02.dynamic system - Dynamic System Dongik Shin Hanyang...

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Dynamic System Dongik Shin Hanyang University Dynamic System Dynamic system o has memory (integrator) has memory (integrator), o responds based on not only current but also past input , o is represented by a differential equation o is represented by a , or o has energy-storing elements (spring or mass). Static system o has no memory (integrator), o responds based on only current input, o is represented by an algebraic equation, or o has only energy-dissipating or -transforming elements. 2 Copyright © Dongik Shin, 2010
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Representation of Dynamic System There are various methods to represent a dynamic system: o Graphical method Graphical methods Free-body diagram Block diagram Signal-flow graph Bond graph (not dealt in this course) o Differential equations Input-output representation State-space representation o Transfer function o Impulse response o Frequency response 3 Copyright © Dongik Shin, 2010 1 1. Dynamic system 2. Linear system and impulse response 3. Linear time-invariant system 4. Causal system Copyright © Dongik Shin, 2010 4
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Integrable Function Space The set of all real-valued integrable function f is a vector space over () or : ft f t  is a vector space over . o Addition f + g of any two functions f , g Î is defined as and is closed: ( )() f gt f tg t += + fg + Î (point-wise addition) o Scalar multiplication a f of any and f Î is defined as and closed ( )() f t aa = (point-wise multiplication) f a Î 5 Copyright © Dongik Shin, 2010 Dirac Delta Function (Impulse Function) Dirac delta function is defined as 0i f if ad 0 0 nd 1 t tt t t dd ¥ ì ï ï = í ï ï = ¥= î ¹ ò area = 1 t t It is the limit function of the rectangle function. d ( t ) 0 l i e ) mrct ( t t d D D = 1 t D area = 1 Note delta functions are integrable. 2 t D - 2 t D rect D t ( t ) 6 Copyright © Dongik Shin, 2010
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Any integral function f ( t ) is a linear combination of Dirac delta functions d ( t t ) parameterized by t Î . 0 t D () ( )rect ( ft fk k t tt ¥ »D - D å ) t k D =-¥ 0 t D ( )( )d f t td t t ¥ =- ò Delta functions as a whole forms a basis of . 7 Copyright © Dongik Shin, 2010 Discrete Analogy ( ) ( 0 ) ( )( 1 ) (1 2 ) (2 ) ( ) ( ) j fn f n f n f n fj n j dd d d ¥ =-¥ =+ - + - + += - å  where d ( n ) is Kronecker delta 0 0 n ì ï ï ¹ 10 n n d = í ï = ï î f ( n ) = f (0) d ( n ) f (1) d ( n -1) f (2) d ( n -2) + + + + 012 n 0 n 1 n 2 n 8 Copyright © Dongik Shin, 2010
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Dynamic System & Ordinary Differential Equation An n -th order ordinary differential equation (lumped dynamic system) where ( ) () ( ) ,( ) , ( ) , , () , , , 0 , nm fyt y t ut ut yu tt t = o y ( t ) is the response or output function o u ( t ) is the input function can be thought as a mapping S from to :  or ) : ( S y Su = Note that the symbol y ( t ) (or u ( t )) has two different readings: o a mapping y : ft i l t t i o a function value at time t .
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This note was uploaded on 10/16/2011 for the course MECHATRONI 111 taught by Professor Jung during the Spring '11 term at Hanyang University.

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02.dynamic system - Dynamic System Dongik Shin Hanyang...

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