APMonitor

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1. Control signal - aileron angle

plt.plot(tm,w,'r:',lw=2,label='Aileron Angle')

The Gekko Optimization Suite is a machine learning and optimization package in Python for mixed-integer and differential algebraic equations.

📄 Gekko Publication and References

The GEKKO package is available in Python with pip install gekko. There are additional example problems for equation solving, optimization, regression, dynamic simulation, model predictive control, and machine learning.

The aircraft optimization problem is based on a simple aircraft model. The mathematical equations are ordinary differential equations (ODEs) that are integrated to final time T. The objective is to minimize the time required to achieve zero angle of attack, pitch angle, and pitch rate. The manipulated variable (control) is the aileron angle.

The problem is transformed to aileron angle 0 to 1 with an equivalent substitution.

$$ \begin{array}{llcl} \min_{x, w, T} & T \\ \mbox{s.t.} & \dot{x}_0 &=& -0.877 \; x_0 + x_2 - 0.088 \; x_0 \; x_2 + 0.47 \; x_0^2 \\ &&& - 0.019 \; x_1^2 - x_0^2 \; x_2 + 3.846 \; x_0^3 \\ &&& - 0.215 \; w + 0.28 \; x_0^2 \; w + 0.47 \; x_0 \; w^2 + 0.63 \; w^3 \\ & \dot{x}_1 &=& x_2 \\ & \dot{x}_2 &=& -4.208 \; x_0 - 0.396 \; x_2 - 0.47 \; x_0^2 - 3.564 \; x_0^3 \\ &&& -20.967 \; w + 6.265 \; x_0^2 \; w + 46 \; x_0 \; w^2 + 61.4 \; w^3 \\ & x(0) &=& (0.4655,0,0)^T \\ & x(T) &=& (0,0,0)^T \\ & w(t) &\in& \{-0.05236,0.05236\} \end{array} $$

(:title Aircraft Optimization:) (:keywords optimization, aircraft, angle, binary, optimal control, benchmark, nonlinear control, dynamic optimization, engineering optimization, Python, GEKKO, differential, algebraic, modeling language:) (:description Aircraft optimization solved with Python Gekko.:)

The aircraft optimization problem is based on a simple aircraft model. The mathematical equations are ordinary differential equations (ODEs) that are integrated to final time T. The objective is to minimize the time required to achieve zero angle of attack, pitch angle, and pitch rate. The manipulated variable (control) is the aeleron angle.

$$ \begin{array}{llcl} \min_{x, w, T} & T \\ \mbox{s.t.} & \dot{x}_0 &=& -0.877 \; x_0 + x_2 - 0.088 \; x_0 \; x_2 + 0.47 \; x_0^2 - 0.019 \; x_1^2 - x_0^2 \; x_2 + 3.846 \; x_0^3 \\ &&& - 0.215 \; w + 0.28 \; x_0^2 \; w + 0.47 \; x_0 \; w^2 + 0.63 \; w^3 \\ & \dot{x}_1 &=& x_2 \\ & \dot{x}_2 &=& -4.208 \; x_0 - 0.396 \; x_2 - 0.47 \; x_0^2 - 3.564 \; x_0^3 \\ &&& -20.967 \; w + 6.265 \; x_0^2 \; w + 46 \; x_0 \; w^2 + 61.4 \; w^3 \\ & x(0) &=& (0.4655,0,0)^T \\ & x(T) &=& (0,0,0)^T \\ & w(t) &\in& \{-0.05236,0.05236\} \end{array} $$

where `x_0` is the angle of attack in radians, `x_1` is the pitch angle, `x_2` is the pitch rate in rad/s, and the control function `w` is the tail deflection angle in radians. Both initial and terminal values of the differential states are fixed.

(:toggle hide solution button show="Aircraft Optimization Source":) (:div id=solution:) (:source lang=python:) from gekko import GEKKO import numpy as np import matplotlib.pyplot as plt

m = GEKKO(remote=False) n = 101; m.time = np.linspace(0,1,n)

ksi = 0.05236 tf = m.FV(value=5, lb=1, ub=10) tf.STATUS = 1 m.Minimize(tf/n)

1. x0 = angle of attack
2. x1 = pitch angle
3. x2 = pitch rate

x0,x1,x2 = m.Array(m.Var,3,value=0,lb=-1,ub=1) x0.value = 0.4655

1. Control signal - aeleron angle

w = m.MV(value=0, lb=0, ub=1); w.STATUS=1

m.Equation(x0.dt()/tf == -0.877*x0 + x2 - 0.088*x0*x2

           + 0.47*x0**2 - 0.019*x1**2 -x0**2*x2 + 3.846*x0**3
           -(0.215*ksi-0.28*x0**2*ksi -0.47*x0*ksi**2 - 0.63*ksi**3)*w
           -(-0.215*ksi + 0.28*x0**2 -0.47*x0*ksi**2 + 0.63*ksi**3)*(1-w))

m.Equation(x1.dt()/tf==x2) m.Equation(x2.dt()/tf == -4.208*x0 - 0.396*x2 - 0.47*x0**2 - 3.564*x0**3

           +20.967*ksi - 6.265*x0**2*ksi + 46*x0*ksi**2 -61.4*ksi**3
           -(20.967*ksi - 6.265*x0**2*ksi - 61.4*ksi**3)*2*w)

m.fix_final(x0,0) m.fix_final(x1,0) m.fix_final(x2,0)

m.options.IMODE=6 m.options.SOLVER=1 m.options.MAX_ITER=1000

m.options.COLDSTART=0 m.solve()

tm = tf.value[0]*m.time plt.figure(figsize=(6,4)) plt.subplot(2,1,1) plt.plot(tm,w,'r:',lw=2,label='Aeleron Angle') plt.grid(); plt.legend() plt.subplot(2,1,2) plt.plot(tm,x0,'r-',lw=2,label='AOA') plt.plot(tm,x1,'b:',lw=2,label='Pitch') plt.plot(tm,x2,'k--',lw=2,label='Pitch Rate') plt.grid(); plt.legend() plt.savefig('aircraft.png',dpi=300); plt.tight_layout(); plt.show() (:sourceend:) (:divend:)

The Gekko Optimization Suite is a machine learning and optimization package in Python for mixed-integer and differential algebraic equations. The GEKKO package is available in Python with pip install gekko. There are additional example problems? for equation solving, optimization, regression, dynamic simulation, model predictive control, and machine learning.