Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
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Example of duality<br />
Primal <strong>L<strong>in</strong>ear</strong> Program<br />
Lagrange dual cost<br />
Dual <strong>L<strong>in</strong>ear</strong> Program<br />
Popt = <strong>in</strong>f<br />
x<br />
c ⊤ x<br />
subject to x ≥ 0, b − Ax = 0<br />
ℓ(y, z) = <strong>in</strong>f c<br />
x ⊤ x − y ⊤ x + z ⊤ (b − Ax)<br />
<br />
b<br />
=<br />
⊤z if c − A⊤z − y = 0<br />
−∞ otherwise<br />
Dopt = sup<br />
z<br />
b ⊤ z<br />
subject to y = c − A ⊤ z ≥ 0<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 36 / 59