topic22 - Topic #22 16.31 Feedback Control Systems • RS...

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Unformatted text preview: Topic #22 16.31 Feedback Control Systems • RS – Robust Stability • NP – Nominal Performance • RP – Robust Performance • Small Gain theorem – test closed-loop systems of the form: Δ( s ) M ( s ) – If M ( s ) and Δ( s ) are stable, and we know that Δ( j ω ) < 1 ∀ ω , then if it can be shown that M ( j ω ) < 1 ∀ ω , then the loop gain (product of the two) is less than 1 for all frequencies and the closed-loop system is guaranteed to be stable. Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 22–1 Nominal Performance • We have seen that a typical performance objective is to shape the sensitivity so that | W s ( s ) S ( s ) | ≤ 1 ∀ ω where W s ( s ) is a stable, min phase approximation of the perfor- mance goal, e.g., W s ( s ) = s/M + ω B s + αω B so that at low frequency | W s | ≈ 1 /α (e.g. α = 0 . 01 1 ), and at high frequency, | W s | ≈ 1 /M (e.g. M = 2 > 1 ) • To visualize this NP problem: we want | S ( j ω ) | < | W s ( j ω ) | − 1 ∀ ω which is the same as | 1 + L ( j ω ) | > | W s ( j ω ) | ∀ ω – Recall that | 1 + L ( j ω ) | is the distance from the critical point ( − 1 , 0) to each point on the Nyquist curve L ( j ω ) . – NP requires that this distance be greater than | W s ( j ω ) | – Test this condition by drawing a circle of radius | W s ( j ω ) | cen- tered at ( − 1 , 0) and check that the graph L ( j ω ) always remains outside these circles. – Must be done at all frequency points. November 18, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 22–2 • Interpretation of the NP condition is that the point L ( j ω ) on the Nyquist plot must lie outside the disc centered at -1, radius | W s ( j ω ) | disc( − 1 , | W s ( j ω ) | ) Figure 1: Visualization of the SISO nominal performance test. • There is a different disk that we must consider for each frequency point on the Nyquist plot. • See difference from previous robust stability test? November 18, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 22–3 • Robust Performance – For NP we required | W s S | < 1 ∀ ω – But for RP , we require that | W s S | < 1 ∀ ω and for all possible models in the model set....
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This note was uploaded on 11/07/2011 for the course AERO 16.31 taught by Professor Jonathanhow during the Fall '07 term at MIT.

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topic22 - Topic #22 16.31 Feedback Control Systems • RS...

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