95  Satellite Control

Users Guide for dyn.Opt, Example 7a, 7b, 7c

95.1  A satellite control problem

Find T (controls) over t in [0; 100 ] to minimize

7c is free end time

7a:

J =
 100 0
((T12+T22+T32)*0.5) dt +
 w12 + w22 + w32 + (e1 − 0.70106)2 + (e2 − 0.0923)2 + (e3 − 0.56098)2 + (e4 − 0.43047)2

7b:

J =
 100 0
((T12+T22+T32)*0.5) dt

7c:

 J = tF

subject to:

 I1 = 1.0e6
 I2 = 833333.0
 I3 = 916667.0
 T1S = 550
 T2S = 50
 T3S = 550

 de1 dt
= 0.5*(w1*e4w2*e3+w3*e2
 de2 dt
= 0.5*(w1*e3+w2*e4w3*e1
 de3 dt
= 0.5*(−w1*e2+w2*e1+w3*e4
 de4 dt
= −0.5*(w1*e1+w2*e2+w3*e3
 dw1 dt
=
 (I2−I3)*w2*w3+T1*T1S I1

 dw2 dt
=
 (I3−I1)*w3*w1+T2*T2S I2

 dw3 dt
=
 (I1−I2)*w1*w2+T3*T3S I3

 e(0) = [0  0  0  1]
 w(0) = [0.01  0.005  0.001]

7b, 7c - x(100) = [0.70106 0.0923 0.56098 NaN 0 0 0]; 7c - free time 7c - -1 <= T <= 1

Reference: [16]

95.2  Problem setup

```toms t

% Starting guess
e1opt = 0;
e2opt = 0;
e3opt = 0;
e4opt = 0;
w1opt = 0;
w2opt = 0;
w3opt = 0;
T1opt = 0;
T2opt = 0;
T3opt = 0;

% Final times
tfs = zeros(3,1);

for i=1:3
if i == 3
toms t_f
runs = [10 20 101];
else
runs = [10 40];
end
if i == 2
e1opt = 0; e2opt = 0;
e3opt = 0; e4opt = 0;
w1opt = 0; w2opt = 0;
w3opt = 0; T1opt = 0;
T2opt = 0; T3opt = 0;
end
for n=runs
```
```        if i == 3
p = tomPhase('p', t, 0, t_f, n);
else
p = tomPhase('p', t, 0, 100, n);
end

setPhase(p);
tomStates e1 e2 e3 e4 w1 w2 w3
tomControls T1 T2 T3

if i == 3
x0 = {t_f == 100};
else
x0 = {};
end
% Initial guess
x0 = {x0; collocate({T1 == T1opt
T2 == T2opt; T3 == T3opt})
icollocate({
e1 == e1opt; e2 == e2opt
e3 == e3opt; e4 == e4opt
w1 == w1opt; w2 == w2opt
w3 == w3opt})};

if i == 3
cbox = {
1 <= t_f <= 1000
-1 <= collocate(T1) <= 1
-1 <= collocate(T2) <= 1
-1 <= collocate(T3) <= 1};
else
cbox = {};
end

% Boundary constraints
cbnd = initial({e1 == 0
e2 == 0;     e3 == 0
e4 == 1;     w1 == 0.01
w2 == 0.005; w3 == 0.001});
```

95.3  Problem 7b and 7c modifications

```        if i ~= 1
cbnd = {cbnd
final({
e1 == 0.70106
e2 == 0.0923
e3 == 0.56098
w1 == 0
w2 == 0
w3 == 0
})};
end

% ODEs and path constraints
I1 = 1.0e6;
I2 = 833333.0;
I3 = 916667.0;
T1Sc = 550;
T2Sc = 50;
T3Sc = 550;
ceq = collocate({
dot(e1) == 0.5*(w1.*e4-w2.*e3+w3.*e2)
dot(e2) == 0.5*(w1.*e3+w2.*e4-w3.*e1)
dot(e3) == 0.5*(-w1.*e2+w2.*e1+w3.*e4)
dot(e4) == -0.5*(w1.*e1+w2.*e2+w3.*e3)
dot(w1) == ((I2-I3)*w2.*w3+T1*T1Sc)/I1
dot(w2) == ((I3-I1)*w3.*w1+T2*T2Sc)/I2
dot(w3) == ((I1-I2)*w1.*w2+T3*T3Sc)/I3});

% Objective
if i == 1
objective = final(w1)^2 + final(w2)^2 + final(w3)^2 +...
(final(e1) - 0.70106)^2 + (final(e2) - 0.0923)^2 + ...
(final(e3) - 0.56098)^2 + (final(e4) - 0.43047)^2 ...
+ integrate((T1.^2+T2.^2+T3.^2)*0.5);
elseif i == 2
objective = integrate((T1.^2+T2.^2+T3.^2)*0.5);
else
objective = t_f;
end
```

95.4  Solve the problem

```        options = struct;
if i == 1
options.name = 'Satellite Control 7a';
elseif i == 2
options.name = 'Satellite Control 7b';
else
options.name = 'Satellite Control 7c';
end
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);

e1opt = subs(e1, solution);
e2opt = subs(e2, solution);
e3opt = subs(e3, solution);
e4opt = subs(e4, solution);
w1opt = subs(w1, solution);
w2opt = subs(w2, solution);
w3opt = subs(w3, solution);
T1opt = subs(T1, solution);
T2opt = subs(T2, solution);
T3opt = subs(T3, solution);
```
```Problem type appears to be: qpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2011-02-05
=====================================================================================
Problem: ---  1: Satellite Control 7a           f_k       0.463944669252504550
sum(|constr|)      0.000000135115457319
f(x_k) + sum(|constr|)      0.463944804367961870
f(x_0)      0.139185999999999230

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv   27 ConJacEv   27 Iter   15 MinorIter  105
CPU time: 0.140625 sec. Elapsed time: 0.140000 sec.
```
```Problem type appears to be: qpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2011-02-05
=====================================================================================
Problem: ---  1: Satellite Control 7a           f_k       0.463944366974706650
sum(|constr|)      0.000000000342695392
f(x_k) + sum(|constr|)      0.463944367317402020
f(x_0)     -0.536077645550793180

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv    3 ConJacEv    3 Iter    2 MinorIter  255
CPU time: 0.218750 sec. Elapsed time: 0.250000 sec.
```
```Problem type appears to be: qpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2011-02-05
=====================================================================================
Problem: ---  1: Satellite Control 7b           f_k      71.411903807059261000
sum(|constr|)      0.000000066784327430
f(x_k) + sum(|constr|)     71.411903873843585000
f(x_0)      0.000000000000000000

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv   20 ConJacEv   20 Iter   17 MinorIter  205
CPU time: 0.140625 sec. Elapsed time: 0.141000 sec.
```
```Problem type appears to be: qpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2011-02-05
=====================================================================================
Problem: ---  1: Satellite Control 7b           f_k      71.411593978634926000
sum(|constr|)      0.000000015292225816
f(x_k) + sum(|constr|)     71.411593993927156000
f(x_0)     71.236662445283258000

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv    5 ConJacEv    5 Iter    3 MinorIter  346
CPU time: 0.421875 sec. Elapsed time: 0.422000 sec.
```
```Problem type appears to be: lpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2011-02-05
=====================================================================================
Problem: ---  1: Satellite Control 7c           f_k      99.185331597188878000
sum(|constr|)      0.000010903612751909
f(x_k) + sum(|constr|)     99.185342500801625000
f(x_0)    100.000000000000000000

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv    8 ConJacEv    8 Iter    6 MinorIter   83
CPU time: 0.078125 sec. Elapsed time: 0.078000 sec.
```
```Problem type appears to be: lpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2011-02-05
=====================================================================================
Problem: ---  1: Satellite Control 7c           f_k      98.945461990937687000
sum(|constr|)      0.000017743233884652
f(x_k) + sum(|constr|)     98.945479734171570000
f(x_0)    100.000000000000000000

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv    5 ConJacEv    5 Iter    4 MinorIter  154
CPU time: 0.093750 sec. Elapsed time: 0.109000 sec.
```
```Problem type appears to be: lpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2011-02-05
=====================================================================================
Problem: ---  1: Satellite Control 7c           f_k      98.834204968061798000
sum(|constr|)      0.000012503587541902
f(x_k) + sum(|constr|)     98.834217471649339000
f(x_0)    100.000000000000000000

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv    7 ConJacEv    7 Iter    6 MinorIter  972
CPU time: 8.406250 sec. Elapsed time: 8.813000 sec.
```
```    end
tfs(i) = subs(final(t),solution);
% We only want to plot the solution from the first problem
if i == 1
tp  = subs(collocate(t),solution);
e1p = subs(collocate(e1),solution);
e2p = subs(collocate(e2),solution);
e3p = subs(collocate(e3),solution);
e4p = subs(collocate(e4),solution);
w1p = subs(collocate(w1),solution);
w2p = subs(collocate(w2),solution);
w3p = subs(collocate(w3),solution);
T1p = subs(collocate(T1),solution);
T2p = subs(collocate(T2),solution);
T3p = subs(collocate(T3),solution);
end
end

disp(sprintf('\nFinal time for 7a = %1.4g',tfs(1)));
disp(sprintf('\nFinal time for 7b = %1.4g',tfs(2)));
disp(sprintf('\nFinal time for 7c = %1.4g',tfs(3)));
```
```Final time for 7a = 100

Final time for 7b = 100

Final time for 7c = 98.83
```

95.5  Plot result

```subplot(3,1,1)
plot(tp,e1p,'*-',tp,e2p,'*-',tp,e3p,'*-',tp,e4p,'*-');
legend('e1','e2','e3','e4');
title('Satellite Control state variables (e)');

subplot(3,1,2)
plot(tp,w1p,'*-',tp,w2p,'*-',tp,w3p);
legend('w1','w2','w3');
title('Satellite Control state variables (w)');

subplot(3,1,3)
plot(tp,T1p,'+-',tp,T2p,'+-',tp,T3p,'+-');
legend('T1','T2','T3');
title('Satellite Control controls');
```