MIT16_30F10_lec06

# MIT16_30F10_lec06 - Topic#6 16.30/31 Feedback Control...

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Topic #6 16.30/31 Feedback Control Systems State-Space Systems What are state-space models? Why should we use them? How are they related to the transfer functions used in classical control design and how do we develop a state-space model? What are the basic properties of a state-space model, and how do we analyze these?

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Fall 2010 16.30/31 6–2 TF’s to State-Space Models The goal is to develop a state-space model given a transfer function for a system G ( s ) . There are many, many ways to do this. But there are three primary cases to consider: 1. Simple numerator (strictly proper) y 1 = G ( s ) = u s 3 + a 1 s 2 + a 2 s + a 3 2. Numerator order less than denominator order (strictly proper) y b 1 s 2 + b 2 s + b 3 N ( s ) u = G ( s ) = s 3 + a 1 s 2 + a 2 s + a 3 = D ( s ) 3. Numerator equal to denominator order (proper) y b 0 s 3 + b 1 s 2 + b 2 s + b 3 = G ( s ) = u s 3 + a 1 s 2 + a 2 s + a 3 These 3 cover all cases of interest September 21, 2010
Fall 2010 16.30/31 6–3 Consider case 1 (specific example of third order, but the extension to n th follows easily) y 1 = G ( s ) = u s 3 + a 1 s 2 + a 2 s + a 3 can be rewritten as the differential equation ... y + a 1 y ¨ + a 2 y ˙ + a 3 y = u choose the output y and its derivatives as the state vector y ¨ x = y ˙ y then the state equations are ... ˙ x = y ¨ y = a 1 1 a 2 0 a 3 0 ¨ y ˙ y + 1 0 u ˙ y 0 1 0 y 0 y ¨ y = 0 0 1 y ˙ + [0] u y This is typically called the controller form for reasons that will become obvious later on. There are four classic (called canonical ) forms observer, con- troller, controllability, and observability. They are all useful in their own way. September 21, 2010

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Fall 2010 16.30/31 6–4 Consider case 2 y u = G ( s ) = b 1 s 2 + b 2 s + b 3 s 3 + a 1 s 2 + a 2 s + a 3 = N ( s ) D ( s ) Let y y v = u v · u where y/v = N ( s ) and v/u = 1 /D ( s ) Then representation of v/u = 1 /D ( s ) is the same as case 1 ...
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