Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
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First examples <strong>in</strong> control<br />
Example 2: Jo<strong>in</strong>t stabilization<br />
Given (A1, B1), . . . , (Ak, Bk), f<strong>in</strong>d F such that<br />
(A1 + B1F ), . . . , (Ak + BkF ) asymptotically stable.<br />
Equivalent to f<strong>in</strong>d<strong>in</strong>g F , X1, . . . , Xk such that for j = 1, . . . , k:<br />
<br />
−Xj<br />
0<br />
0<br />
(Aj + BjF )Xj + Xj(Aj + BjF ) ⊤<br />
<br />
≺ 0 not an LMI!!<br />
Sufficient condition: X = X1 = . . . = Xk, K = F X, yields<br />
<br />
−X 0<br />
<br />
≺ 0 an LMI!!<br />
0 AjX + XA ⊤ j + BjK + K ⊤ B ⊤ j<br />
Set feedback F = KX −1 .<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 49 / 59