002_BME343_Continuous_Time_Systems

002_BME343_Continuous_Time_Systems - Time-Domain Analysis...

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TD analysis of CT systems 2-1 Time-Domain Analysis of Continuous-Time Systems • Neurons • Carcinogenesis • Mechanics of biological tissues • Cancer • Scar tissue formation • Intracellular dynamics • Cell cycle • Arterial diseases • Multi-scale modeling of the heart Examples of Mathematical Biology
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TD analysis of CT systems 2-2 Objectives • Determine the characteristic equation, characteristic roots, and characteristic modes of a linear constant coefficient differential equation • Calculate the impulse response for a linear constant coefficient differential equation • Calculate the zero-input and zero-state response of a linear constant coefficient differential equation • Calculate and plot the convolution between two functions in continuous time • Use the fact that exponentials are eigenfuctions of linear time invariant systems to simplify convolution • Be able to intuitively explain system behavior in terms of the characteristic modes and time constants
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TD analysis of CT systems 2-3 Linear, Time-Invariant, Continuous-Time (LTIC) Systems • Linear differential systems 1 11 1 1 1 () ... ... ( ) NN MM NM N N dy d y d y aa a dt dt dt dx d x d x bb b x t dt dt dt yt b −− + = = ++ + + + + 1 1 ... ) ( ) ( . . ) ( ) ( . N N aD a D a yt bD b D b D b x t D + + + = =+ + + + 1 . . . N N b D b D b PD + + + = 1 (. . ). QD DD a a + + = ( ) ( ) QDyt PDxt = Note that a 0 =1 , i.e., there is no coefficient in front of the first ( D N ) term Note that D is an operator, not a variable
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TD analysis of CT systems 2-4 LTIC Systems • Linear differential systems • Theoretically, M and N can take on any value • However, for practical purposes, M<N – Stability –No ise 1 11 . () . . M M NM N N bD b D b D b PD −− + + ++ + = 1 ( ) ... N N N N aD QD DD a a + + = ( ) ( ) QDyt PDxt = t yt dx d =
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TD analysis of CT systems 2-5 Response of a Linear System 0 00 1 () ( ) ( t C t y tv t R x t x d t t C = + + τ) τ Total response = = zero-input response + + zero-state response zero-state response zero-input response
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TD analysis of CT systems 2-6 Zero-Input Response • Zero-input response y 0 ( t ) – system response to internal conditions (input x ( t ) = 0) • Our goal is to find the solution of this equation Physiological System Input Output Stimulus Response () 0 xt = 0 () yt 0 ( ) 0 QDy t = 1 11 0 ... ( ) 0 NN aD a t DD a y ++ + + =
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TD analysis of CT systems 2-7 Zero-Input Response • The solution can be found using different approaches • We will take a shortcut and suggest that solution should have the following (general) form: 1 11 0 ... ( ) 0 () NN aD a t DD a y ++ + + = 0 t y tc e λ = 0 0 2 22 0 0 2 0 0 t t N Nt N N dy Dy t c e dt dy Dy t c e dt Dyt c e dt == M
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TD analysis of CT systems 2-8 Zero-Input Response • More simple math 1 11 ... ) 0 ( NN t aa ca e λ λλ ++ + + = 1 ... 0 a + + = () 0 Q λ= 12 ( ) ( )( )...( ) 0 N Q λλλλ =− = 2 01 ( ) ... N N t tt y tc e c e c e =+ + 1 0 ... ( ) 0 aD a t DD a y + + = 0 t yt c e = 0 0 2 22 0 0 2 0 0 t t N Nt N N dy Dy t c e dt dy Dy t c e dt Dyt dt == M 1 2 Characteristic Polynomial and Equation 3 λ i : characteristic roots (characteristic values, eigenvalues, natural frequencies) Solution: linear combination of characteristic modes
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This note was uploaded on 09/28/2009 for the course BME 343 taught by Professor Emelianov during the Fall '09 term at University of Texas at Austin.

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002_BME343_Continuous_Time_Systems - Time-Domain Analysis...

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