# Lect_21 - Nonlinear Systems and Control Lecture 21 L 2...

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Unformatted text preview: Nonlinear Systems and Control Lecture # 21 L 2 Gain & The Small-Gain theorem – p.1/14 Theorem 5.4: Consider the linear time-invariant system ˙ x = Ax + Bu, y = Cx + Du where A is Hurwitz. Let G ( s ) = C ( sI − A ) − 1 B + D . Then, the L 2 gain of the system is sup ω ∈ R bardbl G ( jω ) bardbl – p.2/14 Lemma: Consider the time-invariant system ˙ x = f ( x,u ) , y = h ( x,u ) where f is locally Lipschitz and h is continuous for all x ∈ R n and u ∈ R m . Let V ( x ) be a positive semidefinite function such that ˙ V = ∂V ∂x f ( x,u ) ≤ a ( γ 2 bardbl u bardbl 2 −bardbl y bardbl 2 ) , a,γ > Then, for each x (0) ∈ R n , the system is finite-gain L 2 stable and its L 2 gain is less than or equal to γ . In particular bardbl y τ bardbl L 2 ≤ γ bardbl u τ bardbl L 2 + radicalBigg V ( x (0)) a – p.3/14 Proof V ( x ( τ )) − V ( x (0)) ≤ aγ 2 integraldisplay τ bardbl u ( t ) bardbl 2 dt − a integraldisplay τ bardbl y ( t ) bardbl 2 dt V ( x ) ≥ integraldisplay τ bardbl y ( t ) bardbl 2 dt ≤ γ 2 integraldisplay τ bardbl u ( t ) bardbl 2 dt + V ( x (0)) a bardbl y τ bardbl L 2 ≤ γ bardbl u τ bardbl L 2 + radicalBigg V ( x (0)) a – p.4/14 Lemma 6.5: If the system ˙ x = f ( x,u )...
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Lect_21 - Nonlinear Systems and Control Lecture 21 L 2...

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