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lecture8

# lecture8 - MAE 171A Dynamic Systems Control Lecture 8...

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MAE 171A: Dynamic Systems Control Lecture 8: Steady-state Error and System Type Goele Pipeleers

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Recall from Lecture 7 Feedback Control control configuration D ( s ) G ( s ) controller system u y w w : disturbance r : reference u : control signal y : output r v v : sensor noise 1
control objectives r w v y u STABILITY regulation tracking control effort overall system In This Lecture evaluate the asymptotic (i.e. for t → ∞ ) tracking/regulation behavior for certain input signals 2

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System Type: General Definition a system is said to be Type 0 if a step input 1 Type 1 if a ramp input t Type 2 if a parabolic input t 2 Type k if a polynomial input t k results in an output converging to a nonzero constant stability is a prerequisite for this classification 3
Analysis: Laplace Transform and Final Value Theorem consider stable system H ( s ) : Y ( s ) = H ( s ) U ( s ) polynomial input signal u ( t ) = t k k ! L -→ U ( s ) = 1 s k +1 steady-state value of y ( t ) = L - 1 H ( s ) s k +1 via final value theorem: y ss = lim t →∞ y ( t ) = lim s 0 sY ( s ) = lim s 0 H ( s ) s k hence: H ( s ) is Type k lim s 0 H ( s ) s k = c 6 = 0 < 4

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Additional Conditions example: H ( s ) = s ( s + 1) s 2 + 2 s + 5 lim s 0 H ( s ) = 0 lim s 0 H ( s ) s = lim s 0 s + 1 s 2 + 2 s + 5 = 1 5 Type 1 lim s 0 H ( s ) s 2 = lim s 0 s + 1 s ( s 2 + 2 s + 5) =
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lecture8 - MAE 171A Dynamic Systems Control Lecture 8...

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