NEO DEFLECTION THROUGH A MULTI-MIRROR SYSTEM · Optimal Control Problem Minimise: 9/2/2012 O L C L...

FINITE ELEMENTS IN TIME and GAUSS PSEUDO-SPECTRAL METHODS

Massimiliano Vasile

Deptartment of Mechanical & Aerospace Engineering

University of Strathclyde, Glasgow, UK

9/2/2012

Direct Transcription with GPSM

Some Examples

Finite Perturbative

Elements

Direct Transcription

with FET

Basics of Nonlinear

Programming

OPTIMAL CONTROL

Optimal Control Problem

Minimise:

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

J t t t t L t t t dt x x x u

And algebraic constraints

nitttFtx

,,...,1 ),),(),(()( ux

0( ( ), ( ), ) 0, 1,..., 2i f ft t t i k n x x

fi tttmitttG 0 ,,...,1 ,0)),(),(( ux

Subject to a set of differential constraints

Boundary constraints

Example of Optimal Control Problem

Minimise:

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And algebraic constraints

Subject to a set of differential constraints

Boundary constraints

T Dv g

2 max -

3 0 0 0

( ( ), ( ), ) ( ) 0

( ( ), ( ), ) ( )

f f f f

t t t h t

t t t v t

t t t m t m

t t t h t h

0 0( ), ( ), , ( )f f fJ t t t t v t x x

Pontryagin Maximum Principle

Consider the optimal control problem:

If is optimal, with response , being and the spaces of piecewise continuous and piecewise continuous and differentiable functions respectively. Then, there exist and such that:

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min ( , ) ( ), ( ), , ( ( ), ( ), )

( ) ( , ( ), ( ))

( ( ), ( )) 0

f f fU

J t t t t t L t t t dt

subject to

t t t t

uu x x x u

x F x u

ψ x x

0 0ˆ( , ) [ , ] [ , )q

ft C t T t T u * 1

0ˆ [ , ]nC t Tx C

1* * 1

0 0ˆ, ,

nC t T

* * * * *

* * * *

( ), ( ) [0,0],

( ) 0, ( ) ( ( ), ( ), ( ), ( )), . . [ , ]

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

arg min ( ( ), ( ), ( ), ( ))

( ) 0, (

t t t t t

t t H t t t t a e t t

with H t t t t L t t

H t t t t

t const H

λ x u λ

x u λ x u λ F x u

u x u λ

* * * *

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

t f f f t tt t

f x xt t

t t t t

u λ ν ψ

λ ν ψ

ν ψ ν

* bnν

First Order Necessary Optimality Conditions

Euler-Lagrange Equations:

with optimality condition

and transfersality conditions:

First order conditions only guarantee that the solution is locally stationary.

Legendre-Clebsch Condition:

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),,(),,( 00 ff

ff tt xxxx

),,( tFH

H Li ,...,n

0 1,...,T

H Lj l

00, uu fH t t t

Example 2: quadratic control

Minimise

Subject to:

With boundary conditions:

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J u dt

Hamiltonian

with adjoint equations:

optimality condition:

And transversality conditions:

From the adjoint equations and the optimal control condition we get:

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2x vH u v x u

( ) sin cos

cos sin

t A t B t

t B t A t

u A t B t

and if one replaces the control law into the dynamic equation:

which is a harmonic oscillator with a periodic forcing term:

The solution is of the form:

And the four constants can be determined so that the boundary conditions are satisfied.

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cos sin

v x A t B t

cos sinx x A t B t

cos sin cos sin2 2

A Bx C t D t t t

Optimal Control Problem with Mixed Path Constraints

Form the Lagrangian:

with Lagrangian multipliers m

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min ( , ) ( ), ( ), , ( ( ), ( ), )

( ) ( , ( ), ( ))

( ( ), ( )) 0

( , ( ), ( )) 0

f f fU

subject to

t t t t

uu x x x u

x F x u

ψ x x

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

H t t t t L t t

u μ G

x u λ x u λ F x u

Maximum Principle with Maxed Path Constraints

If is optimal, with response , being and the spaces of piecewise continuous and piecewise continuous and differentiable functions respectively. Then, there exist , and such that:

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* * * *

* * * * *

* * * * * *

( , ) 0 0

* * * *

( ), ( ), ( ) [0,0],

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

( ) ( ( ), ( ), ( ), ( ), ( )), . . [ , ]

arg min ( ( ), ( ), ( ), ( ))

t t t t t t

t t t t t t t

t t t t t t a e t t

H t t t t

λ x u λ μ

x x u λ μ

u x u λ

* * * * *

| ( , , ) 0

( ) 0; 0;

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

t f f f t tt t

f x xt t

t const H t t t t

u G x u

μ G μ

x u λ ν ψ

λ ν ψ

ν ψ ν

0 0ˆ( , ) [ , ] [ , )q

ft C t T t T u C1C

* bnν

1* * 1

0 0ˆ, ,

nC t T

0ˆ [ , ]nC t Tx

mC t Tμ

First Order Necessary Optimality Conditions with Mixed Path Constraints

Euler-Lagrange equations:

with Optimality condition:

and second order condition:

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),,(),,( 00 ff

ff tt xxxx

( , , )i i

T - ; 1T

i i i i

Li ,...,n

x x x x m

0 1,...,T T

j j j j

u u u u m

Transversality conditions

Example of Solution

The extended Hamiltonian is

where only one of the two conditions on the thrust is true at any one time. The dynamic equations are:

and the adjoint equations are:

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T Dv g

1 2 max

( )x v m

T D Tv g T T T

m g I m m

1 1( ) ; ( ) =- ; ( ) = x v v x v m v

D D T Dt t t

x m x v m v m m

Example of Solution

The optimal control condition writes

where again only one of the two m is true at any one time, and the transversality conditions are:

If one looks at the optimality condition can define the switching function as:

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spT m g I

x f x x

2 1 max

0 0, 0

0 0, 0 0

0 singular arc

I T Tm g I

I Tm g I

Im g I

Treatment of Singular Arcs

Along singular arcs we have that:

Is satisfied for any admissible control. A solution is to differentiate a minimum of p times with respect to time until one obtains:

The second equation provides the required control law while p-1 is the order of the singular arc.

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Optimal Control Problem with State Path Constraints

In this case the path constraint can not be directly used to form a Lagrangian function. In fact suppose that G≥0 is a path constraint on the position. This constraint does not provide any information on the velocity and acceleration at the contact point G=0. As a consequence there is no information on the correct control to be applied. This is particularly important when pure state path constraints are introduced via a direct approach.

The correct way to introduce a pure state path constraint is to differentiate p times with respect to time till the control appears explicitly:

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min ( , ) ( ), ( ), , ( ( ), ( ), )

( ) ( , ( ), ( ))

( ( ), ( )) 0

( , ( )) 0

f f fU

subject to

t t t t

uu x x x u

x F x u

ψ x x

Optimal Control Problem with State Path Constraints

Once the control appears explicitly in the constraints, one can form the Lagrangian (indirect adjoining approach):

For example for p=1 and G≥0 one has:

:At a contact point t the second constraints causes a discontinuity in adjoint variables and Hamiltonian:

with q another Lagrange multiplier

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

H t t t t L t t

x u λ x u λ F x u

( , )0

( , ) 0

( ) ( ) ( ) ( , )

t t q t t

Maximum Principle with State Path Constraints If is optimal, with response , being and the spaces of piecewise continuous and piecewise continuous and differentiable functions respectively. Then, there exist , , , and such that:

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* * * *

* * * * *

* * * * * *

( , ) 0 0

* * * *

( ), ( ), ( ) [0,0],

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

( ) ( ( ), ( ), ( ), ( ), ( )), . . [ , ]

arg min ( ( ), ( ), ( ), ( ))

t t t t t t

t t t t t t t

t t t t t t a e t t

H t t t t

λ x u λ η

x x u λ η

u x u λ

* * * * *

) | ( , , ) 0, ( , , ) 0

( ) 0; 0; 0

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

t f f f t tt t

f x xt t

dt t t

t const H t t t t

Gu x u G x u

η G η η

x u λ ν ψ

λ ν ψ

ν ψ ν

0 0ˆ( , ) [ , ] [ , )q

ft C t T t T u C1C

* bnν

1* * 1

0 0ˆ, ,

nC t T

0ˆ [ , ]nC t Tx

mC t Tη ( ) cnt θ

( ) ( ) ( ) ( , )

( ) ( , ( )) 0; ( ) 0; ( ) ( )

t t t t

t t t q t q t t

λ λ θ G x

θ G x

Some References

Bryson A E, Jr. & Ho Y C. Applied optimal control: optimization, estimation, and control. Waltham, MA: Blaisdell, 1969. 481 p.

Betts J.T. Practical Methods for Optimal Control and Estimation Using Nonlinear Programming. Advances in Design and Control, SIAM 2001

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FINITE ELEMENTS IN TIME

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Previous Works on Finite Elements in Time for Optimal Control

Since early ’90s Finite Elements in Time used for Indirect Transcription: • Hodges, D. H. and Bless, R. R., “Weak Hamiltonian Finite Element Method for

Optimal Control Problems,” Journal of Guidance, Control, and Dynamics, Vol. 14, No. 1, 1991, pp. 148–156.

• Hodges, D. H., Bless, R. R., Calise, A. J., and Leung, M., “Finite Element Method for Optimal Guidance of an Advanced Launch Vehicle,” Journal of Guidance, Control, and Dynamics, Vol. 15, No. 3, 1992, pp. 664–671.

• Bottasso, C. and Ragazzi, A., “Finite Element and Runge-Kutta Methods for Boundary-Value and Optimal Control Problems,” Journal of Guidance, Control, and Dynamics, Vol. 23, No. 4, 2000, pp. 749–751.

Late ’90s Finite Elements in Time on Spectral Basis for Direct Transcription:

• Vasile M., Bottasso C.L., Finzi A.E., Lunar Orbital Dynamics by Finite Element in Time Method, Aerotecnica Missili e Spazio Vol. 75. Numero 3/4, Luglio-Dicembre 1996.

• Finzi A., Vasile M. Numerical Solutions for Lunar Orbits. IAF-97-A.5.08, 48th International Astronautical Congress, October 6-10, 1997/Turin, Italy

• Vasile, M. and Finzi, A., “Direct Lunar Descent Optimisation by Finite Elements in Time Approach,” Journal of Mechanics and Control, Vol. 1, No. 1, 2000.

• Vasile, M. and Bernelli-Zazzera, F., “Optimizing Low-Thrust and Gravity Assist Maneuvres to Design Interplanetary Trajectories,” The Journal of the Astronautical Sciences, Vol. 51, No. 1, 2003, January-March 2003.

ackgro

Direct Finite Element Transcription (DFET)

Strong solution of the differential equations:

Weak solution of the differential equations:

The solution x and control u satisfy the differential equations in a weak sense, i.e. with respect to the test functions w.

Boundary conditions are not exactly satisfied but are satisfied with respect to the test functions w.

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dt x F x u x

( , ) ( )

dt w x F x u w x x

Strong vs. Weak solution of the differential equations:

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dt x F x u x 0

( , ) ( )

T w w w T w b

dt w x F x u w x x

0( )wx t

Partial integration of the differential terms:

The integral can be solved by polynomial representation of the states x, control u and test functions w and Gauss quadrature.

The choice of the test function w can be such that the projection is orthogonal.

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T T T T b

T T T b

w x w F x u w x w x

w x w x w F x u w x

w x w F x u w x

S IN T

Polynomial representation of x, u and w:

Where xs, us and ws are discrete nodes and fs and gs are polynomials of order p-1 and p respectively.

Note that in principle fs and gs can be any arbitrary function generated on any arbitrary base.

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T T T b

dt w x w F x u w x

S IN T

1 1 1 1 0

( , , ) ( ) ( )

T Tt p p p

s s s s s s s s p p f t t

s s st

g f g t dt w g t w g t

w x w F x u x x

The equation must hold true for any arbitrary ws , hence:

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( ) ( ) ( ) ( , , )

( ) ( ) ( , , ) 0

( ) ( ) ( ) ( , , )

s s s s t

j s s j s s

p s s p s s t

g t f t g t t dt

g t f t g t dt

g t f t g t t dt

x F x u x

x F x u

x F x u x

1 1 1 1 0

( , , ) ( ) ( )

T Tt p p p

s s s s s s s s p p f t t

s s st

g f g t dt w g t w g t

w x w F x u x x

Now assume we wanted to use Gauss quadrature formulas to solve the integrals, then we would have:

Where tk are Gauss points and tk are Gauss weights.

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( ) ( ) ( ) ( , , )

( ) ( ) ( ) ( , , ) 0

( ) ( ) ( ) ( , , )f

k k s s k k s s k t

k j k s s k j k s s k

k p k s s k p k s s k t

t t t t

x F x u x

x F x u

x F x u x

Direct Finite Element Transcription (DFET) Discretised Weak solution of the differential equations:

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0( )wx t

T T T b

dt w x w F x u w x

Spectral basis: Gauss points used for integration and discretisation

Example of DFET Integration Let us consider the simple linear system with a constant u:

In weak form we can write:

And assume we use polynomials of order 0 for states and controls and of order 1 for the test functions:

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x v xdt

v x u v

01 2 1 2

0 0 0 0

( ) ( ) 1 11; ;

( ) ( ) ( ) ( )

f f f f

f w w w w w wt t t t

t t t t t t t t

Example of DFET Integration The weak solution becomes:

Because they need to be true for any arbitrary weight w we can derive the two equations:

We can now take Gauss integration formulas to solve the integral:

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01 2 1 2 2 1

0 0 0 0 0

( ) ( )1 1

( ) ( ) ( ) ( )

f b bt

s s sf f f ft f

x v x xw w w w dt w w

v x u v v

t t t t

t t t t t t t t

( ) ( )

s s sf ft

s s sf ft f

x v xdt

v x u v

x v xdt

v x u v

t t t t

s s s f

x v xt

v x u vt t

x v xt

v x u vt t

Example of DFET Integration The Gauss weight for a single point is equal to 2, therefore we have:

And if we sum the second to the first we get:

Note that for nonlinear function this is an implicit scheme.

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1 / 22

s s s f

x v xt t

v x u vt t

x v xt t

v x u vt t

s s s f

x v xt

v x u v

x v xt

v x u v

v v x u

• The Time domain is decomposed in finite elements leading to a polynomial development of the solution on spectral basis (Gauss Points)

• Differential constraints are expressed in weak form leading to discontinuities at boundaries

• Gauss formulas for the solution of the integral: high Integration order 2n = 2k+2

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Boundary

Boundary gap db

),(11 iii

( , ) ( )

dt w x F x u w x x

S IN T

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• The controls are discretised using the Guass points used for integration.

• The controls can be discontinuous at the boundaries.

• The integration error within an element is absorbed into the discontinuity at the boundaries.

S IN T

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DFET in Summary

Representation of controls, states and weights on spectral basis.

Same bases for the discretisation of objective function and algebraic constraints:

Differential equations are transformed into algebraic equations:

2],),(),([),,(

b tLtJ

ttt puxxx

)( ; )(1i

)()()()( 111

txwxwFwxw tttt

0)),(),(( kksks uxG

S IN T

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DFET Assembling Process

S IN T

( ) ( ) ( ) ( , , )

( ) ( ) ( , , ) 0

( ) ( ) ( ) ( , , )

s s s s i

j s s j s s

p s s p s s i

g t f t g t t dt

g t f t g t dt

g t f t g t t dt

x F x u x

x F x u

x F x u x

( ) ( ) ( ) ( , , )

( ) ( ) ( , , ) 0

( ) ( ) ( ) ( , , )

s s s s i

j s s j s s

p s s p s s i

g t f t g t t dt

g t f t g t dt

g t f t g t t dt

x F x u x

x F x u

x F x u x

The end boundary node of one element

must be equal to the beginning boundary

node of the following element

The last equation of one element

is matched to the first equation of

the following element.

Boundary nodes disappear except

for the two extremal nodes

Example of DFET Assembly Assume we have two elements:

The end of the first element must be equal to the beginning of element 2:

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x v xt

v x u v

x v xt

v x u v

s s s f

x v xt

v x u v

x v xt

v x u v

1 1 2 2

1 1 1 2 2 2

s s s s

s s s s s s

s s s f

x v xt

v x u v

x v x vt t

v x u v x u

x v xt

v x u v

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The optimal control problem is transformed into a nonlinear programming problem

b dtLtJ ),,(),,,( 0 puxpxx

0),,,( tpuxFx

0),,,( tpuxG

0),,,(0

b tpxx

)J( min y

ul byb

y=[xs,us,xb

f,t0,tf,p]

S IN T

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Optimal Control

Problem DFET

NLP Problem

min J(y)

where y=[x,u,ti ,tf]

Sparse Sequential

Quadratic

Programming

ul byb

iisiss

0)),(),((

)()()()(

xwxwFwxw

S IN T

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Multi-Phase DFET

Discontinuous states can be inserted at the boundary of two phases.

The end states of one phase are matched to the beginning states of another phase through some interphase constraints or...

…phases can be in parallel.

),,( tJ lux

Interphase-link Constraints

),,( tluxFx

0),,( tluxG

0))(()(

),,,( l

link ttxx

Phase l

Assembly

NLP Solver

0),,,(0

l tpxx

S IN T

The KKT Optimality Conditions

Lagrangian function of the NLP problem:

According to Karush-Kuhn-Tucker first order optimality conditions:

02/09/2012 41

With transversality conditions:

Which are equivalent to the maximum principle transversality conditions.

KKT Optimality conditions on controls and costates:

If one assumes that:

Then one gets the integral forms of the optimality conditions:

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Example: Minimum quadratic control with path constraints

Simple linear system with minimization of the integral of the square of the control action:

Linear differential constraints

Path constraint:

Boundary conditions:

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Discontinuity at the Boundaries

From the theory of Delfour et al. 1981 on Galerkin methods for ODEs:

And a maximum convergence rate of:

Delfour also demonstrated the accuracy of FET developed on Gauss Legendre, Radau and Lobatto points.

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From the theory of Delfour et al. 1981 on Galerkin methods for ODEs:

And a maximum convergence rate of:

Simple double integrator:

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The expected convergence of the discontinuity at the boundary is O(hp+2):

Example

Elliptical orbit motion:

Therefore one could use the following indicator:

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h-p Adaptivity Strategies

p-adaptivity

h-adaptivity

Given the residual on the error at each element:

Solve the following linear quadratic optimization problem:

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If the residuals are not zero, then all

the elements with a nonzero residual

are split in two.

Examples: Minimum Time to Orbit

The problem is to minimize the time to reach a given altitude with a given velocity (Bryson and Ho 1979):

With the dynamic constraints:

And the boundary conditions:

From Pontryagin maximum principle one gets the control law:

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Examples: Step-by-Step Adaptation Process

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Examples: Step-by-Step Adaptation Process

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Given a solution of the NLP problem, one has C(x)=0.

By linearizing in a neighbourhood of x one can get:

Where the matrix is the Jacobian of the constraints whose components are the gradients with respect to states and controls.

If the uncertainty on the initial state is given and no variation on the final state is required, one can solve the following linear quadratic optimisation problem:

Robust Control

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11 12 13

J U J X J X

If the control is considered to be fixed, the Jacobian matrix can be used to derive the state transition matrix from initial states to final states:

Starting from the bottom of the matrix one can compute:

And then proceed with::

Up to:

Transition Matrix

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• Vasile M. Finite elements in Time: A Direct Transcription Method for

Optimal Control Problems. AAS, Astrodynamics Specialist Conference,

Toronto 1-5 August 2010.

Some References

02/09/2012 57

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GAUSS PSEUDO-SPECTRAL METHODS (GPSM)

Consider again the differential equations:

And the polynomial representation of states and controls:

Pseudo-spectral Transcription

( , , ) 0t x F x u

With the first derivatives:

Let’s assume now that the nodes xs are chosen to be the zeros ts of

orthogonal polynomial, like Legendre polynomials, in the interval [-1,1].

Then f(t) is different from zero at each t=ts .

Then we can collocate the differential equations as follows:

Now consider the stong integral form:

( , , ) 0; s 1,...,2

s k k s s s

x F x u

And assume the integral term is integrated numerically with a Gauss

quadrature formula:

Then one can choose a Lagrange interpolating polynomial that

interpolates x0 plus the M Gauss points xs to represent the states (the

controls repsectively).

0( ) ( ) ( , , )

t t t dt x x F x u

( ) ( ) ( , , )2

f s s s s

tt t t

x x F x u

The complete set of equations becomes:

( , , ) 0; s 1,...,2

( ) ( ) ( , , )2

s k k s s s

f s s s s

x F x u

x x F x u

0( )x t

The controls are collocated only at the Gauss points:

( , , ) 0; s 1,...,2

( ) ( ) ( , , )2

s k k s s s

f s s s s

x F x u

x x F x u

( )fx t

0( )x t

As for DFET one can partition the time domain in segments and use a PS

transcription on each segment:

Multiple Phases Pseudo-spectral Transcription

1 , 1 0,

, 1 0,

( ) ( ); 1,..., 1j f j j j segment

t t j n

It is now clear that in order to restore the continuity from one segment to

the next one has to add a matching condition:

The matching condition restores the continuity on the states but not on the

control that can remain discontinuous.

Benson, D. A., Huntington, G. T., Thorvaldsen, T. P., and Rao, A. V., "Direct

Trajectory Optimization and Costate Estimation via an Orthogonal

Collocation Method," Journal of Guidance, Control and Dynamics, Vol. 29,

No. 6, November–December, 2006, pp. 1435–1440

Benson, D. A., A Gauss Pseudospectral Transcription for Optimal Control,

Ph.D. Thesis, Dept. of Aeronautics and Astronautics, Massachusetts

Institute of Technology, November 2004.

http://dspace.mit.edu/handle/1721.1/28919

Huntington, G. T., Advancement and Analysis of a Gauss Pseudospectral

Transcription for Optimal Control, Ph.D. Thesis, Dept. of Aeronautics and

Astronautics, Massachusetts Institute of Technology, May 2007.

http://dspace.mit.edu/handle/1721.1/42180

Some References

9/2/2012

BASICS OF NONLINEAR PROGRAMMING

•Find x* solution of the nonlinear problem c(x*)=0

•Given a first guess x and a linear expansion of c in x

)()( xxdx

xdcxcxc

pxxcdx

•Quadratic convergence if cC1and cxx0

•Problem: computing the derivative of c

The value of x is updated with the Newton’s step p given by

Simple 1D Newton Method

•Find x that minimises F(x)

•Given a value x, the function F is expanded up to the second order:

))(('')(2

1))((')()( xxxFxxxxxFxFxF

))(('')('0)(')(

xxxFxFxFdx

Sufficient condition for optimality

First order necessary condition is to have:

0)('' xF

Simple 1D Minimisation

•Find the vector x=[x1,…,xn]T such that:

where G is the Jacobian matrix of nonlinear functions c

Newton’s method in N dimensions B

•Find x=[x1,…,xn]T that minimises F(x):

Where H is the Hessiana matrix of F

First order necessary conditions for optimality

Newton’s method for minimisation in N dimensions B

•Build an estimate B* of matrix B:

),(* xch BB

In this case the convergence is no more quadratic but superlinear

Update the Hessian matrix with the BFGS (Broyden-Fletcher-

Goldfarb-Shanno) formula:

Approximation of the Hessian and Jacobian matrix B

•find x that minimises F(x) subject to:

0xc )(

Suffcicient condition for optimality is that the Hessian of the L is:

First order necessari conditions for optimality (Kuhn-Tuker):

A scalar function called Lagrangian is defined as:

T cFFL1

)()()(),( xxcxx

0vHv L

Constrained optimisation: Equality Constraints B

1)( xxxF

02)( 21 xxxc

)1,1(* x

Minimise:

Subject to

And solution

At the solution point

Equivalent necessary condition is that the projection of the gradient of the

function F on the constraint is zero:

Equivalent sufficient condition is that the projection of the Hessiana is

positive defined:

Constrained optimisation: Equality Constraints B

•Find x that minimises F(x) subject to:

0)( xc

For x=x*:

•Some constraints are equal to zero

•Others are strictly satisfied

set active the tobelongs i.e. for 0)( Aici x

sconstraint inactive ofset the tobelong i.e. 'for 0)( Aici x

Necessari condition for optimality is that lagrangian multipliers at the solution

Constrained optimisation: Inequality Constraints B

•Two different approaches:

•active set method

•Interior point method

Active Set

This methods looks for the active set at every optimisation step

Interior Point

Inequality constraints are penalised in the objective function

through a barrier function dependent on a penality parameter m:

)(ln)( xx cF m

The solution is forced to stay within the feasible region by the

barrier function lnc(x)

Constrained optimisation: Inequality Constraints

The solution of the constrained optimisation problem satisfies the following

first order conditions:

)()()(),,( xcxcxxiTeTFL mm

Slack variables are introduced to insert inequality

constraints as equality constraints:

sxcxsx

])([)(),,( TFL

Constrained optimisation: The K-T system

•The direction of steepest descent is computed solving the K-T system:

)( scxGs dd T Tdd sxp ,

•The solution of the K-T system gives the direction of steepest

descent and the step. The step length is corrected by a factor a:

•The value a is such that the bounds on slacks s and

unknowns x are not violated

Constrained optimisation: The K-T system

HxxxgTT

Given the quadratic objective function

Subject to linear constraints

Solution method: starting from a first guess x0 and an estimate of the

active set A0

1)at step k solve the K-T system with the constraints in Ak as equalities

2)Maximum step length along the direction p which does not violate active

constraints

3) if the step crosses on inactive constraints the step is reduced and the

constraint is inserted among active constraints, then go to step 1 otherwise

check the sign of the associated lagrangian multipliers

4) If all the multiplier are positive then stop, otherwise if erase from the active

set the constraint associated to the most negative multipliers.

Sequential Quadratic Programming

C2>0 Feasible region

1)( xxxF

02)( 211 xxxc

24)( 212 xxxc

Step x1 x2 Active set

0 4 0 A0=c2

1 2.77 1.85 2=-5.53 erase c2

2 1.2 0.8 a=0.5 add c1

3 1 1 A*=c1

Minimise

Subject to

Example

NOTICE!

Speed of convergence and robustness improve if Jacobian ed Hessin

are analytical

Robustness of convergence: local convergence must be

acchieved from every starting point in the solution space:

1)(),,( scscscsx TTFM

Unidimensional minimisation in the direction of steepest descent

using a quadratic or cubic model modello of the merit function:

)(),,( min a MM sx

Betts 2000

where is a weight matrix

The merit function

• Betts J.T. Practical Methods for Optimal Control and Estimation

Using Nonlinear Programming. Advances in Design and Control,

SIAM 2001.

• Fletcher, R. Practical Methods of Optimization, Wiley, 1987.

• Gill, P.E, W. Murray and M. Wright, Practical Optimization,

Academic Press, 1981.

• Nocedal, J. and S. Wright, Numerical Optimization, Springer, 1998

Some References

9/2/2012

SOME SPACE APPLICATIONS

9/2/2012 ● 82

ulti-G

ravity

BepiColombo •3000 variables and constraints for the NLP problem •4 to 7 swingbys •resonant orbits •more than 20 switching points

9/2/2012 ● 83

ulti-G

ravity

Earth-Europa Transfer

• 6000-7000 variables and constraints for the NLP problem

• 14 swingbys

• resonant orbits

• variable thrust

• Variable reference frames

9/2/2012 ● 84

ulti-G

ravity

WSB Transfers

• 1000-1500 variables and constraints

• Highly nonlinear and unstable dynamics

• Impulsive manoeuvres

9/2/2012 ● 85

Some References

ulti-G

ravity

• Vasile M., Campagnola S. Design of Low-Thrust Gravity Assist Trajectories to Europa.

Journal of the British Interplanetary Society, Vol. 62, No.1, pp. 15-31, 8 January 2009.

http://arxiv.org/ftp/arxiv/papers/1105/1105.1823.pdf

• Vasile M., Bernelli-Zazzera F., Targeting a heliocentric orbit combining low-thrust

propulsion and gravity assist manoeuvres, Operational Research in Space and Air,

Kluwer Academy Press, Ed. 1-4020-1218-7, 2003.

• Vasile M., Bernelli-Zazzera F., Optimising low-thrust and gravity assist manoeuvres to

design interplanetary trajectories, The Journal of Astronautical Sciences, vol. 51(1),

2003. http://arxiv.org/ftp/arxiv/papers/1105/1105.1829.pdf

• Vasile M. Bernelli-Zazzera F., Direct Averaging for Multiple Revolution Trajectory

Optimisation. 2nd International Symposium on Low Thrust Trajectories (LOTUS2),

Toulouse 18-20 June 2002.

• Vasile M., Bernelli-Zazzera F., Combining Low-Thrust and Gravity Assist Manoeuvres to

Reach Planet Mercury. AAS/AIAA Astrodynamics Specialist Conference,30 Jul-2 Aug

2000,Quebec City, Canada

• Vasile M., Bernelli Zazzera F., Jehn R., Janin G., Optimal Interplanetry Trajectories

Using a Combination of Low-Thrust and Gravity Assist Manoeuvres. IAF-00-A.5.07,

51st International Astronautical Congress,2-6 Oct,2000/Rio de Janeiro, Brazil.

• Vasile, M. and Finzi, A., “Direct Lunar Descent Optimisation by Finite Elements in Time

Approach,” Journal of Mechanics and Control, Vol. 1, No. 1, 2000.

9/2/2012

FINITE PERTURBATIVE ELEMENTS

9/2/2012

Equations of Motion

Non-singular Equinoctial elements:

No singularities for zero-inclination

and zero-eccentricity orbits.

tan sin2

tan cos2

Gauss planetary equations in Equinoctial elements, under a perturbing acceleration ε in

the r-t-h frame:

11 2 1 2

22 1 1

2sin cos cos cos cos sin

cos cos cos 1 sin cos sin ( cos sin ) sin

cos cos cos 1 sin cos sin (

da a pP L P L

dt h r

dP r p pL P L P Q L Q L

dt h r r

dP r p pL P L P Q

dt h r r

b a b a

b a b a b

b a b a

2 211 2

2 221 2

cos sin ) sin

1 sin sin2

1 cos sin2

cos sin sin

dQ rQ Q L

dL rQ L Q L

dt a h

Finite P

ATIVE ELEM

9/2/2012

Low Thrust Two-Points Boundary Value Problem

Objective:

Where the control vector u is:

Satisfying Gauss’ Planetary Equations and the boundary constraints:

and with:

Assumptions:

Perturbing acceleration ε is very small compared to the local gravitational acceleration:

Constant modulus and direction in the radial-tangential reference frame.

( ), ( ), ( )T

u t t t a b

V t dt

, , const a b

max( )t

Finite P

ATIVE ELEM

9/2/2012

From Time to True Longitude

A system of differential equations in time is translated into a system of differential equations in true longitude:

It is possible to rewrite Gauss Planetary Equations with respect to Longitude by applying the chain rule:

Where , with the hypothesis of small, constant, perturbing acceleration:

Thus obtaining also the time equation:

d d dt

dL dt dL

1 21 sin cos

dt r h

dL h P L P L

Finite P

ATIVE ELEM

( , , , , )ddt

f L a bX X ( , , , , )ddL

f L a bX X

9/2/2012

First order expansion of Equations of Motion (1)

0 1X X X

' ' ' 0 1X X X

The aim is that of obtaining a first-order expansion of the variation of

Equinoctial elements in the form with respect to a reference orbit:

With the manipulations described above, one obtains a set of

equations in the form:

Which could be integrated between L0 and L.

Finite P

ATIVE ELEM

9/2/2012

Excursus on Perturbation Theory

Consider the following example:

And the first order expanded solution:

If this expanded solution is substituted in the original equation one gets:

Finite P

ATIVE ELEM

2x x x

0 1 0 1 0 1

2 2 2 2

0 1 0 1 0 0 1 1

x x x x x x

x x x x x x x x

9/2/2012

Excursus on Perturbation Theory

Now assuming we are interested only in first order terms we can

collect terms with similar order of :

The first equation is a simple harmonic oscillator and once its solution is

substituted in the second equation one has a simple harmonic oscillator

with periodic forcing term.

Finite P

ATIVE ELEM

9/2/2012

First order expansion of Equations of Motion (2)

A first order expansion could be obtained by simply integrating the ODEs with respect

to L, which could be done in closed form.

This requires finding the primitives of the integrals in the form:

1 sin cos

I L dLP L P L

LI L dL

P L P L

LI L dL

P L P L

For example:

2010 20

2 2 2 2

10 10 20 10 20 10 20 10

3/22 2

2 2 2 2

10 10 20 10 20 10 20 10

3/22 2

20 10 20

1 sin cos

ln 1 cos 1 sin 1

1 cos sin 1

LI L dL

P L P L

P P P P L P L P P P

P L P L P

10 20 20 101 cos sin 1P P P L P L

Finite P

ATIVE ELEM

9/2/2012

Analytical Solution of the Equations of Motion

Thus the first order approximate solution of perturbed Keplerian motion takes the

2 2 2 2

0 1 0 0 0 20 2 0 10 2 0 0 0 11 0

1 10 11

2 20 21

1 10 11

2 20 21

( ) 2 cos cos ( , ) ( , ) 22 cos sin ( , )

s ca L a a a h a P I L L P I L L h a I L L

P L P P

Q L Q Q

t L t t

b a b a

A complete set of analytic equations parameterised on the Longitude is thus available

to propagate the perturbed orbital motion, in the form:

, , , ,0 0(L L) f (L ) L a b X X

Finite P

ATIVE ELEM

9/2/2012

Some remarks on accuracy

The accuracy of the approximation is dependent on:

The ratio between the local gravitational acceleration and the perturbation. Thus:

Error increases according to the thrust level works better with small

thrust magnitude (<1 N).

Error increases with the distance from the central body works better very

close to the attractor.

The amplitude of the trajectory arc. Error increases superlinearly with the

amplitude of the propagation interval.

Tests have revealed that in most practical applications thrust and local gravitational

force still present a favourable ratio.

The error in amplitude is mitigated by dividing a single arc into sub-arcs of smaller

amplitude.

Finite P

ATIVE ELEM

9/2/2012

FPET method

A Direct Transcription Method based on Finite Perturbed Elements in Time (FPET) has been designed using the Perturbative approach.

Each transfer trajectory is divided into n subarcs:

, , , ,2

Amplitude of arc is ∆L. Perturbed motion propagated

using analytical solution. Constant thrust vector in the r-

t-h reference frame. Reference node for propagation

is the midpoint of the arc. Motion is propagated

analytically backwards and forwards by from the midpoint to obtain boundary nodes. (better accuracy compared to a single sided propagation)

Reference

Upper boundary

Lower boundary

Forward

propagation

Backward

propagation

Finite P

ATIVE ELEM

9/2/2012

FPET method

The subarcs are then matched to each other at the boundaries to obtain the complete trajectory.

Conceptually similar to Sims & Flanagan Direct Transcription Method which used Keplerian arcs with ∆V discontinuities at the boundaries.

In the FPET method, thrust is continuous, albeit constant within each element.

In both cases, orbit propagation is analytical.

Finite P

ATIVE ELEM

9/2/2012

Accuracy and CPU time (1)

To assess accuracy and CPU cost, orbital motion under constant thrust was propagated within a given time interval.

A range of values for the number sub-arcs was considered.

Accuracy was calculated in terms of relative error on final state with respect to numerical integration of the exact equations of motion.

CASE 1, Heliocentric orbital motion: 0.5 N continuous thrust on a 2000 kg spacecraft. ε=2.5 10-5 m/s2

Departure from Earth. 1.5 years.

Finite P

ATIVE ELEM

9/2/2012

The LT Boundary problem with FPET

CASE 2, Geocentric orbital motion: 0.5 N continuous thrust on a 2000 kg spacecraft. ε=2.5 10-5 m/s2

Departure from LEO 5 days (≈50 revolutions).

FPET method is 2 orders of magnitude faster than numerical integration. Even with few elements per revolution, accuracy is high for geocentric orbits. In heliocentric orbits, accuracy is still adequate. The FPET method is particularly advantageous with a relatively low number of

sub-arcs.

Finite P

ATIVE ELEM

9/2/2012

The Low Thrust Two-Points Boundary Value Problem with FPET

The FPET transcription method is used to solve the LT boundary problem:

Decision variables for each of the n FPET:

Position of the reference point (5 scalars).

Acceleration magnitude, azimuth and elevation (3 scalars).

8n decision variables and 5(n+1)+1 scalar constraints.

The problem is efficiently solve with a gradient-based local optimizer (fmincon active-set).

. . , 2, , 1 0

, 1, ,

i i FPET

i max FPET

Fixed time of flight

Arrival conditions

Departure conditions

Inter-element matching conditions

Total ∆V

Upper bounds on acceleration

Finite P

ATIVE ELEM

9/2/2012

Earth-Mars Direct transfer (1)

Boundary problem:

Departure from Earth at 5600 MJD2000.

Rendezvous with Mars after a transfer time of 3 years.

2 complete revolutions.

Maximum acceleration: 2.5 10-5 m/s2.

40 FPET.

Initial guess for the local optimizer: constant thrust profile.

Finite P

ATIVE ELEM

9/2/2012

Earth-Mars Direct transfer (2)

Results:

Total ∆V: 5.63 km/s.

Relative error 10-3.

Solution found with DITAN: 5.71 km/s.

Hohmann Transfer: 5.49 km/s.

Finite P

ATIVE ELEM

9/2/2012

Earth-Mars Direct transfer: Multi-Objective Problem (1)

Bi-objective optimization problem: ∆V and Time of Flight are minimised.

Each time to objective function is called, a boundary problem needs to be solved.

Solved with EPIC, a hybrid-memethic stochastic optimizer.

8000 function evaluations.

Lower Upper

T0 [MJD2000] 5000 5779.94

ToF [days] 100 1500

nrev 1 3

Finite P

ATIVE ELEM

9/2/2012

Earth-Mars Direct transfer: Multi-Objective Problem (2)

Bi-objective optimization problem: ∆V and Time of Flight are minimised.

An interesting comparison could be made with an analogous Biimpulsive Transfer to Mars.

Finite P

ATIVE ELEM

9/2/2012

Some References

Zuiani, F., Vasile, M., Avanzini, G., Palmas A. Direct transcription of low-thrust trajectories with finite trajectory elements. (2012) Acta Astronautica, 72. pp. 108-120. ISSN 0094-5765

http://www.sciencedirect.com/science/article/pii/S0094576511002852

http://strathprints.strath.ac.uk/33788/

Zuiani F., Vasile M. Preliminary Design of Debris Removal Missions by Means of Simplified Models for Low-Thrust, Many-Revolution Transfers. International Journal of Aerospace Engineering, Volume 2012 (2012), Article ID 836250, 22 pages, doi:10.1155/2012/836250

http://www.hindawi.com/journals/ijae/2012/836250/

Finite P

ATIVE ELEM

9/2/2012 ● 106

Massimiliano Vasile

massimiliano.vasile@strath.ac.uk

Dept. of Mechanical & Aerospace Engineering

University of Strathclyde, Glasgow, UK

http://www.strath.ac.uk/space/research/optimisation/

NEO DEFLECTION THROUGH A MULTI-MIRROR SYSTEM · Optimal Control Problem Minimise: 9/2/2012 O L C L...

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