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# topic6 - Topic#6 16.31 Feedback Control Systems State-Space...

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Topic #6 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 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? 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].

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Fall 2007 16.31 6–1 SS Introduction State space model: a representation of the dynamics of an N th order system as a first order differential equation in an N -vector, which is called the state . Convert the N th order differential equation that governs the dynamics into N first-order differential equations Classic example: second order mass-spring system m ¨ p + c ˙ p + kp = F Let x 1 = p , then x 2 = ˙ p = ˙ x 1 , and ¨ x 2 = ¨ p = ( F c ˙ p kp ) /m = ( F cx 2 kx 1 ) /m ˙ p ¨ p = 0 1 k/m c/m � � p ˙ p + 0 1 /m u Let u = F and introduce the state x = x 1 x 2 = p ˙ p ˙ x = A x + Bu If the measured output of the system is the position, then we have that y = p = 1 0 p ˙ p = 1 0 x 1 x 2 = c x September 17, 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 6–2 Most general continuous-time linear dynamical system has form ˙ x ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) y ( t ) = C ( t ) x ( t ) + D ( t ) u ( t ) where: t ∈ R denotes time – x ( t ) ∈ R n is the state (vector) – u ( t ) ∈ R m is the input or control – y ( t ) ∈ R p is the output A ( t ) ∈ R n × n is the dynamics matrix B ( t ) ∈ R n × m is the input matrix C ( t ) ∈ R p × n is the output or sensor matrix D ( t ) ∈ R p × m is the feedthrough matrix Note that the plant dynamics can be time-varying. Also note that this is a multi-input / multi-output (MIMO) system. We will typically deal with the time-invariant case Linear Time-Invariant (LTI) state dynamics ˙ x ( t ) = A x ( t ) + B u ( t ) y ( t ) = C x ( t ) + D u ( t ) so that now A, B, C, D are constant and do not depend on t . September 17, 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].

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Fall 2007 16.31 6–3 Basic Definitions Linearity – What is a linear dynamical system? A system G is linear with respect to its inputs and output u ( t ) G ( s ) y ( t ) iff superposition holds: G ( α 1 u 1 + α 2 u 2 ) = α 1 Gu 1 + α 2 Gu 2 So if y 1 is the response of G to u 1 ( y 1 = Gu 1 ), and y 2 is the response of G to u 2 ( y 2 = Gu 2 ), then the response to α 1 u 1 + α 2 u 2 is α 1 y 1 + α 2 y 2 A system is said to be time-invariant if the relationship between the input and output is independent of time. So if the response to u ( t ) is y ( t ) , then the response to u ( t t 0 ) is y ( t t 0 ) x ( t )
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