Advanced process monitor (APMonitor) is a modeling language for
differential algebraic (
DAE) 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 relationships. Linear programming is ...
,
integer programming
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective ...
,
nonlinear programming
In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or st ...
, nonlinear mixed integer programming, dynamic simulation,
moving horizon estimation Moving horizon estimation (MHE) is an optimization approach that uses a series of measurements observed over time, containing noise (random variations) and other inaccuracies, and produces estimates of unknown variables or parameters. Unlike determi ...
, and nonlinear
model predictive control Model predictive control (MPC) is an advanced method of process control that is used to control a process while satisfying a set of constraints. It has been in use in the process industries in chemical plants and oil refineries since the 1980s. In r ...
. APMonitor does not solve the problems directly, but calls
nonlinear programming
In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or st ...
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 eat ...
, and
SNOPT. The APMonitor API provides exact first and second derivatives of continuous functions to the solvers through
automatic differentiation
In mathematics and computer algebra, automatic differentiation (AD), also called algorithmic differentiation, computational differentiation, auto-differentiation, or simply autodiff, is a set of techniques to evaluate the derivative of a function ...
and in
sparse matrix
In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as sparse b ...
form.
Programming language integration
Julia
Julia is usually a feminine given name. It is a Latinate feminine form of the name Julio and Julius. (For further details on etymology, see the Wiktionary entry "Julius".) The given name ''Julia'' had been in use throughout Late Antiquity (e ...
,
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementa ...
,
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
Object-oriented programming (OOP) is a programming paradigm based on the concept of " objects", which can contain data and code. The data is in the form of fields (often known as attributes or ''properties''), and the code is in the form of ...
modeling language
A modeling language is any artificial language that can be used to express information or knowledge or systems in a structure that is defined by a consistent set of rules. The rules are used for interpretation of the meaning of components in the ...
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
Dynamic simulation (or dynamic system simulation) is the use of a computer program to model the time-varying behavior of a dynamical system. The systems are typically described by ordinary differential equations or partial differential equations. ...
, nonlinear
model predictive control Model predictive control (MPC) is an advanced method of process control that is used to control a process while satisfying a set of constraints. It has been in use in the process industries in chemical plants and oil refineries since the 1980s. In r ...
,
moving horizon estimation Moving horizon estimation (MHE) is an optimization approach that uses a series of measurements observed over time, containing noise (random variations) and other inaccuracies, and produces estimates of unknown variables or parameters. Unlike determi ...
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
In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or st ...
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 = sol x2'
Similar interfaces are available for
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementa ...
and
Julia
Julia is usually a feminine given name. It is a Latinate feminine form of the name Julio and Julius. (For further details on etymology, see the Wiktionary entry "Julius".) The given name ''Julia'' had been in use throughout Late Antiquity (e ...
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). 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 cultures
*
chemical reactor
A chemical reactor is an enclosed volume in which a chemical reaction takes place. In chemical engineering, it is generally understood to be a process vessel used to carry out a chemical reaction, which is one of the classic unit operations in che ...
s
*
cogeneration (power and heat)
*
distillation columns
*
drilling automation
*
essential oil steam distillation
*
friction stir welding
Friction stir welding (FSW) is a solid-state joining process that uses a non-consumable tool to join two facing workpieces without melting the workpiece material. Heat is generated by friction between the rotating tool and the workpiece material ...
* 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
*
EMSO
*
GEKKO
*
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementa ...
*
Modelica
Modelica is an object-oriented, declarative, multi-domain modeling language for component-oriented modeling of complex systems, e.g., systems containing mechanical, electrical, electronic, hydraulic, thermal, control, electric power or process-o ...
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