MIT6_241JS11_lec12

MIT6_241JS11_lec12 - 6.241 Dynamic Systems and Control...

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6.241 Dynamic Systems and Control Lecture 12: I/O Stability Readings: DDV, Chapters 15, 16 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology March 14, 2011 E. Frazzoli (MIT) Lecture 12: I/O Stability Mar 14, 2011 1 / 12
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± Introduction Last week, we looked at notions of stability for state-space systems, with no inputs. Now we want to consider notions of stability under the effect of a (forcing) input. Central to the discussion is the notion of norm of a signal—which is just the same we already discussed, when considering signals as inFnite-dimensional vectors. In the following, let w : T R n , with w ( t ) = w 1 ( t ) w 2 ( t ) ... w n ( t ) . E. Frazzoli (MIT) Lecture 12: I/O Stability Mar 14, 2011 2 / 12
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norms -norm: Peak magnitude w = sup t T w ( t ) = sup t T max i =1 ... n | w i ( t ) | 2-norm: (Square root of the) Energy w 2 2 = k Z w [ k ] w [ k ] = k Z w [ k ] 2 2 (DT) ± −∞ w ( t ) w ( t ) dt = ± −∞ w ( t ) 2 2 dt (CT) Power (NOT a norm!) P w = ρ 2 w = lim N + 1 2 N N ² k = N w [ k ] w [ k ] = lim N + 1 2 N N ² k = N w [ k ] 2 2 (DT) lim T + 1 2 T ± T T w ( t ) w ( t ) dt = lim T + 1 2 T ± T T w ( t ) 2 2 dt (CT) 1-norm: Action w 1 = k Z w [ k ] 1 (DT) ± −∞ w ( t ) 1 dt (CT) E. Frazzoli (MIT)
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MIT6_241JS11_lec12 - 6.241 Dynamic Systems and Control...

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