Optimal Control Theory SF 2852

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1 university-logo Optimal Control Theory SF 2852 Ulf Jönsson Optimization and Systems Theory, Department of Mathematics Royal Institute of Technology (KTH) Spring 2011

2 university-logo Optimal Control Theory SF Course information and course logistics 2 Course overview and application examples

3 Course Information and Course Logistics This is a course on optimal control Optimization problems involving difference or differential equations Many engineering problems are naturally posed as optimal control problems Economics and logistics Aerospace systems Automotive industry Autonomous systems and robotics Bio-engineering Process control Power systems

4 Optimal control is an important branch of mathematics Roots in the calculus of variations Dynamic programming Pontryagin minimum principle (PMP) Linear quadratic control Model predictive control (MPC)

5 Teachers Ulf Jönsson ( Hildur Æsa Oddsdóttir (

6 Course Material Optimal Control: Lecture notes Exercise notes on optimal control theory Some complementary material will be handed out. It will also be posted on the course homepage.

7 university-logo Homework Three optional homework sets HW1 Discrete dynamic programming and model predictive control. Due on Tuesday April 12, at HW2 Dynamic Programming and PMP: Due on Friday April 29, at HW3 PMP and related topics. Due on Thursday May 19, at Each homework has 3-5 problems. Grading: Each homework may give maximum 2 bonus credits for the exam.

8 university-logo Final Written Exam The final exam takes place on Wednesday May 25, 2011 at You must register for the exam during April 18 to May 8, Grade A B C D E FX Total credit (points) Total credit = exam score + homework score. The maximum exam score = 50. Maximum bonus from the homework sets = 6. You may use Mathematics Handbook and a formula sheet found on the course homepage.

9 university-logo Course Overview and Application Examples 1 Discrete time optimal control theory 2 Continuous time optimal control theory

10 university-logo Discrete Time Optimal Control minφ(x N ) + Σ N 1 k=0 f 0(k, x k, u k ) subj. to x k+1 = f(k, x k, u k ) x 0 given, x k X k u k U(k, x k ) Sequential decision problem which contain as special cases Graph search problems Combinatorial optimization Discrete time optimal control

11 Shortest Path problem university-logo Initial node Terminal node Stage 0 Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Find the shortest path from the initial node at stage 0 to the terminal node at stage 5

12 The Knapsack Problem university-logo n max p j x j j=1 subject to n w j x j c, x j = 0 or 1, j = 1,...,n. j=1

13 Discrete Optimal Control Theory Discrete optimal control problem Dynamic programming Optimization theory Global theory Feedback solutions Local theory Open loop control Model predictive control Feedback solution Online optimization university-logo

14 Continuous Time Optimal Control tf minφ(t 1, x(t 1 )) + f 0 (t, x(t), u(t))dt subj. to t 0 x(t) ẋ(t) = f(t, x(t), u(t)) x(t 0 ) S 0, x(t 1 ) S 1 u U(x) S 0 S 1 S 0 and S 1 are manifolds (intersection of surfaces). university-logo

15 Optimal Control of Car y v θ b y R φ v x θ x Problem: Shortest path. ẋ(t) = v cos(θ(t)), x(0) = 0, x(t) = x ẏ(t) = v sin(θ(t)), y(0), y(t) = ȳ min T subj. to θ(t) = ω(t), θ(0) = 0, θ(t) = θ ω(t) v/r, T 0 university-logo

16 Orbit Transfer of Satellite Problem: Orbit transfer thrust speed ξ3 u ξ1 ξ2 gravitational center

17 Orbit Transfer of Satellite university-logo Problem: max x 1 (T) s.t. u ẋ 1 = c3 2x 2 ẋ 2 = x 3 2 x 1 1 x1 2 ẋ 3 = c2 3 x 2x 3 x 1 + sin u 1/c 1 c 2 c 3 t + c 3 cos u 1/c 1 c 2 c 3 t x(0) = (1 0 1) T S f = {x R 3 x 2 = 0, x 3 1 x 1 = 0}

18 Resource Allocation university-logo T max u( ) 0 (1 u(t))x(t)dt + x(t) ẋ(t) = αu(t)x(t), (0 < α < 1) subj. to x(0) = x 0 > 0 u(t) [0, 1], t A portion u(t) of the production rate x(t) is invested in the factory (and increases the production capacity) The rest (1 u(t))x(t) is stored in a warehouse. Maximimize the sum of the total amount of goods stored in the warehouse and the final capacity of the factory.

19 Optimal Control Theory Optimal control problem Dynamic programming Pontryagin minimum principle Global theory Feedback solutions Local theory Open loop control Feedback control u(t) = ψ(t, x(t)) (depends on state) Open loop control u(t) = ψ(t)

Optimal Control Theory SF 2852

Optimal Control Theory SF 2852 Johan Karlsson Optimization and Systems Theory, Department of Mathematics Royal Institute of Technology (KTH) Spring 2017 Optimal Control Theory SF 2852 1 Course information