆ( ) ( ) ( ) ( ) ( ) ( )

ˆ( ) ( ) ( ) ( ) ( ) ( )

ˆ ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( )ˆ( )

(ˆ( )

x t T t T t T t T t T t

x t T t T t T t T t T t

T t T t T t T t T tx t

T t T t T t T t T tx t

T tx t

= ɺ ɺ ɺ ɺ

0 0 0

1 1 1

2 2 2

3 3 3

4 4 41 4 2 4 3 4 4 4

,

) ( ) ( ) ( ) ( )

a a b

a a b

a a bf

a a b

a a bT t T t T t T t

Differentiation matrix

3. Approximating the differential equation

(Example)

: An algebraic constraint



4. Discretizing the integral equation - Gauss Quadratures

1

1

ˆ ˆ( ( ), ( )) ( ( ), ( ))i i i

L x t u t dt w L x t u t

−

≈∑∫

A quadrature rule is an approximation of the definite integral

of a function, usually stated as a weighted sum of function

values at specified points within the domain of integration.

=

−≅ = + ∑x u x u

0

0

ˆ ˆ ˆ ˆ( ( ), ( )) ( ( ), ( ))2

NfN

N N n n n

n

t tJ J E t t w L t t



Finally,

The optimal control problem has been simplified to a lower dimensional nonlinear programming problem.

�φ

=

=

−= +

= ≤ ≤

=

∑

∑ ɺ

A number

x x x u

a x u

x x

x u

0

0

0

0

0

ˆ ˆ ˆ ˆ( ( ), ( )) ( ( ), ( ))2

ˆ ˆ( ) ( ( ), ( )) 0

ˆ ˆ( ( ), ( )) 0

ˆ ˆ( ( )

,

,

int ,

int , , (

NfN

N n n n

n

N

n n

n

n

n

n

n

n

N

Minimize

Subject to

with end po conditions

t tJ E t t w L t t

f n N

e t t

and path cons

t t t

htra t ts ≤ ≤ ≤)) 0 0 n N

φ=

= ∑0

( ( )ˆ )N

n

n

nu t tbφ

=

= ∑0

( ( )ˆ )N

n

n

nx t ta

A Toy ProblemA Toy ProblemA Toy ProblemA Toy Problem

Dr. Radhakant PadhiDr. Radhakant PadhiDr. Radhakant PadhiDr. Radhakant PadhiAssociate Professor





Toy Problem

� Step 1. Decide on which polynomial we are planning as the trail

function

�(i) Chebyshev Polynomials of the first kind. Say take N=4

(First 5 polynomials)

1

2

0

( )

( ) ( ) ( )

( (0), (1)) (1, )

( ) 1

J u t dt

x t x t u t

Minimize

Subject to

Boundary conditio

u

n x x e

t

=

= +

=

≤

∫ɺ

T t T t t T t t T t t t T t t t2 3 4 2

0 1 2 3 4( ) 1, ( ) , ( ) 2 1 ( ) 4 3 , ( ) 8 8 1= = = − = − = − +



�(ii) Shifted Chebyshev polynomials for the interval [0,1]T t

T t t

T t t t

T t t t t

T t t t t t

0

1

2

2

3 2

3

4 3 2

4

( ) 1

( ) 2 1

( ) 8 8 1

( ) 32 48 18 1

( ) 128 256 160 32 1

=

= −

= − +

= − + −

= − + − +

Step 2. Define the collocation (grid) points

i

it a b a b i N

N

1 1( ) ( )(cos( )); 0,1,..., 1

2 2 1

π= + + − = −

−

0 0 .1 4 6 5 0 .5 0 .8 5 3 5 1T im e ( s e c s )

Toy Problem (Contd.)

�Shifted



Step 3. Approximate x(t) and u(t) using trial function

n n

n

n n

n

x t a T t

u t b T t

4

0

4

0

ˆ ( ) ( )

ˆ ( ) ( )

=

=

=

=

∑

∑

d

dt

0 2 0 6 0

0 0 8 0 16

0 0 0 12 0

0 0 0 0 16

0 0 0 0 0

=

Step 4. Find the differentiation matrix and equate the

state equation at the grid points

k k k

dx t x t u t

dtˆ ˆ ˆ( ( )) ( ) ( )= +




Step 5. Compute Tn(t) matrix for ti and equate thestate equation, i = 0,1,2,3,4

T

1 1 1 1 1

1 1/ 2 0 1/ 2 1

1 0 1 0 1

1 1/ 2 0 1/ 2 1

1 1 1 1 1

− −

− − −=

− −

x t a

x t a

aT Dx t

ax t

ax t

00

1 1

22

33

44

ˆ( )

ˆ( )

( )ˆ( )

ˆ( )

ˆ( )

= ×

ɺ

ɺ

ɺ

ɺ

ɺ

Finally, we have a b

a b

a bT T D I

a b

a b

1

0 0

1 1

2 2

3 3

4 4

( )−

× − =




Step 6. Apply boundary conditions

This gives,

x x e( (0), (1)) (1, )=

a a a a a

a a a a a

0 1 2 3 4

0 1 2 3 4

1

2.7182

− + − + =

+ + + + =

a

x t a

x t a

x t a

e a

0

1 1

2 2

3 3

4

1 1 1 1 11

( ) 1 1/ 2 0 1/ 2 1

( ) 1 0 1 0 1

( ) 1 1/ 2 0 1/ 2 1

1 1 1 1 1

− − − − −=

− −

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Time(secs)

State, x(t)

Fix x(0) = 1

Fix x(1) = e




Step 7. Approximate the integral equation

{ }iw b a

n( ) 1.57 0.7854 0.5236 0.3927 0.3142

1

π= − =

+

N

k k

k

J w u t4

2

0

1ˆ ( )

2 =

≅ = ∑J u t dt

1

2

0

( )= ∫

where




Finally, we have

Apply any KKT (or any other static optimization technique) for solving the optimal control problem

42

0

4 42

0 0

4

1 2

0

1ˆ ( )

2

ˆ ˆ ˆ( ) ( ) ( ) 0

ˆ ( ) 1 0 0 ,1 , 2 , 3 , 4

1ˆ ˆ ˆ ˆ( ) ( ( ) ( ) ( ) )

2

ˆ ˆ ˆ( ( ) 1 ) ( (0 ) 1 ) ( ( ) 1 )

N

k k

k

k k k

k

N T

k k k k k k

k k

T

k k

k

J w u t

x t u t x t

u t f o r k

J w u t x t u t x t

u t x x e

=

= =

=

=

+ − =

− ≤ =

= + + − +

− + − + −

∑

∑ ∑

∑

ɺ

ɺλ

µ ν ν

D e f in e th e a u g m e n te d c o s t fu n c t io n ,




0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

State, x(t)

Time (secs)

Interpolated

Collocation point

Exact

x(1) =2.7182

x(0) =1

a a a a a0 1 2 3 41.7534 0.8503 0.1052 0.0088 0.0005= = = = =

Toy Problem: Results



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2

-1

0

1

2

3

4x 10

-3

Time (secs)

Control, u(t)<=1

b b b b b0 1 2 3 40.0014 0.0004 -0.0009 0.0006 -0.0016= = = = =

Toy Problem: Results

MinimumMinimumMinimumMinimum----time Fronttime Fronttime Fronttime Front----totototo----back Turning of an Airback Turning of an Airback Turning of an Airback Turning of an Air----

Launched Missile using PseudoLaunched Missile using PseudoLaunched Missile using PseudoLaunched Missile using Pseudo----Spectral MethodSpectral MethodSpectral MethodSpectral Method




Full Paper: Proceedings of 2012 IFAC EGNCA Workshop, Bangalore



Reference

Vinita Chellappan and Radhakant Padhi,

Flight Path Angle Reversal of an Air-to-

Air Missile in Minimum Time Using

Pseudo-spectral Method, IFAC

Workshop on Embedded Guidance,

Navigation and Control (EGNCA), 2012,

Bangalore, INDIA.



The problem

� Launch a missile from a aircraft in the forward direction and attack a target in the rear hemisphere.

� Missile has to reverse the flight path angle as quicklyas possible so as to intercept the target.

� The problem is to reverse the flight path angle from an initial value (around 00) to -1800 in minimum time.

� After turning it should also have the required velocity (a constraint on the final Mach number) to intercept the target.

Aircraft

Missile

Target



The problem –

Cost function and boundary conditions

0

ft

J dt= ∫Minimize

subject to the constraint

γ γ= = −

= =

0 0

0

0

( ) known (around 0 ), ( ) 180

( ) known, ( ) 0.8

f

f

t t

M t M t

The problem is to reverse the flight path angle from 00 to -1800

maintaining a final Mach number of 0.8 in minimum time.



System Dynamics

The point mass equations of motion:

( ) sin cos( )

1( ) ( cos sin( ))

( ) sin

( ) cos

b

b

D W TV t

m m m

t L W TmV

h t V

x t V

γ α δ

γ γ α δ

γ

γ

= − − + +

= − + +

=

=

ɺ

ɺ

ɺ

ɺ



System DynamicsRef: Han et al., “State constrained Agile Missile Control with

Adaptive Critic Based Neural Networks”, JGCD, 2002

The non-dimensional parameters were considered as:

2

2

2

2

'( ) sin cos( )

1'( ) ( cos sin( ))

sin'( )

cos'( )

w D w b

w L w b

M S M C T

S M C TM

a Mh

g

a Mx

g

τ γ α δ

γ τ γ α δ

γτ

γτ

= − − + +

= − + +

=

=

2

= ; = ; = ; =2

w w

g T a S VT S M

at mg mg a

ρτ

Non-dimensional point mass equations of motion:



Minimum time problem

� Free final time problem - state constrained problem, with bounds on control.

� Equations of motion are reformulated using flight path angle as the independent variable.

� Leads to a fixed final condition.

� Assumption: Flight path angle is a monotonic continuous function with respect to time.



as independent variable

Modified state equations with respect to flight path angle

0 0 0

2( cos sin( ))

f f ft

w L w at

dt aMJ dt d d

d g S M C T

γ γ

γ γ

γ γγ γ α δ

= = =− − + +∫ ∫ ∫

' 1 ' '= ; = ; = ; =

' ' ' '

dM M dt dh h dh x

d d d dγ γ γ γ γ γ γ γ

Modified cost function

γ

The minimum time problem Hard constraint problem( ) 0.8

fM γ =



Nonlinear programming problem

γ

γ α γ δ γ γ

γ α γ δ γ γγ

γ α γ δ γ γ

=

=

−=

+ + −

− + + − = + + −

∑

∑ ɺ

0

2

0

2

2

0

ˆ ( )

ˆ2 ˆ ˆ( ( ) sin( ( ) ( )) cos )

ˆˆ ˆ( ) cos( ( ) ( )) sin )( )

ˆˆ ˆ(

Minimize

Sub( )

ject to

Subjec

sin( ( ) ( )

t

co

t

) s )

NfN k

k

kw k L w k a k k

Nw k D w k a k k

k k k

kw k L w k a k k

t t aMJ w

g S M C T

S M C Ta T

g S M C T

α γ δ γ

γ

γ

≤ ≤

= = ∈

= =

0 0

0 0 0 0 0

o path constraints

and

with end point con

ˆˆ( ) 20 ( ) 72

ˆ( ) ( ) where [0.3,0.8]

ˆ(

diti

) ( ) 8

on

.

s

0

k k

f f N

e x M M M

e x M

γ γ α γ γ δ γ γ

γ γ γ γ

γ

= = =

= =

= = =

= =

=

∑ ∑ ∑

∑ ∑

0 0 0

0 0

0 0

ˆˆ ˆ( ) ( ); ( ) ( ); ( ) ( )

ˆ ˆ( ) ( ); ( ) ( )

Grid points, , 0,...,

N N N

k k k k k k k k k k k k

k k k

N N

k k N k k N

k k

k

M a T b T c T

M a T M a T

k N

0 1 2 3 4 50.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time, sec.

Mach n

um

ber,

M

-400 -300 -200 -100 0 100 200 300 4004200

4300

4400

4500

4600

4700

4800

4900

5000

Range, m

He

igh

t, m

0.8

0.7

0.6

0.5

0.4

0.3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-21

-20

-19

-18

-17

-16

-15

-14

-13

-12

-11

Time(sec)

AO

A,

α,

deg

0.8 0.7 0.6 0.5 0.4 0.3

0 1 2 3 4 5-74

-72

-70

-68

-66

-64

-62

-60

-58

-56

Time, sec

δb,

deg

M=0.8



Mach and AOA Vs Flight path angle

-180 -160 -140 -120 -100 -80 -60 -40 -20 00.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Flight path angle, γ, deg

Mach n

um

ber,

M

0.8

0.7

0.6

0.5

0.4

0.3

-180 -160 -140 -120 -100 -80 -60 -40 -20 0-21

-20

-19

-18

-17

-16

-15

-14

-13

-12

-11

Flight path angle, γ (deg)

AO

A,

α,

deg

0.8

0.7

0.6

0.5

0.4

0.3



Flight path angle history

0 1 2 3 4 5-180

-160

-140

-120

-100

-80

-60

-40

-20

0

Time, sec

Flig

ht

path

angle

, γ,

deg

0.8

0.7

0.6

0.5

0.4

0.3



Effect of reducing the AOA

0 1 2 3 4 5 6-88

-86

-84

-82

-80

-78

-76

-74

-72

-70

Time, sec

Sh

ear

angle

, δ

v,

de

g

α=-30

α=-25

α=-20

α=-15

α=-10

0 1 2 3 4 5 6-40

-30

-20

-10

0

Time(sec)

AO

A,

α,

deg

0 1 2 3 4 5 6-80

-70

-60

-50

-40

Time, sec

δb,

deg



Effect of reducing the AOA on Mach number

along with the flight path angle

0 1 2 3 4 5 60.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

Time, sec.

Mach n

um

ber,

M

α=-30

α=-25

α=-20

α=-15

α=-10

0 1 2 3 4 5 6-180

-160

-140

-120

-100

-80

-60

-40

-20

0

Time, sec

Flig

ht

pa

th a

ngle

, γ,

deg

α=-30

α=-25

α=-20

α=-15

α=-10



Effect of the number of grids

-180 -160 -140 -120 -100 -80 -60 -40 -20 00.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

Flight path angle, γ, deg

Mach n

um

ber,

M

30

20

12

10

5

-180 -160 -140 -120 -100 -80 -60 -40 -20 0-21

-20

-19

-18

-17

-16

-15

-14Angle of attack Vs γ

Flight path angle, γ (deg)

AO

A,

α,

deg

30

20

12

10

5



Selection of number of grids

No.ofgrids

Computational time

5 0.899679

10 1.022969

12 1.102450

20 1.652399

30 3.314869

Intel® Core™ 2 Quad CPU

Q6600 @2.40 GHz, 1.98GB RAM

Software: MATLAB 7.4

• The number of grids for analysis: 20

• 20 grids are comparable with 30

Note: Real-time implementation in “C” or “Assembly language” isexpected to be much (at least 50 times) faster than the MATLAB Code



Comparison of Chebyshev and Legendre

approximation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

Time, sec.

Mach n

um

ber,

M

Legendre

Chebyshev

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-22

-20

-18

-16

-14

Time(sec)

AO

A,

α,

deg

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-66

-64

-62

-60

Time, sec

δb,

deg

Legendre

Chebyshev

Both lead to identical results!



Conclusions: Missile-turning Problem

� Real-hemisphere engagement is feasible (no need of “dog fight”!)

� Minimum-time flight path angle reversal is feasible with “realistic control force”

� Promising numerical results

• Computationally very efficient & is a viable tool for optimal guidance

• Chebyshev and Legendre approximations lead to identical results (serves as a verification)



References on Pseudo-spectral

Methods for Optimal Control

� I. M. Ross and M. Karpenko, A Review of Pseudospectral Optimal Control: From Theory to Flight, Annual Reviews in Control, Vol. 36, 2012, pp.182–197.

� Gong, Q., Fahroo, F., and Ross, I.M., Spectral Algorithm for Pseudospectral Methods in Optimal Control, Journal of Guidance, Control, and Dynamics, Vol. 31, No. 3, 2008.

� F. Fahroo and I. M. Ross, Direct Trajectory Optimization via a Chebyshev Pseudospectral Method, Journal of Guidance, Control, and Dynamics, Vol. 25, 2002, pp. 160–166.

� F. Fahroo and I. M. Ross, Costate Estimation by a Legendre Pseudospectral Method, Journal of Guidance, Control and Dynamics, Vol.24, No.2, March–April 2001, pp.270–277.



Thanks for the Attention….!!

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