Ve216LectureNotesChapter2Part2

Ve216LectureNotesChapter2Part2 - Ve216 Lecture Notes...

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Ve216 Lecture Notes Dianguang Ma Spring 2009
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2.9 Differential/Difference Equations An important class of continuous-time/discrete- time systems is that for which the input and output are related through a linear constant- coefficient differential/difference equation. = = = = - = - = M k k N k k M k k k k N k k k k k n x b k n y a t x dt d b t y dt d a 0 0 0 0 ] [ ] [ ) ( ) (
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2.9 Differential/Difference Equations The order refers to the highest derivative of the output y(t) appearing in the equation. In the case when N=0, y(t) is an explicit function of x(t) and its derivatives. For N>0, the equation specifies the output implicitly in terms of the input. In order to obtain an explicit expression, we must solve the differential equation. = = = M k k k k N k k k k t x dt d b t y dt d a 0 0 ) ( ) (
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2.9 Differential/Difference Equations The solution y(t) consists of two parts – a particular solution (or the forced response of the system, a signal of the same form as the input) plus a homogeneous solution (or the natural response of the system). ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( t y t y t y t y t y t y t y n h f p h p = = + =
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2.9 Differential/Difference Equations The differential equation does not completely specify the output in terms of the input, and we need to identify auxiliary conditions to determine completely the input- output relationship for the system. Basically the time origin is defined as the time instant at which x(t) is applied to the system, i.e., we assume that x(t)=0 for t<0. And we are required to determine the response for t>0 from the differential equation with the initial conditions - - - - = - - = = = 0 1 1 0 2 2 0 0 ) ( , , ) ( , ) ( , ) ( t N N t t t t y dt d t y dt d t y dt d t y
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2.10 Solving Differential and Difference Equations The homogeneous solution is the solution of the homogeneous equation 0 ) ( 0 ) ( = = N k h k k k t y dt d a The homogeneous solution is of the form = = N i t r i h i e c t y 1 ) ( ) ( where r i are the N roots of the characteristic equation
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2.10 Solving Differential and Difference Equations (The characteristic equation of the system) 0 0 = = N k k k r a The form of the homogeneous solution changes slightly when the characteristic equation has repeated roots. If a root r j is repeated p times terms ) ( terms ) 1 ( ) 1 ( 1 0 ) ( ) ( p N p t r p p j t r j t r j h j j j e t c te c e c t y - - - + + + =
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2.10 Solving Differential and Difference Equations The particular solution represents any solution of the differential equation for the given input. A particular solution is usually obtained by assuming an output of the same general form as the input ) sin( ) cos( ) cos( ) cos( 1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Solution Particular Input 2 1 2 1 t c t c t c t ce e c t c t c at at ϖ ϕ φ + = + + + - -
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2.10 Solving Differential and Difference Equations
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Ve216LectureNotesChapter2Part2 - Ve216 Lecture Notes...

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