Laplace Transforms & Transfer Functions

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  • M.D. Bryant ME 344 notes 03/25/08

    1

    Laplace Transforms & Transfer Functions

    Laplace Transforms: method for solving differential equations, converts differential equations in time t into algebraic equations in complex variable s Transfer Functions: another way to represent system dynamics, via the s representation gotten from Laplace transforms, or excitation by est

  • M.D. Bryant ME 344 notes 03/25/08

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    Laplace Transforms Purpose: Converts linear differential equation in t into algebraic equation in s

    Forward transform, t s: f(t) F(s)

    F(s) = t= 0

    f(t) e- st dt = L {f(t); t s }

    Convergence/existence of integral: |f(t)| < M e- at , a + Re(s) > 0

    so that e-[a + Re(s)]t finite as t

    Inverse transform, s t: F(s) f(t)

    f(t) = 12j

    = c- j

    c+j F(s) e st ds = L-1{F(s); s t }

    Integral in complex plane, rarely do

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    Procedure:

    Convert Linear Differential Equations

    !

    x = Ax + f (t)

    !

    dn

    x

    dtn

    + an"1

    dn"1

    x

    dtn"1

    +!+ a2

    d2

    x

    dt2+ a

    1

    d x

    dt+ a

    0x = f (t)

    in time t into algebraic equations in complex variable s, via forward Laplace transforms:

    X(s)=L {x(t); t s }, F(s)=L {f(t); t s }

    Solve resulting algebraic equations in s,

    for solution X = X(s)

    Convert solution X(s) into time function x(t), via inverse Laplace transform:

    x(t) = L-1{X(s); s t }

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    Example: Solve initial value problem, differential equation with initial condition:

    x + 2 n x + 2n x = Ho us(t)

    x(0) = xo , x (0) = x1

    Apply Laplace transform

    L= t= 0

    { . } e- st dt to all terms in equation:

    !

    x (t)e" st

    t= 0

    #

    $ dt + 2%&n x (t)e

    " st

    t= 0

    #

    $ dt + &n

    2

    x(t)e" st

    dtt= 0

    #

    $

    =

    t= 0

    Ho us(t) e- st dt

    factor constants and apply definition for us(t) :

    t= 0

    d2x

    dt2 e- st dt + 2n

    t= 0

    dxdt e- st dt +

    2n

    t= 0

    x e- st dt

    = Ho t= 0

    e- st dt

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    Define X(s) = t= 0

    x(t) e- st dt

    Integrate by parts

    dxdt e- st|t=0 +s

    t= 0

    dxdt e- st dt +2n {x(t) e- st|t=0

    +s t= 0

    x(t) e- st dt } +

    2n X(s) =

    !

    HO

    ("s) e- st|

    t=0

    Rearrange, note

    !

    limt"#e

    $ st

    = 0

    - x (0) + s (- x(0) + s X(s) ) + 2n {- x(0) +

    s X(s) } + 2n X(s) =

    Ho s

    Note: L {dx/dt} = - x(0) + s X(s)

    L {d2x/dt2} = L {d

    !

    x /dt} =-

    !

    x (0) + sL {dx/dt}

    Result: algebraic equation in s

    !

    (s2

    + 2"#n

    s+#n

    2

    )X(s) =H

    O

    s+ x

    1+ (s+ 2"#

    n

    )x0

    Note: x (0) = x1 , x(0) = xo

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    Solve

    X(s) =

    Ho s + x1 + ( s + 2 n) xo

    s2 + 2 n s + 2n

    Note:

    o characteristic equation in denominator o terms from initial conditions x1 ,

    xo grouped with excitation term

    !

    HO

    s

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    Apply inverse Laplace: x(t) = L-1{X(s); s t }

    x(t) = Ho

    2n

    L-1{

    2n

    s (s2 + 2 n s + 2n)

    }

    + xo

    2n

    L-1{s

    2n

    s2 + 2 n s + 2n

    }

    + x1 + 2 n xo

    2n

    L-1{

    2n

    s2 + 2 n s + 2n

    }

    o Use Laplace transform tables for L-1:

    x(t) = Ho

    2n

    {1 - e- n t sin(n 1 - 2 t + arccos )

    1 - 2 }

    + xo

    2n

    { -

    2n e- n t sin(n 1 - 2 t - arccos )

    1 - 2 }

    + x1 + 2 n xo

    2n

    { n e- n t sin(n 1 - 2 t )

    1 - 2 }

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    Transfer Functions Method to represent system dynamics, via

    s representation from Laplace transforms. Transfer functions show flow of signal through a system, from input to output.

    Transfer function G(s) is ratio of output x

    to input f, in s-domain (via Laplace trans.):

    !

    G(s) =X(s)

    F(s) Method gives system dynamics

    representation equivalent to

    Ordinary differential equations State equations

    Interchangeable: Can convert transfer function to differential equations

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    Transfer Function G(s) Describes dynamics in operational sense Dynamics encoded in G(s) Ignore initial conditions (I.C. terms are

    transient & decay quickly) Transfer function, for input-output

    operation, deals with steady state terms

    h(t)

    H(s)

    x(t)

    X(s)

    G(s)

    Example: Speed of automobile

    o Output: speed x(t) expressed as X(s) o Output determined by input, system

    dynamics & initial conditions o Input: gas pedal depression h(t) o Dynamics: fuel system delivery,

    motor dynamics & torque, transmission, wheels, car translation, mass, wind drag, etc.

    o Initial conditions (initial speed) influence on current speed diminishes over time, thus ignore

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    Transfer Function Example: 2nd order system

    Differential equation at steady state (can ignore initial conditions)

    x + 2 n x + 2n x = h(t)

    Apply Laplace transform, define

    X(s) =L{x(t); t s}, H(s) =L{h(t); t s }

    Result: algebraic equation

    (s2 + 2n s + 2n )X(s) = H(s)

    Transfer function G(s): ratio of output

    X(s) to input H(s)

    G(s) = X(s)H(s) = 1

    s2 + 2 n s + 2n

    Note: o characteristic equation in denominator o denominator roots => eigenvalues o transfer functions eigenvalues called

    poles

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    Example: First order system

    !

    " x + x = f (t)

    Apply Laplace transform, define

    X(s) =L{x(t); t s}, F(s) =L{f(t); t s }

    L{ x ; t s} +L{x(t); t s} =L{f(t); t s }

    Result: algebraic equation sX(s) + X(s) = F(s)

    Transfer function

    G(s) =X(s)/F(s) =

    !

    1

    "s+1

    Note: o characteristic equation in denominator o transfer functions eigenvalues = poles o p1 = 1 = - 1/

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    Transfer Function from State Equations

    Matrix form

    !

    x = Ax + f (t)

    Explicit equation form

    !

    x k= a

    jkj=1

    n

    " xj+ f

    k(t)

    Define

    Xk(s)=L {xk(t); t s }, Fk(s)=L {fk(t); t s }

    Apply Laplace transform

    !

    sX(s) = AX(s)+F(s)

    or

    !

    sXk(s) = a

    jkj=1

    n

    " Xj(s)+F

    k(s)

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    Rearrange matrix form:

    !

    sI " A[ ]X(s) = F(s)

    Solve, via Cramers rule:

    !

    Xk(s) =

    det sI " A[ ]kth column replaced by F ( s )

    { }det sI " A[ ]

    Transfer function, define which output

    Xk(s) and which input Fj(s):

    !

    Gkj(s) =

    Xk(s)

    Fj(s)

    Solve, via previous result with all

    components of F(s) zero, except Fj(s)

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    Example: differential equations from transfer function:

    !

    G(s) =X(s)

    F(s)=

    2s+ 5

    s4

    + 3s3

    + 2s2

    + 2s+1

    Cross multiply ratios:

    !

    s4 + 3s 3 + 2s 2 + 2s+1( )X(s) = 2s+ 5( )F(s)

    !

    s4

    X + 3s3

    X + 2s2

    X + 2sX + X = 2sF + 5F Treat: s => d/dt

    !

    d4x

    dt4

    + 3d3x

    dt3

    + 2d2x

    dt2

    + 2dx

    dt+ x = 2

    df

    dt+ 5 f (t)

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    Block Diagrams

    h(t)

    H(s)

    x(t)

    X(s)

    G(s)

    Another way to represent system dynamics pictorially

    Weakness: lacks causality information

    Shows signal flow through system

    Transfer function G(s) inside block

    Output:

    X(s) = G(s) H(s)

    transfer function times input H(s)

    Can assemble blocks into system model:

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    Cascaded Blocks

    H(s) X1(s)

    G1(s)X2(s)

    G2(s)X(s)

    G3(s)

    Output = input to next Models stringing of components

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    Example: stereo system

    CD player amplifier

    speaker

    CD player amplifier speakers

    CD player:

    !

    G1(s) =

    X1(s)

    H (s)

    Input: H(s) from CD laser reader

    Output: CD voltage X1(s)

    Power amplifier:

    !

    G2(s) =

    X2(s)

    X1(s)

    Input: CD output voltage X1(s) Output: amp voltage X2(s)

    Speakers:

    !

    G3(s) =

    X(s)

    X2(s)

    Input: amp voltage X2(s) Output: sound, acoustic

    pressure X(s)

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    H(s) X1(s)

    G1(s)X2(s)

    G2(s)X(s)

    G3(s)

    Overall transfer function is product of block transfer functions:

    !

    G(s)