LMIs in Control/pages/Hurwitz detectability Hurwitz detectability is a dual concept of Hurwitz stabilizability and is defined as the matrix pair ( A , C ) {\displaystyle (A,C)} , is said to be Hurwitz detectable if there exists a real matrix L {\displaystyle L} such that A + L C {\displaystyle A+LC} is Hurwitz stable. x  ̇ ( t ) = A x ( t ) + B u ( t ) , y ( t ) = C x ( t ) + D u ( t ) x ( 0 ) = x 0 {\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\x(0)&=x_{0}\\\end{aligned}}} where x ( t ) ∈ R n {\displaystyle x(t)\in \mathbb {R} ^{n}} , y ( t ) ∈ R m {\displaystyle y(t)\in \mathbb {R} ^{m}} , u ( t ) ∈ R q {\displaystyle u(t)\in \mathbb {R} ^{q}} , at any t ∈ R {\displaystyle t\in \mathbb {R} } . The matrices A , B , C , D {\displaystyle A,B,C,D} are system matrices of appropriate dimensions and are known. There exist a symmetric positive definite matrix P {\displaystyle P} and a matrix W {\displaystyle W} satisfying A T P + P A + W T C + C T W < 0 {\displaystyle A^{T}P+PA+W^{T}C+C^{T}W<0} There exists a symmetric positive definite matrix P {\displaystyle P} satisfying N c T ( A T P + P A ) N c < 0 {\displaystyle N_{c}^{T}(A^{T}P+PA)N_{c}<0} with N c {\displaystyle N_{c}} being the right orthogonal complement of C {\displaystyle C} . There exists a symmetric positive definite matrix P {\displaystyle P} such that A T P + P A < γ C T C {\displaystyle A^{T}P+PA<\gamma C^{T}C} for some scalar γ > 0 {\displaystyle \gamma >0} Matrix pair ( A , C ) {\displaystyle (A,C)} , is Hurwitz detectable if and only if following LMI holds A T P + P A + W T C + C T W < 0. {\displaystyle A^{T}P+PA+W^{T}C+C^{T}W<0.} N c T ( A T P + P A ) N c < 0 {\displaystyle N_{c}^{T}(A^{T}P+PA)N_{c}<0} A T P + P A − γ C T C < 0 {\displaystyle A^{T}P+PA-\gamma C^{T}C<0} Thus by proving the above conditions we prove that the matrix pair ( A , C ) {\displaystyle (A,C)} is Hurwitz Detectable. Find the MATLAB implementation at this link below Hurwitz detectability Links to other closely-related LMIs LMI for Hurwitz stability LMI for Schur stability Schur Detectability A list of references documenting and validating the LMI. https://en.wikibooks.org/wiki/LMIs_in_Control
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