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

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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].
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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 (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].
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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].
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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. Consider multiplicative uncertainty G Δ = (1 + W i Δ) G N , σ [Δ( j ω )] 1 ω Then L Δ = G Δ G c = (1 + W i Δ) G N G c = (1 + W i Δ) L N Define S Δ = (1 + L Δ ) 1 as the set of possible sensitivity TF for this uncertain system. For RP , we need | W s S Δ | < 1 ω and all possible Δ s Can rewrite this condition as RP | W s | < | 1 + �L Δ | ∀ ω and all possible Δ s so we need all possible L Δ ( j ω ) to remain outside the disc( 1 , | W s ( j ω ) | ) But we already know that L Δ ( j ω ) will remain inside the disc( L N ( j ω ) , | W i ( j ω ) L N ( j ω ) | ) which follows from our previous analysis (21–6) of the
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