lecture_10 - 2.160 System Identification, Estimation, and...

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2.160 System Identification, Estimation, and Learning Lecture Notes No. March 13, 2006 6 Model Structure of Linear Time Invariant Systems 6.1 Model Structure In representing a dynamical system, the first step is to find an appropriate structure of the model. Depending on the choice of model structure, efficiency and accuracy of modeling are significantly different. The following example illustrates this. Consider the impulse response of a stable, linear time-invariant system, as shown below. Impulse Response is a generic representation that can represent a large class of systems, but is not necessarily efficient, i.e. it often needs a lot of parameters for representing the same input-output relationship than other model structures. u(t) y(t) G(q) A linear time-invariant system is completely characterized by its impulse g (1) response k g ) ( g (2) 1, a, a 2 ,… ( ( ) k q G ) = q k g k = 1 k g ) ( g (1), g (2) ,… Too many parameters although truncated. 1 2 3 4 k Can we represent the system with fewer parameters? ( k 1 Consider k g ) = a k = ,... 3 , 2 , 1 ( k 1 k q G ) = a q k = 1 a k k 1 = a k 1 k ( 1 Multiplying a : q G ) = q a q = q G ) ( q q k = 1 k = 2 q 1 a 1 ( ( 1 q G ) = q G ) = q = 1 q q 1 aq 1 q a Therefore, q G ) is represented by only one parameter: one pole when using a rational ( function. The number of parameters reduces if one finds a proper model structure. The following section describes various structures of linear time-invariant systems. 1 10
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6.2 A Family of Transfer Function Models 6.2.1 ARX Model Structure Consider a rational function for q G ) : ( ( ( ( t y ) = q B ) tu ) (1) q A ) ( ( ( where q A ) and q B ) are polynomials of q : ( q A ) 1 + q a 1 + ... + a n q n a , (2) 1 a n b q B ) q b 1 + ... + q b n ( 1 b ( ( ( t y ) ( t u ) q G ) = q B ) q A ) ( The input-output relationship is then described as ( ( ( t y ) + t y a 1) + ... + t y a n ) 1 n a = t u b + ... + t u b n ) (3) 1 ( n b ( b See the block diagram below. ( t e ) t e b ( ) t e ) t u ) a ( ( t y ) + + + ( 1 q b 1 + _ + 1 1 q q b 2 + + a 1 + + 1 q 1 q + a 2 + + 1 nb b q + a na eXogenous input Auto Regression 2
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Now let us consider an uncorrelated noise input t e ) entering the system. As long as ( ( the noise enters anywhere between the output t y ) and the block of b 1 , i.e. ( ), ( ( t e t e ), te ) in the above block diagram, the dynamic equation remains the same a b and is given by: ( ( ( t y ) + t y a 1) + ... + t y a n ) 1 n a ( n b ( b ( = t u b + ... + t u b n ) + t e ) (4) 1 Including the noise term, this model is called “Auto Regressive with eXogenous ( ( input” model, or ARX Model for short. Using the polynomials q A ) and q B ) , (4) reduces to ( ) ( ( ) ( ( t y q A ) = t u q B ) + ) (5) The adjustable parameters involved in the ARX model are T θ = ( a , a 2 ..., , a , b , b 2 ,..., b ) (6) 1 n a 1 n b Comparing (5) with (11) of Lecture Notes 9 yields ( ( 1 ( q G , ) = q B ) q H , ) = (7) q A ) q A ) ( ( See the block diagram below.
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.160 taught by Professor Harryasada during the Spring '06 term at MIT.

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lecture_10 - 2.160 System Identification, Estimation, and...

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