1. Dynamic Optimization Dr
Dynamic Optimization Dr. Abebe Geletu Ilmenau University of Technology Department of Simulation and Optimal Processes (SOP)

2. 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.

3. Course Content Chapters Introduction Mathematical Preliminaries
Numerical Methods of Differential Equations
Modern Methods of Nonlinear Constrained Optimization Problems
Direct Methods for Dynamic Optimization Problems
Introduction of Model Predictive Control (Optional)
References:

4. 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

5. Space and Flight Industries
Dynamic Processes:
Start up
Landing
Trajectory control

6. Chemical Industries Dynamic Processes: Start-up Chemical reactions
Change of Products
Feed variations
Shutdown
Chemical Industries

7. Industrial Robot Dynamische Processes: Positionining Transportation

8. 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.

9. Strategies for Systems Analysis
System analysis requires system modeling and simulation 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

10. 1.3.1A. 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.

11. 1.3.2. Examples of dynamic models
Linear Differential Equations
Example RLC circuit (Ohm‘s and Kirchhoff‘s Laws)

12. 1.3.2. Examples of dynamic models
Nonlinear Differential equations

13. 1.3.2. Examples of dynamic models
Nonlinear Differential equations

14. 1.3.1B 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

15. 1.3.1B. 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.

16. 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?
Methods of Numerical Optimization
Optimal Control

17. Optimal Control of a space-shuttle1 2
Initial States:
The shuttle has a drive engine for
both launching and landing.
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?

18. Optimal Control of a space-shuttle1 2
Model Equations:
Then
Hence
Objectives of the optimal control:
Minimization of the error:
Minimization of energy:

19. 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 so that the shuttel can be halted at the desired position?

20. Optimal Operation of a Batch Reactor

21. 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:

22. Mathematical Formulation:
Objective of the optimization:
Model equations:
Process constraints:
Initial states:
Time interval:
This is a nonlinear dynamic optimization problem.

23. 1.5 Optimization of Dynmaic Systems
General form of a dynamic optimization problem
a DAE system

24. State and control discretization
1.4. Solution strategies for dynamic optimization problems
Solution Strategies
Indirect Methods
Direct Methods
Dynamic
Programming
Sequential
Method
Maximum Principle
Simultaneous Method
State and control discretization
Nonlinear Optimization
Solution Nonlinear Optimization Algorithms

25. 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

26. 1.5. Nonlinear Optimization formulation of dynamic optimization problem
After appropriate renaming of variables we obtain a non-linear programming problem (NLP)
Collocation Method