Dynamic Optimization
Dr. Abebe Geletu
Ilmenau University of Technology
Department of Simulation and Optimal Processes
(SOP)
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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.
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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:
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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
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Space and Flight Industries
Dynamic Processes:
• Start up
• Landing
• Trajectory control
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Dynamic Processes:
• Start-up
• Chemical reactions
• Change of Products
• Feed variations
• Shutdown
Chemical Industries
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Industrial Robot
Dynamische Processes:
• Positionining
• Transportation
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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.
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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
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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.
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1.3.2. Examples of dynamic models
• Linear Differential Equations

x  Ax  Bu
• Example RLC circuit (Ohm‘s and Kirchhoff‘s Laws)
    R
 1  i   1 
i
L
L    L v
 
v   
v   1
0

 C  C
 C   0 
•
 i 
x   ,
 vC 
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1 
 R
  

1 
i
L
L


, B   L , u  v
x   , A
 0 
1
v 
0


 
 C
 C


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1.3.2. Examples of dynamic models
• Nonlinear Differential equations

x(t )  f ( x(t ), u (t ), t )
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1.3.2. Examples of dynamic models
• Nonlinear Differential equations

x(t )  f ( x(t ), u (t ), t )
5i1  15(i1  i3 )  220V
R(i2  i3 )  5i2  10i2  0
20i3  R(i3  i2 )  15(i3  i1 )  0
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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
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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.
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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
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Methods of
Numerical
Optimization
Optimal Control of a space-shuttle
x1 (t ) : Position
x2 (t ) : Speed
u (t ) : PropulsiveForce
0
1
2
The shuttle has a drive engine for
both launching and landing.
m : Mass (m  1 kg)
Initial States:
x1 (0)  2 m, x2 (0)  1 m/s
Objective: To land the space vehicle at a given position , say
position „0“, where it could be brought halted after landing.
Target states: Position x S  0 , Speed x S  0
1
2
What is the optimal strategy to bring the space-shuttle to the
desired state with a minimum energy consumption?
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Optimal Control of a space-shuttle
x1 (t ) : Position
x2 (t ) : Speed
0
1
u (t ) : PropulsiveForce
m : Mass (m  1 kg)
2
Model Equations:
x1 (t )  x2 (t )
Then
Hence
u (t )  m a(t )  m x2 (t )
 x1  0 1  x1  0
 x   0 0  x   1 u
  2  
 2 
x1 (t )  x2 (t )
x  Ax  Bu
x2 (t ) 
Objectives of the optimal control:
• Minimization of the error:
•Minimization of energy:
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x1S  x1 (t ); x2S  x2 (t )
u(t )
www.tu-ilmenau.de/simulation
1
u (t )
m
Problem formulation:
 

Performance function:
1
min
x1S  x1 (t )
u (t ) 2
0

2

 2 x2S  x2 (t )
Model (state ) equations:
 x1  0 1  x1  0
 x   0 0  x   1 u
  2  
 2 
Initial states:
x1 (0)  2; x2 (0)  1
Desired final states:
x1S  0; x2S  0

2

 u(t ) dt
2
How to solve the above optimal control problem in order to achieve
the desired goal? That is, how to determine the optimal trajectories
x1* (t ), x2* (t ) that provide a minimum energy consumption u * (t ) so
that the shuttel can be halted at the desired position?
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Optimal Operation of a Batch Reactor
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Optimal Operation of a Batch Reactor
Tsoll
Some basic operations of a
batch reactor
TC
• feeding Ingredients
• adding chemical catalysts
• Raising temprature
• Reaction startups
• Reactor shutdown
Chemical ractions:
2nd order
A  B
1st order
 C
CA (0)  1 mol/l,CB (0)  0, CC (0)  0
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: 298 K  T  398 K
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Mathematical Formulation:
Objective of the optimization:
Model equations:
max C B (t f )
T (t )
dCA
 k1 (T )C A2
dt
dCB
 k1 (T )C A2  k 2 (T )C B
dt
dCC
 k 2 (T )CB
dt
 E 
k1 (T )  k10 exp  1 
 RT 
 E 
k 2 (T )  k 20 exp  2 
 RT 
Process constraints:
Initial states:
298 K  T  398 K
CA (0)  1 mol/l,CB (0)  0, CC (0)  0
0  t  tf
Time interval:
This is a nonlinear dynamic optimization problem.
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1.5 Optimization of Dynmaic Systems
General form of a dynamic optimization problem
min J ( x, u , p )
wit h

x  f ( x, u , p ), x(t0 )  x0
g1 ( x, u , p )  0
g 2 ( x, u , p )  0
u min  u  u max
t0  t  t f .
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a DAE system
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
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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
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1.5. Nonlinear Optimization formulation of
dynamic optimization problem
• After appropriate renaming of variables we obtain a
non-linear programming problem (NLP)
min E ( X , U , p )
U
with
FCollocation
( X , U ,Method
p)  0
G ( X ,U , p)  0
u min  U  u max .
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