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

Generative AI Learning

Use these prompts as you work the benchmark set. Predict before you solve - the value of a benchmark is knowing what the answer should look like.

Tip: Benchmark 4 (batch reactor, maximize intermediate B with temperature bounds) is mathematically the modern battery fast-charging problem: maximize charge delivered while a temperature-like state stays under a degradation limit. After solving it, ask the AI to map every variable of your solution onto the fast-charging problem and identify which constraint plays the 398 K role.

What to Turn In

Submit a short report (PDF, 1-2 pages) that curates your results into a demonstration of what you learned. You may use Generative AI to help write the report, but you must guide it to the correct visualizations, justifications, and assumptions. Answer these questions:

1. Tabulate your objective values for the benchmark exercises and flag any that disagree with published/literature values - with your diagnosis of each disagreement.
2. For one benchmark: show state and control profiles, identify where each control bound or path constraint is active, and explain the switching structure physically.
3. Rerun that benchmark from two different initializations (or node counts). Same solution? Show the evidence and state what it implies about local minima or discretization error.
4. From the audit prompt: what defects did you find in the AI-generated formulation, and which one would have changed the answer the most?