21_loop_shaping_2018_10_09_blank-1.pdf - Robust Performance and Loop Shaping MECH 412 System Dynamics and Control Prof James Richard Forbes McGill

21_loop_shaping_2018_10_09_blank-1.pdf - Robust Performance...

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Robust Performance and Loop Shaping MECH 412 - System Dynamics and Control Prof. James Richard Forbes McGill University, Department of Mechanical Engineering October 9, 2018 1/37
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Uncertainty - A Fact of Life I Remember, “All models are wrong , but some are useful !” I This statement applies to everything; to dynamics, circuits, heat transfer, fluids, manufacturing, economics, finance, . . . everything! I In some cases, it is possible to characterize uncertainty parametrically , P ( s ) = b m s m + b m - 1 s m - 1 + · · · + b 1 s + b 0 s n + a n - 1 s n - 1 + · · · + a 1 s + a 0 , ¯ b i b i ¯ b i , ¯ a j a j ¯ a j . I Unfortunately, in practice it can be hard to find the bounds ¯ b i , ¯ b i , ¯ a j , and ¯ a j , especially if m and n are large. I Or, if bounds ¯ b i , ¯ b i , ¯ a j , and ¯ a j can be found, sometimes it’s impractical or over-conservative to apply tools, such as Kharitonov’s Theorem, that exploit such information. 2/37
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Uncertainty Characterization I Consider multiplicative uncertainty of the form P t ( s ) 2 P , P = { P ( s ) ( 1 + Δ ( s ) W 2 ( s )) | | Δ ( |! ) | 1 , \ Δ ( |! ) 2 [ - , ] } . I P t ( s ) is the “true” plant, which exists but is never known, and P ( s ) is the nominal plant model. I P is a set of models that is defined by a nominal plant model, P ( s ) , an uncertainty generator, Δ ( s ) , and an uncertainty bound, W 2 ( s ) . I Δ ( s ) and W 2 ( s ) are both BIBO stable transfer functions. I Δ ( s ) has no physical interpretation, but simplifies finding a solution to the “robust control design” problem (to come). P t ( s ) P ( s ) System Uncertainty Uncertainty Model Figure: Nominal plant P ( s ) , true plant P t ( s ) , system uncertainty, uncertainty model. (Modified from [1].) 3/37
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The Nonparametric Uncertainty Bound W 2 ( s ) I The uncertainty bound, W 2 ( s ) , is a problem dependent transfer function. It “encodes” model confidence. I Smaller | W 2 ( |! ) | means we are confident in the model. I Larger | W 2 ( |! ) | means we are not confident in the model. I In general, | W 2 ( |! ) | starts small when ! is small (i.e., less model uncertainty at low frequency) and increases when ! increases (i.e., more model uncertainty at high frequency). I It is implicitly assumed that the “true” plant and the nominal plant have the same number of poles. P t ( s ) P ( s ) System Uncertainty Uncertainty Model Figure: Nominal plant P ( s ) , true plant P t ( s ) , system uncertainty, uncertainty model. (Modified from [1].) 4/37
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I Let P ( s ) be the nominal plant, and P k ( s ) , k = 1 , 2 , . . . , N be “possible” plants. I P ( s ) is your “best guess” at what the true plant is. I Each P k ( s ) is another possible plant model, different than the nominal. I For example, each P k ( s ) could come from different models derived under different assumptions (e.g., different constants, different linearization point(s), etc.), or I each P k ( s ) could be from a different set of experimental data. I Consider P k ( s ) = P ( s ) ( 1 + Δ ( s ) W 2 ( s )) , 1 W 2 ( s ) P k ( s ) P ( s ) - 1 = Δ ( s ) .
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