sos.htm

来自「国外专家做的求解LMI鲁棒控制的工具箱,可以相对高效的解决LMI问题」· HTM 代码 · 共 459 行 · 第 1/2 页

double(lower)
<font color="#000000">
 ans =</font></pre>
          <pre><font color="#000000">&nbsp;&nbsp;&nbsp;  0.75</font></pre>
          </td>
        </tr>
      </table>
      <p>Multiple SOS constraints can also be used. Consider the following problem of 
      finding the smallest possible <b>t</b> such that the polynomials <b>t(1+xy)<sup>2</sup>-xy+(1-y)<sup>2</sup></b> 
      and <b>(1-xy)<sup>2</sup>+xy+t(1+y)<sup>2</sup> </b>are both 
      sum of squares.</p>
      <table cellPadding="10" width="100%">
        <tr>
          <td class="xmpcode">
          <pre>sdpvar x y t
p1 = t*(1+x*y)^2-x*y+(1-y)^2;
p2 = (1-x*y)^2+x*y+t*(1+y)^2;
F = set(sos(p1))+set(sos(p2));
solvesos(F,t);</pre>
          <pre>double(t)
<font color="#000000">
 ans = 

   0.2500</font></pre>
          <pre>sdisplay(sosd(F(1)))</pre>
          <pre><font color="#000000">ans = </font></pre>
          <pre><font color="#000000">    '-1.102+0.95709y+0.14489xy'
    '-0.18876-0.28978y+0.47855xy'</font></pre>
          <pre>sdisplay(sosd(F(2)))</pre>
          <pre><font color="#000000">ans = </font></pre>
          <pre><font color="#000000">    '-1.024-0.18051y+0.76622xy'
    '-0.43143-0.26178y-0.63824xy'
    '0.12382-0.38586y+0.074568xy'</font></pre>
          <pre>&nbsp;</pre>
          </td>
        </tr>
      </table>
      <p>
          <img border="0" src="demoicon.gif" width="16" height="16">If you have 
		parametric variables, bounding the feasible region typically 
		improves numerical behavior. Having lower bounds will 
			additionally decrease the size of the optimization problem 
			(variables bounded from below can be treated as translated cone 
			variables in <a href="dual.htm#dualization">dualization</a>, which 
			is used by <a href="reference.htm#solvesos">
			solvesos</a>).</p>
		<p>
          <img border="0" src="demoicon.gif" width="16" height="16"> One of the 
			most common mistake people make when using the sum of squares module 
			is to forget to declare some parametric variables. This will 
			typically lead to a (of-course erroneous) huge sum of squares 
			problem which easily freezes MATLAB and/or give strange error 
			messages (trivially infeasible, nonlinear parameterization, etc). 
			Make sure to initially run the module in verbose mode to see how 
			YALMIP characterizes the problem (most importantly to check the 
			number of parametric variables and independent variables).</p>
		<p>
          <img border="0" src="demoicon.gif" width="16" height="16">When you use 
			a kernel representation (<code>sos.model=1</code> and typically the case also 
			for <code>sos.model=0</code>), YALMIP will derive and solve a 
			problem which is related to the dual of your original problem. This 
			means that warnings about infeasibility after solving the SDP actually means 
		unbounded objective, and vice versa. If you use <code>sos.model=2</code>, 
			a primal problem is solved, and YALMIP error messages are directly related to 
			your problem.</p>
		<p>
          <img border="0" src="demoicon.gif" width="16" height="16">The quality 
			of the SOS approximation is typically improved substantially if the 
			tolerance and precision options of the semidefinite solver is 
			decreased. As an example, having <code>sedumi.eps</code> 
			less than 10<sup>-10 </sup> when solving sum of squares problems is 
			typically recommended for anything but trivial problems. There is a 
			higher likelihood that the semidefinite solver will complain about 
			numerical problems in the end-phase, but the resulting solutions are 
			typically much better. This seem to be even more important in 
			parameterized problems.</p>
		<p>
          <img border="0" src="demoicon.gif" width="16" height="16">To evaluate 
			the quality and success of the sum of squares decomposition, do not 
			forget to study the discrepancy between the decomposition and the 
			original polynomial.<checksetche&nbsp; The quality 
			of the SOS approximation is typically improved substantially if the 
			tolerance and precision options of the semidefinite solver is 
			decreased. As an example, having <code> No problems in the 
			semidefinite solver is no guarantee for a successful decomposition 
			(due to numerical reasons, tolerances in the solvers etc.)</p>
		<checksetche&nbsp; The quality 
			of the SOS approximation is typically improved substantially if the 
			tolerance and precision options of the semidefinite solver is 
			decreased. As an example, having <code>
		<h3>Polynomial parameterizations</h3>
      <p>A special feature of the sum of squares package in YALMIP is the 
      possibility to work with nonlinear SOS parameterizations, i.e. SOS problems 
      resulting in PMIs (polynomial matrix inequalities) instead of LMIs. The following piece of code solves a 
      nonlinear control <i>synthesis</i> problem using sum of squares. Note 
      that this requires the solver <a href="solvers.htm#penbmi">PENBMI</a>.</p>
      <table cellPadding="10" width="100%">
        <tr>
          <td class="xmpcode">
          <pre>clear all
yalmip('clear');

% States...
sdpvar x1 x2
x = [x1;x2];

% Non-quadratic Lyapunov z'Pz 
z = [x1;x2;x1^2];
P = sdpvar(3,3);
V = z'*P*z;

% Non-linear state feedback
v = [x1;x2;x1^2];
K = sdpvar(1,3);
u = K*v;

% System x' = f(x)+Bu
f = [1.5*x1^2-0.5*x1^3-x2; 3*x1-x2];
B = [0;1];

% Closed loop system, u = Kv
fc = f+B*K*v;

% Stability and performance constraint dVdt &lt; -x'x-u'u
% NOTE : This polynomial is bilinear in P and K
F = set(sos(-jacobian(V,x)*fc-x'*x-u'*u));

% P is positive definite, bound P and K for numerical reasons
F = F + set(P&gt;0)+set(25&gt;P(:)&gt;-25)+set(25&gt;K&gt;-25);

% Minimize trace(P)
% Parametric variables P and K automatically detected
% by YALMIP since they are both constrained
solvesos(F,trace(P));</pre>
          </td>
        </tr>
      </table>
      <h3><a name="lowrank"></a>Low-rank sum-of-squares (<font color="#FF0000">experimental!</font>)</h3>
		<p>By using the capabilities of the solver <a href="solvers.htm#LMIRANK">LMIRANK</a>, 
		we can pose sum-of-squares problems where we search for decompositions 
		with few terms (low-rank Gramian). Consider the following problem where 
		a trace heuristic leads to an effective rank of 5, perhaps 6. </p>
      <table cellPadding="10" width="100%" id="table2">
        <tr>
          <td class="xmpcode">
          <pre>x = sdpvar(1,1);
y = sdpvar(1,1);
f = 200*(x^3 - 4*x)^2+200 * (y^3 - 4*y)^2+(y - x)*(y + x)*x*(x + 2)*(x*(x - 2)+2*(y^2 - 4));
g = 1 + x^2 + y^2;
p = f * g;
F = set(sos(h));
[sol,v,Q] = solvesos(F,[],sdpsettings('sos.traceobj',1));

eig(Q{1})
<font color="#000000">ans =</font></pre>
			<pre><font color="#000000">  1.0e+003 *</font></pre>
			<pre><font color="#000000">    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0000
    0.0001
    0.0124
    0.3977
    3.3972
    3.4000
    6.7972</font></pre>
          </td>
        </tr>
      </table>
      <p>We solve the problem using <a href="solvers.htm#LMIRANK">LMIRANK</a> 
		instead, and aim for a rank less than or equal to 4. The desired rank is 
		specified easily in the <a href="reference.htm#sos">sos</a> construct.</p>
      <table cellPadding="10" width="100%" id="table3">
        <tr>
          <td class="xmpcode">
          <pre>x = sdpvar(1,1);
y = sdpvar(1,1);
f = 200*(x^3 - 4*x)^2+200 * (y^3 - 4*y)^2+(y - x)*(y + x)*x*(x + 2)*(x*(x - 2)+2*(y^2 - 4));
g = 1 + x^2 + y^2;
p = f * g;
F = set(sos(h,<font color="#FF0000">4</font>));
[sol,v,Q] = solvesos(F,[],sdpsettings('lmirank.eps',0));

eig(Q{1})
<font color="#000000">ans =</font></pre>
			<pre><font color="#000000">  1.0e+003 *</font></pre>
			<pre><font color="#000000">   -0.0000
   -0.0000
   -0.0000
   -0.0000
   -0.0000
   -0.0000
   -0.0000
   -0.0000
    0.0000
    0.0000
    0.4634
    4.2674
    4.6584
    7.1705</font></pre>
          </td>
        </tr>
      </table>
      <p>The resulting rank is indeed effectively 4. Note though that <a href="solvers.htm#LMIRANK">LMIRANK</a> 
		works on the dual problem side, and can return slightly infeasible 
		solutions (in terms of positive definiteness.) Keep in mind though that 
		sum-of-squares decompositions <i>almost always</i> be slightly wrong, in 
		one way or the other. If a dual (also called image) approach is used 
		(corresponding to <font color="#0000FF">sos.model=2</font>), positive 
		definiteness may be violated, and if a primal approach (also called 
		kernel) approach is used (corresponding to <font color="#0000FF">
		sos.model=1</font>), there is typically a difference between the 
		polynomial and its decomposition. This simply due to the way SDP solvers 
		work. See more in the example <font color="#0000FF">sosex.m</font></p>
		<p>Remember that <a href="solvers.htm#LMIRANK">LMIRANK</a> is a local 
		solver, hence there are no guarantees that it will find a low rank 
		solution even though one is known to exist. Moreover, note that <a href="solvers.htm#LMIRANK">LMIRANK</a> 
		does not support objective functions.</td>
    </tr>
  </table>
</div>

</body>

</html>