LMI 转化与Matlab工具箱

LMI 转化与Matlab工具箱

目录

• 一些结论
• • (1) [1]中给出下面两结论
• lmivar
• • case1
  • case2
  • case3
  • case4
• 参考

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>ATPA−P<0
⇔ ∃ P > 0 , G , s . t . [ P ∗ G A G + G T − P ] > 0 Leftrightarrowexists P>0,;G,;s.t.;begin{bmatrix}P&ast\GA&G+G^T-Pend{bmatrix}>0 ⇔∃P>0,[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>(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 Leftrightarrowexists P>0,;G,;L,;K=LG^{-1},;;s.t.;begin{bmatrix}P&AG+BL\ast&G+G^T-Pend{bmatrix}>0 ⇔∃P>0,G,L,K=LG−[P∗​AG+BLG+GT−P​]>0
S T U S − V < 0 ⇔ [ V ∗ G S G + G T − U ] > 0 S^TUS-V<0Leftrightarrowbegin{bmatrix}V&ast\GS&G+G^T-Uend{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 ntimes n n×n矩阵.

case3

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