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MIT6_241JS11_lec13

# MIT6_241JS11_lec13 - 6.241 Dynamic Systems and Control...

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6.241 Dynamic Systems and Control Lecture 13: I/O Stability Readings: DDV, Chapters 15, 16 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology March 14, 2011 E. Frazzoli (MIT) Lecture 13: I/O Stability Mar 14, 2011 1 / 6

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L 2 -induced norm Theorem ( H norm is the L 2 -induced norm) The L 2 -induced norm of a causal, CT, LTI, stable system S with impulse response H ( t ) and transfer function H ( s ) is S 2 , ind = sup ω R σ max [ H ( j ω )] = H . From Parseval’s equality, y 2 = 1 + Y ( j ω ) Y ( j ω ) d ω. 2 2 π −∞ Hence, 1 2 2 σ max ( H ( j ω )) 2 U ( j ω ) U ( j ω ) d ω sup σ max [ H ( j ω )] 2 2 . y 2 2 π R ω u To show the bound is tight, pick (SISO case) u ( t ) = exp ( t + j ω 0 t ), i.e., U ( s ) = 1 / ( s j ω 0 ), with � < 0. Then, y 2 = | H ( + j ω 0 ) | 2 u 2 2 2 As 0, by the continuity of H on the imaginary axis, the gain approaches | H ( j ω 0 ) | . E. Frazzoli (MIT) Lecture 13: I/O Stability Mar 14, 2011 2 / 6
Computation of norm H Theorem Let H ( s ) = C ( sI A )

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