l12 - above But what if there is a zero Then for example...

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16.06 Lecture 12 State Space Modeling John Deyst October 1, 2003 To day’s Topics 1. State space model for nth order D.E.s 2. State space models for transfer functions 3. Examples 1

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We want a method for creating an n dimensional state space model from any set of LTI ordinary diﬀerential equations of order n . Sup- pose an output w (scalar) is related to in input r (scalar) by the LTI ordinary D.E. We deﬁne states as w and it’s n 1 derivatives So we have n states (same as the order of the D.E.). Now we diﬀer- entiate x , through x ( n 1) 2
We still need the derivative of x n .I t i s This one we obtain from the original D.E. Then we substitute from our state deﬁnations We thus have n ﬁrst order diﬀerential equations in terms of states and the input. In vector matrix

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Also, since our output is w Since we know A, B, C and D we have a state space model. We can draw a block diagram of this system as follows 4
Example- Then there are two states(second order equation) So, from the original D.E. we have and the vector/matrix equations become and w = x 1 is the output so 5

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zeros. So Now suppose, instead of a D.E. we have a transfer function with no This is just the transfer function of our original D.E. so we immedi- ately have the state space model for this transfer function. It’s the one derived

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Unformatted text preview: above! But what if there is a zero? Then, for example Break this down into two transfer functions, one with only poles the other with only zeros 6 The rst block can simply be represented as the state space model we developed earlier. However, now the output is a linear combination of derivatives of x 1 . In particular So, in the time domain we have or nally Hence the zeros are created by the elements of the C matrix. 7 Example Break G ( s ) down into two T.F.s Obtain the state space model for the rst block. Its D.E. is Dene states as Then from the D.E. So our state D.E. is 8 For the second block we have or in the time domain so Lets look at this system in terms of our block diagram The zero is created by feeding the x 2 state forward. The feed forward combination is created by the C matrix. 9...
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l12 - above But what if there is a zero Then for example...

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