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Unformatted text preview: 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 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 R max [ H ( j )] = H . From Parsevals equality, y 2 = 1 + Y ( j ) Y ( j ) d . 2 2 Hence, 1 2 2 max ( H ( j )) 2 U ( j ) U ( j ) d sup max [ H ( j )] 2 2 . y 2 2 R u To show the bound is tight, pick (SISO case) u ( t ) = exp ( t + j t ), i.e., U ( s ) = 1 / ( s j ), with < 0. Then, y 2 = | H ( + j ) | 2 u 2 2 2 As 0, by the continuity of H on the imaginary axis, the gain approaches | H ( j ) | ....
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This note was uploaded on 01/12/2012 for the course ECE 6.241j taught by Professor Prof.emiliofrazzoli during the Spring '11 term at MIT.
- Spring '11