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04.state-space representation

# 04.state-space representation - State-Space State-Space...

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State-Space Representation Dongik Shin 1 1. Motive 2. State-space representation 3. Relationship with transfer function o Non-uniqueness of state variables Copyright © Dongik Shin, 2010 2

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Motive An n -th order differential equation has n integrators. (1 ) () n n - The independent combinations of integrated values y , , …, y ( n -1) are states 11 0 n yy y a ay u a - ++ = y . -a 0 -a 1 -a 2 y u y y  n y 3 Copyright © Dongik Shin, 2010 An n -th order differential equation can be written in n first-order differential equations of n state variables . Let then 1 xy = 12 xx = 2 ( 2) n - = 23 = = 1 ) n n n - - = = 1 01 1 nn n xa x a x a x u - - =- - - - + Or in matrix form, 010 0 0 éù é ù é ù 22 001 0 0 d u êú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú =+ ê ú ê ú   012 1 d 000 1 0 1 n a a x t a -- - ê ú ê ú ê ú ê ú ê ú ê ú --- - ê ú ê ú ëû ë û ë û 4 Copyright © Dongik Shin, 2010
The output is é 1 2 10 00 x x y éù ê ú ê ú ê ú ê ú = êú 1 n n x x - ëû ê ú ê ú ê ú ë The state variables describe the present configuration of a system and can be used to determine the future response, given the excitation inputs and the equations describing the dynamics inputs and the equations describing the dynamics. 5 Copyright © Dongik Shin, 2010 Example: Spring-Mass-Damper () My t by t ky t u t ++=  Let 1 xt y t = Then, the state equation is 2 y t = 12 1 kb = Th t t ti 21 2 u t MMM =- - + The output equation 1 yt x t = 6 Copyright © Dongik Shin, 2010

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Or in matrix form, 11 22 01 0 () 1 d d xt ut kb t éù é ù êú ê ú =+ ê ú ê úê ú ê ú -- ëû 1 MM M ê ú ë û é ù 2 1 ) 0 ( yt = ê ú ë û 7 Copyright © Dongik Shin, 2010 Example: R-L-C Circuit State equation d d cL t Cv t u t i t =- 12 u t CC + 1 2 c L xv xi = = d d LL c t Li t R i t v t + 212 1 R 2 O v t Rx t == Output equation 8 Copyright © Dongik Shin, 2010
Or in matrix form, 11 1 1 0 () d xt C ut C éù êú - = + 22 1 0 d R LL t ëû - 1 2 0 yt R = 9 Copyright © Dongik Shin, 2010 Example: Nonlinear System 2 sin c T g l l qq +=  Let ml 12 , xx == 1 sin g T = =+ Nonlinear system cannot be written in a matrix 21 2 c l ml = - + 2 1 1 d 0 x x g ê 2 1 2 d sin c x T x l l t m - 10 Copyright © Dongik Shin, 2010

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State-Space Representation: LTI System Generally, a state space representation of the linear time-invariant differential equation is () ttt =+ xA xB u yC xD u where x ( t ) is called state vector and its components are state variables. We assume that n state variables, m input variables, and p output variables 1 1 state vector input vector T n T m tx t x t tu t u t éù = êú ëû = x u 1 output±vector T p ty t y t = y 11 Copyright © Dongik Shin, 2010 The matrices have their own names: ( ) state matrix nn t ´ A ( ) input matrix output matrix nm p n t t ´ ´ B C ( ) direct transition m tr x ai pm t ´ D Note that for SISO ( m = 1, p = 1) system, o B ( t ) is a column vector o C ( t ) is a row vector o D ( t ) is a scalar.
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04.state-space representation - State-Space State-Space...

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