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02.dynamic system

# 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) (integrator), o responds based on not only current but also past input , o is represented by a differential equation or , 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 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 I l 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
Integrable Function Space The set of all real-valued integrable function f is a vector space over ( ) or : f t f t . o Addition f + g of any two functions f , g Î is defined as ( )( ) ( ) ( ) f f ( i i ddi i ) and is closed: g t f t g t + = + f g + Î (point-wise addition) o Scalar multiplication a f of any a Î and f Î is defined as and closed ( )( ) ( ) f t f t a a = (point-wise multiplication) f a Î 5 Copyright © Dongik Shin, 2010 Dirac Delta Function (Impulse Function) Dirac delta function is defined as 0 if ( ) ( ) if a d 0 0 n d 1 t t t t t d d ¥ 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 ( ) li e ) m r ct ( t t 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 ( ) f t f k k t t t ¥ » D - D å t k D =-¥ 0 t D ( ) ( ) ( )d f t f t t d 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 f n f n f n f n f j n j d d d d ¥ =-¥ = + - + - + + = - å where d ( n ) is Kronecker delta 0 0 n ì ï ï ¹ ( ) 1 0 n n d = í ï = ï î f ( n ) = f (0) d ( n ) f (1) d ( n -1) f (2) d ( n -2) + + + + 1) 2) 0 1 2 n 0 n 1 n 2 n 8 Copyright © Dongik Shin, 2010
Dynamic System & Ordinary Differential Equation An n -th order ordinary differential equation (lumped dynamic system) ( ) ( ) ( ) where , ( ), ( ), ( ), ( ) ( ), , ( ), 0 , n m f y t y t u t u t y u t t 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 S u = N h h b l ( ) ( ( )) h diff di Note that the symbol y ( t ) (or u ( t )) has two different readings: o a mapping y : f ti l t ti t o a function value at time .

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

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