Dynamic Optimization Dr Abebe Geletu Ilmenau University of

Dynamic Optimization Dr. Abebe Geletu Ilmenau University of Technology Department of Simulation and Optimal Processes (SOP) Seite 1 www. tu-ilmenau. de/simulation

Chapter 1: Introduction 1. 1 What is a system? "A system is a self-contained entity with interconnected elements, process and parts. A system can be the design of nature or a human invention. " A system is an aggregation of interactive elements. • A system has a clearly defined boundary. Outside this boundary is the environment surrounding the system. • The interaction of the system with its environment is the most vital aspect. • A system responds, changes its behavior, etc. as a result of influences (impulses) from the environment. Seite 2 www. tu-ilmenau. de/simulation

Course Content Chapters 1. Introduction 2. Mathematical Preliminaries 3. Numerical Methods of Differential Equations 4. Modern Methods of Nonlinear Constrained Optimization Problems 5. Direct Methods for Dynamic Optimization Problems 6. Introduction of Model Predictive Control (Optional) References: Seite 3 www. tu-ilmenau. de/simulation

1. 2. Some examples of systems • Water reservoir and distribution network systems • Thermal energy generation and distribution systems • Solar and/or wind-energy generation and distribution systems • Transportation network systems • Communication network systems • Chemical processing systems • Mechanical systems • Electrical systems • Social Systems • Ecological and environmental system • Biological system • Financial system • Planning and budget management system • etc Seite 4 www. tu-ilmenau. de/simulation

Space and Flight Industries Dynamic Processes: • Start up • Landing • Trajectory control Seite 5 www. tu-ilmenau. de/simulation

Dynamic Processes: • Start-up • Chemical reactions • Change of Products • Feed variations • Shutdown Chemical Industries Seite 6 www. tu-ilmenau. de/simulation

Industrial Robot Dynamische Processes: • Positionining • Transportation Seite 7 www. tu-ilmenau. de/simulation

1. 3. Why System Analysis and Control? 1. 3. 1 Purpose of systems analysis: • study how a system behaves under external influences • predict future behavior of a system and make necessary preparations • understand how the components of a system interact among each other • identify important aspects of a system – magnify some while subduing others, etc. Seite 8 www. tu-ilmenau. de/simulation

Strategies for Systems Analysis • System analysis requires system modeling and simulation • A model is a representation or an idealization of a system. • Modeling usually considers some important aspects and processes of a system. • A model for a system can be: • a graphical or pictorial representation • a verbal description • a mathematical formulation indicating the interaction of components of the system Seite 9 www. tu-ilmenau. de/simulation

1. 3. 1 A. Mathematical Models • The mathematical model of a system usually leads to a system of equations describing the nature of the interaction of the system. • These equations are commonly known as governing laws or model equations of the system. • The model equations can be: time independent steady-state model equations time dependent dynamic model equations In this course, we are mainly interested in dynamical systems. Sytems that we evolove with time are known as dynamic systems. Seite 10 www. tu-ilmenau. de/simulation

1. 3. 2. Examples of dynamic models • Linear Differential Equations • Example RLC circuit (Ohm‘s and Kirchhoff‘s Laws) • Seite 11 www. tu-ilmenau. de/simulation

1. 3. 2. Examples of dynamic models • Nonlinear Differential equations Seite 12 www. tu-ilmenau. de/simulation

1. 3. 2. Examples of dynamic models • Nonlinear Differential equations Seite 13 www. tu-ilmenau. de/simulation

1. 3. 1 B Simulation • studies the response of a system under various external influences – input scenarios • for model validation and adjustment – may give hint for parameter estimation • helps identify crucial and influential characterstics (parameters) of a system • helps investigate: instability, chaotic, bifurcation behaviors in a systems dynamic as caused by certain external influences • helps identify parameters that need to be controlled Seite 14 www. tu-ilmenau. de/simulation

1. 3. 1 B. Simulation. . . • In mathematical systems theory, simulation is done by solving the governing equations of the system for various input scenarios. This requires algorithms corresponding to the type of systems model equation. Numerical methods for the solution of systems of equations and differential equations. Seite 15 www. tu-ilmenau. de/simulation

1. 4 Optimization of Dynamic Systems • A system with degrees of freedom can be always manuplated to display certain useful behavior. • Manuplation possibility to control • Control variables are usually systems degrees of freedom. We ask: What is the best control strategy that forces a system display required characterstics, output, follow a trajectory, etc? Optimal Control Seite 16 www. tu-ilmenau. de/simulation Methods of Numerical Optimization

Optimal Control of a space-shuttle 0 1 2 The shuttle has a drive engine for both launching and landing. Initial States: Objective: To land the space vehicle at a given position , say position „ 0“, where it could be brought halted after landing. Target states: Position , Speed What is the optimal strategy to bring the space-shuttle to the desired state with a minimum energy consumption? Seite 17 www. tu-ilmenau. de/simulation

Optimal Control of a space-shuttle 0 1 2 Then Hence Objectives of the optimal control: • Minimization of the error: • Minimization of energy: Seite 18 www. tu-ilmenau. de/simulation Model Equations:

Problem formulation: Performance function: Model (state ) equations: Initial states: Desired final states: How to solve the above optimal control problem in order to achieve the desired goal? That is, how to determine the optimal trajectories that provide a minimum energy consumption that the shuttel can be halted at the desired position? Seite 19 www. tu-ilmenau. de/simulation so

Optimal Operation of a Batch Reactor Seite 20 www. tu-ilmenau. de/simulation

Optimal Operation of a Batch Reactor Some basic operations of a batch reactor • feeding Ingredients • adding chemical catalysts • Raising temprature • Reaction startups • Reactor shutdown Chemical ractions: Initial states: Objective: What is the optimal temperature strategy, during the operation of the reactor, in order to maximize the concentration of komponent B in the final product? Allowed limits on the temperature: Seite 21 www. tu-ilmenau. de/simulation

Mathematical Formulation: Objective of the optimization: Model equations: Process constraints: Initial states: Time interval: This is a nonlinear dynamic optimization problem. Seite 22 www. tu-ilmenau. de/simulation

1. 5 Optimization of Dynmaic Systems General form of a dynamic optimization problem a DAE system Seite 23 www. tu-ilmenau. de/simulation

1. 4. Solution strategies for dynamic optimization problems Solution Strategies Indirect Methods Dynamic Programming Direct Methods Maximum Principle Sequential Method Simultaneous Method State and control discretization Nonlinear Optimization Solution Nonlinear Optimization Algorithms Seite 24 www. tu-ilmenau. de/simulation

Solution strategies for dynamic optimization problems Indirect methods (classical methods) • Calculus of variations ( before the 1950‘s) • Dynamic programming (Bellman, 1953) • The Maximum-Principle (Pontryagin, 1956)1 Lev Pontryagin Direct (or collocation) Methods (since the 1980‘s) • Discretization of the dynamic system • Transformation of the problem into a nonlinear optimization problem • Solution of the problem using optimization algorithms Verfahren Seite 25 www. tu-ilmenau. de/simulation

1. 5. Nonlinear Optimization formulation of dynamic optimization problem • After appropriate renaming of variables we obtain a non-linear programming problem (NLP) Collocation Method Seite 26 www. tu-ilmenau. de/simulation