walkthrough.html

This demo shows how to use MATLAB, Optimization Toolbox, and Genetic Algorithm and Direct Search Too

<span class="comment">% Modify options setting</span>
<span class="comment">% Note: You can use the defaults as well, this will speed up the solution</span>
options = optimset(options,<span class="string">'Display'</span> ,<span class="string">'iter'</span>);
options = optimset(options,<span class="string">'TolFun'</span> ,0.1);
options = optimset(options,<span class="string">'LargeScale'</span> ,<span class="string">'off'</span>);
<span class="comment">% Request plots, add my custom plot to the mix</span>
options = optimset(options,<span class="string">'OutputFcn'</span> ,{ @updatePlot });
options = optimset(options,<span class="string">'PlotFcns'</span> ,{  @optimplotx @optimplotfval });
tic
[x_x0] = fmincon(@(x) wbObjFun(x,time,idealProfile),x0,Aineq,bineq,<span class="keyword">...</span>
        [],[],lb,ub,[],options);
toc
x_x0
</pre><pre class="codeoutput">
                                max     Line search  Directional  First-order 
 Iter F-count        f(x)   constraint   steplength   derivative   optimality Procedure 
    0     16      8.37961     -0.01086                                         
    1     39      7.07475     -0.01458      0.00781          221          101   
    2     60      6.31931     -0.08378       0.0313         13.2         9.91   
    3     78      4.63623      -0.1544         0.25        -2.12         5.63   
    4     94      2.82522    -0.006032            1         1.71         6.79   
    5    110       1.7788    2.22e-016            1       -0.313         13.4   
    6    126      48.3345      -0.5437            1         99.1           69   
    7    142      27.5839      -0.4806            1         20.4         14.6   
    8    158      9.69029  -4.441e-016            1         29.3           15   
    9    175      9.19751      -0.1622          0.5         24.6         36.5   
   10    192      7.11708      -0.1193          0.5         11.1         33.3   
   11    208      2.28753            0            1         8.86           24   
   12    225     0.326541            0          0.5         1.36         23.9   
   13    246     0.273352  -4.441e-016       0.0313        0.166         19.6   
Optimization terminated: magnitude of directional derivative in search
 direction less than 2*options.TolFun and maximum constraint violation
  is less than options.TolCon.
Active inequalities (to within options.TolCon = 1e-006):
  lower      upper     ineqlin   ineqnonlin
                          8           
Elapsed time is 292.878329 seconds.

x_x0 =

   14.8410
   12.5973
   18.3707
    5.6539
   12.1541
   18.3603
   -2.5993
   15.0912
   11.2407
    6.4329
   -1.8655
   15.5438
    7.6907
   12.5108
   12.5000

</pre><img vspace="5" hspace="5" src="walkthrough_05.png"> <img vspace="5" hspace="5" src="walkthrough_06.png"> <h2>Solve the Problem Using a Different Start Point<a name="8"></a></h2>
         <p>At this point, I&#8217;d like take a look at our problem from a different angle.  As you may know, gradient based solvers, such
            as the one we used here, tend to fail if the objective function does not have smooth derivatives or the problem is ill-defined.
             Let's change the start point of this problem, and see if <tt>fmincon</tt> can find a solution.
         </p><pre class="codeinput">load <span class="string">wbPSOpt</span>
tic
<span class="keyword">try</span>
    [x_x0c] = fmincon(@(x) wbObjFun(x,time,idealProfile),x0c,Aineq,bineq,<span class="keyword">...</span>
       [],[],lb,ub,[],options);
<span class="keyword">catch</span>
    disp(<span class="string">'We caught the following error'</span>)
    nowork = lasterror;
    nowork.message
<span class="keyword">end</span>
toc
</pre><pre class="codeoutput">
                                max     Line search  Directional  First-order 
 Iter F-count        f(x)   constraint   steplength   derivative   optimality Procedure 
    0     16      15.4198   1.554e-015                                         
    1     33      393.237      -0.3751          0.5          940          109   
    2     50      275.985      -0.1875          0.5         -234         55.5   
    3     66      163.538   1.776e-015            1         -632         47.1   
We caught the following error

ans =

Error using ==&gt; sim
Error originates in Mechanical block DoubleWishbone/Double Wishbone/Double Wish/shock abs/Spherical1. A constraint has been violated. Check constraint solver type and tolerances in the Machine Environment block and Simulink solver and tolerances in Configuration Parameters. Check for kinematic singularities.

Elapsed time is 78.577761 seconds.
</pre><img vspace="5" hspace="5" src="walkthrough_07.png"> <img vspace="5" hspace="5" src="walkthrough_08.png"> <p>The error message is from our Simulink model and basically tells us we put in a bad definition for geometry that our model
            could not handle.
         </p>
         <p>Just to be clear, the problem is not with the <tt>fmincon</tt> solver, but with my knowledge of the design space in which we are looking to explore. <tt>fmincon</tt> did what I asked it to do, it just found a point that my model could not handle.  To use <tt>fmincon</tt> we need to add constraints to our problem definition or redefine our objective function to prevent this situation from happening.
         </p>
         <p>I&#8217;m pointing this out because in practice you may often encounter problems where you just don&#8217;t have enough information to
            completely define your problem or you may have an objective function that does not have well defined gradients, or even holes
            in the design space. This often renders traditional gradient based solvers, such as the one used here, ineffective. This is
            where algorithms in Genetic Algorithm and Direct Search Toolbox are useful.
         </p>
         <h2>Solve the Problem Using Genetic Algorithm and Direct Search Toolbox<a name="10"></a></h2>
         <p>We can use pattern search to solve this problem without having to redefine our objective function, constraints, or the Simulink
            model.  Let's define our problem as before and solve.  We loaded in the problem formulation in the structure <tt>wbPSOpt</tt> in the previous step.  Let's see if pattern search can find a solution.
         </p>
         <p><i>Note: You can set up your problem in the <tt>psearchtool</tt> GUI analogously to the <tt>optimtool</tt> GUI.  File --&gt; Import Options --&gt; wbPSOpt, set up the problem definition, and then click on <b>Start</b>.</i></p><pre class="codeinput"><span class="comment">% psearchtool % uncomment this line to run interactively</span>
<span class="comment">%</span>
<span class="comment">% Command line equivalent (comment out this section if using psearchtool)</span>
<span class="comment">% -----------------------</span>
<span class="comment">% Start with default options</span>
options = psoptimset;
<span class="comment">% Modify some parameters</span>
<span class="comment">% Note: You can use the defaults as well, this will speed up the solution</span>
options = psoptimset(options,<span class="string">'TolFun'</span> ,0.06);
options = psoptimset(options,<span class="string">'MeshContraction'</span> ,0.25);
options = psoptimset(options,<span class="string">'MaxMeshSize'</span> ,1 );
options = psoptimset(options,<span class="string">'PollingOrder'</span> ,<span class="string">'success'</span>);
options = psoptimset(options,<span class="string">'SearchMethod'</span> ,@GPSPositiveBasisNp1);
options = psoptimset(options,<span class="string">'Display'</span> ,<span class="string">'iter'</span>);
options = psoptimset(options,<span class="string">'Cache'</span> ,<span class="string">'on'</span>);
options = psoptimset(options,<span class="string">'CacheTol'</span> ,2.2204e-010);
<span class="comment">% Request plots, add my custom plot to the mix</span>
options = psoptimset(options,<span class="string">'OutputFcns'</span> ,{ { @updatePlot } });
options = psoptimset(options,<span class="string">'PlotFcns'</span> ,{  @psplotbestf @psplotbestx });
<span class="comment">%%Run PATTERNSEARCH</span>
tic
[x_x0c_ps] = patternsearch(@(x) wbObjFun(x,time,idealProfile),x0c,<span class="keyword">...</span>
    Aineq,bineq,[],[],lb,ub,[],options);
toc
x_x0c_ps
</pre><pre class="codeoutput">

Iter     f-count          f(x)      MeshSize     Method
    0        1        15.4198             1      
    1        4        4.73299             1     Successful Search
    2        9         4.5874             1     Successful Search
    3       10        4.38583             1     Successful Search
    4       11        4.06205             1     Successful Search
    5       12        3.66635             1     Successful Search
    6       18        3.29301             1     Successful Search
    7       49        3.29301          0.25     Refine Mesh
    8       50        3.12307          0.25     Successful Search
    9       61        2.94817          0.25     Successful Search
   10       63        2.58323          0.25     Successful Search
   11       64        2.38127          0.25     Successful Search
   12       69        2.31824          0.25     Successful Search
   13       70        2.29935          0.25     Successful Search
   14       84        2.20802          0.25     Successful Search
   15       86        1.93259          0.25     Successful Search
   16       92        1.73249          0.25     Successful Search
   17       93        1.56504          0.25     Successful Search
   18       94        1.45678          0.25     Successful Search
   19       95        1.43817          0.25     Successful Search
   20      126        1.43817        0.0625     Refine Mesh
   21      137        1.29085        0.0625     Successful Search
   22      143        1.26478        0.0625     Successful Search
   23      144        1.24756        0.0625     Successful Search
   24      145        1.24026        0.0625     Successful Search
   25      156        1.19536        0.0625     Successful Search
   26      162        1.14688        0.0625     Successful Search
   27      163        1.10743        0.0625     Successful Search
   28      164        1.07901        0.0625     Successful Search
   29      165         1.0635        0.0625     Successful Search
   30      166        1.06245        0.0625     Successful Search

Iter     f-count        f(x)       MeshSize      Method
   31      177        1.02141        0.0625     Successful Search
   32      183       0.966209        0.0625     Successful Search
   33      184       0.924368        0.0625     Successful Search
   34      185       0.899389        0.0625     Successful Search
   35      186       0.894236        0.0625     Successful Search
   36      203       0.893817        0.0625     Successful Search
   37      210       0.880355        0.0625     Successful Search
   38      237       0.880355       0.01563     Refine Mesh
Optimization terminated: change in the function value less than options.TolFun.
Elapsed time is 255.561565 seconds.

x_x0c_ps =

   12.1238
   12.5998
   16.7597
    9.1324
   12.0842
   17.0282
   -1.6750
   14.4139
    9.0207
    6.0950
   -1.6124
   14.8704
    7.1561
   11.2301
   11.5000

</pre><img vspace="5" hspace="5" src="walkthrough_09.png"> <img vspace="5" hspace="5" src="walkthrough_10.png"> <p>Pattern search found a solution.  We set the function tolerance for both <tt>fmincon</tt> and <tt>patternsearch</tt> to limit the total number of iterations required to satisfy our convergence criteria.  Decreasing this value would allow
            the solution to converge more tightly on the 'Ideal' profile, but would require longer solution times.
         </p>
         <h2>Summary<a name="12"></a></h2>
         <p>We used two approaches to optimize the design of a double wishbone suspension system.  The first approach illustrated traditional
            gradient base techniques that work well for well-defined problems.  We then found a condition where the gradient based method
            did not find a solution, since we do not have a completely defined problem and were unlucky in the choice of start point.
             The second approach, pattern search, was used to illustrate non-gradient based methods that work well for problems that are
            ill-defined or do not have well define gradients or objective functions.
         </p>
         <p class="footer"><br>
            Published with MATLAB&reg; 7.3<br></p>
      </div>
      <!--
##### SOURCE BEGIN #####
%% Optimization of a Double Wishbone Suspension System
% This demo shows how to use MATLAB, Optimization Toolbox, and Genetic
% Algorithm and Direct Search Toolbox to optimize the design of a double
% wishbone suspension system.
%
% _Note: You will need to have the following products installed in order to
% run this demo: MATLAB, Simulink, Optimization Toolbox, Genetic Algorithm
% and Direct Search Toolbox, and SimMechanics. Optional: Virtual Reality
% Toolbox._
%
%% Introduction to the Problem
% We wish to optimize the response of a double wishbone suspension system.