# state_space_model02 - 28-29. State-space modeling 28-29.6...

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28-29. State-space modeling28-29.6me 375 - cmkState-space/output block diagram and transfer functionConsider aLINEARsystem with an input!u(t)(of lengthp), state vector!x(t)(of lengthn) and output vector!y(t)(of lengthq). The state space equations and output equationsare written in the general form of:!"x(t)=A!x(t)+B!u(t)(1)!y(t)=C!x(t)+D!u(t)(2)whereAis ann×nmatrix,Bis ann×pmatrix,Cis anq×nmatrix andDis ann×pmatrix. The block diagram for the system governed by equations (1) and (2) canbe drawn as follows:Recall from our earlier work with scalar systems having an inputu(t)and an outputy(t),the Laplace transform of the outputY(s)was related to the Laplace transform of the inputU(s)through the transfer functionG(s)as:Y(s)=G(s)U(s)where the above Laplace transforms are found using zero initial conditions. Recall that thepoles and zeros of the transfer function defined for us the nature of the expected responseof the system for a given input.Here, for our general state-state systems, we will seek the “transfer function matrix”G(s)relating the Laplace transform of the input!U(s)to the Laplace transform of theoutput!Y(s):!Y(s)=G(s)!U(s)where, again, the Laplace transforms are found through zero initial conditions. To this end,let’s first take the Laplace transform of the state-space/output equations (1) and (2):s!X(s)=A!X(s)+B!U(s)(1a)!Y(s)=C!X(s)+D!U(s)(1b)inputC!u(t)B!"x(t)!x(t)!y(t)ADoutputintegra,onstate
28-29. State-space modeling28-29.7me 375 - cmkFrom equation (1a), we have:s IA!X(s)=B!U(s)!X(s)=s IA1B!U(s)(1c)whereIis then×n“identity” matrix2andE1is the inverse of the square matrixE(later on, we will discuss how to find the inverse of a matrix). Substitution of (1c) intoequation (2b) gives:!Y(s)=Cs IA1B!U(s)+D!U(s)=Cs IA1B+D!U(s)=G(s)!

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Term
Spring
Professor
CagriA.Savran(P)
Tags
Complex number, 28 29 State space modeling
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