Dynamic Optimization Dr. Abebe Geletu Winter … · Dynamic Optimization Dr. Abebe Geletu ... a verbal description ... renaming of variables we obtain a non-linear programming problem

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Dynamic Optimization

Dr. Abebe Geletu Winter Semester 2011/2012

Ilmenau University of Technology Department of Simulation and Optimal Processes

(SOP)

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Course Content Topics 1. Introduction 2. Mathematical Preliminaries - review of calculus of several variables, - numerical methods of linear and nonlinear equations 3. Numerical Methods of Differential and Differential Algebraic Equations - Euler methods, Rung Kuttat Methods, Collocation on finite elments 4. Modern Methods of Nonlinear Constrained Optimization Problems - necessary Optimality Conditions (KKT conditions) - the sequential quadratic programming (SQP) method - the interior point method (Optional) 5. Direct Methods for Dynamic Optimization Problems - An overview of the maximum principle - Direct methods – Collocation on finite elements 6. Introduction to Model Predictive Control (Optional) Prerequisites: Programming under MATLAB, (Knowledge of C/C++ is advantageous)


Course Content… References: • J. T. Betts: Practical Methods for Control Using Nonlinear Programming. SIAM 2001. • R. D. Rabinet III, et al. Applied Dynamic Programming for Optimization of Dynamical

Systems. SIAM 2005. • M. Papageorgiou: Optimierung. Oldenburg. 1996. • J. Nocedal, S. J. Wright: Numerical Optimization, Springer 2006. • D. E. Kirk: Optimal Control Theory, McGraw-Hill, 1992. • Chiang: Elements of Dynamic Optimization, McGraw-Hill, 1992. Additional references will be cited for individual topics. Software and Resources • The Matlab ODE Toolbox • The Matlab Optimization Toolbox • The Open Modelica Simulation Environment: http://www.openmodelica.org • General Pseudospectral Optimal Control Software (GPOS): http://www.gpops.org • GAMS

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Chapter 1: Introduction


"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.

1.1 What is a system?

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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


Space and Flight Industries

Dynamic Processes: • Start up • Landing • Trajectory control


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


Industrial Robot

Dynamic Processes:

• Positionining

• Transportation


1.3. Why System Analysis and Control?

• 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.

1.3.1 Purpose of systems analysis:


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


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.

• 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.

• These equations are commonly known as governing laws or model equations of the system.

Sytems that evolove with time are known as dynamic systems.


• Linear Differential Equations • Example RLC circuit (Ohm‘s and Kirchhoff‘s Laws)

•

BuAxx +=•

Examples of Dynamic models – RLC Circuit

vuLBC

LLR

Avi

vi

CC

=

=

−−=

=

= •

••

,0

1 ,

01

1 ,x , x

vLvi

C

LLR

vi

CC

+

−−=

•

•

0

1

01

1

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• Nonlinear Differential equations

Example: Cart mounted inverted one-bar pendulum position of the cart : position of the cart center the angle Nonlinear Model Equations (Using Newton and D‘Lambert‘s Laws)


)),(),(()( ttutxftx =•

Examples of Dynamic Models – Inverted Pendulum

x11, yx

1ϕ

( )2

111

11112

111111

2

11111111

34

sincos

sincos)(

lmI

glmlmIxlm

lmFlmxmm

=

=++

+=++

••••

•••

ϕϕϕ

ϕϕϕ


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


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.


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 to display required characterstics, output, follow a trajectory, etc?

Optimal Control Methods of Numerical Optimization


Optimal Control of a space-shuttle

0 1 2 Force Propulsive:)(

Speed:)(Position:)(

2

1

tutxtx

kg) 1( Mass: =mm

Initial States: m/s1)0( m,2)0( 21 == xx

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 halted after landing. Target states: Position , Speed 01 =Sx 02 =SxWhat is the optimal strategy to bring the space-shuttle to the desired state with a minimum energy consumption?


Optimal Control of a space-shuttle

0 1 2

Force Propulsive:)(Speed:)(Position:)(

2

1

tutxtx

kg) 1( Mass: =mmModel Equations:

)()()()()(

2

21

txmtamtutxtx

===

Then

)(1)(

)()(

2

21

tum

tx

txtx

=

=

uxx

xx

+

=

10

0010

2

1

2

1

Hence BuAxx +=Objectives of the optimal control:

• Minimization of the error: )();( 2211 txxtxx SS −−•Minimization of energy: )(tu


Problem formulation:

Performance function: [ ] [ ] [ ]{ }∫∞

+−+−0

2222

211)(

)()(2)(21min dttutxxtxx SS

tu

Model (state ) equations: uxx

xx

+

=

10

0010

2

1

2

1

Initial states:

0;0

1)0(;2)0(

21

21

==

==SS xxxx

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?

)(),( *2

*1 txtx )(* tu


Optimal Operation of a Batch Reactor


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: CBAorder1st order 2nd→→

Initial states: 0)0(,0)0(mol/l,1)0( === CBA CCC

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: KTK 398298 ≤≤


Mathematical Formulation: Objective of the optimization: )(max

)( fBtTtC

BC

BAB

AA

CTkdt

dC

CTkCTkdt

dC

CTkdt

dC

)(

)()(

)(

2

22

1

21

=

−=

−=

−=

−=

RTEkTk

RTEkTk

2202

1101

exp)(

exp)(

KTK 398298 ≤≤

0)0(,0)0(mol/l,1)0( === CBA CCC

ftt ≤≤0

Model equations:

Process constraints: Initial states: Time interval: This is a nonlinear dynamic optimization problem.


1.5 Optimization of Dynmaic Systems

( )( )( )

0 0

1

2

min max

0 f

min J(x, u)

x(t) f x(t), u(t) , x(t ) x

g x(t), u(t) 0

g x(t), u(t) 0u u ut t t .

with•

= =

=

≤

≤ ≤≤ ≤

a DAE system

General form of a dynamic optimization problem

DynOpt


1.6. Solution strategies for dynamic optimization problems

Solution Strategies

Indirect Methods Direct Methods

Dynamic Programming

Maximum Principle

Simultaneous Method Sequential

Method

State and control discretization

Nonlinear Optimization

Solution Nonlinear Optimization Algorithms


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 resulting problem using optimization algorithms


1.7. Nonlinear Optimization formulation of dynamic optimization problem

u

min max

min f (x,u)

withF(x,u) 0G(x,u) 0u u u .

=≤

≤ ≤

• After discretization of DynOpt and appropriate renaming of variables we obtain a non-linear programming problem (NLP)

Dynamic Optimization Dr. Abebe Geletu Winter … · Dynamic Optimization Dr. Abebe Geletu ... a verbal description ... renaming of variables we obtain a non-linear programming problem

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