APMonitor

Last updated
APMonitor
Developer(s) APMonitor
Stable release
v1.0.1 / January 31, 2022 (2022-01-31)
Repository https://github.com/APMonitor/
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 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, integer programming, nonlinear programming, nonlinear mixed integer programming, dynamic simulation, [2] moving horizon estimation, [3] and nonlinear model predictive 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

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.

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 [5] 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.

!filesavedashs71.apmVariablesx1=1,>=1,<=5x2=5,>=1,<=5x3=5,>=1,<=5x4=1,>=1,<=5EndVariablesEquationsminimizex1*x4*(x1+x2+x3)+x3x1*x2*x3*x4>25x1^2+x2^2+x3^2+x4^2=40EndEquations

The problem is then solved in Python by first installing the APMonitor package with pip install APMonitor or from the following Python code.

# Install APMonitorimportpippip.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 problemfromAPMonitor.apmimport*# Solve optimization problemsol=apm_solve("hs71",3)# Access solutionx1=sol["x1"]x2=sol["x2"]

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). 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. [6] 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)

ModelpendulumParametersm=1g=9.81s=1End ParametersVariablesx=0y=-sv=1w=0lam=m*(1+s*g)/2*s^2End VariablesEquationsx^2+y^2=s^2$x=v$y=wm*$v=-2*x*lamm*$w=-m*g-2*y*lamEnd EquationsEnd Model

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. 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 parametersjl=1000*jm! load inertia (1000 times the rotor)bl=1.0e-3! load damping (friction)k=1.0e2! spring constant for motor shaft to loadb=0.1! spring damping for motor shaft to loadEnd ParametersVariablesi=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 VariablesEquationslm*$i-v=-rm*i-kb*$th_mjm*$dth_m=kt*i-(bm+b)*$th_m-k*th_m+b*$th_l+k*th_ljl*$dth_l=b*$th_m+k*th_m-(b+bl)*$th_l-k*th_ldth_m=$th_mdth_l=$th_lEnd 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.Parametersp1=0.068! 1/minp2=0.037! 1/minp3=0.000012! 1/minp4=1.3! mL/(min·µU)p5=0.000568! 1/mLp6=0.00006! 1/(min·µmol)p7=0.03! 1/minp8=4.5! mL/(min·µU)k1=0.02! 1/mink2=0.03! 1/minpF2=0.17! 1/minpF3=0.00001! 1/minn=0.142! 1/minVolG=117! dLVolF=11.7! L! basal parameters for Type-I diabeticIb=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 rateu1=3! µU/min! glucose uptake rateu2=300! mg/min! external lipid infusionu3=0! mg/minEnd parametersIntermediatesp9=0.00021*exp(-0.0055*G)! dL/(min*mg)End IntermediatesVariablesI=IbX=XbG=GbY=YbF=FbZ=ZbEnd variablesEquations! 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

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References

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  6. ↑ Harney, D. (2013). "Numerical evaluation of the stability of stationary points of index-2 differential-algebraic equations: Applications to reactive flash and reactive distillation systems". Computers & Chemical Engineering. 49: 61–69. doi:10.1016/j.compchemeng.2012.09.021.
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