Exercise for Optimal control – Week 1 Choose two problems to solve. Exercise 1 (Fundamental lemma of CoV). Let f be a real valued function defined on open interval (a, b) and f satisfies ∫ b a f(x)h(x)dx = 0 for all h ∈ Cc(a, b), i.e., h is continuous on (a, b) and its support, i.e., the closure of {x : h(x) ̸= 0} is contained in (a, b). 1) Show that f is identically zero if f is continuous. If f

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2023/ex1.pdf - 2025-07-08

Exercise for Optimal control – Week 3 Choose 1.5 problems to solve. Exercise 1. Consider a harmonic oscillator ẍ + x = u whose control is constrained in the interval [−1, 1]. Find an optimal controller u which drives the system at initial state (x(0), ẋ(0)) = (X1, X2) to the origin in minimal time. Draw the phase plot. Exercise 2. Consider a rocket, modeled as a particle of constant mass m movin

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2023/ex3.pdf - 2025-07-08

Exercise for Optimal control – Week 4 Choose one problem to solve. Disclaimer This is not a complete solution manual. For some of the exercises, we provide only partial answers, especially those involving numerical problems. If one is willing to use the solution manual, one should judge whether the solutions are correct or wrong by him/herself. Exercise 1. Use tent method to derive the KKT conditi

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2023/ex4-sol.pdf - 2025-07-08

Exercise for Optimal control – Week 5 Choose 2 problems to solve. Exercise 1. A public company has in year k profits amounting to xk SEK. The management then distributes uk to the shareholders and invests xk − uk in the company itself. Each SEK invested in such way will increase the company profit by θ > 0 the following year so that xk+1 = xk + θ(xk − uk). Suppose x0 ≥ 0 and 0 ≤ uk ≤ xk so that xk

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2023/ex5.pdf - 2025-07-08

Exercise for Optimal control – Week 6 Choose 1.5 problems to solve. Disclaimer This is not a complete solution manual. For some of the exercises, we provide only partial answers, especially those involving numerical problems. If one is willing to use the solution manual, one should judge whether the solutions are correct or wrong by him/herself. Exercise 1. Derive the policy iteration scheme for t

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2023/ex6_sol.pdf - 2025-07-08

6 Lecture 6. Final step of the proof of MP and a start of DP 6.1 The proof of the maximum principle (finally!) In our previous lecture, we started proving the maximum principle for the Mayer problem ẋ = f(x, u) with cost J = φ(x(tf )) under the constraint u(t) ∈ U , x(tf ) ∈ M . The basic tool for the proof is the method of tent. To that end, we defined the following tents: Ω0 = {x1} ∪ {x : φ(x)

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2023/lec6.pdf - 2025-07-08

PII: S0005-1098(99)00171-5 Automatica 36 (2000) 363}378 Drum-boiler dynamicsq K.J. As stroK m!,*, R.D. Bell" !Department of Automatic Control, Lund Institute of Technology, Box 118, S-221 00 Lund, Sweden "Department of Computing, School of Mathematics, Physics, Computing and Electronics, Macquarie University, New South Wales 2109, Australia Received 2 October 1998; revised 7 March 1999; received i

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/PhysicalModeling/Lectures/Astrom-Bell.pdf - 2025-07-08

Compartment Models K. J. Åström 1. Introduction 2. Compartment Models 3. Flow Systems 4. Measurement of Volumes and Flows 5. Summary 6. References Introduction ◮ Early work by Teorell and Widmark on propagation of alcohol in the body 1920 ◮ Teorell coined the term compartment model around 1937 ◮ Extensive application in pharmacokinetics Dost 1953 Models required for FDA approval of new drugs Shepp

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/PhysicalModeling/Lectures/Compartmentseight.pdf - 2025-07-08

Fluid Dynamics Modeling K. J. Åstr{öm 1. Introduction 2. Review of Fluid Dynamics 3. Simple Water Tank 4. Simple Gas Tank 5. Tanks, Pipes and Turbines 6. Summary Historical Remarks ◮ Hydroelectric power ◮ Control of dams and turbines ◮ Founded in civil engineering A not so well recognized base of automatic control Evangelisti (an IFAC founder) in Italy Many others in civil engineering Vattenfalls

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/PhysicalModeling/Lectures/Fluidseight.pdf - 2025-07-08

Ships and Aerospace Karl Johan Åström Department of Automatic Control LTH Lund University Ships and Aerospace K. J. Åström 1. Introduction 2. A Little History 3. Sensing and actuation 4. Stability and Manoevrability 5. Autopilots 6. Dynamic Modeling Ships 7. Dynamic Modeling Aircrafts 8. Summary Ships and Aerospace ◮ Cutting edge technology ◮ Technology driver ◮ Driving forces: Emerging technologi

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/PhysicalModeling/Lectures/L06ShipsAndAerospaceeight.pdf - 2025-07-08

A Modeling Methodology 1. Introduction 2. Representation of Models 3. Units 4. Schematic Diagrams 5. A Water Tank 6. Electrical Circuits 7. Summary A Modeling Methodology ◮ Purpose of modeling: understanding, control design, diagnostics, ... ◮ Cut a system into subsystems ◮ Write mass, momentum and energy balances for each subsystem ◮ Discretize partial differential equations ◮ Add constitutive eq

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/PhysicalModeling/Lectures/L3-Methodologyeight.pdf - 2025-07-08

Differential Algebraic Equations Contents: 1. Introduction 2. Differential Algebraic Equations 3. Linear DAE 4. The Notion of Index 5. Numerical Methods 6. Summary Goal: ◮ To develop a basic understanding of differential-algebraic equations Introduction ◮ Cut a system into subsystems ◮ Use object orientation to structure the system ◮ Write mass, momentum and energy balances for each subsystem ◮ As

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/PhysicalModeling/Lectures/L4-DAEeight.pdf - 2025-07-08

Mechanical Systems 1. Introduction 2. Astronomy 3. Newton, Lagrange, Hamilton and Jacobi 4. Pendulum on a Cart 5. Furuta Pendulum 6. Ball and Beam 7. Summary Natural Science and Engineering Science Many similarities but also many differences Natural Phenomena Insight Understanding Analysis Isolation Fundamental Laws Technical Systems Insight Understanding Synthesis Interaction System Principles Fe

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/PhysicalModeling/Lectures/L5-Mechanicaleight_01.pdf - 2025-07-08

Friction Models and Friction Compensation Karl J. Åström Department of Automatic Control Lund University 1. Introduction 2. Friction Models 3. The LuGre Model 4. Effects of Friction on Control Systems 5. Friction Compensation 6. Summary Introduction ◮ Essential in Motion Control ◮ Classics Leonardo da Vinci (1452-1519 Amontons 1699 Coulomb 1785 ◮ Tribology ◮ Control ◮ Physics AFM ◮ Surface force a

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/PhysicalModeling/Lectures/L6-FrictionModelseight.pdf - 2025-07-08

Bicycle Dynamics and Control Karl Johan Åström Department of Automatic Control LTH, Lund University Thanks to Richard Klein and Anders Lennartsson Why Model? ◮ Insight and understanding ◮ Analysis, Simulation, Virtual reality ◮ Design optimization ◮ Control design ◮ Implementation The internal model principle A process model is part of the controlller ◮ Operator training ◮ Hardware in the loop sim

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/PhysicalModeling/Lectures/L9A-Bikeseight.pdf - 2025-07-08

MTK for control Physical modeling in Julia For those about to control Acknowledgement This presentation contains an assortment of content contributed by multiple people ● Chris Rackauckas ● Yingbo Ma ● Probably more, thank you! Outline ● X Differential equations ● Equation-based modeling ○ Symbolics ○ ModelingToolkit (MTK) ○ Tools on top of MTK ● MTK Standard library ● Current status ● Project ide

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Optimal Control of RLCT Networks Circuit Theory Richard Pates Who cares? ...classical theory of passive network synthesis–a beautiful subject that reached its zenith around 1960, only to decline steadily thereafter as an active research interest... –Malcolm Smith Who cares? ...classical theory of passive network synthesis–a beautiful subject that reached its zenith around 1960, only to decline ste

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PowerPoint Presentation Model-Based Policy Learning CS 285: Deep Reinforcement Learning, Decision Making, and Control Sergey Levine Class Notes 1. Homework 3 is out! Due next week • Start early, this one will take a bit longer! 1. Last time: model-based reinforcement learning without policies 2. Today: model-based reinforcement learning of policies • Learning global policies • Learning local polic

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/StudyCircleDeepReinforcementLearning/CS285-Lecture12-ModelBasedPolicyLearning.pdf - 2025-07-08

PowerPoint Presentation Reframing Control as an Inference Problem CS 285: Deep Reinforcement Learning, Decision Making, and Control Sergey Levine Class Notes 1. Homework 3 is out! Due Oct 21 • Start early, this one will take a bit longer! Today’s Lecture 1. Does reinforcement learning and optimal control provide a reasonable model of human behavior? 2. Is there a better explanation? 3. Can we deri

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/StudyCircleDeepReinforcementLearning/CS285-Lecture14-ControlAsInference.pdf - 2025-07-08

PowerPoint Presentation Inverse Reinforcement Learning CS 285: Deep Reinforcement Learning, Decision Making, and Control Sergey Levine Today’s Lecture 1. So far: manually design reward function to define a task 2. What if we want to learn the reward function from observing an expert, and then use reinforcement learning? 3. Apply approximate optimality model from last week, but now learn the reward

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/StudyCircleDeepReinforcementLearning/CS285-Lecture15-InverseReinforcementLearning.pdf - 2025-07-08