Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
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Ma<strong>in</strong> LMI problems<br />
1 LMI feasibility problem: Test whether there exists x1, . . . , xn such<br />
that F (x) ≺ 0.<br />
2 LMI optimization problem: M<strong>in</strong>imize f(x) over all x for which the<br />
LMI F (x) ≺ 0 is satisfied.<br />
How is this solved?<br />
F (x) ≺ 0 is feasible iff m<strong>in</strong>x λmax(F (x)) < 0 and therefore <strong>in</strong>volves<br />
m<strong>in</strong>imiz<strong>in</strong>g the function<br />
x ↦→ λmax (F (x))<br />
Possible because this function is convex!<br />
There exist efficient algorithms (Interior po<strong>in</strong>t, ellipsoid).<br />
Siep Weiland and Carsten Scherer (DISC) <strong>L<strong>in</strong>ear</strong> <strong>Matrix</strong> <strong>Inequalities</strong> <strong>in</strong> <strong>Control</strong> Class 1 43 / 59
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