Dynamic Optimization

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Optimal Control Problem (OCP) solvers are a powerful tool for designing control strategies that optimize a given objective function subject to constraints. They are widely used in engineering, economics, finance, and management, and are applied to a wide range of problems to find optimal solutions that minimize costs, maximize profits, reduce risk, or achieve other desired outcomes. The control strategy is typically represented as a function of time that maps the control input (manipulated variable) to the state of the system. The state of the system is determined by a set of differential equations, and the objective function is a measure of the system performance over a time horizon.

Objective: Set up and solve several OCP benchmarks. Create a program to optimize and display the results. Estimated Time (each): 30 minutes

Hanging Chain

(:title OCP Benchmarks:)

Objective: Set up and solve several optimal control problem (OCP) benchmarks. Create a program to optimize and display the results. Estimated Time (each): 30 minutes

Batch Reactor

Bryson-Denham Problem

Integral Objective (Luus)

Maximize Profit (Commercial Fishery)

Minimize Final Time (Jennings)

Batch Reactor

plt.plot(m.time,x1.value,'k:',lw=2,label=r'$x_1$') plt.plot(m.time,x2.value,'b-',lw=2,label=r'$x_2$') plt.plot(m.time,x2.value,'k-',lw=2,label=r'$x_3$')

plt.plot(m.time,u.value,'r--',lw=2,label=r'$u$') plt.plot(t,-np.sin(t),'k:',lw=2,label='Exact')

(:title Optimal Control Benchmark Problems II:)

1. Beal, L.D.R., Hill, D., Martin, R.A., and Hedengren, J. D., GEKKO Optimization Suite, Processes, Volume 6, Number 8, 2018, doi: 10.3390/pr6080106. Article

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

• Solve the following nonlinear and constrained problem.

Objective: Set up and solve several optimal control benchmark problems. Create a program as shown in reference [1] to optimize and display the results. Estimated Time (each): 30 minutes

Aly-Chan Singular Control Problem

• Solve the following nonlinear and constrained problem as shown in reference [2].

Bryson-Denham Problem

Maximize Profit (Commercial Fishery)

Commercial Fishery

Minimize Final Time (Jennings)

(: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:)

(: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.:)

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

Integral Objective (Luus)

<iframe width="560" height="315" src="https://www.youtube.com/embed/WIe31dTaW6g" frameborder="0" allowfullscreen></iframe>

(:html:) <iframe width="560" height="315" src="https://www.youtube.com/embed/yuj217itTa8" frameborder="0" allowfullscreen></iframe> (:htmlend:)

Luus Problem

Luus Problem in MATLAB and Python

<iframe width="560" height="315" src="https://www.youtube.com/embed/yuj217itTa8" frameborder="0" allowfullscreen></iframe>

Economic Optimization of Fishery in MATLAB and Python

(:html:) <iframe width="560" height="315" src="https://www.youtube.com/embed/MA53XPp-a7I" frameborder="0" allowfullscreen></iframe> (:htmlend:)

Jennings Problem

Jennings Problem in MATLAB and Python

<iframe width="560" height="315" src="https://www.youtube.com/embed/MA53XPp-a7I" frameborder="0" allowfullscreen></iframe>

Commercial Fishery

Dynamic Optimization Fishery Solution in MATLAB and Python

(:html:) <iframe width="560" height="315" src="https://www.youtube.com/embed/jB7WWGHFNIw" frameborder="0" allowfullscreen></iframe> (:htmlend:)

See estimation example using the same model to explain estimator objectives.

<iframe width="560" height="315" src="https://www.youtube.com/embed/cd0TkFtXfWs" frameborder="0" allowfullscreen></iframe>

Aly-Chan Singular Control Problem²

Aly Singular Control Problem

Aly Singular Control Solution in MATLAB and Python

Catalyzed Reaction

Dynamic Optimization Catalyst Solution in MATLAB and Python

(:html:) <iframe width="560" height="315" src="https://www.youtube.com/embed/jB7WWGHFNIw" frameborder="0" allowfullscreen></iframe> (:htmlend:)

<iframe width="560" height="315" src="https://www.youtube.com/embed/jB7WWGHFNIw" frameborder="0" allowfullscreen></iframe>

(:htmlend:)

Catalyzed Reaction

Solution to Catalyzed Reaction Problem

Dynamic Optimization Catalyst Solution in MATLAB and Python

(:html:)

(:html:) <iframe width="560" height="315" src="https://www.youtube.com/embed/tY8kyJq4BEY" frameborder="0" allowfullscreen></iframe> (:htmlend:)

Aly-Chan Problem²

• The objective is to minimize final state x₃(pi/2) by adjusting the value of u.

• 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).

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

Dynamic Optimization Aly-Chan Solution in MATLAB and Python

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.