Objective: Set up and solve the Luus optimal control benchmark problem. Create a program to optimize and display the results. Estimated Time: 30 minutes
The Luus optimal control problem has an integral objective function. Integrals are a natural expression of many systems where the accumulation of a quantity is maximized or minimized.
Integral expressions are reformulated in differential and algebraic form by defining a new variable.
$$x_2 = \frac{1}{2} \int_0^2 x_1^2(t) \, dt$$
The new variable and integral are differentiated and included as an additional equation. The problem then becomes a minimization of the new variable `x_2` at the final time.
The initial condition of the integral starts at zero and becomes the integral in the time range of 0 to 2. The value that is minimized is at the final point in the time horizon of the optimal control problem.
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