APMonitor

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 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 = 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.

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 (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

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

• cell cultures
• chemical reactors
• cogeneration (power and heat)
• distillation columns
• drilling automation
• essential oil steam distillation
• friction stir welding
• hydrate formation in deep-sea pipelines
• infectious disease spread
• oscillators
• severe slugging control
• solar thermal energy production
• solid oxide fuel cells
• 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 m^2) 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