EML Spring 2012

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1 Home Page 1 of 2 1/10/2012 Search EML Spring 2012 Optimal Control Home Instructor Anil V. Rao Office Hours: M, W, F 2:00 PM to 3:00 PM Office: MAE-A, Room 314 Tel: (352) anilvrao@ufl.edu Teaching Assistant Kathryn Schubert Office Hours: TBD Office: MAE-A, Room 304 Tel: (352) kschuber@ufl.edu Required Textbook Kirk, D. E., Optimal Control Theory: An Introduction, Dover Publications, Syllabus: Calculus of Variations Optimal Control Theory Numerical Methods for Solving Optimal Control Problems Homework Assignments Assignment #1 Assignment #2 Assignment #3 Assignment #4 Assignment #5 Course Project The project for the course is as follows. Choose an optimal control problem in your area of

2 Home Page 2 of 2 1/10/2012 interest. The problem must be nonlinear, contain a minimum of a four-dimensional state and a two-dimensional control, and must not have an analytic solution. Formulate the complete set of first-order optimality conditions for the problem. Then, solve your problem using at least two different numerical methods that we study in the course. In solving your problem, you must utilize at least one indirect and one direct method. Once you have applied your numerical methods, analyze the solutions you obtain. In your analysis, consider the following questions. Can you determine the proximity of your numerical approximations to the true optimal solution? If you are unable to ascertain how close your solution is to the true optimal, how do you know you have obtained a reasonable approximation? What is the computational efficiency of your method? What are the limitations of the methods you have chosen? Given your analysis, what would seek in a numerical method to overcome any deficiencies in the methods you chose? It is recommended that you think of a problem as early in the course as possible and not wait until the last minute! The course project is due by the start of class on 25 April Exams (All Exams Are Take-Home) Take-Home First Examination Given Out: Friday 24 February 2011 (via ) Due: Monday 27 February 2011 in Class Take-Home Second Examination Given Out: Monday 23 April 2011 (via ) Due: Wednesday 25 April 2011 in Class Grading: Exam 1: 30 percent Exam 2: 30 percent Project: 40 percent Grading Policy No make-up exams and no rescheduling of exams to other times. Academic Honesty Policy Your examinations must be completed independently. If anyone is caught having worked together on an exam or having used an unauthorized source, the penalty is an automatic failure in the course and the case will be reported through the University of Florida academic honesty violation system Anil Rao Contact Me

3 Question 1 Let A R n n be a real symmetric n n matrix. Prove the following: The eigenvalues of A are real The eigenvectors corresponding to distinct eigenvalues are orthogonal Assume now that in the case of repeated eigenvalues the eigenvectors are still orthogonal (i.e., for λ i = λ j it is still the case that the corresponding eigenvectors v i and v j are orthogonal). Prove the following properties of the eigenvector matrix V =[v 1 v n ]: V is an orthoogonal matrix, i.e., V 1 = V T V T AV = Λ where Λ is a diagonal matrix of eigenvalues Question 2 Let A R m n be a real matrix. Prove the following results: The general solution of the linear system Ax = b is x = x h +x p where x h is the homogeneous solution and satisfies Ax h = 0 and x p is the particular solution and satisfies Ax p = b InthecasewhereA is square, the unique solution to the linear system Ax = b is x = A 1 b and this solution exists if and only if the matrix A is nonsingular. Question 3 Let A, B R n n be constant n n matrices. Furthermore, let a(s) and b(s) each be arbitrary polynomials in s of degree m. Prove the following result: a(a)b(b) = b(b)a(a) AB = BA Question 4 Let A R n n be a constant n n matrix. Prove that the matrix exponential e A can be written as a linear combination of powers of A of degree n 1 or less, i.e., where α k, (k =0,...,n 1) are real numbers. n 1 e A = α k A k k=0 Assignment #1 Due Date: 14 January 2008 Page 1

4 Question 5 Let B R n n be an n n real matrix. Furthermore, let L(A) =AB + BA be a matrix operator whose input argument is A. Show that L is a linear operator. Question 6 Let A R n n be an n n real square matrix. Furthermore, let x be an eigenvector of A with eigenvalue λ. Prove that A k x = λ k x for all k 1. Question 7 Let p(s) be a polynomial of degree n in s C. Furthermore, let A R n n be an n n real square matrix. Finally, let λ be an eigenvalue of A with eigenvector x. Prove that p(λ) is an eigenvalue of p(a) with eigenvector x. Question 8 Let A R n n be an n n real matrix. Suppose that the eigenvectors of A form a complete set, i.e., span(v 1 v n )=R n (i.e., the set of eigenvectors (v 1,...,v n ) form a linearly independent set that spans R n ). Prove the following result: D = V 1 AV where V =[v 1 v n ] is the matrix of eigenvectors and D = diag(λ 1,...,λ n ) is a diagonal matrix whose diagonal elements are the eigenvalues of A. Question 9 Let p(s) be a polynomial in s C of degree n. Furthermore, let A R n n be an arbitrary square matrix. Finally, let T R n n be a nonsingular square matrix. Prove that p(t 1 AT)=T 1 p(a)t Question 10 Prove the following properties of the state transition matrix Φ(t, τ): (a) Φ(t, τ) =Φ(t, t 1 )Φ(t 1,τ) Assignment #1 Due Date: 14 January 2008 Page 2

5 (b) Φ(t, τ) =Φ 1 (τ,t) (c) Φ(t, τ) t = A(t)Φ(t, τ) Φ(t, τ) (d) = Φ(t, τ)a(τ) τ (e) Φ(t, τ) is nonsingular t and τ Question 11 Let ẋ = Ax be a linear time-invariant system with initial condition x(0) = x 0. The solution of this system can be written implicitly in integral form as x(t) =x(0) + t 0 Ax(τ)dτ Suppose now that we use the following iterative procedure to solve the problem: x (k+1) (t) =x(0) + t 0 Ax (k) (τ)dτ where x (k) (t) is the solution of the k th iteration (k 0). Assuming that the zeroth iteration is x (0) = x 0, prove the following result: lim k x(k) (t) =x(t) =e At x 0 Repeat the procedure for x (0) compared to the first case? = x x 0. How does your result differ in the second case as Assignment #1 Due Date: 14 January 2008 Page 3

6 Question 1 Using the approach for calculus of variations developed in class (not the approach used in Kirk s book), derive the necessary conditions for optimality that minimize the integral J = tf t 0 L[x(t), ẋ(t),t]dt for the following sets of boundary conditions: t 0 fixed, x(t 0 )=x 0 fixed; t f fixed, x(t f )=x f fixed t 0 fixed, x(t 0 )=x 0 fixed; t f fixed, x(t f )=x f free t 0 fixed, x(t 0 )=x 0 fixed; t f free, x(t f )=x f fixed t 0 fixed, x(t 0 )=x 0 fixed; t f free, x(t f )=x f free Question 2 Prove the following theorem (known as the fundamental lemma of variational calculus). Given a continuous function f(t) on the time interval t [t 0,t f ] and that tf f(t)δx(t) =0 t 0 for all continuous functions δx(t) on t [t 0,t f ] such that δx(t 0 )=δx(t f )=0.Thenδx(t) must be identically zero on t [t 0,t f ]. Question 3 (Kirk 1998) Determine the extremal solutions of the following two functionals: (1) J(x(t)) = 1 0 [x2 (t) ẋ 2 (t)] dt, x(0) = 1, x(1) = 1 (2) J(x(t)) = 2 0 [x2 (t) 2ẋ(t)x(t)+ẋ 2 (t)] dt, x(0) = 1, x(2) = 3 Question 4 Determine the extremal curve x(t) of the functional J(x(t)) = tf 0 1+ẋ2 (t)dt with boundary conditions x(0) = 5 x 2 (t f )+(t f 5) 2 4 = 0 where t f is free. Plot the extremal solution and give a geometric interpretation of the result. Assignment #2 Due Date: 28 January 2008 Page 1

7 Question 5 The Brachistochrone Problem is one of the earliest and most famous problems in the calculus of variations. The brachistochrone problem is stated as follows. Let x(t) and y(t) measure the horizontal and vertically downward components of position of a particle of mass m in an inertially fixed Cartesian coordinate system. Starting at the the point (x(t 0 ),y(t 0 )) = (0, 0), determine the path along which a particle of mass m must move under the influence of constant gravity g such that it reaches the point x(t f ),y(t f )=(1, 1) in minimum time. Assume in your solution that t 0 =0. Assignment #2 Due Date: 28 January 2008 Page 2

8 Question 1 Solve Problem 5-1 in Kirk. Question 2 Solve Problem 5-2 in Kirk. Question 3 Solve Problem 5-5 in Kirk. Question 4 Solve Problem 5-6 in Kirk. Question 5 Solve Problem 5-8 in Kirk. Question 6 Solve Problem 5-16 in Kirk. Assignment #3 Due Date: 15 Februay 2008 Page 1

9 Optimal Control Assignment #4 Question 1 Solve Problem 5-3 in Kirk. Question 2 Solve Problem 5-7 in Kirk. Question 3 Solve Problem 5-10 in Kirk. Question 4 Solve Problem 5-14 in Kirk. Question 5 Solve Problem 5-15 in Kirk. Question 6 Consider the following optimal control problem. The dynamics are given as ẋ = V cos θ + u ẏ = V sin θ where V is a constant and θ is the control. The boundary conditions are given as x(t ) = x(0) y(t ) = y(0) Assuming that T is fixed, determine the optimal trajectory (x(t),y(t)) and the optimal control θ(t) that maximizes the area enclosed by the region that the aircraft flies over the interval from t [0,T]. Knowing that the area enclosed by the trajectory is given as J = T determine the optimal solution (x (t),y (t),u (t)) 0 yẋdt (1) Anil V. Rao University of Florida Page 1

10 Optimal Control Assignment #5 For the problems given in Assignment #3, implement the following: The standard shooting method The multiple-shooting method In implementing the multiple-shooting method, you will need to break the interval into subintervals. Experiment with various numbers of fixed-width subintervals to see how the solution changes. In order to complete this assignment, you will need to read up on root finding in a numerical analysis book. This assignment will take some time, so do not rush through it. Anil V. Rao University of Florida Page 1

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STUDENT NAME: STUDENT SIGNATURE: STUDENT ID NUMBER: SECTION NUMBER RECITATION INSTRUCTOR: MA262 FINAL EXAM SPRING 2016 MAY 2, 2016 TEST NUMBER 01 INSTRUCTIONS: 1. Do not open the exam booklet until you are instructed to do so. 2. Before you open the booklet fill in the information below and