Satellite Guidance & Control I

Southeast Satellite Operations

Bidder’s Conference

March 2005

Disclosure• I have been hired by Southeast Satellite

Operations to cover this basic material.

• Engineering Competence is demonstrated by – Knowing this material completely– Going beyond this material demonstrates

competence relative to other competitors

• What goes on in this conference stays in this conference (feel free to ask questions)

UnderstandingDimensions

• Knowledge– Extent to which unschooled

perspectives have been transcended

• Methods– Process knowledge to produce

additional knowledge

• Purposes– Explain, reinterpret, operate on

world

• Forms– Process by which understanding

is communicated

Levels• Master

– Integrative, creative, thinking

• Apprentice– Disciplinary knowledge

and modes of thinking

• Novice– Rituals and

mechanisms of testing in school

• Naïve– directly available in the

world

Simple Satellite Model

2

( ) ( )0 1 0( )

(0) (0) 1( ) ( )

( )1( )

( ) (

1( ) ( )

0( )

)J t u

t tdu t

dt t t

t

t

s U sJs

y t J t

•Know these models.•Know how to get from one to any other.•Know what everything means.

Alternative SS Models

• OCF

• CCF

• Where does J go?– options

• SIDF

• All have the same transfer function

• How are the SS models related?

Perspective• Friction?• Spring constant?

( ) ( ) ( ) ( )

( ) 0 1 ( ) 0( )

( ) ( )

Mx t Bx t Kx t ku t

x t x tdu t

x t B M K M x t k Mdt

( ) ( ) ( ) ( )

( ) 0 1 ( ) 0( )

( ) ( )

J t B t K t ku t

t tdu t

t B J K J t k Jdt

Recall Solution to SS Models

( )

0

( ) ( ) ( ), (0)

( ) (0) ( )t

At A t

x t Ax t Bu t x

x t e x e Bu d

It is assumed that the SS model is committed to memory. In addition it is assumed that the solution and the derivation of the solution is committed to memory.

Guidance Problem• Find an input function that will “guide” the

state from its initial state, to a desired final state in specified time.– This is a design problem

• It involves making choices

• The answer is not unique

• Some choices are better than others

• An engineer is expected to evaluate options and choose best option for client/boss

A Few Example Inputs

1

1 2

2

( )

(

0

)

( ) 0

( )T

f

T A

M t

u t t

M t t

u m b

u ae b

u B e v

e

0 ft

Use your imagination to generate others.

Example Guidance Problem( )(0)

(0) , ( )( )(0)

0 1 1,

0 0 0 1

f

ff

At

tx x t

t

tA e

Expected to know how to derive eAt.Expected to know how to verify eAt.

Begin with solution to state equations.

Choose Parameters for first Input (1)

0

0

0

0

2

0

3 2

2

0

0

1 0( )

0 1 1

( )1

/ 3 / 2

f f

f f

f

f

f

f

f

At At

f

At A

t

t

t

f f

f f

t

f

At

f

At

f

m b d

m b d

m bd

m b

mt

x e x e

x e x e

e x x

e x xbt

mt bt

Choose Parameters for first Input (2)

3 2

20

0

0

1

3 2

2

0

/ 3 / 2

/ 3 / 2

/ 2

f

f

f

f

At

f

At

f

At

f

A

f f

f f

f f

f

f

f

t

mt bt

mt bt

t t m

t

e x x

e x x

e

M

x x

e

t b

mM

b

xb

xm

Choose Parameters for first Input (3)

10

fAt

f

me xM x

b

( )u m b

What step could have caused failure?

Matlab functions for 1st Input

function xdot = AxpB1(t,x,A,B,m,b)u=m*t+b; %u1 is defined herexdot=A*x+B*u;

function [m,b] = u1parms(x0,xf,tf,A)% compute parameters m and b for the input function% u(t)=mt+b% Syntax [m,b] = u1(x0,xf,tf,A)% x0 = initial state; xf = final state; tf = final time% A is 2 by 2 A matrix in state equations% assumptions: n = 2, CCFM=[-tf^3/3 -tf^2/2 tf^2/2 tf ];% Should check for invertibility of Mx = M^(-1)*(expm(-A*tf)*xf-x0);m = x(1); b = x(2);

Results>> [t,x]=ode45(@AxpBu1,[0 2],x0,[],A,B,m,b);

>> plot(t,x)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1response to u1

time, seconds

blue

is x

1, g

reen

is x

2

Choose Parameters for 4th Input (1)

0

0

2

0

3

0

0

2

0

02

1 0 1 00 1

0 1 1 1

11

1

1 1

3 21

2

f

f

f f

f f

f

f

f

At At

f

At At

f

At

f

A

t

t

t

f f

f f

t

f

x e x e

x e x e

e x x

vd

d v

d v

t te x x v

t t

Choose Parameters for 4th Input (2)

3 2

02

10

1 1

3 21

2

f

f

f f

f

At

f

At

f

f

t t

t t

ve x x

e xW xv

( )TT Au B e v

Failure?

Matlab functions for 4th Inputfunction v=u4parms(x0,xf,tf,A)%need comments, syntax, definitions, etcW=[tf^3/3 -tf^2/2 -tf^2/2 tf];v=W^(-1)*(expm(-A*tf)*xf-x0);

function xprime = AxpBu4(t,x,A,B,v)u=B'*expm(-A'*t)*v; % u4 herexprime=A*x+B*u;

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1response to u4

time, seconds

blue

is x

1(t)

, gr

een

is x

2(t)

>> [t,x]=ode45(@AxpBu4,[0 2],x0,[],A,B,v);

>> plot(t,x)

Choose Parameters for Additional Inputs

• Left for the entertainment of potential bidders

• Reduce to Computer code.– This is one way to clearly demonstrate to SSO

that you have an algorithm with specified inputs that produces the specified outputs.

• Test/verify algorithm (computer code)

Evaluate Performance for various inputs

• Use one of the ODE? functions to solve the DE– ode23, ode45, ode113, ode15s, ode23s, ode23t,

ode23tb – Define Matlab functions for the input functions– Define Matlab functions for the DE with

specified input function– Plots– Balance performance and fuel consumption– Exercise judgement

Compare inputs

21 4

0

( ) ( ) ( ) ( ) .5ft t

t

E u u t dt E u E u

Energy used?

1

4

( ) 0( ) .5 .5

1 0 0 0( ) 0 1 1 .5

1 .5 .5

TT A t

u t mt b t

u t B e v tt

What could cause failure?• How can you guarantee success?

• Can you easily predict failure?

• CONTROLLABILITY or CONTROLLABLE

• Matrices that had to be inverted involved e-AtB and/or its transpose

• Inverses fail to exist when ranks are too low

• Rank of integral of [(e-AtB)u(t)] =?

• Cayley-Hamilton theorem

Loose Ends to tie up

• Many state space models can share the same transfer function. – How are these related?

• Under what conditions can an input be found that guides an arbitrary initial state to an arbitrary final state?