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topic20 - Topic#20 16.31 Feedback Control Systems...

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Topic #20 16.31 Feedback Control Systems Closed-loop system analysis Bounded Gain Theorem Robust Stability 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 20–1 SISO Performance Objectives Basic setup: G c ( s ) G ( s ) r u y d i d o n control error: e = y r = S r T n + SG d i + S d o where L = GG c , S = ( I + L ) 1 , T = L ( I + L ) 1 For good tracking performance of r (typically low frequency), re- quire | S ( j ω ) | small 0 ω w 1 To reduce impact of sensor noise n (typically high frequency), require | T ( j ω ) | small ω w 2 Since T ( s ) + S ( s ) = I s , then we cannot make both | S ( j ω ) | and | T ( j ω ) | small at the same frequencies. Design constraint 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 20–2 SISO Design Approaches There are two basic approaches to design: Indirect that works on L itself Direct that works on S and T Indirect. Note that If | L ( j ω ) | � 1 S L 1 ( | S | � 1 ), T 1 If | L ( j ω ) | � 1 S 1 , T L ( T 1) So we can convert the performance requirements on S , T into specifications on L (A) High loop gain Good command following & dist rejection (B) Low loop gain Attenuation of sensor noise. Must be careful when | L ( j ω ) | ≈ 1 , require that arg L = ± 180 to maintain stability 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 20–3 Direct approach works with S and T . Since e = y r , then for perfect tracking, we need e 0 want S 0 since e = Sr + . . . Suﬃcient to discuss the magnitude of S because the only re- quirement is that it be small. Direct approach is to develop an upper bound for | S | and then test if | S | is below this bound. 1 | S ( j ω ) | < W s ( j ω ) ω ? | | or equivalently, whether | W s ( j ω ) S ( j ω ) | < 1 , ω Typically pick simple forms for weighting functions (first or second order), and then cascade them as necessary. Basic one: s/M + ω B W s ( s ) = s + ω B A Figure 1: Example of a standard performance weighting filter. Typically have A 1 , M > 1 , and | 1 /W s | ≈ 1 at ω B 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 20–4 Example: Simple system with G c = 1 150 G ( s ) = (10 s + 1)(0 . 05 s + 1) 2 Require ω B 5 , a slope of 1, low frequency value less than A = 0 . 01 and a high frequency peak less than M = 5 .

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topic20 - Topic#20 16.31 Feedback Control Systems...

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