LMI 转化与Matlab工具箱

一些结论

(1) [1]中给出下面两结论

      ∃ P > 0 ,    s . t .    A T P A − P < 0 \ \ \ \ \ \exists P>0,\;s.t.\;A^TPA-P<0      ∃P>0,s.t.ATPA−P<0
⇔ ∃ P > 0 ,    G ,    s . t .    [ P ∗ G A G + G T − P ] > 0 \Leftrightarrow\exists P>0,\;G,\;s.t.\;\begin{bmatrix}P&\ast\\GA&G+G^T-P\end{bmatrix}>0 ⇔∃P>0,G,s.t.[PGA​∗G+GT−P​]>0
      ∃ P > 0 ,    s . t .    ( A + B K ) T P ( A + B K ) − P < 0 \ \ \ \ \ \exists P>0,\;s.t.\;{(A+BK)}^TP(A+BK)-P<0      ∃P>0,s.t.(A+BK)TP(A+BK)−P<0
⇔ ∃ P > 0 ,    G ,    L ,    K = L G − 1 ,      s . t .    [ P A G + B L ∗ G + G T − P ] > 0 \Leftrightarrow\exists P>0,\;G,\;L,\;K=LG^{-1},\;\;s.t.\;\begin{bmatrix}P&AG+BL\\\ast&G+G^T-P\end{bmatrix}>0 ⇔∃P>0,G,L,K=LG−1,s.t.[P∗​AG+BLG+GT−P​]>0
我进行了可能会有用的推广 S T U S − V < 0 ⇔ [ V ∗ G S G + G T − U ] > 0 S^TUS-V<0\Leftrightarrow\begin{bmatrix}V&\ast\\GS&G+G^T-U\end{bmatrix}>0 STUS−V<0⇔[VGS​∗G+GT−U​]>0

lmivar

case1

X1 = lmivar(1,[n 0]);
X 1 = δ I n X_1=\delta I_n X1​=δIn​

case2

X2=lmivar(1,[n 1]);
X 2 = X 2 T X_2=X_2^T X2​=X2T​是 n × n n\times n n×n矩阵.

case3

X3 = lmivar(2,[m n]);
X 3 X_3 X3​是 m × n m\times n m×n矩阵.

case4

[X1,n,sX1] = lmivar(2,[2 3]);
[X2,n,sX2] = lmivar(2,[3 2]);
[X,n,sX] = lmivar(3,[sX1,zeros(2);zeros(3),sX2]);

参考

[1]M.C. de Oliveira and J. Bernussou and J.C. Geromel. A new discrete-time robust stability condition[J]. Systems & Control Letters, 1999.