Dynamic Optimization

Exercise

Objective: Solve the dynamic optimization benchmark problems² and more dynamic optimization benchmark problems. Complete the 9 exercises as shown in the Jupyter Notebook link below. For each problem, create a program to optimize and display the results. Estimated Time (each): 10-30 minutes

DynOpt Benchmarks on GitHub or on Google Colab.

DynOpt Benchmark Solutions on GitHub or on Google Colab.

Example 1a

• Nonlinear, unconstrained, minimize final state

$$\min_{u(t)} x_2 \left( t_f \right)$$ $$\mathrm{subject \; to}$$ $$\frac{dx_1}{dt}=u$$ $$\frac{dx_2}{dt}=x_1^2 + u^2$$ $$x(0) = [1 \; 0]^T$$ $$t_f=1$$

Example 1b

• Nonlinear, unconstrained, minimize final state with terminal constraint

$$\min_{u(t)} x_2 \left( t_f \right)$$ $$\mathrm{subject \; to}$$ $$\frac{dx_1}{dt}=u$$ $$\frac{dx_2}{dt}=x_1^2 + u^2$$ $$x(0) = [1 \; 0]^T$$ $$x_1 \left( t_f \right)=1$$ $$t_f=1$$

Solutions to Benchmarks 1a and 1b

Dynamic Optimization Benchmark 1a and 1b in MATLAB and Python

Example 2

• Nonlinear, constrained, minimize final state

$$\min_{u(t)} x_4 \left( t_f \right)$$ $$\mathrm{subject \; to}$$ $$\frac{dx_1}{dt}=x_2$$ $$\frac{dx_2}{dt}=-x_3 \, u + 16 \, t - 8$$ $$\frac{dx_3}{dt}=u$$ $$\frac{dx_4}{dt}=x_1^2+x_2^2+0.0005 \left(x_2 + 16 \, t -8 -0.1x_3\,u^2\right)^2$$ $$x(0) = [0 \; -1 \; -\sqrt{5} \; 0]^T$$ $$-4 \le u \le 10$$ $$t_f=1$$

Solution to Benchmark 2

Dynamic Optimization Benchmark 2 in MATLAB and Python

Example 3

• Tubular reactor with parallel reaction

$$\max_{u(t)} x_2 \left( t_f \right)$$ $$\mathrm{subject \; to}$$ $$\frac{dx_1}{dt}=-\left(u+0.5u^2\right) x_1$$ $$\frac{dx_2}{dt}=u \, x_1$$ $$x(0) = [1 \; 0]^T$$ $$0 \le u \le 5$$ $$t_f=1$$

Solution to Benchmark 3

Dynamic Optimization Benchmark 3 in MATLAB and Python

Example 4

• Batch reactor with consecutive reactions A->B->C

$$\max_{T(t)} x_2 \left( t_f \right)$$ $$\mathrm{subject \; to}$$ $$\frac{dx_1}{dt}=-k_1 \, x_1^2$$ $$\frac{dx_2}{dt}=k_1 \, x_1^2 - k_2 \, x_2$$ $$k_1 = 4000 \, \exp{\left(-\frac{2500}{T}\right)}$$ $$k_2 = 6.2e5 \, \exp{\left(-\frac{5000}{T}\right)}$$ $$x(0) = [1 \; 0]^T$$ $$298 \le T \le 398$$ $$t_f=1$$

Solution to Benchmark 4

Dynamic Optimization Benchmark 4 in MATLAB and Python

Example 5

• Catalytic reactor with A<->B->C

$$\max_{u(t)} \left(1 - x_1 \left( t_f \right) - x_2 \left( t_f \right) \right)$$ $$\mathrm{subject \; to}$$ $$\frac{dx_1}{dt}=u \left(10 \, x_2 - x_1 \right)$$ $$\frac{dx_2}{dt}=-u \left(10 \, x_2 - x_1 \right)-\left(1-u\right) x_2$$ $$x(0) = [1 \; 0]^T$$ $$0 \le u \le 1$$ $$t_f=12$$

Solution to Benchmark 5

Dynamic Optimization Benchmark 5 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. M. Čižniar, M. Fikar, M.A. Latifi: A MATLAB Package for Dynamic Optimisation of Processes, 7th International Scientific – Technical Conference – Process Control 2006, June 13 – 16, 2006, Kouty nad Desnou, Czech Republic. Article