Mechatronics notes

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Mechatronic Systems 2024 2 3 Index. NON-LINEAR PROGRAMMING. 7 Introduction to NLP . 7 Unconstrained NLP . 7 Equality constrained NLP. 8 Equality and inequality constrained NLP . 9 Karush-Kuhn-Tucker (KKT) Conditions. 10 Convex problem. 12 CALCULUS OF VARIATIONS. 14 First order necessary condition for optimality: Euler-Lagrange equation. 14 Statement. 14 Proof. 15 Why is it a first order condition if it contains second order derivative terms? 18 Why is it a local condition? 19 Hamilton’s interpretation of Lagrangian mechanics. 19 Second order necessary condition for optimality: Legendre equation. 20 DYNAMIC PROGRAMMING. 22 Introduction to the Dynamic Optimization Problem. 22 DiXerence between Optimal Control Problem and Calculus of Variations. 22 First order necessary conditions: case of fixed final time 𝒕𝟏 and free final point 𝒙𝒕𝟏. 22 The conditions in other terms: Euler-Lagrange equations. 29 Hamilton’s Principle. 30 First order necessary conditions: case of free final time 𝒕𝟏 and free final point 𝒙𝒕𝟏. 32 How to deal with constraints. 44 Equality constraint on the final state: terminal constraint. 44 Equality constraint throughout the journey of the system. 50 Inequality constraint throughout the journey of the system. 52 Numerical methods. 55 Steepest-descent algorithm. 55 Cart on a track: double integrator plant. 56 PONTRYAGIN MAXIMUM PRINCIPLE. 61 Introduction to Pontryagin Maximum Principle (PMP). 61 DiMerence with Dynamic Optimization Programming. 61 The principle: a family of theorems. 62 Case of free end-time, fixed end-point problem. 62 Example of the double integrator plant. 63 Case of fixed end-time, free end-point problem. 68 DiXerences between conditions stated in free-end time, fixed end-point and fixed-end time, free end-point scenarios. 69 BELLMAN PRINCIPLE OF…