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8/4/2019 Exposicion Control 2 44042035
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FROM A TRANSFER
FUNCTION PASS STATEVARIABLE
Carlos Ivn Mesa, 44042035
UNISALLE
Ing. Diseo & Automatizacin Electrnica
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TRANSFORMATIONS BETWEENMODELSWe have seen the usual expressions ofmathematical models of continuous-time systems.
The general model, valid for linear and nonlinearsystems is the differential equation model.
The linear model of state support and, if constant
coefficients, the transfer function model or, moregenerally, the model called polynomialrepresentation.
The question that immediately arises is that, giventhe representation of a system in one of the
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TRANSFER FUNCTION EQUATIONS OF STATEREALIZATION
Obtaining state model from the transfer matrixmodel is called realization.
This designation is that the information providedby the state model, which contains data about theinternal structure of the system is closer to reality
than that given by the transfer function model onlyrefers to the relationship between input and outputof the system.
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MATRICES A, B, C, D
As discussed below,from the matrices A, B,
C, D, that define a givenstate model can beobtained immediatelycalled simulation
diagram.
This diagram containsall the information
needed to build aphysical system analog
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IE
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SUBSTITUTING
In the state space, Substituting into the modelstate x by a model of state get transformed:
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STATE SPACE
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SIMILARITY TRANSFORMATIONOf the infinite possible realizationsof a given transfer matrix G (s),each defined by their matrices A, B,
C, D, there are some, calledcanonical, which are speciallyshaped and correspond to certainconfigurations physical systemname associated with them. Theyare called controller canonical
forms, observer, controllability,observability and diagonal.
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GIVEN TRANSFER FUNCTION
Let's see how are the canonical formscorresponding to a given transfer function
Suppose the characteristic polynomial of G (s) ismonic, ie, the coefficient of the n-th degree is 1,and G(s) is strictly their own. Then the transfer
function can be written as:
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MATRIX CONTROLLERCANONICALNote that the coefficients ai, bj the numerator anddenominator polynomials appear in reverse order
than usual.
The matrices that define the controller canonicalform G (s) are:
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OBSERVER CANONICAL
The matrices that define the observer canonicalform of G (s) are:
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OBSERVABILITY CANONICAL
The matrices that define the observabilitycanonical form of G (s) are
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CANONICAL FORMS
The numbers B1, B2, ... BN, last appear in the twocanonical forms, controllability and observability,
are called Markov parameters, whose value isgiven by
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DIAGONAL CANONICAL FORM
Last, thematrices of the
diagonalcanonical formfor aperformance
defined by theirmatrices A, B, C,D, is defined bythe
transformation
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THANKS