Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
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Why are LMI’s <strong>in</strong>terest<strong>in</strong>g?<br />
Reason 1: LMI’s def<strong>in</strong>e convex constra<strong>in</strong>ts on x, i.e.,<br />
S := {x | F (x) ≺ 0} is convex.<br />
Reason 2: Solution set of multiple LMI’s<br />
F1(x) ≺ 0, . . . , Fk(x) ≺ 0<br />
is convex and representable as one s<strong>in</strong>gle LMI<br />
⎛<br />
F1(x) 0 . . . 0<br />
⎜<br />
F (x) = ⎝<br />
.<br />
0 .. 0<br />
⎞<br />
⎟<br />
⎠ ≺ 0<br />
0 . . . 0 Fk(x)<br />
Reason 3: Incorporate aff<strong>in</strong>e constra<strong>in</strong>ts such as<br />
F (x) ≺ 0 and Ax = b<br />
F (x) ≺ 0 and x = Ay + b for some y<br />
F (x) ≺ 0 and x ∈ S with S an aff<strong>in</strong>e set.<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 45 / 59
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