Fall 2001 16.3119-1 Bounded gain There exist very easy ways of testing(analytically) whether S Gu)< SISO Bounded Gain Theorem: Gain of generic stable system A r+ Bu C r+ Du is bounded in the sense that Gmax=sup GG)=sup C(jwI-A)B+D <y if and only if (iff 2. The hamiltonian matrix A+b(I-D'D)-D C B(- DID)-IBT C(+D(I-DD-DCT-A-CDOI-D'D-B has no eigenvalues on the imaginary axis Note that with D=0. the Hamiltonian matrix is 4 BB Eigenvalues of this matrix are symmetric about the real and imaginary axis (related to the SRLFall 2001 16.31 19–1 Bounded Gain • There exist very easy ways of testing (analytically) whether |S(jω)| < γ, ∀ω • SISO Bounded Gain Theorem: Gain of generic stable system x˙ = Ax + Bu y = Cx + Du is bounded in the sense that Gmax = sup ω |G(jω)| = sup ω |C(jωI − A) −1 B + D| < γ if and only if (iff) 1. |D| < γ 2. The Hamiltonian matrix H =
A + B(γ2I − DTD)−1DTC B(γ2I − DT D)−1BT −CT (I + D(γ2I − DTD)−1DT )C −AT − CTD(γ2I − DTD)−1BT has no eigenvalues on the imaginary axis. • Note that with D = 0, the Hamiltonian matrix is H = A 1 γ2BBT −CTC −AT  – Eigenvalues of this matrix are symmetric about the real and imaginary axis (related to the SRL)