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Unformatted text preview: And therefore K = bracketleftbig 0 1 bracketrightbig 1 3 bracketleftbigg 3 1 1 bracketrightbiggbracketleftbigg 3 9 12 bracketrightbigg = bracketleftbig 3 4 bracketrightbig Check: ( det [ sI F + GK ]) = det parenleftbigg sI bracketleftbigg 2 4 3 4 bracketrightbiggparenrightbigg = s 2 + 2 s + 4 2. Calculate the estimator for the system. The poles of the dynamics of the estimation error should be chosen in λ 1 = 2 , λ 2 = 4 . Due to the asked error dynamics, the estimator characteristic polynomial should be: α e ( s ) = ( s + 2)( s + 4) = s 2 + 6 s + 8 The observer gains can be simply directly calculated with Ackermann’s estimator formula. L = α e ( F ) O 1 bracketleftbigg 1 bracketrightbigg We compute: O = bracketleftbigg H HF bracketrightbigg = bracketleftbigg 1 3 4 bracketrightbigg ⇒ O 1 = 1 3 bracketleftbigg 4 1 3 bracketrightbigg and α e ( F ) = F 2 +6 F +8 I = bracketleftbigg 1 3 4 bracketrightbiggbracketleftbigg 1 3 4 bracketrightbigg +6 bracketleftbigg 1 3 4 bracketrightbigg +8 bracketleftbigg 1 0 0 1 bracketrightbigg = bracketleftbigg 3 3 0 bracketrightbigg And therefore L = bracketleftbigg 3 3 0 bracketrightbigg 1 3 bracketleftbigg 4 1 3 bracketrightbiggbracketleftbigg 1 bracketrightbigg = bracketleftbigg 1 1 bracketrightbigg Check: ( det [ sI F + LH ]) = det parenleftbigg sI bracketleftbigg 1 1 3 5 bracketrightbiggparenrightbigg = s 2 + 6 s + 8 Question 5 The system G ( s ) has a transfer function: G ( s ) = s + τ ( s + 2) where the parameter τ is not exactly known, but bounded by 1 ≤ τ ≤ 2 This system can be described as a nominal system G ( s ) = ( s + 1) ( s + 2) with multiplicative uncertainty l ( s ) . 1. Describe the unstructured multiplicative uncertainty l ( s ) as a function of the uncertain τ . G ( s ) = s + τ ( s + 2) G ( s ) = ( s + 1) ( s + 2) From G ( s ) = G ( s )(1 + l ( s )) it follows l ( s ) = G ( s ) /G ( s ) 1 = s + τ ( s + 2) ( s + 2) ( s + 1) 1 = τ 1 ( s + 1) 2. Show that  l ( jω )  ≤ 1 for all frequencies. Find maximum max ω  l ( jω )  = max ω τ 1 √ 1 + ω 2 = τ 1 √ 1 + 0 2 = τ 1 For all 1 ≤ τ ≤ 2 we find  l ( jω )  ≤ 1 3. Prove that the controller D ( s ) = 1 s + 1 robustly stabilizes the system. To prove that the controller D ( s ) robustly stabilizes the system, we have to prove that  T 1 ( jω )  >  l ( jω )  for all ω Compute T ( s ) for G ( s ) = ( s +1) ( s +2) and D ( s ) = 1 s +1 : T ( s ) = G D 1 + G D = ( s +1) ( s +2) 1 s +1 1 + (...
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 Spring '10
 Taufik
 Steady State, Laplace

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