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topic7 - Topic#7 16.31 Feedback Control Systems State-Space...

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Unformatted text preview: Topic #7 16.31 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 con- trol 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? 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 7–1 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 u = G ( s ) = 1 s 3 + a 1 s 2 + a 2 s + a 3 2. Numerator order less than denominator order (strictly proper) 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 ) 3. Numerator equal to denominator order (proper) y u = G ( s ) = b s 3 + b 1 s 2 + b 2 s + b 3 s 3 + a 1 s 2 + a 2 s + a 3 • These 3 cover all cases of interest September 22, 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 7–2 • Consider case 1 (specific example of third order, but the extension to n th follows easily) y u = G ( s ) = 1 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 x = ⎡ ⎣ ¨ y ˙ y y ⎤ ⎦ then the state equations are ˙ x = ⎡ ⎣ ... y ¨ y ˙ y ⎤ ⎦ = ⎡ ⎣ − a 1 − a 2 − a 3 1 1 ⎤ ⎦ ⎡ ⎣ ¨ y ˙ y y ⎤ ⎦ + ⎡ ⎣ 1 ⎤ ⎦ u y = 0 0 1 ⎡ ⎣ ¨ y ˙ y y ⎤ ⎦ + [0] u • This is typically called the controller form for reasons that will become obvious later on....
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topic7 - Topic#7 16.31 Feedback Control Systems State-Space...

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