Modelado de sistemas en espacio de estados

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EECE 360 Lecture 7

Outline
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Previously
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State Equation Representation of Dynamic Systems
Dr. Oishi Electrical and Computer Engineering University of British Columbia

Transferfunctions vs. state-space models

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Today
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Linear algebra review State-space models --> transfer functions Closed-form solution to state-space models

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Next time
! http://courses.ece.ubc.ca/360 eece360.ubc@gmail.com
EECE 360 v2.4

Transfer function --> state-space models
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Chapter 3.1 - 3.5
1 EECE 360 v2.4

State-space models
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The State DifferentialEquation
system matrix input matrix input vector

To get a state-space model:
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Start with a high-order differential equation Convert to a set of 1st order coupled differential equations Writein state-space form (A,B,C,D)

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Now that we have a state-space model:
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State differential equation: Output equation:
output matrix

How does this relate to transfer functions? Howcan we find a transfer function from a given statespace model?

EECE 360 v2.4

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EECE 360 v2.4

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Key results: State-space to T.F.
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Linear Algebra Review
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With zero initialconditions: Y (s) = C(sI " A)"1 B + D U(s) With non-zero initial conditions:

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X(s) = (sI " A)"1 BU(s) + (sI " A)"1 x(0) ! Y (s) = C(sI " A)"1 BU(s) + C(sI " A)"1 x(0) + DU(s)
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!  !  ! ** Need to find

(sI-A)-1.
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To manipulate state-space representations of transfer functions, we need specific tools from linear algebra Matrix properties Matrix operations Matrix exponential...See Appendix E (online) from Dorf and Bishop.
EECE 360 v2.4 6

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EECE 360 v2.4

Matrix Inversion
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Summary: Linear Algebra Rev.
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Find A-1 such that Example:
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Basicmatrix operations 2 x 2 matrix determinant

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First find determinant Then find coefficients of the adjoint matrix (!11, …, !33)

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2 x 2 matrix inverse

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**Know specific...
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