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APMonitor

 

APMonitor

APMonitor
Developer(s) APMonitor
Stable release v0.7.0 / 20 Jan 2015
Operating system Cross-platform
Type Technical computing
License Proprietary, BSD
Website APMonitor product page

Advanced process monitor (APMonitor), is a modeling language for differential algebraic (DAE) equations.[1] It is a free web-service for solving representations of physical systems in the form of implicit DAE models. APMonitor is suited for large-scale problems and allows solutions of linear programming, integer programming, nonlinear programming, nonlinear mixed integer programming, dynamic simulation,[2] moving horizon estimation,[3] and nonlinear control.[4] APMonitor does not solve the problems directly, but calls nonlinear programming solvers such as APOPT, BPOPT, IPOPT, MINOS, and SNOPT. The APMonitor API provides exact first and second derivatives of continuous functions to the solvers through automatic differentiation and in sparse matrix form.

Contents

  • High Index DAEs 1
    • Pendulum motion (index-3 DAE form) 1.1
  • Interfaces to scripting languages 2
  • Applications in APMonitor Modeling Language 3
    • Direct current (DC) motor 3.1
    • Blood glucose response of an insulin dependent patient 3.2
  • See also 4
  • References 5
  • External links 6

High Index DAEs

The highest order of a derivative that is necessary to return a DAE to ODE form is called the differentiation index. A standard way for dealing with high-index DAEs is to differentiate the equations to put them in index-1 DAE or ODE form (see Pantelides algorithm). However, this approach can cause a number of undesirable numerical issues such as instability. While the syntax is similar to other modeling languages such as gProms, APMonitor solves DAEs of any index without rearrangement or differentiation.[5] As an example, an index-3 DAE is shown below for the pendulum motion equations and lower index rearrangements can return this system of equations to ODE form (see Index 0 to 3 Pendulum example).

Pendulum motion (index-3 DAE form)

Model pendulum
  Parameters
    m = 1
    g = 9.81
    s = 1
  End Parameters

  Variables
    x = 0
    y = -s
    v = 1
    w = 0
    lam = m*(1+s*g)/2*s^2
  End Variables

  Equations
    x^2 + y^2 = s^2
    $x = v
    $y = w
    m*$v = -2*x*lam
    m*$w = -m*g - 2*y*lam
  End Equations
End Model

Interfaces to scripting languages

Python and MATLAB are two mathematical programming languages that have APMonitor integration. Using integration with scripting and programming languages as a web-service has a number of advantages and disadvantages. The advantages include an alternative to the built-in optimization toolboxes, processing of optimization solutions is simplified, server-side upgrades are transparent to the user, and improved cross-platform availability. Some of the disadvantages are that users are generally reluctant to use a web-service with proprietary models or data, a persistent internet connection is required, and the calculation techniques are not open to inspection as with open-source packages.

Applications in APMonitor Modeling Language

Many physical systems are naturally expressed by differential algebraic equation. Some of these include:

Models for a direct current (DC) motor and blood glucose response of an insulin dependent patient are listed below.

Direct current (DC) motor

Model motor
  Parameters
    ! motor parameters (dc motor)
    v   = 36        ! input voltage to the motor (volts)
    rm  = 0.1       ! motor resistance (ohms)
    lm  = 0.01      ! motor inductance (henrys)
    kb  = 6.5e-4    ! back emf constant (volt·s/rad)
    kt  = 0.1       ! torque constant (N·m/a)
    jm  = 1.0e-4    ! rotor inertia (kg m²)
    bm  = 1.0e-5    ! mechanical damping (linear model of friction: bm * dth)

    ! load parameters
    jl = 1000*jm    ! load inertia (1000 times the rotor)
    bl = 1.0e-3     ! load damping (friction)
    k = 1.0e2       ! spring constant for motor shaft to load
    b = 0.1         ! spring damping for motor shaft to load
  End Parameters

  Variables
    i     = 0       ! motor electric current (amperes)
    dth_m = 0       ! rotor angular velocity sometimes called omega (radians/sec)
    th_m  = 0       ! rotor angle, theta (radians)
    dth_l = 0       ! wheel angular velocity (rad/s)
    th_l  = 0       ! wheel angle (radians)
  End Variables

  Equations
    lm*$i - v = -rm*i -    kb *$th_m
    jm*$dth_m =  kt*i - (bm+b)*$th_m - k*th_m +     b *$th_l + k*th_l
    jl*$dth_l =             b *$th_m + k*th_m - (b+bl)*$th_l - k*th_l

    dth_m = $th_m
    dth_l = $th_l 
  End Equations
End Model

Blood glucose response of an insulin dependent patient

! Model source:
! A. Roy and R.S. Parker. “Dynamic Modeling of Free Fatty 
!   Acids, Glucose, and Insulin: An Extended Minimal Model,”
!   Diabetes Technology and Therapeutics 8(6), 617-626, 2006.
Model human
  Parameters
    p1 = 0.068       ! 1/min
    p2 = 0.037       ! 1/min
    p3 = 0.000012    ! 1/min
    p4 = 1.3         ! mL/(min·µU)
    p5 = 0.000568    ! 1/mL
    p6 = 0.00006     ! 1/(min·µmol)
    p7 = 0.03        ! 1/min
    p8 = 4.5         ! mL/(min·µU)

    k1 = 0.02        ! 1/min
    k2 = 0.03        ! 1/min
    pF2 = 0.17       ! 1/min
    pF3 = 0.00001    ! 1/min
    n = 0.142        ! 1/min
    VolG = 117       ! dL
    VolF = 11.7      ! L

    ! basal parameters for Type-I diabetic
    Ib = 0           ! Insulin (µU/mL)
    Xb = 0           ! Remote insulin (µU/mL)
    Gb = 98          ! Blood Glucose (mg/dL)
    Yb = 0           ! Insulin for Lipogenesis (µU/mL)
    Fb = 380         ! Plasma Free Fatty Acid (µmol/L)
    Zb = 380         ! Remote Free Fatty Acid (µmol/L)

    ! insulin infusion rate
    u1 = 3           ! µU/min

    ! glucose uptake rate
    u2 = 300         ! mg/min

    ! external lipid infusion
    u3 = 0           ! mg/min
  End parameters

  Intermediates
    p9 = 0.00021 * exp(-0.0055*G)  ! dL/(min*mg)
  End Intermediates

  Variables
    I = Ib
    X = Xb
    G = Gb
    Y = Yb
    F = Fb
    Z = Zb
  End variables

  Equations
    ! Insulin dynamics
    $I = -n*I  + p5*u1

    ! Remote insulin compartment dynamics
    $X = -p2*X + p3*I

    ! Glucose dynamics
    $G = -p1*G - p4*X*G + p6*G*Z + p1*Gb - p6*Gb*Zb + u2/VolG

    ! Insulin dynamics for lipogenesis
    $Y = -pF2*Y + pF3*I

    ! Plasma-free fatty acid (FFA) dynamics
    $F = -p7*(F-Fb) - p8*Y*F + p9 * (F*G-Fb*Gb) + u3/VolF

    ! Remote FFA dynamics
    $Z = -k2*(Z-Zb) + k1*(F-Fb)
  End Equations
End Model

See also

References

  1. ^
  2. ^
  3. ^
  4. ^
  5. ^
  6. ^
  7. ^
  8. ^
  9. ^
  10. ^
  11. ^
  12. ^

External links

  • APMonitor home page
  • Dynamic optimization course with APMonitor
  • APMonitor documentation
  • Online solution engine with IPOPT
  • Comparison of popular modeling language syntax
  • Download MATLAB Interface to APMonitor
  • Download Python Interface to APMonitor
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