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MIT16_30F10_lec05

# MIT16_30F10_lec05 - Topic#5 16.30/31 Feedback Control...

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Topic #5 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 5–2 SS Introduction State space model: a representation of the dynamics of an N th order system as a Frst order diﬀerential equation in an N -vector, which is called the state . Convert the N th order diﬀerential equation that governs the dy- namics into N Frst-order diﬀerential equations Classic example: second order mass-spring system mp ¨ + cp ˙ + kp = F Let x 1 = p , then x 2 = p ˙ = x ˙ 1 , and x ˙ 2 = p ¨ = ( F cp ˙ kp ) /m = ( F cx 2 kx 1 ) /m ± ± ± ± p ˙ 0 1 p 0 = + u p ¨ k/m c/m p ˙ 1 /m Let u = F and introduce the state x 1 p x = = x ˙ = A x + Bu x 2 p ˙ If the measured output of the system is the position, then we have that ² ³ p ² ³ x 1 y = p = 1 0 = 1 0 = C x p ˙ x 2 September 21, 2010
Fall 2010 16.30/31 5–3 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 21, 2010

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Fall 2010 16.30/31 5–4 Basic Defnitions 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 G u 1 + α 2 G u 2 So if y 1 is the response of G to u 1 ( y 1 = G u 1 ), and y 2 is the response of G to u 2 ( y 2 = G u 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 ) Example: the system x ˙( t ) = 3 x ( t ) + u ( t ) y ( t ) = x ( t ) is LTI, but x ˙( t ) = 3 t x ( t ) + u ( t ) y ( t ) = x ( t ) is not.
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MIT16_30F10_lec05 - Topic#5 16.30/31 Feedback Control...

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