x  ̇ ( t ) = A x ( t ) + B p p ( t ) + B w w ( t ) , z ( t ) = C z x ( t ) p i ( t ) = φ i ( q i ( t ) ) , i = 1 , ... , n p q = C q x , 0 ≤ σ φ i ( σ ) ≤ σ 2   ∀ σ ∈ R {\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{p}p(t)+B_{w}w(t),\\z(t)&=C_{z}x(t)\\p_{i}(t)&=\phi _{i}(q_{i}(t)),i=1,\dots ,n_{p}\\q&=C_{q}x,\\0&\leq \sigma \phi _{i}(\sigma )\leq \sigma ^{2}\ \forall \sigma \in \mathbb {R} \end{aligned}}} The matrices A , B p , B w , C q , C z {\displaystyle A,B_{p},B_{w},C_{q},C_{z}} . The following feasibility problem should be solved as sufficient condition for the stability of the above Lur'e system. Find P > 0 , Λ = d i a g ( λ 1 , ... , λ n p ) ⪰ 0 , T = d i a g ( τ 1 , ... , τ n p ) ⪰ 0 : [ A ⊤ P + P A P B p + A ⊤ C q ⊤ Λ + C q ⊤ T B p ⊤ P + Λ C q A + T C q Λ C q B p + B p ⊤ C q ⊤ Λ − 2 T ] ≺ 0 {\displaystyle {\begin{aligned}{\text{Find}}\;&P>0,\Lambda =diag(\lambda _{1},\dots ,\lambda _{n_{p}})\succeq 0,T=diag(\tau _{1},\dots ,\tau _{n_{p}})\succeq 0:\\&{\begin{bmatrix}A^{\top }P+PA&PB_{p}+A^{\top }C_{q}^{\top }\Lambda +C_{q}^{\top }T\\B_{p}^{\top }P+\Lambda C_{q}A+TC_{q}&\Lambda C_{q}B_{p}+B_{p}^{\top }C_{q}^{\top }\Lambda -2T\end{bmatrix}}\prec 0\\\end{aligned}}} https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/9030b507b39f658fb34fef8563ce9130d5dc2baa https://codeocean.com/capsule/0232754/tree If the feasibility problem with LMI constraints has solution, then the Lure's system is stable. The LMI is only a sufficient condition for the existence of a Lur’e Lyapunov function that proves stability of Lur'e system . It is also necessary when there is only one nonlinearity, i.e., when n p = 1 {\displaystyle n_{p}=1} .
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