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More Optimal Control Problems

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March 07, 2018, at 06:55 PM by 10.37.83.132 -
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Commercial Fishery

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Jennings Problem

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(:toggle hide gekko button show="Show GEKKO (Python) Code":) (:div id=gekko:) (:source lang=python:) import numpy as np import matplotlib.pyplot as plt from gekko import GEKKO

m = GEKKO()

nt = 1001 t = np.linspace(0,np.pi/2,nt) m.time = t

  1. Variables

x1 = m.Var(value=0) x2 = m.Var(value=1) x3 = m.Var(value=0)

u = m.MV(value=0,ub=1,lb=-1) u.STATUS = 1 u.DCOST = 0

p = np.zeros(nt) p[-1] = 1.0 final = m.Param(value=p)

  1. Equations

m.Equation(x1.dt()==x2) m.Equation(x2.dt()==u) m.Equation(2*x3.dt()==x2**2-x1**2)

  1. Objective Function

m.Obj(x3*final)

m.options.IMODE = 6 m.options.NODES = 4 m.solve()

plt.figure(1) plt.subplot(2,1,1) plt.plot(m.time,x1.value,'k:',LineWidth=2,label=r'$x_1$') plt.plot(m.time,x2.value,'b-',LineWidth=2,label=r'$x_2$') plt.plot(m.time,x2.value,'k-',LineWidth=2,label=r'$x_3$') plt.subplot(2,1,2) plt.plot(m.time,u.value,'r--',LineWidth=2,label=r'$u$') plt.plot(t,-np.sin(t),'k:',LineWidth=2,label='Exact') plt.legend(loc='best') plt.xlabel('Time') plt.ylabel('Value') plt.show() (:sourceend:) (:divend:)

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(:title More Optimal Control Problems:) (:keywords nonlinear control, optimal control, dynamic optimization, engineering optimization, MATLAB, Python, GEKKO, differential, algebraic, modeling language, university course:) (:description More optimal control problems solved with Dynamic Optimization in MATLAB and Python.:)

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Objective: Set up and solve several dynamic optimization benchmark problems. Create a program1 to optimize and display the results. Estimated Time (each): 30 minutes

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Objective: Set up and solve several optimal control benchmark problems. Create a program1 to optimize and display the results. Estimated Time (each): 30 minutes

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Luus Problem

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February 10, 2016, at 07:04 AM by 174.148.96.184 -
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February 10, 2016, at 06:39 AM by 174.148.96.184 -
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Luus Problem

February 10, 2016, at 06:36 AM by 174.148.96.184 -
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February 10, 2016, at 06:11 AM by 174.148.96.184 -
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Jennings Problem

February 10, 2016, at 12:59 AM by 174.148.92.107 -
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February 10, 2016, at 12:59 AM by 174.148.92.107 -
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February 10, 2016, at 12:39 AM by 174.148.92.107 -
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Commercial Fishery

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February 09, 2016, at 01:22 PM by 174.148.118.172 -
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See estimation example using the same model to explain estimator objectives.

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February 09, 2016, at 12:49 PM by 174.148.118.172 -
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Aly-Chan Problem2

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Aly-Chan Singular Control Problem2

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Catalyzed Reaction

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February 09, 2016, at 06:33 AM by 174.148.220.36 -
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Solution to Catalyzed Reaction Problem

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February 09, 2016, at 06:31 AM by 174.148.220.36 -
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Solution to Aly-Chan Problem

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Catalyzed Reaction

Solution to Catalyzed Reaction Problem

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Aly-Chan Problem

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Aly-Chan Problem2

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  • The objective is to minimize final state x_3_(pi/2) by adjusting the value of u.
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  • The objective is to minimize final state x3(pi/2) by adjusting the value of u.
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  • Nonlinear, constrained, minimize final state
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  • Solve the following nonlinear and constrained problem.
  • The objective is to minimize final state x_3_(pi/2) by adjusting the value of u.
  • Compare to the exact solution of u(t)= -sin(t).
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Exercise

Objective: Set up and solve several dynamic optimization benchmark problems. Create a program1 to optimize and display the results. Estimated Time (each): 30 minutes

Aly-Chan Problem

  • Nonlinear, constrained, minimize final state

Solution to Aly-Chan Problem


References

  1. Hedengren, J. D. and Asgharzadeh Shishavan, R., Powell, K.M., and Edgar, T.F., Nonlinear Modeling, Estimation and Predictive Control in APMonitor, Computers and Chemical Engineering, Volume 70, pg. 133–148, 2014. Article
  2. Aly G.M. and Chan W.C. Application of a modified quasilinearization technique to totally singular optimal problems. International Journal of Control, 17(4): 809-815, 1973.