Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
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Computer tools support<br />
• Matlab ® Robust <strong>Control</strong> Toolbox<br />
• Public doma<strong>in</strong> SDP-solvers (Matlab ® based):<br />
– SP (Boyd & Vanderberghe) –old,<br />
– SeDuMi (Sturm, now – McMaster University),<br />
• Public doma<strong>in</strong> preprocessors<br />
– YALMIP (Löfberg <strong>in</strong> ETH Zurich):<br />
• Many other possibilities (see YALMIP www<br />
page)<br />
KSO materiały do wykładu 2008/09 19<br />
Scope of the lecture<br />
• <strong>Control</strong> – classical vs. modern<br />
• Optimisation approach<br />
• Computer tools support<br />
• LMI primer<br />
• Basic LMI tools for control<br />
• Analysis<br />
• Synthesis<br />
• Conclusions<br />
KSO materiały do wykładu 2008/09 20<br />
F(<br />
x)<br />
LMI Primer (1) - def<strong>in</strong>ition<br />
LMI is a matrix dependent on vector x<br />
F<br />
0<br />
m<br />
i 1<br />
T<br />
z Fz<br />
x i<br />
F i<br />
0,<br />
z<br />
0<br />
where m<br />
x R , Fi<br />
Positive def<strong>in</strong>ite matrix F is such that<br />
0, z<br />
n<br />
R .<br />
n n<br />
If LMI has a solution (is feasible) then the solution is a<br />
non empty set. The solution set is convex.<br />
x|<br />
F(<br />
x)<br />
0<br />
KSO materiały do wykładu 2008/09 21<br />
R<br />
7
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