PROPT Satellite Control: Difference between revisions

From TomWiki
Jump to navigationJump to search
(Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Users Guide for dyn.Opt, Example 7a, 7b, 7c ==A satellite control problem== Find T (controls) over t in [...")
 
 
(One intermediate revision by one other user not shown)
Line 12: Line 12:


<math> J = \int_0^{100} ((T1^2+T2^2+T3^2)*0.5) \mathrm{d}t + </math>
<math> J = \int_0^{100} ((T1^2+T2^2+T3^2)*0.5) \mathrm{d}t + </math>
<math> w_1^2 + w_2^2 + w_3^2 + (e_1 - 0.70106)^2 + (e_2 - 0.0923)^2 + (e_3 - 0.56098)^2 + (e_4 - 0.43047)^2 </math>
<math> w_1^2 + w_2^2 + w_3^2 + (e_1 - 0.70106)^2 + (e_2 - 0.0923)^2 + (e_3 - 0.56098)^2 + (e_4 - 0.43047)^2 </math>


7b:
7b:


<math> J = \int_0^{100} ((T1^2+T2^2+T3^2)*0.5) \mathrm{d}t </math>
<math> J = \int_0^{100} ((T1^2+T2^2+T3^2)*0.5) \mathrm{d}t </math>


7c:
7c:


<math> J = t_F </math>
<math> J = t_F </math>


subject to:
subject to:


<math> I1 = 1.0e6 </math>
<math> I1 = 1.0e6 </math>
<math> I2 = 833333.0 </math>
<math> I2 = 833333.0 </math>
<math> I3 = 916667.0 </math>
<math> I3 = 916667.0 </math>
<math> T1S = 550 </math>
<math> T1S = 550 </math>
<math> T2S = 50 </math>
<math> T2S = 50 </math>
<math> T3S = 550 </math>
<math> T3S = 550 </math>


<math> \frac{de_1}{dt} = 0.5*(w_1*e_4-w_2*e_3+w_3*e_2) </math>
<math> \frac{de_1}{dt} = 0.5*(w_1*e_4-w_2*e_3+w_3*e_2) </math>
<math> \frac{de_2}{dt} = 0.5*(w_1*e_3+w_2*e_4-w_3*e_1) </math>
<math> \frac{de_2}{dt} = 0.5*(w_1*e_3+w_2*e_4-w_3*e_1) </math>
<math> \frac{de_3}{dt} = 0.5*(-w_1*e_2+w_2*e_1+w_3*e_4) </math>
<math> \frac{de_3}{dt} = 0.5*(-w_1*e_2+w_2*e_1+w_3*e_4) </math>
<math> \frac{de_4}{dt} = -0.5*(w_1*e_1+w_2*e_2+w_3*e_3) </math>
<math> \frac{de_4}{dt} = -0.5*(w_1*e_1+w_2*e_2+w_3*e_3) </math>
<math> \frac{dw_1}{dt} = \frac{(I2-I3)*w_2*w_3+T1*T1S}{I1} </math>
<math> \frac{dw_1}{dt} = \frac{(I2-I3)*w_2*w_3+T1*T1S}{I1} </math>
<math> \frac{dw_2}{dt} = \frac{(I3-I1)*w3*w1+T2*T2S}{I2} </math>
<math> \frac{dw_2}{dt} = \frac{(I3-I1)*w3*w1+T2*T2S}{I2} </math>
<math> \frac{dw_3}{dt} = \frac{(I1-I2)*w1*w2+T3*T3S}{I3} </math>
<math> \frac{dw_3}{dt} = \frac{(I1-I2)*w1*w2+T3*T3S}{I3} </math>


<math> e(0) = [0 \ 0 \ 0 \ 1] </math>
<math> e(0) = [0 \ 0 \ 0 \ 1] </math>
<math> w(0) = [0.01 \ 0.005 \ 0.001] </math>
<math> w(0) = [0.01 \ 0.005 \ 0.001] </math>


7b, 7c - x(100) = [0.70106 0.0923 0.56098 NaN 0 0 0]; 7c - free time 7c - -1 <= T <= 1
7b, 7c - x(100) = [0.70106 0.0923 0.56098 NaN 0 0 0]; 7c - free time 7c - -1 <= T <= 1
Line 399: Line 418:


[[File:satelliteControl_01.png]]
[[File:satelliteControl_01.png]]
[[Category:PROPT Examples]]

Latest revision as of 05:30, 14 February 2012

Notice.png

This page is part of the PROPT Manual. See PROPT Manual.

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

A satellite control problem

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

7c is free end time

7a:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_1^2 + w_2^2 + w_3^2 + (e_1 - 0.70106)^2 + (e_2 - 0.0923)^2 + (e_3 - 0.56098)^2 + (e_4 - 0.43047)^2 }


7b:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J = \int_0^{100} ((T1^2+T2^2+T3^2)*0.5) \mathrm{d}t }


7c:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J = t_F }


subject to:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I1 = 1.0e6 }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I2 = 833333.0 }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I3 = 916667.0 }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T1S = 550 }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T2S = 50 }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T3S = 550 }


Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{de_1}{dt} = 0.5*(w_1*e_4-w_2*e_3+w_3*e_2) }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{de_2}{dt} = 0.5*(w_1*e_3+w_2*e_4-w_3*e_1) }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{de_3}{dt} = 0.5*(-w_1*e_2+w_2*e_1+w_3*e_4) }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{de_4}{dt} = -0.5*(w_1*e_1+w_2*e_2+w_3*e_3) }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dw_1}{dt} = \frac{(I2-I3)*w_2*w_3+T1*T1S}{I1} }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dw_2}{dt} = \frac{(I3-I1)*w3*w1+T2*T2S}{I2} }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dw_3}{dt} = \frac{(I1-I2)*w1*w2+T3*T3S}{I3} }


Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e(0) = [0 \ 0 \ 0 \ 1] }

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 

% Copyright (c) 2007-2008 by Tomlab Optimization Inc.

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});

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

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
Time for symbolic processing: 0.41532 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Satellite Control 7a           f_k       0.463944669252504660
                                       sum(|constr|)      0.000000135115456869
                              f(x_k) + sum(|constr|)      0.463944804367961540
                                              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: 0.031200 sec. Elapsed time: 0.040000 sec. 


Problem type appears to be: qpcon
Time for symbolic processing: 0.4063 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Satellite Control 7a           f_k       0.463944366974705650
                                       sum(|constr|)      0.000000000342695654
                              f(x_k) + sum(|constr|)      0.463944367317401300
                                              f(x_0)     -0.536077645550791850

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: 0.078000 sec. Elapsed time: 0.082000 sec. 


Problem type appears to be: qpcon
Time for symbolic processing: 0.37013 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Satellite Control 7b           f_k      71.411903807059332000
                                       sum(|constr|)      0.000000066784327540
                              f(x_k) + sum(|constr|)     71.411903873843656000
                                              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: 0.031200 sec. Elapsed time: 0.041000 sec. 


Problem type appears to be: qpcon
Time for symbolic processing: 0.37064 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Satellite Control 7b           f_k      71.411593978635054000
                                       sum(|constr|)      0.000000015292228483
                              f(x_k) + sum(|constr|)     71.411593993927283000
                                              f(x_0)     71.236662445283315000

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: 0.156001 sec. Elapsed time: 0.155000 sec. 


Problem type appears to be: lpcon
Time for symbolic processing: 0.44253 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Satellite Control 7c           f_k      99.185331597188878000
                                       sum(|constr|)      0.000010903612748681
                              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: 0.031200 sec. Elapsed time: 0.024000 sec. 


Problem type appears to be: lpcon
Time for symbolic processing: 0.4436 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Satellite Control 7c           f_k      98.945461990937730000
                                       sum(|constr|)      0.000017743233813344
                              f(x_k) + sum(|constr|)     98.945479734171542000
                                              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: 0.046800 sec. Elapsed time: 0.041000 sec. 


Problem type appears to be: lpcon
Time for symbolic processing: 0.44408 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Satellite Control 7c           f_k      98.834204968061897000
                                       sum(|constr|)      0.000012503591248200
                              f(x_k) + sum(|constr|)     98.834217471653147000
                                              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: 5.756437 sec. Elapsed time: 2.873000 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

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');

SatelliteControl 01.png

Retrieved from "http://tomwiki.com/index.php?title=PROPT_Satellite_Control&oldid=2749"