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History of Real Time Systems

History of Real Time Systems History of Real Time Systems Gautham Department of Automatic Control, Lund University 1/14 Gautham: History of Real Time Systems Overview Introduction 1940s 1950s 1960s RTOS A look at RTSS Cloud. The future? 2/14 Gautham: History of Real Time Systems Real Time Systems I Real Time Systems describes hardware and software systems subject to a ”real-time constraint”, for e

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/HistoryOfControl/2016/hoc_Gautham.pdf - 2025-06-29

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1 Lecture 1: Introduction (Karl Johan) 2 Lecture 2: Calculus of variation (CoV) and the Maximum principle In this lecture, we are going to learn the maximum principle. The MP is a type of CoV, so we will first study the classical theory of CoV. Then we will try to move from the classical CoV theory to the optimal control setting, there we will immediately encounter some essential difficulties that

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2023/Lec2.pdf - 2025-06-29

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Optimal Control Introduction Karl Johan Åström Department of Automatic Control LTH Lund University Optimal Control K. J. Åström 1. Introduction 2. Calculus of Variations 3. Optimal Control 4. Computations 5. Stochastic Optimal Control 6. Conclusions Theme: Subspecialities A Brief History Early beginning: Bernoulli, Newton, Euler, Lagrange The Golden Era 1930-39: Department of Mathematics at Univer

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2023/OptimalControlIntroeight.pdf - 2025-06-29

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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 - 2025-06-29

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Exercise for Optimal control – Week 2 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 (Insect control). Let w(t) and r(t) denote

https://www.control.lth.se/fileadmin/control/Education/DoctorateProgram/Optimal_Control/2023/ex2-sol.pdf - 2025-06-29

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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 - 2025-06-29

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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 - 2025-06-29

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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 - 2025-06-29

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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 - 2025-06-29

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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 - 2025-06-29