Advanced process monitor (APMonitor) is a modeling language for
differential algebraic (
DAE
DAE or Dae may refer to:
As an acronym
* DAE (chemotherapy), a chemotherapy regimen consisting of Daunorubicin, Ara-C (cytarabine) and Etoposide
* Daporijo Airport, the IATA code for an airport in India
* Daxing Airport Express, the airport trans ...
) equations. It is a free web-service or local server for solving representations of physical systems in the form of implicit DAE models. APMonitor is suited for large-scale problems and solves
linear programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, li ...
,
integer programming,
nonlinear programming, nonlinear mixed integer programming, dynamic simulation,
moving horizon estimation, and nonlinear
model predictive control. APMonitor does not solve the problems directly, but calls
nonlinear programming solvers such as
APOPT,
BPOPT,
IPOPT,
MINOS
In Greek mythology, Minos (; grc-gre, Μίνως, ) was a King of Crete, son of Zeus and Europa. Every nine years, he made King Aegeus pick seven young boys and seven young girls to be sent to Daedalus's creation, the labyrinth, to be eaten ...
, and
SNOPT
SNOPT, for Sparse Nonlinear OPTimizer, is a software package for solving large-scale nonlinear optimization problems written by Philip Gill, Walter Murray and Michael Saunders. SNOPT is mainly written in Fortran, but interfaces to C, C++, Pyth ...
. The APMonitor API provides exact first and second derivatives of continuous functions to the solvers through
automatic differentiation and in
sparse matrix form.
Programming language integration
Julia,
MATLAB,
Python are mathematical programming languages that have APMonitor integration through web-service APIs. The
GEKKO Optimization Suite is a recent extension of APMonitor with complete Python integration. The interfaces are built-in optimization toolboxes or modules to both load and process solutions of optimization problems. APMonitor is an
object-oriented modeling language and optimization suite that relies on programming languages to load, run, and retrieve solutions. APMonitor models and data are compiled at run-time and translated into objects that are solved by an optimization engine such as
APOPT or
IPOPT. The optimization engine is not specified by APMonitor, allowing several different optimization engines to be switched out. The simulation or optimization mode is also configurable to reconfigure the model for
dynamic simulation, nonlinear
model predictive control,
moving horizon estimation or general problems in
mathematical optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
.
As a first step in solving the problem, a mathematical model is expressed in terms of variables and equations such as the Hock & Schittkowski Benchmark Problem #71 used to test the performance of
nonlinear programming solvers. This particular optimization problem has an objective function
and subject to the inequality constraint
and equality constraint
. The four variables must be between a lower bound of 1 and an upper bound of 5. The initial guess values are
. This mathematical model is translated into the APMonitor modeling language in the following text file.
! file saved as hs71.apm
Variables
x1 = 1, >=1, <=5
x2 = 5, >=1, <=5
x3 = 5, >=1, <=5
x4 = 1, >=1, <=5
End Variables
Equations
minimize x1*x4*(x1+x2+x3) + x3
x1*x2*x3*x4 > 25
x1^2 + x2^2 + x3^2 + x4^2 = 40
End Equations
The problem is then solved in Python by first installing the APMonitor package with pip install APMonitor or from the following Python code.
# Install APMonitor
import pip
pip.main( install','APMonitor'
Installing a Python is only required once for any module. Once the APMonitor package is installed, it is imported and the apm_solve function solves the optimization problem. The solution is returned to the programming language for further processing and analysis.
# Python example for solving an optimization problem
from APMonitor.apm import *
# Solve optimization problem
sol = apm_solve('hs71', 3)
# Access solution
x1 = sol x1'x2 = solx2'
X, or x, is the twenty-fourth and third-to-last letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''"ex"'' (pronounced ), ...
Similar interfaces are available for
MATLAB and
Julia with minor differences from the above syntax. Extending the capability of a modeling language is important because significant pre- or post-processing of data or solutions is often required when solving complex optimization, dynamic simulation, estimation, or control problems.
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 Pantelides algorithm in mathematics is a systematic method for reducing high-index systems of differential-algebraic equations to lower index. This is accomplished by selectively adding differentiated forms of the equations already present in the s ...
). 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. 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 (se
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
Applications in APMonitor Modeling Language
Many physical systems are naturally expressed by
differential algebraic equation. Some of these include:
*
cell culture
Cell culture or tissue culture is the process by which cells are grown under controlled conditions, generally outside of their natural environment. The term "tissue culture" was coined by American pathologist Montrose Thomas Burrows. This te ...
s
*
chemical reactors
*
cogeneration (power and heat)
*
distillation columns
*
drilling automation
*
essential oil
An essential oil is a concentrated hydrophobic liquid containing volatile (easily evaporated at normal temperatures) chemical compounds from plants. Essential oils are also known as volatile oils, ethereal oils, aetheroleum, or simply as the o ...
steam distillation
*
friction stir welding
* hydrate formation in deep-sea pipelines
*
infectious disease spread
*
oscillators
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
* severe slugging control
* solar thermal energy production
*
solid oxide fuel cell
A solid oxide fuel cell (or SOFC) is an electrochemical conversion device that produces electricity directly from oxidizing a fuel. Fuel cells are characterized by their electrolyte material; the SOFC has a solid oxide or ceramic electrolyte.
A ...
s
*
space shuttle launch simulation
*
Unmanned Aerial Vehicles (UAVs)
Models for a direct current (DC) motor and blood glucose response of an insulin dependent patient are listed below. They are representative of differential and algebraic equations encountered in many branches of science and engineering.
Direct current (DC) 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
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.
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
See also
*
APOPT
*
ASCEND
ASCEND is an open source, mathematical modelling chemical process modelling system developed at Carnegie Mellon University since late 1978. ASCEND is an acronym which stands for Advanced System for Computations in Engineering Design. Its main us ...
*
EMSO
*
GEKKO
*
MATLAB
*
Modelica
References
External links
APMonitor home pageDynamic optimization coursewith APMonitor
APMonitor documentationAPMonitor citationsOnline solution enginewith IPOPT
of popular modeling language syntax
* Downloa
APM MATLABAPM Python o
APM Juliaclient for APMonitor
* Downloa
APMonitor Server (Windows)
* Downloa
APMonitor Server (Linux)
{{DEFAULTSORT:Apmonitor
Numerical programming languages
Mathematical optimization software