EML 6934 – Optimal Control Course Project Fall 2017 1 Overview of Course Project The project for the course is structured as follows. You are required to choose two problems from the two lists shown below. One of the problems must be from Section 2 while the second problem must be from Section 3. For the problem chosen from Section 2 you must compute either the analytic optimal solution or must compute a numerical solution using an indirect method (that is, by deriving the first-order optimality conditions and solving the Hamiltonian boundary-value problem). You must then solve the problem using one of the direct methods studied in the course. A comparison between the analytic/indirect solution and the direct so- lution is then required. In your analysis you must explain the key issues encountered when solving the problem using the indirect and direct approaches. The problems from Section 3 do not have analytic solutions and cannot be easily solved us- ing a simple indirect method (for example, indirect shooting). Consequently, it is necessary to employ a more sophisticated approach to solve these problems. Your task is to solve your cho- sen probem using either one of the indirect methods or one of the direct methods studied in the course. Regardless of the approach you choose, you must provide the following analysis of your numerical solutions. First, you must assess the proximity of your numerical solution to the “true” optimal solution (knowing full well that you do not have the “true” optimal solution to your chosen problem). In other words, how do you know you have obtained a good approxima- tion to the true optimal? Next, you must provide a study of both the computational efficiency and robustness of your chosen method in determining the numerical approximation. In your analysis, explain how good an initial guess you need to provide in order to solve the problem and how efficiently you are able to solve the problem. In addition, analyze the limitations of the method you have chosen? Given your analysis, explain if the method you chose is the best one for this type of problem? If not, which method would be more preferable? As as part of your analysis you can check the numerical solutions you obtain against the solu- tions obtained using the MATLAB optimal control software GPOPS − II. A license of GPOPS − II is available at no charge for use at the University of Florida and can be obtained by registering on the GPOPS − II website by clicking here. GPOPS − II implements a variable-order Legendre- Gauss-Radau quadrature method and the details of the method can be found in the journal article that is soon to appear in the ACM Transactions on Mathematical Software. The final version of the journal article can be found by clicking here. Finally, you must provide a comprehensive report detailing all of the results you obtained and what you learned about the numerical methods you employed. You must also provide all code used to generate your results. Please note, you must implement all numerical computations 1 yourself and cannot use any canned software to solve your problems. The course project is due on the last day of Spring 2014 classes. 2 Elementary Optimal Control Problems Every problem found in this list has an analytic solution. You must choose one problem from this Section for your project. For any problem you choose in this Section you are required to derive the optimal solution using the optimal control theory learned in the course. Next, you are required to solve your chosen problem using one of the indirect methods and one direct method studied in the course. You must then compare the quality of the numerical solutions obtained against the optimal solution. You must then analyze the quality of your numerical approximations and assess the key computational issues you encountered when trying to solve this problem using your chosen indirect and direct approach. 2.1 Hyper-Sensitive Problem Consider the following optimal control problem taken from Ref. 1. Minimize the cost functional 1 Z tf J = (x2 + u2)dt (1) 2 0 subject to the dynamic constraint x_ = −x + u; (2) the boundary conditions x(0) = 1; (3) x(tf ) = 1:5; and tf fixed. Solve this optimal control problem for the following values of tf : 10, 20, 50, 100, 500, and 1000. Do you notice anything interesting in your ability to compute the both the analytic so- lution on your computer and the numerical solution as tf increases? Describe your observation. 2.2 Linear Tangent Steering Problem Consider the following optimal control problem. Minimize the cost functional J = tf (4) subject to the dynamic constraints x_ 1 = x3; x_ = x ; 2 4 (5) x_ 3 = a cos u; x_ 4 = a sin u; 2 and the boundary conditions x1(0) = 0; x2(0) = 0; x3(0) = 0; x4(0) = 0; (6) x2(tf ) = 5; x3(tf ) = 45; x4(tf ) = 0: 2.3 Ground Mobile Robot Problem The following optimal control problem is originally found in Ref. 2 and corresponds to the min- imum time transfer of a ground mobile between a given initial and terminal state. Minimize the cost functional J = tf (7) subject to the dynamic constraints x_ = cos(θ); y_ = sin(θ); (8) θ_ = u; and the boundary conditions x(0) = 0; y(0) = 0; θ(0) = −π; (9) x(tf ) = 0; y(tf ) = 0; θ(tf ) = π: 2.4 Moon Lander Problem This optimal control problem was originally posed by Meditch.3 The objective is to attain a soft landing on moon during vertical descent from an initial altitude and velocity above the lunar surface. The problem is stated as follows. Minimize the cost functional Z tf J = udt (10) 0 subject to the dynamic constraints h_ = v; (11) v_ = −g + u; the boundary conditions h(0) = 10; v(0) = −2; (12) h(tf ) = 0; v(tf ) = 0; 3 and the control inequality constraint umin ≤ u ≤ umax; (13) where umin = 0, umax = 3, g = 1:5, and tf is free. 2.5 Bryson-Denham Problem Consider the following optimal control minimum-energy optimal control problem with an in- equality state constraint taken from Ref. 4. Minimize the cost functional 1 Z 1 J = a2dt (14) 2 0 subject to the dynamic constraints x_ = v; (15) v_ = a; (16) the boundary conditions x(0) = x(1) = 0; (17) v(0) = −v(1) = 1; (18) and the constraint x(t) ≤ `. 3 Advanced Optimal Control Problems No problem in this Section has an analytic solution. As a result, every problem must be solved numerically. As with the elementary problems found in Section 2, the problem you choose in this Section must be solved using either one of the indirect methods or one of the direct meth- ods studied in the course. You must then provide the following analysis of your numerical approximations. First, provide an assessment of the proximity of your numerical solutions to the optimal solution (which, as stated, is not known for these problems). How do you know you have obtained a reasonable approximation? Next, what is the computational efficiency of the numerical methods you used to solve your problem? What are the limitations of the methods you have chosen on your problem. Given your analysis, what numerical method would you seek in order to overcome the deficiencies you found with the methods you chose? 3.1 Crossrange Maximization During Entry of a Reusable Launch Vehicle The reusable launch vehicle entry problem is taken from Ref. 5. The objective of the problem is to maximize the crossrange subtended by the vehicle during entry, where the entry starts at the edge of the sensable atmosphere and terminates at the start of the terminal area energy management (TAEM) phase. The problem is stated as follows. Maximize the objective functional J = φ(tf ) (19) 4 subject to the dynamic constraints r_ = v sin γ; v cos γ sin θ_ = ; r cos φ v cos γ cos φ_ = ; r D (20) v_ = − − g sin γ; m L cos σ g v γ_ = − − cos γ; mv v r L sin σ v cos γ sin tan φ _ = + ; mv cos γ r and the boundary conditions h(0) = 79248 km ; h(tf ) = 24384 km; θ(0) = 0 deg ; θ(tf ) = Free; φ(0) = 0 deg ; φ(tf ) = Free; (21) v(0) = 7802:88 m/s ; v(tf ) = 762:0 m/s; γ(0) = −1 deg ; γ(tf ) = −5 deg; (0) = 90 deg ; (tf ) = Free; where r = h + Re is the geocentric radius, h is the altitude, Re is the polar radius of the Earth, θ is the longitude, φ is the latitude, v is the speed, γ is the flight path angle, and is the azimuth angle. Furthermore, the aerodynamic and gravitational forces are computed as 2 D = ρv SCD=2; 2 L = ρv SCL=2; (22) g = µ/r2; where ρ = ρ0 exp(−h=H) is the atmospheric density, ρ0 is the density at sea level, H is the density scale height, S is the vehicle reference area, CD is the coefficient of drag, CL is the coefficient of lift, and µ is the gravitational parameter. The coefficient of lift and drag are computed, respec- tively, as 2 CD = CD0 + CD1α + CD2α ; (23) CL = CL0 + CL1α; (24) where α is the angle of attack. Table 1 provides the constants used in this problem. 3.2 Maximization of Mass-to-Orbit for a Multiple-Stage Launch Vehicle The problem considered in this section is the ascent of a multiple-stage launch vehicle. The objective is to maneuver the launch vehicle from the ground to the target orbit while maximizing the remaining fuel in the upper stage.