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Unformatted text preview: Topic #11 16.31 Feedback Control Systems StateSpace Systems Statespace model features Observability Controllability Minimal Realizations Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 111 StateSpace Model Features There are some key characteristics of a statespace model that we need to identify. Will see that these are very closely associated with the concepts of pole/zero cancelation in transfer functions. Example: Consider a simple system G ( s ) = 6 s + 2 for which we develop the statespace model Model # 1 x = 2 x + 2 u y = 3 x But now consider the new state space model x = [ x x 2 ] T Model # 2 x = 2 1 x + 2 1 u y = 3 0 x which is clearly different than the first model, and larger. But lets looks at the transfer function of the new model: G ( s ) = C ( sI A ) 1 B + D = 3 0 sI 2 1 1 2 1 = 3 0 2 s +2 1 s +1 = 6 s + 2 !! October 3, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 112 This is a bit strange, because previously our figure of merit when comparing one statespace model to another (page 78) 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? The clue is that the dynamics associated with the second state of the model x 2 were eliminated when we formed the product G ( s ) = 3 0 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 polezero cancelation in the transfer function G ( s ) 6 s + 2 = 6( s + 1) ( s + 2)( s + 1) G ( s ) Note that model #2 is one possible statespace model of G ( s ) (has 2 poles) For this system we say that the dynamics associated with the sec ond state are...
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This note was uploaded on 11/07/2011 for the course AERO 16.31 taught by Professor Jonathanhow during the Fall '07 term at MIT.
 Fall '07
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