Exposicion Control 2 44042035

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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Exposicion Control 2 44042035