MIT16_30F10_lec09

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

This preview shows pages 1–4. Sign up to view the full content.

Topic #9 16.30/31 Feedback Control Systems State-Space Systems State-space model features Observability Controllability Minimal Realizations

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
± 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
± 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).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 11

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online