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<title>YALMIP Example : Rank constrained LMIs</title>
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<h2>Rank constrained LMIs</h2>
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<p>
<img border="0" src="exclamationmark.jpg" align="left" width="16" height="16">This
example requires the solver <a href="solvers.htm#lmirank">LMIRank</a> (and a
semidefinite solver)</p>
<p>A lot of problems, in particular in control theory, can be
addressed using rank constraints. Unfortunately, adding a
simple rank constraint to a matrix in a standard LMI problem leads
to an NP-hard optimization problem. Nevertheless, with a reasonable
initial guess, a local search can be efficient for finding a
low-rank solution in some cases. The solver <a href="solvers.htm#lmirank">LMIRank</a>
performs such a local search and is interfaced by YALMIP. Details on
the algorithm <a href="solvers.htm#lmirank">LMIRank</a>
can be found in
<a href="readmore.htm#ORSI">[Orsi et. al.]</a>.</p>
<h3>Dynamic controller design using rank constraints</h3>
<p>To illustrate rank constraints in YALMIP, and how to use
the solver <a href="solvers.htm#lmirank">LMIRank</a>, we solve one
of the examples in
<a href="readmore.htm#ORSI">[Orsi et. al.]</a>. The problem is to design a
dynamic output feedback controller for the system <strong>x' = Ax+Bu,
y=Cx</strong></p>
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<pre>A = [0.2 0 1 0;0 0.2 0 1;-1 1 0.2 0;1 -1 0 0.2];
B = [0 0 1 0]';
C = [0 1 0 0];</pre>
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<p>Define matrices <b>X</b> and <b>Y</b> (essentially modeling a Lyapunov
matrix and its inverse) </p>
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<pre>X = sdpvar(4,4);
Y = sdpvar(4,4);</pre>
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<p>Without going into the theoretical background, the following LMI
constraints are necessary for the existence of a controller. </p>
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<pre>Bp = null(B')';
Cp = null(C)';
W = eye(3)*1e-6;</pre>
<pre>F = set(Bp*(A*X+X*A')*Bp' < -W) + set(Cp*(Y*A+A'*Y)*Cp' < -W);</pre>
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<p>The rank constraint enter when we want to couple the matrices X
and Y, and ensure the existence of a dynamic controller with at most 2
states. Once again stating the equations without any theoretical
justification, the resulting constraints are</p>
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<pre>F = F + set([X eye(4);eye(4) Y] >= 0); % not needed, see below
F = F + set(rank([X eye(4);eye(4) Y]) <= 4+2);</pre>
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<p>In the current implementation in YALMIP, a rank constraint
automatically appends a positive semidefiniteness constraint on the
involved matrix (i.e. a non-strict constraint, so these appended cones are
not affected by the option <code>shift</code>.) Hence, the first constraint in the previous piece
of code can (and should) be removed. Also note the use of a
non-strict rank constraint. A strict inequality can be used, but will
currently be interpreted as non-strict, hence it should not be used
in order to avoid confusion. At the moment, the rank operator is
implemented rather straightforwardly using a <a href="extoperators.htm">nonlinear operator</a>,
but requires some dedicated code internally (read hack) to work when
actually calling a solver. Hence, it should be emphasized that the current
implementation is experimental and can be subject to changes in
order to make the operator better integrated.</p>
<p>At this point, we are ready to solve the problem. The solver <a href="solvers.htm#lmirank">LMIRank</a>
needs an initial guess, and this can be supplied either by the user
(by initializing variables and using the <code>'usex0'</code> option
in standard fashion) or
automatically by YALMIP. YALMIP computes an initial guess by
removing the rank constraint and minimizing the trace of the rank
constrained matrix (or sum of traces of all rank constrained
matrices). Note that <a href="solvers.htm#lmirank">LMIRank</a> does not
support objective functions, but only solves feasibility problems.
If you want to optimize an objective function, you need to, e.g.,
perform a bisection. The following call will automatically select an
SDP solver to find the initial guess, solve the initial problem, and
then call <a href="solvers.htm#lmirank">LMIRank</a> (if installed) to search for a
low-rank solution.</p>
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<pre>solvesdp(F);</pre>
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<p>The following call ensures that the solver for the initial guess
is <a href="solvers.htm#sedumi">SeDuMi</a>, and uses an accuracy recommended by the author of <a href="solvers.htm#lmirank">LMIRank</a>.</p>
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<pre>solvesdp(F,[],sdpsettings('lmirank.solver','sedumi','sedumi.eps',0))</pre>
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<p>Note that these computations only give a feasible pair <b>X</b>
and <b>Y</b>. To actually compute the controller (called reconstruction) requires some additional steps. </p>
<p>If we compute the eigenvalues of our rank
constrained matrix, we immediately see that we effectively have a
rank of 6.</p>
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<pre>eig(double([X eye(4);eye(4) Y]))</pre>
<pre><font color="#000000">ans =
-0.0000
-0.0000
1.6780
2.5342
2.7754
2.9674
6.2289
6.8932</font></pre>
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<p>indeed, this is the rank reported by YALMIP.</p>
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<pre>rank([X eye(4);eye(4) Y])
<font color="#000000">Linear scalar (real, derived, 1 variables, current value : 6)</font></pre>
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<p>Note that rank computations are inherently hard from a numerical
point of view, hence it may easily happen that the reported rank is
higher than the desired rank, even though <a href="solvers.htm#lmirank">LMIRank</a>
claimed feasibility. In such cases, a more detailed analysis using
tolerances might be necessary (YALMIP always uses the default
tolerance.)</p>
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<pre>rank(double([X eye(4);eye(4) Y]))</pre>
<pre><font color="#000000">ans =
6</font></pre>
<pre>rank(double([X eye(4);eye(4) Y]),1e-6)</pre>
<pre><font color="#000000">ans =
6</font></pre>
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