Dilation
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Matrix inequalities can be dilated in order to obtain a larger matrix inequality. This can be a useful technique to separate design variables in a BMI (bi-linear matrix inequality), as the dilation often introduces additional design variables.
A common technique of LMI dilation involves using the projection lemma in reverse, or the "reciprocal projection lemma." For instance, consider the matrix inequality
[ P A + A T P − P P ∗ − P ] < 0 , {\displaystyle {\begin{bmatrix}{\mathbf {PA}}+{\mathbf {A^{T}P}}-{\mathbf {P}}&{\mathbf {P}}\\*&{\mathbf {-P}}\end{bmatrix}}<0,}
where P ∈ § n × m {\displaystyle {\mathbf {P}}\in \S ^{n\times m}} , A ∈ R n × n {\displaystyle {\mathbf {A}}\in \mathbb {R} ^{n\times n}} , with P > 0. {\displaystyle {\mathbf {P}}>0.} This can be rewritten as
[ A T 1 0 1 0 1 ] [ 0 P 0 ∗ − P 0 ∗ ∗ − P ] [ A 1 1 0 0 1 ] < 0. {\displaystyle {\begin{bmatrix}{\mathbf {A^{T}}}&{\mathbf {1}}&{\mathbf {0}}\\{\mathbf {1}}&{\mathbf {0}}&{\mathbf {1}}\end{bmatrix}}{\begin{bmatrix}{\mathbf {0}}&{\mathbf {P}}&{\mathbf {0}}\\*&{\mathbf {-P}}&{\mathbf {0}}\\*&*&{\mathbf {-P}}\end{bmatrix}}{\begin{bmatrix}{\mathbf {A}}&{\mathbf {1}}\\{\mathbf {1}}&{\mathbf {0}}\\{\mathbf {0}}&{\mathbf {1}}\end{bmatrix}}<0.} (1)
Then since P > 0 , {\displaystyle {\mathbf {P}}>0,}
[ − P 0 ∗ − P ] < 0 , {\displaystyle {\begin{bmatrix}{\mathbf {-P}}&{\mathbf {0}}\\*&{\mathbf {-P}}\end{bmatrix}}<0,}
which is equivalent to
[ 0 1 0 0 0 1 ] [ 0 P 0 ∗ − P 0 ∗ ∗ − P ] [ 0 0 1 0 0 1 ] < 0. {\displaystyle {\begin{bmatrix}{\mathbf {0}}&{\mathbf {1}}&{\mathbf {0}}\\{\mathbf {0}}&{\mathbf {0}}&{\mathbf {1}}\end{bmatrix}}{\begin{bmatrix}{\mathbf {0}}&{\mathbf {P}}&{\mathbf {0}}\\*&{\mathbf {-P}}&{\mathbf {0}}\\*&*&{\mathbf {-P}}\end{bmatrix}}{\begin{bmatrix}{\mathbf {0}}&{\mathbf {0}}\\{\mathbf {1}}&{\mathbf {0}}\\{\mathbf {0}}&{\mathbf {1}}\end{bmatrix}}<0.} (2)
These expanded inequalities (1) and (2) are now in the form of the strict projection lemma, meaning they are equivalent to
Φ ( P ) + G ( A ) V H T + H V T G T ( A ) , {\displaystyle {\mathbf {\Phi }}({\mathbf {P}})+{\mathbf {G}}({\mathbf {A}}){\mathbf {VH^{T}}}+{\mathbf {HV^{T}G^{T}}}({\mathbf {A}}),} (3)
where N ( G T ( A ) ) = R ( N G ( A ) ) , N ( H T ) = R ( N H ) , {\displaystyle N({\mathbf {G^{T}}}({\mathbf {A}}))=R({\mathbf {N}}_{G}({\mathbf {A}})),N({\mathbf {H^{T}}})=R({\mathbf {N}}_{H}),} and V ∈ R n × n . {\displaystyle V\in \mathbb {R} ^{n\times n}.} By choosing
G ( A ) = [ − 1 A T 1 ] , H = [ 1 0 0 ] , {\displaystyle {\mathbf {G}}({\mathbf {A}})={\begin{bmatrix}{\mathbf {-1}}\\{\mathbf {A^{T}}}\\{\mathbf {1}}\end{bmatrix}},{\mathbf {H}}={\begin{bmatrix}{\mathbf {1}}\\{\mathbf {0}}\\{\mathbf {0}}\end{bmatrix}},}
we can now rewrite the inequality (3) as
[ − ( V + V T ) V T A + P V T ∗ − P 0 ∗ ∗ − P ] < 0 , {\displaystyle {\begin{bmatrix}-({\mathbf {V}}+{\mathbf {V^{T}}})&{\mathbf {V^{T}A}}+{\mathbf {P}}&{\mathbf {V^{T}}}\\*&{\mathbf {-P}}&{\mathbf {0}}\\*&*&{\mathbf {-P}}\end{bmatrix}}<0,}
which is the new dilated inequality.
Examples
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Some useful examples of dilated matrix inequalities are presented here.
Example 1
Consider matrices A , G ∈ R n × n , Δ ∈ R m × m , P ∈ § n , δ 1 , δ 2 , a , b ∈ R > 0 , {\displaystyle {\mathbf {A,G}}\in \mathbb {R} ^{n\times n},{\mathbf {\Delta }}\in \mathbb {R} ^{m\times m},{\mathbf {P}}\in \S ^{n},\delta _{1},\delta _{2},a,b\in \mathbb {R} _{>0},} where P > 0 {\displaystyle {\mathbf {P}}>0} and b = a − 1 . {\displaystyle b=a^{-1}.} The following matrix inequalities are equivalent:
A P + P A T + δ 1 P + δ 2 A P A T + P Δ T Δ P < 0 ; {\displaystyle {\mathbf {AP}}+{\mathbf {PA^{T}}}+\delta _{1}{\mathbf {P}}+\delta _{2}{\mathbf {APA^{T}}}+{\mathbf {P\Delta ^{T}\Delta P}}<0;}
[ 0 − P P 0 P Δ T ∗ 0 0 − P 0 ∗ ∗ − δ 1 − 1 P 0 0 ∗ ∗ ∗ − δ 2 − 1 P 0 ∗ ∗ ∗ ∗ − 1 ] + H e ( [ A 1 0 0 0 ] G [ 1 − b 1 b 1 1 b Δ T ] ) < 0. {\displaystyle {\begin{bmatrix}{\mathbf {0}}&{\mathbf {-P}}&{\mathbf {P}}&{\mathbf {0}}&{\mathbf {P\Delta ^{T}}}\\*&{\mathbf {0}}&{\mathbf {0}}&{\mathbf {-P}}&{\mathbf {0}}\\*&*&-\delta _{1}^{-1}{\mathbf {P}}&{\mathbf {0}}&{\mathbf {0}}\\*&*&*&-\delta _{2}^{-1}{\mathbf {P}}&{\mathbf {0}}\\*&*&*&*&{\mathbf {-1}}\\\end{bmatrix}}+He({\begin{bmatrix}{\mathbf {A}}\\{\mathbf {1}}\\{\mathbf {0}}\\{\mathbf {0}}\\{\mathbf {0}}\\\end{bmatrix}}{\mathbf {G}}{\begin{bmatrix}{\mathbf {1}}&-b{\mathbf {1}}&b{\mathbf {1}}&{\mathbf {1}}&b{\mathbf {\Delta }}^{T}\end{bmatrix}})<0.}
Example 2
Consider matrices A , V ∈ R n × n , P , X ∈ § n , B ∈ R n × m , C ∈ R p × n , D ∈ R p × m , R ∈ § m , {\displaystyle {\mathbf {A,V}}\in \mathbb {R} ^{n\times n},{\mathbf {P,X}}\in \S ^{n},{\mathbf {B}}\in \mathbb {R} ^{n\times m},C\in \mathbb {R} ^{p\times n},{\mathbf {D}}\in \mathbb {R} ^{p\times m},{\mathbf {R}}\in \S ^{m},} and S ∈ § p , {\displaystyle {\mathbf {S}}\in \S ^{p},} where P , R , S , X > 0. {\displaystyle {\mathbf {P,R,S,X}}>0.} The matrix inequality
[ − V − V T V A + P V B 0 V ∗ − 2 P + X 0 C T 0 ∗ ∗ − R D T 0 ∗ ∗ ∗ − S 0 ∗ ∗ ∗ ∗ − X ] < 0 {\displaystyle {\begin{bmatrix}-{\mathbf {V}}-{\mathbf {V^{T}}}&{\mathbf {VA}}+{\mathbf {P}}&{\mathbf {VB}}&{\mathbf {0}}&{\mathbf {V}}\\*&-2{\mathbf {P}}+{\mathbf {X}}&{\mathbf {0}}&{\mathbf {C^{T}}}&{\mathbf {0}}\\*&*&-{\mathbf {R}}&{\mathbf {D^{T}}}&{\mathbf {0}}\\*&*&*&-{\mathbf {S}}&{\mathbf {0}}\\*&*&*&*&-{\mathbf {X}}\\\end{bmatrix}}<0}
implies the inequality
[ P A + A T P P B C T ∗ − R D T ∗ ∗ − S ] {\displaystyle {\begin{bmatrix}{\mathbf {PA}}+{\mathbf {A^{T}P}}&{\mathbf {PB}}&{\mathbf {C^{T}}}\\*&-{\mathbf {R}}&{\mathbf {D^{T}}}\\*&*&-{\mathbf {S}}\end{bmatrix}}}
Example 3
Consider matrices A , V ∈ R n × n , Q , X ∈ § n , B ∈ R n × m , C ∈ R p × n , D ∈ R p × m , R ∈ § m , {\displaystyle {\mathbf {A,V}}\in \mathbb {R} ^{n\times n},{\mathbf {Q,X}}\in \S ^{n},{\mathbf {B}}\in \mathbb {R} ^{n\times m},C\in \mathbb {R} ^{p\times n},{\mathbf {D}}\in \mathbb {R} ^{p\times m},{\mathbf {R}}\in \S ^{m},} and S ∈ § p , {\displaystyle {\mathbf {S}}\in \S ^{p},} where Q , R , S , X > 0. {\displaystyle {\mathbf {Q,R,S,X}}>0.} The matrix inequality
[ − V − V T V T A T + Q 0 V T C V T ∗ − 2 Q + X B 0 0 ∗ ∗ − R D T 0 ∗ ∗ ∗ − S 0 ∗ ∗ ∗ ∗ − X ] < 0 {\displaystyle {\begin{bmatrix}-{\mathbf {V}}-{\mathbf {V^{T}}}&{\mathbf {V^{T}A^{T}}}+{\mathbf {Q}}&{\mathbf {0}}&{\mathbf {V^{T}C}}&{\mathbf {V^{T}}}\\*&-2{\mathbf {Q}}+{\mathbf {X}}&{\mathbf {B}}&{\mathbf {0}}&{\mathbf {0}}\\*&*&-{\mathbf {R}}&{\mathbf {D^{T}}}&{\mathbf {0}}\\*&*&*&-{\mathbf {S}}&{\mathbf {0}}\\*&*&*&*&-{\mathbf {X}}\\\end{bmatrix}}<0}
implies the inequality
[ A Q + Q A T B Q C T ∗ − R D T ∗ ∗ − S ] {\displaystyle {\begin{bmatrix}{\mathbf {AQ}}+{\mathbf {QA^{T}}}&{\mathbf {B}}&{\mathbf {QC^{T}}}\\*&-{\mathbf {R}}&{\mathbf {D^{T}}}\\*&*&-{\mathbf {S}}\end{bmatrix}}}
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