lecture_17 - 2.160 System Identification Estimation and...

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2.160 System Identification, Estimation, and Learning Lecture Notes No. 1 7 April 24, 2006 12. Informative Experiments 12.1 Persistence of Excitation Informative data sets are closely related to “ Persistence of Excitation”, an important concept used in adaptive and learning controls. See the block diagram of an indirect adaptive control system below. The control system monitors input-output data in order to identify the plant model in real, and modifies the feedback control as the plant dynamics vary; hence the control system is adaptive to varying plant dynamics. Adaptation Law Model Plant Feedback Control ) ( t y ) ( t u - + (Indirect) Adaptive Control Success of this adaptive control system hinges on the data. The central question is whether the input-output data obtained in real-time are informative enough to identify the plant model uniquely. This is often questionable, since the control system tends to drive the plant to a specific set point or to follow a specific trajectory. The trajectory may not be rich enough to excite the system. The following theory of persistent excitation and informative experiment are fundamental to these questions. Definition 4 A quasi-stationary signal { } ) ( t u , with spectrum ) ( ω u Φ , is said to be persistently exciting of order , n if the condition: ( ) 0 ) ( 2 Φ u i n e M ( 1 ) implies ( ) 0 i n e M ( 2 ) where is an arbitrary linear filter of form: ) ( q M n n n n q m q m q m q M + + + = " 2 2 1 1 ) ( (3) 1
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Remarks: 1. Note () ) ( 2 ω u i n e M Φ is the power spectrum of ) ( ) ( ) ( t u q M t v = . Therefore, a signal u(t) that is persistently exciting of order cannot be filtered to zero by any ( n -1) st order moving average filter (3), hence it is called persistently exciting. n ) ( q M ) ( t u ) ( t v 2. Consider function , associated with ) ( ) ( 1 z M z M n n ( ) 2 i n e M . 11 2 1 2 12 1 2 ( ) ( ) ( ) ( ) nn n n M zM z mz mm z m z z m z −− −+ =++ + + + + + ++ + "" (4) If a+ib is a zero of , a-ib is also a zero, since the function has all real coefficients. Also, if a+ib is a zero, then its reciprocal ) ( ) ( 1 z M z M n n 2 2 b a ib a + is also a zero of the function since the function is symmetric with respect to the unit circle, . See the figure below. This function can have at most ( n -1) zeros on the unit circle. Therefore, 1 z z ( ) ( ) ( ) i n i n i n e M e M e M = 2 may be zero for at most ( n -1) different frequencies. In consequence, if 0 ) ( Φ u for at least n different frequencies; π < < n , , " 1 then u ( t ) is persistently exciting.
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lecture_17 - 2.160 System Identification Estimation and...

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