MIT16_30F10_lec09

MIT16_30F10_lec09 - Topic #9 16.30/31 Feedback Control...

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Topic #9 16.30/31 Feedback Control Systems State-Space Systems State-space model features Observability Controllability Minimal Realizations
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± ² ³ ´ µ ± Fall 2010 16.30/31 9–2 State-Space Model Features There are some key characteristics of a state-space model that we need to identify. Will see that these are very closely associated with the concepts of pole/zero cancelation in transfer functions. x x Example: Consider a simple system 6 G ( s ) = s + 2 for which we develop the state-space model Model # 1 x ˙ = 2 x + 2 u y = 3 x But now consider the new state space model ¯ T ¯ = [ x x 2 ] 2 0 2 ˙ ¯ x y = 3 0 Model # 2 = + u 1 ¯ x 0 1 which is clearly different than the Frst model, and larger. But let’s looks at the transfer function of the new model: G ¯ ( s ) = C ( sI A ) 1 B + D ±· 1 ² ³ = ´ 3 0 µ sI 2 0 1 0 1 2 2 ´ µ s +2 6 = 3 0 = !! 1 s + 2 s +1 September 30, 2010
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± Fall 2010 16.30/31 9–3 This is a bit strange, because previously our fgure oF merit when comparing one state-space model to another (page 6– ?? ) was whether they reproduced the same same transFer Function Now we have two very different models that result in the same transFer Function Note that I showed the second model as having 1 extra state, but I could easily have done it with 99 extra states!! So what is going on? A clue is that the dynamics associated with the second state oF the model x 2 were eliminated when we Formed the product ¯ G ( s ) = ² 3 0 ³ 2 s +2 1 s +1 because the A is decoupled and there is a zero in the C matrix Which is exactly the same as saying that there is a pole-zero cancelation in the transFer Function G ˜ ( s ) 6 6( s + 1) = G ˜ ( s ) s + 2 ( s + 2)( s + 1) Note that model #2 is one possible state-space model oF G ˜ ( s ) (has 2 poles) ±or this system we say that the dynamics associated with the second state are unobservable using this sensor (defnes C matrix).
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This note was uploaded on 02/03/2012 for the course AERO 16.30 taught by Professor Ericferon during the Fall '04 term at MIT.

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MIT16_30F10_lec09 - Topic #9 16.30/31 Feedback Control...

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