Overview on Mathematical Modeling

Overview onMathematical Modeling

Prof. Ing. Michele MICCIO

•Dept. of Industrial Engineering (Università di Salerno)

•Prodal Scarl (Fisciano)

Rev. 6 of May 25, 2016

Models are used not only in the natural sciences(such as physics, biology, earth science,meteorology) and engineering/architecturedisciplines, but also in the social sciences (such aseconomics, psychology, sociology and politicalscience).

Here is a list:

 Physical Models

 Analogic Models

 Provisional Theories(e.g., molecular and atomic models)

 Maps and Drawings(e.g., PI&D, geographycal maps, etc.)

 Mathematical and symbolic models

TYPES OF MODELS

 § 1.4 in Himmelblau D.M. e Bischoff K.B., “Process Analysis andSimulation”, Wiley & Sons Inc., 1967

MATHEMATICAL MODELINGDefinitions

A mathematical model is a representation, inmathematical terms, of certain aspects of anon-mathematical system.

Aris, 1999

A mathematical model is a set of mathematicalequations that are intended to capture theeffect of certain system variables on certainother system variables.

Graham C. Goodwin, Stefan F.Graebe, Mario E. Salgado,Control System Design, Ch. 3,Prentice Hall

A model may be prescriptive or illustrative,but, above all, it must be useful !

Wilson, 1991

MATHEMATICAL MODELINGDefinitions

A mathematical model is a description of asystem using mathematical concepts andlanguage. The process of developing amathematical model is termed mathematicalmodelling. Mathematical models are used notonly in the natural sciences (such as physics,biology, earth science, meteorology) andengineering disciplines (e.g. computer science,artificial intelligence), but also in the social sciences (such as economics, psychology,sociology and political science); physicists,engineers, statisticians, operations researchanalysts and economists use mathematicalmodels most extensively.

…

A mathematical model usually describes asystem by a set of variables and a set ofequations that establish relationships betweenthe variables.

WHY MATH MODELINGFOR PROCESS SYSTEMS?

•Understand the problem: Why doesone need a model?

•Is it:

to design a controller?

to analyze the performance of theprocess?

to understand the process better?

to simplify the complexity of a system

etc.

Introduction to Process ControlRomagnoli & Palazoglu

“First principle Models”

 Basic models

 Models involving trasport phenomena

 Models based on the “populationbalance approach”

Empirical or “fitting” models

Dynamic models

•dynamic models with “input-outputrappresentation”

•state-space dynamic models

•black box dynamic models

Time Series

Statistical models

(on the base ofthe approach adopted for model development)

MATHEMATICAL MODELS1st CLASSIFICATION

Deterministic Models

 rielaborated from § 1.4 in Himmelblau D.M. e Bischoff K.B.,“Process Analysis and Simulation”, Wiley & Sons Inc., 1967

First principles Models

orFundamental Models( Palazoglu & Romagnoli, 2005)orMechanistic or White Box Models( Roffel & Betlem, 2006)

Modelli basati sul “bilancio di popolazione”

Modelli empirici o di “fitting”

Modelli dinamici

•Modelli dinamici “con rappresentazioneingresso-uscita”

•Modelli dinamici “con rappresentazione nellospazio di stato”

•Modelli dinamici a scatola nera (black box)

Serie temporali

Modelli statistici

MATHEMATICAL MODELS1st CLASSIFICATION

(on the base ofthe approach adopted for model development)

GENERAL CONSERVATION LAW

It is the most well known example of aFundamental Principle:

Input - Output + Net Generation = Accumulation

NB: GEN > 0  formation

GEN < 0  disappearance

SIMPLER CASES

For a chemical element:

IN - OUT = ACC

At steady-state:

IN - OUT + GEN = 0

At steady-state and without chemical/biochemicalreactions:

IN - OUT = 0

FIRST PRINCIPLES MODELS

The description of such a system andthe development of such a modelrequire the concept of:

• control volume

• system boundary

 This holds also for the case ofpopulation balance models

“First principles Models”

 Basic models

 Models involving transportphenomena

 Models based on the “populationbalance approach”

Empirical or “fitting” models

Dynamic models

•dynamic models with “input-outputrepresentation”

•state-space dynamic models

•black box dynamic models

Time Series

Statistical models

MATHEMATICAL MODELS1st CLASSIFICATION

 rielaborated from § 1.4 in Himmelblau D.M. e BischoffK.B., “Process Analysis and Simulation”, Wiley & Sons Inc., 1967

(on the base ofthe approach adopted for model development)

FIRST PRINCIPLES MATHEMATICALMODELS.Malthusian Growth Model

An Early and Very Famous Population Model

In 1798 the Englishman Thomas R. Malthus posited amathematical model of population growth. He model, thoughsimple, has become a basis for most future modeling ofbiological populations. His essay, "An Essay on the Principle ofPopulation," contains an excellent discussion of the caveats ofmathematical modeling and should be required reading for allserious students of the discipline. Malthus's observation wasthat, unchecked by environmental or social constraints, itappeared that human populations doubled every twenty-fiveyears, regardless of the initial population size. Said anotherway, he posited that populations increased by a fixed proportionover a given period of time and that, absent constraints, thisproportion was not affected by the size of the population.

By way of example, according to Malthus, if a population of 100individuals increased to a population 135 individuals over thecourse of, say, five years, then a population of 1000 individualswould increase to 1350 individuals over the same period oftime.

Malthus's model is an example of a model with one variableand one parameter. A variable is the quantity we are interestedin observing. They usually change over time. Parameters arequantities which are known to the modeler before the model isconstructed. Often they are constants, although it is possible fora parameter to change over time. In the Malthusian model thevariable is the population and the parameter is the populationgrowth rate.

Example No. 1Malthusian Growth Model

PARAMETERS

STATE VARIABLES

CLASSIFICATION

DYNAMIC AUTONOMOUS MATHEMATICAL MODEL

made by 1 ODE (Order =1) ,

LINEAR, CONSTANT COEFFICIENT MODEL

GENERAL CONSERVATION LAWapplied to No. of individuals in a “closedsystem”:ACC = GEN

GEN = birth - death

INITIAL CONDITION:

Example No. 2Logistic Model

Differently from the Malthusian model …

•developed by Belgian mathematician Pierre Verhulst(1838)

HYPOTHESES

•resources are limited for growth: N(t)≤Nsat

GENERAL CONSERVATION LAWapplied to No. of individuals in a “closedsystem”:ACC = GEN GEN = birth - death

PARAMETERS

STATE VARIABLES

CLASSIFICATION

Dynamic autonomous mathematical model

made by 1 ODE (Order =1) ,

non linear, constant coefficient model

Example No. 2Logistic Model

•resources are limited for growth: N(t)≤Nsat

CLOSED-FORM SOLUTION

N0≠0

Nsat

“First principle Models”

 Basic models

 Models involving transportphenomena

 Models based on the “populationbalance approach”

Empirical or “fitting” models

Dynamic models

•dynamic models with “input-outputrappresentation”

•state-space dynamic models

•black box dynamic models

Time Series

Statistical models

MATHEMATICAL MODELS1st CLASSIFICATION

 rielaborated from § 1.4 in Himmelblau D.M. e Bischoff K.B.,“Process Analysis and Simulation”, Wiley & Sons Inc., 1967

(on the base ofthe approach adopted for model development)

TRANSPORT PHENOMENA

Examples:

•Newton law of viscosity

•Fourier law of heat conduction

•1st Fick law of diffusion

General Law

TRANSPORT AND KINETIC RATEEQUATIONS

Momentum, Heat and Mass transfers

Heat

Mass

Momentum

Flux

τxy

Diffusion transfer (microscopic scale)

Law

Fourier’s law

1st Fick’s law

Newton’s law

Driving force

(gradient)

dT/dx

dCA/dx

dvy/dx

Keyparameter

(diffusioncoefficient)

(Thermalconductivity)

DAB(Diffusivity)

(Viscosity)

Convection transfer (macroscopic scale)

Driving force

∆T

∆CA

∆P

VARIABILI

x: ascissa, m

t: tempo, s

u: velocità, m/s

C: concentrazione del reagente, mol/m3

D: diffusività assiale del gas, m2/s

k: cost. reazione 2° ordine, m3/moli/s

Example No. 3The Tubular Reactor

An example of a model involving transport phenomena.

HYPOTHESES

•Isothermal reactor

•2nd order homogeneous chemical reaction

A → P

•Flat velocityprofile

•Axial diffusion only

E' un'equazione differenziale alle derivateparziali (PDE) quasi-lineare, parabolica delsecondo ordine,integrabile per assegnate condizioni iniziali econdizioni al contorno

Example No. 3The Tubular Reactor

(DIFFERENTIAL) MASS BALANCE

ACC = IN – OUT + GEN

initial condition

boundary conditions

Example No. 3The Tubular Reactor

x = 0

x = L

Boundary and initial conditions

“First principle Models”

 Basic models

 Models involving transportphenomena

 Models based on the “populationbalance approach”

Empirical or “fitting” models

Dynamic models

•dynamic models with “input-outputrappresentation”

•state-space dynamic models

•black box dynamic models

Time Series

Statistical models

MATHEMATICAL MODELS1st CLASSIFICATION

 rielaborated from § 1.4 in Himmelblau D.M. e Bischoff K.B.,“Process Analysis and Simulation”, Wiley & Sons Inc., 1967

(on the base ofthe approach adopted for model development)

MODELS BASED ON A “POPULATIONBALANCE APPROACH”

Formulation

The Population Balance Equation wasoriginally derived in 1964, when two groups ofresearchers studying crystal nucleation andgrowth recognized that many problemsinvolving change in particulate systems couldnot be handled within the framework of theconventional conservation equations only, seeHulburt & Katz (1964) and Randolph (1964).

They proposed the use of an equation for thecontinuity of particulate numbers, termedpopulation balance equation, as a basis fordescribing the behavior of such systems.

This balance is developed from the generalconservation equation:

Input - Output + Net Generation = Accumulation

 § 4.4 in Himmelblau D.M. and Bischoff K.B., “Process Analysis and Simulation”,John Wiley & Sons Inc., 1967 (Collocazione: 660.281 HIM 1)

ELEMENTS NECESSARYFOR A POPULATION BALANCE MODEL

ENTITY DISTRIBUTION FUNCTION

Physical meaning

is the fraction of entities at time t that are:

•contained in the infinitesimal volumedV=dxdydz

•characterized by values of properties in theranges ζ1÷ζ1+dζ1, . . . , ζm÷ζm+dζm

MATHEMATICAL MODELS2nd CLASSIFICATION

Structured Models (or Theoretical orWhite Box Models)

Unstructured Models (or Empirical orBlack Box Models)

Hybrid Models (or Gray Box Models)

(on the base of an approach typical ofprocess engineering)

 Ogunnaike B.A. and Ray W.H., “ProcessDynamics, Modeling and Control”, Oxford Univ.Press, 1994

 Romagnoli & Palazoglu, "Introduction toProcess Control"

STRUCTURED MODELS

Characteristics:

Mathematical equations describing the process system canbe:

a) Conservation equations e.g. total mass, mass ofindividual chemical species, total energy, momentumand number of particles.

b) Rate equations e.g. transport rate and reaction rateequation.

c) Equilibrium equation e.g. reaction and phaseequilibrium.

d)Equations of state e.g. ideal gas law

e)Other constitutive relationships, e.g., control valveflow equation, PID control law, etc.

 Advantages:

They provide fundamental and general knowledge of theprocess

 Disadvantages:

They require a good and reliable understanding of thephysical and chemical phenomena that underlie theprocess

Accuracy of models depends on the assumptions andmathematical ability of the person constructing themodel.

white box

Model

UNSTRUCTURED MODELS

Characteristics:

They rely on measured or observed data availability:

a) Vary input variables and then measure the value of theoutput variables.

b) Perform enough measurements or observations(experiments)

The equations describing a process system in this way consistin mathematical relationships among the variablesrepresenting the available data (fitting).

Such models that rely solely on empirical information aretermed as black-box models.

Example: A black-box model simulating the internalcombustion engine operation and implemented into thecontrol box of a modern car

 Advantages:

a) No or very limited assumptions required.

b) Just focus on the range of process operating conditions.

 Disadvantages:

a) They require an actual operating process and may becomevery time-consuming and costly

b) They don’t allow extrapolation, generally

c) They never give the engineer a fundamental understandingof the process.

black box

Model

where:

y(t) is the measured process variable

u(t) is the controller output signal

The important parameters that result are:

–Steady State Process Gain, KP

–Overall Process Time Constant, P

–Apparent Dead Time, P or td

 The FOPDT model is low order and linear:so it can only approximate the behavior of real processes

Example 4FOPDT model

adapted from:

D. Cooper, "Practical Process Control using Loop-ProSoftware", PDF textbook

An example of Black-Box Dynamic Model

Modeling means fitting a first order plus dead time(FOPDT) dynamic process model to the data set:

STRUCTURED AND UNSTRUCTUREDMODELS:A comparison

Structured Modelstheory-rich and data-poor

Unstructured Modelsdata-rich and theory-poor

“HYBRID” MODELS

•Hybrid Model is developed byincorporating empirical knowledge intothe fundamental understanding of theprocess.

Such models blending fundamental andempirical knowledge are referred to asgray-box models.

Example

An example can be the use of mass andenergy balances to develop a reactor modeland the rate of the reaction will be based onexpressions obtained from laboratoryexperiments.

Introduction to Process ControlRomagnoli & Palazoglu

gray box

Model

“First principle Models”

 Basic models

 Models involving transportphenomena

 Models based on the “populationbalance approach”

Empirical or “fitting” models

Dynamic models

•dynamic models with “input-outputrappresentation”

•state-space dynamic models

•black box dynamic models

Time Series

Statistical models

MATHEMATICAL MODELS1st CLASSIFICATION

 rielaborated from § 1.4 in Himmelblau D.M. e Bischoff K.B.,“Process Analysis and Simulation”, Wiley & Sons Inc., 1967

(on the base ofthe approach adopted for model development)

EMPIRICAL OR “FITTING”MODELS

Mathematical models obtained by curve fitting

Curve fitting is the process of constructing a curve, ormathematical function, that has the best fit to a series ofdata points, possibly subject to constraints.

Curve fitting can involve either interpolation, where anexact fit to the data is required, or smoothing, in which a"smooth" function is constructed that approximately fitsthe data.

A related topic is regression analysis, which focusesmore on questions of statistical inference such as howmuch uncertainty is present in a curve that is fit to dataobserved with random errors.

Fitted curves can be used as an aid for datavisualization, to infer values of a function where no dataare available, and to summarize the relationshipsamong two or more variables.

Extrapolation refers to the use of a fitted curve beyondthe range of the observed data, and is subject to agreater degree of uncertainty since it may reflect themethod used to construct the curve as much as itreflects the observed data.

EMPIRICAL OR “FITTING”MODELS

The interpolation problem

The smoothing or

regression problem

“First principle Models”

 Basic models

 Models involving transportphenomena

 Models based on the “populationbalance approach”

Empirical or “fitting” models

Dynamic models

•input-output dynamic models

•input-state-output dynamic modelsor state-space models

•dynamic black box models

Time Series

Statistical models

MATHEMATICAL MODELS1st CLASSIFICATION

 rielaborated from § 1.4 in Himmelblau D.M. e Bischoff K.B.,“Process Analysis and Simulation”, Wiley & Sons Inc., 1967

(on the base ofthe approach adopted for model development)

DYNAMIC MODELS:definitions

•A dynamical system is a state space S , a set of times Tand a rule R for evolution, R:S×T→S that gives theconsequent(s) to a state s∈S .

A dynamical system consists of an abstract phase space orstate space, whose coordinates describe the state at anyinstant, and a dynamical rule that specifies the immediatefuture of all state variables, given only the present values ofthose same state variables.

For example the state of a pendulum is its angle and angularvelocity, and the evolution rule is Newton's equation F = ma.

•A dynamical mathematical model can be considered to bea tool describing the temporal evolution of an actualdynamical system.

Dynamical systems are deterministic if there is a uniqueconsequent to every state, or stochastic or random if thereis a probability distribution of possible consequents (theidealized coin toss has two consequents with equalprobability for each initial state).

A dynamical system can have discrete or continuous time.

DYNAMIC MODELS:definitions

Order of a dynamical system

It is defined as the order of the dynamicalmathematical model representing the system

Order of a single differential equation

It is defined as the order of the highestderivative in the differential eq.

Ex.: n-th order ODE:

Order of a system of differential equations

It is defined as the sum of the orders of theindividual differential eqs.

Mathematical Models3rd classification



Ch.3 of Himmelblau D.M. and Bischoff K.B., “Process Analysis and Simulation”,Wiley, 1967

http://www.csd.newcastle.edu.au//chapters/chapter3.html

on the base of

•mathematical features of equations and variables,

•also affecting type or easiness of solution

Dynamical models

LINEAR LUMPEDDETERMINISTIC MODELS

 Romagnoli & Palazoglu, “Introduction to Process Control”

“The theory of control is well developed forlinear, deterministic, lumped-parametermodels.”

Nonlinear

Distributed

Stochastic

Linear

Lumped

Deterministic

Simple!! But notnecessarily the best…

“First principle Models”

 Basic models

 Models involving transportphenomena

 Models based on the “populationbalance approach”

Empirical or “fitting” models

Dynamic models

•input-output dynamic models

•input-state-output dynamic modelsor state-space models

•dynamic black box models

Time Series

Statistical models

MATHEMATICAL MODELS1st CLASSIFICATION

 rielaborated from § 1.4 in Himmelblau D.M. e Bischoff K.B.,“Process Analysis and Simulation”, Wiley & Sons Inc., 1967

(on the base ofthe approach adopted for model development)

DYNAMIC MODELS:input-output representation

MATHEMATICALMODEL

Input output

 typical of automatic process control

 Roffel & Betlem, 2006

 Giua &Seatzu, “Analisidei sistemidinamici”, 2006

•An input-output equation reveals the direct relationshipbetween the output variable desired, and the input variable.

•This relationship often includes the derivatives of one orboth variables.

•Even with a higher order equation, the advantage is thatthe output variable is independent of (uncoupled from) anyother variables except the inputs.

INPUT-OUTPUT REPRESENTATIONVariables

Basic Automatic Controlprof. Guariso

“First principle Models”

 Basic models

 Models involving transportphenomena

 Models based on the “populationbalance approach”

Empirical or “fitting” models

Dynamic models

•input-output dynamic models

•input-state-output dynamic modelsor state-space models

•dynamic black box models

Time Series

Statistical models

MATHEMATICAL MODELS1st CLASSIFICATION

 rielaborated from § 1.4 in Himmelblau D.M. e Bischoff K.B.,“Process Analysis and Simulation”, Wiley & Sons Inc., 1967

(on the base ofthe approach adopted for model development)

DYNAMIC MODELS:input-state-output representation

MATHEMATICALMODEL

Input state output

 Giua &Seatzu,“Analisi deisistemidinamici”,2006

Behavioral models

 “internal” model of the process

 Roffel & Betlem, 2006

Intuitively, the state of a system describes enoughabout the system to determine its future behavior.

Example 5WATER OPEN TANK

h(t)

IN :

STATE: h(t)

OUT :

VARIABLES

Initial Condition: h(0) = h0

State equation:

Output transformation:

ACC = IN – OUT

 see alsoEx. 6.1 pag. 118 and Ex. 6101 pag. 175in Stephanopoulos

An example of input-state-output dynamic model

Input-state-output model

Basic Automatic Controlprof. Guariso

Such a “memory”, required to define the present-time condition of the system, is called the state of thesystem at the time instant when the input is applied.

The variables x(t) defining the state are called statevariables.

The state of a dynamic system is such that theknowledge of the state variables at t = t0, together withthe knowledge of the input variables for t  t0,completely determines the behavior of the system forany time t  t0.

State Variables

•Quantities describing the present state of a systemenough well to determine its future behavior.

•They provide a means to keep memory of theconsequences of the past inputs to the system.

•They may be partly coincident with output variables.

•The minimum number of state variables required torepresent a given system, n, is usually equal to theorder of the system's defining differential equation.

•The choices for variables to include inthe state:

–is highly dependent on the fidelity of themodel and the type of system

–is not unique

–as a HINT, the state variables are often givenby the same variables appearing within theinitial conditions of the system’s ODEs

Basic Automatic Controlprof. Guariso

http://www.cds.caltech.edu/~murray/amwiki/index.php/FAQ:_What_is_a_state%3F_How_doe s_one_determine_what_is_a_state_and_what_is_not%3F

State Variables

Basic Automatic Controlprof. Guariso

STATE-SPACE MODELS

Basic Automatic Controlprof. Guariso

where:

•The space spanned x(t) n is the state-space orphase-space

•nN is the order or size of the model

•The n components of x(t) are termed phases

•The function f:+ ·n ·m →n is termed vectorfield

Linear State-space Models.Terminology

A is the state matrix [n·n]

B is the input matrix [n·m]

C is the ??? matrix [p·n]

D is the ??? matrix [p·m]

v is the time vector [n·1]

Initial Condition:

State equation:

Output

transformation:

A·x(t)

[n·n][n·1] = [n·1]

B·u(t)

[n·m][m·1] = [n·1]

v·t

[n·1][1·1] = [n·1]

“First principle Models”

 Basic models

 Models involving transportphenomena

 Models based on the “populationbalance approach”

Empirical or “fitting” models

Dynamic models

•input-output dynamic models

•state-space autonomous dynamicmodels

•dynamic black box models

Time Series

Statistical models

MATHEMATICAL MODELS1st CLASSIFICATION

 rielaborated from § 1.4 in Himmelblau D.M. e Bischoff K.B.,“Process Analysis and Simulation”, Wiley & Sons Inc., 1967

(on the base ofthe approach adopted for model development)

Autonomous System(definition based onthe input-state-output representation)

A dynamical system is autonomous ifthe state not depending on input

the input is constant with time.

adapted from “notecontrolli_PRATTICHIZZO.pdf”

NB:

For a dynamic model described by asystem of ODEs, an autonomous systemis also time invariant

Autonomous System(definition based onthe shape of the vector field)

A dynamical system is autonomous ifits vector field depends on the state x(t),but does not explicitly depend on time.

Examples of 2nd order systems:

Autonomous system Non-autonomous system

withx(t)n

t +

ORBITS OR TRAJECTORIES OF A DYNAMIC SYSTEM

The forward orbit or trajectory of a statevariable x is the time-ordered collection of statesthat x follow from an initial state x0 using thesystem evolution rule.

•When both state space and time are continuous,the forward orbit is a curve x(t) parametric in t≥0

•Different initial states result in differenttrajectories

Example for a system of order n=2

x0=[x10, x20]

PHASE PORTRAIT

The set of of all trajectories plotted in thestate space forms the phase portrait of adynamical system.

•It is parametric in time

•in practice, only representative trajectories areconsidered, indicating the forward orientation byan arrow

•the phase portrait is usually built by numericalmethods for nonlinear systems

•The analysis of phase portraits provides anextremely useful way for visualizing andunderstanding qualitative features of solutions.

•The phase portrait is particolarly useful for scalar(1st order) and planar systems (2nd order)

http://www.mathgrapher.com/images/phase01.gif

TRANSIENTS AND REGIMES

•The orbits may have a bounded orunbounded asymptotic behavior (for t ± ).

•If an orbit has a bounded asymptoticbehavior for t  + , the part of theorbit describing such an asymptoticbehavior is referred to as regime andthe remaining part is referred to astransient.

•The simplest and most well knownregime (bounded) is the equilibriumpoint (steady-state point), which ispossible for every system order n

REGIMES

STEADY STATE

Example:

Stable equilibrium point for a 2nd orderdynamical system

Orbit

Phase portrait

Equilibriumpoint

Time chart

Time trajectory

DYNAMIC REGIME

Example:

Stable periodic regime for a 2nd order dynamicalsystem

Limit cycle

Phase portrait

Time chart

LIMIT CYCLE

Definition:

A closed asymptotic orbit in the phase space thatresults in a periodic oscillation of the statevariables (phases), with amplitude and frequencydepending on the properties of the non-lineardynamic system only and not on the chosen initialconditions.

 in order to have a Limit Cycle as an asymptoticorbit, the non-linear dynamic (continuous time)system must be of order 2 at least.

 the Limit Cycle cannot exist as an asymptoticorbit for linear dynamic system.

LIMIT CYCLE

n = 2

n = 3

2nd ORDER, NON-LINEAR,AUTONOMOUS SYSTEM

dx1/dt = f1(x1(t), x2(t))

dx2/dt = f2(x1(t), x2(t))

1st IC: x1(t0)=x10; x2(t0)=x20

2nd IC: x1(t0)=x11; x2(t0)=x21

Example of Phase Portrait

x0=[x10, x20]

x00=[x11, x21]

THE DIABATIC CSTRExample No. 6

 Diabatic CSTR by Prof. B.W. Bequette:

http://www.rpi.edu/dept/chem-eng/WWW/faculty/bequette/education/links_mods/cstr/



According to the 1st classification,

It is a “first principles” math model

An example of 2nd order, non-linear,autonomous system

AUTONOMOUS SYSTEM

SOLUTION OR BIFURCATIONDIAGRAM

A solution diagram is a diagram in whichthe asymptotic regimes (at least the simplestof them) of a single state variable arereported as a function of a single parameter.

Therefore this diagram allows to representthe possible regimes for each parametervalue.

Furthermore, it also shows the bifurcationvalues: in fact, they correspond to a changein type, number or stability of the asymptoticsolutions.

MATHEMATICAL MODELS4th CLASSIFICATION

(based on the type of description adoptedfor time as the independent variable)

 VALID for DYNAMICAL SYSTEMS only

1. continuous time models

2. discrete time models

CONTINUOUS-TIME SYSTEMS

The time changes continuously:

it is not possible to define a minimumtime interval

it is always possible to consider asmaller time interval.

Accordingly, the theory of continuous-time systems has been defined andstudied, where the variables of thesystem are continuous-time functions,that is, at every time instant t, it ispossible to define and assign thevariable value.

Continuous-time systems: t  ℜ

Ingresso uscita

Input output

DISCRETE-TIME SYSTEMS

input – output representation

MATHEMATICALMODEL

input – state – output representation

MATHEMATICALMODEL

Ingresso stato uscita

Input state output

DISCRETE-TIME SYSTEMS

EXPLICIT TYPE

“IMPLICIT” TYPE

General representation

of input–state–output discrete-time models

SCALAR SYSTEMS:x(k) is a scalar variable

VECTOR SYSTEMS:x(k) is a vector variable of size n

 A deterministic system with discrete time isoften defined a map.

MODEL ORDER

The order n is an integer number obtained asthe product between the size of the spacevector x and the time gap in the implicitmodel representation.

 The time gap is the max time distance inthe implicit model representation

Example 1:

autonomous discrete-time system of order 1 (scalarmodel)

Example 2:

autonomous discrete-time system of order 2 (vectorialmodel)

FIXED POINTSfor autonomous systems x(k+1) = f(x(k))

DEFINITION of a fixed point:

x(k+1) = x(k)

 x(k) may be a scalar or vector variable

The fixed points are determined as thesolutions of the algebraic eq. (scalar) orsystem of eqns.(vector):

x*= f (x*)

A fixed point can be stable or unstable.

A stable fixed point is an attractor forthe trajectories originating in areasonable neighborhood of it.

PERIODIC REGIMES

x(k+1) = f (x(k))

x(k+j)=x(k)

x(k+q)≠x(k)

with q N and q < j

periodical regimes with period “j”

in an autonomous discrete-time model

x(k) =x(k+1) = f (x(k))

 The fixed point can be viewed as aperiodical regime with period “1”

k0 … … … k k+1 … k+q … k+j … … … k+2j …

ORBITS OR TRAJECTORIES

Example:

discrete-time, non-linear, autonomous model of order 2

fixedpoint

periodicregime

Phase Portrait

LOGISTIC MODEL

•a natural saturation level Nsat exists for thepopulation at which the growth rate becomes zero

Search for Equilibrium Points

LOGISTIC MAP

adimen-

siona-lization

discreti-zation

LOGISTIC MAP WITH 1 PARAMETER

x(k+1) = r [1 - x(k)] x(k)

x(k0) = x0

0 ≤ x(k) ≤ 1

x(k+1)→y x(k)→x

Parabola that is

•passing through the origin

•having a vertical axis

•having a downward concavity

•having vertex coordinates: (-b/2a; -Δ/4a)

Diagram of Generalized Eq. y = f(x)

r = growth rate parameter

LOGISTIC MAP:Analytical determination of fixed points

x* = r x* (1 - x*)

 

1st solution2nd solution

x* = 0 x* ≠ 0

1 = r (1 - x*)

x* = 1 – 1/r

 x* only for r>1

x(k+1)

x(k)

fixed points x*= f (x*) by intersection

LOGISTIC MAP:Graphical determination of fixed pointsExample

Logistic map.: x(k+1)=3.55 x(k)(1-x(k))

Bisector of 1st quadrant: x(k+1) = x(k) locus offixed points

COBWEB or STAIRCASECONSTRUCTIONExample

From Wikipedia, the free encyclopedia:http://en.wikipedia.org/wiki/Cobweb_plot

•logistic map: x(k+1)= r x(k)(1-x(k))

•bisector of 1st quadrant: x(k+1) = x(k)

•initial condition: x0 = 0.08

•r ≈ 2.6 (estimated)

THE LOGISTIC MAP IN MATLAB

Useful scripts:

cobweb.m

logistic_time_trajectory.m

logistic_solution_diagram.m

1)r=0.75;

2)r=1.5;

3)r=2.5;

4)r=3.2;

5)r=3.5;

6)r=3.55;

7)r=3.8;

8)r=3.83;

9)r=3.9

 try the script cobweb.m

with the following values for r:

LOGISTIC MAP

r=0.75x0=0.5

Cobweb

vertex r/4

stable fixed point (“trivial solution”: x = 0)

LOGISTIC MAP

r=1.5x0=0.05

Cobweb

vertex r/4

stable fixed point (1 – 1/r)

LOGISTIC MAP

r=2.5x0=0.05

Cobweb

vertex r/4

stable fixed point (1 – 1/r)

LOGISTIC MAP

Time Trajectories for r = 2.5

1) x0=0.05 (in blu)

2) x0=0.06 (in red)

LOGISTIC MAP

r=3.2x0=0.05

Cobweb (vertex r/4)

Time Trajectory

unstable fixed point (1 – 1/r)

period 2 dynamical regime: x= 0.513, 0.799

LOGISTIC MAP

r=3.5x0=0.05

Cobweb (vertex r/4)

unstable fixed point (1 – 1/r)

period 4 dynamical regime

Time Trajectory

LOGISTIC MAP

unstable fixed point (1 – 1/r)

period 8 dynamical regime

r=3.55x0=0.36

Cobweb (vertex r/4)

Time Trajectory

LOGISTIC MAP:switch to deterministic chaos

Cobweb: x0=0.05100 steps

r=3.569945673

r=3.569945671

LOGISTIC MAP

r=3.8 x0=0.17

Cobweb (vertex r/4)

chaotic solution(aperiodic dynamical regime)

Time Trajectory

LOGISTIC MAP:Deterministic caos

Time Trajectories for r=3.8:

1) x0=0.17 (in blu)

2) x0=0.18 (in red)

 sensible dependence from initialconditions

LOGISTIC MAP

r=3.83 x0=0.33

Cobweb (vertex r/4)

period 3

$H:\Documenti condivisi\work...MatLab\esercitazioni TSPC 2009-2010\logistic map\trajectory_r=3.83_x0=0.95_step=200.jpg$

LOGISTIC MAP

period 3

x0=0.95

Time Trajectories

r=3.83

x0=0.33

LOGISTIC MAP

r=3.9x0=0.05

Cobweb (vertex r/4)

aperiodic Regime

Time Trajectory

LOGISTIC MAP

Cobweb

x0=0.2

r =1  4

From Wikipedia, the free encyclopedia:http://en.wikipedia.org/wiki/File:LogisticCobwebChaos.gif

LOGISTIC MAP:remarks on Fixed points

•for 0 < r < 1 one fixed point x*=0 exists, i.e., theorigin, and is stable

•for 1 < r < 3 two fixed points exist:one is unstable (x*=0) and the other one x*=(r-1)/ris stable

•for r > 3 the fixed point becomes unstable, but anew stable periodic regime arises with a period 2k andk ε N (period doubling cascade)

•the periodic regime holds up to a particular valuerc, = 3.569945672.., beyond which the map seems tooscillate among an infinite number of points withoutany regular law: deterministic chaos

•for certain values rc < r < 4 there are still periodicwindows



a stable fixed point is referred to as attractor

a stable periodic regime is referred to as periodicattractor

LOGISTIC MAP.Solution diagram

x0=0.05

x0=0.3

LOGISTIC MAP:regime patterns

The table shows the type of cycles that can be observed and the valueof the growth parameter r for which that cycle appears:

Notice that the period doubling happens faster and faster (8, 16, 32, ...)and then suddenly ends after the so-called accumulation point andchaos appears.

If we call rn+1 the value of r for which the 2n period becomes unstable andif we determine the limit, as n goes to infinity, of ( rn+1 - rn ) / ( rn+2 - rn+1 )we obtain a value of 4.66920160...

Surprisingly, that limit value is the same for all functions of this type ( 1Dmaps with only one quadratic maximum) which approach chaos byperiod doubling.

In fact, that value is one of the universal constants - the Feigenbaumconstant - approximately equal to 4.669. It was discovered byFeigenbaum in 1975 and it characterizes the transition to chaos.

3.449490

3.544090

3.564407

3.568750

3.56969

128

3.56989

256

3.569934

512

3.569943

1024

3.5699451

2048

3.569945557

infinity

accumulation point

3.569945672

from http://to-campos.planetaclix.pt/fractal/caose.html

Modelli a “principi primi” (“First principle Models”)

Modelli basati sul “bilancio di popolazione”

Modelli empirici o di “fitting”

Dynamic models

•input-output dynamic models

•input-state-output dynamic models orstate-space models

•dynamic black box models

Serie temporali

Modelli statistici

MATHEMATICAL MODELS1st CLASSIFICATION

(on the base ofthe approach adopted for model development)

The model structures vary in complexity and order dependingon the flexibility you need to account for the dynamics and onpossible consideration of noise in your system.

The simplest black-box structures are:

1.Linear polynomial model, which is the simplest input-outputmodel (e.g., ARX model)

2.Transfer function, with a given number of adjustable polesand zeros.

3.State-space model, with unknown system matrices, whichyou can estimate by specifying the number of model states

4.Non-linear parameterized functions

$y(t)+ {\alpha_1} y(t-1)+ ...+ {\alpha_n} y(t-n) = {\beta_0} u(t)+ {\beta_1} u(t-1)+...+ {\beta_n} u(t-n) \,$

Black-Box Model Structures

Input output

BLACK BOX

MODEL

SYSTEM IDENTIFICATION

System identification is a methodology for buildingmathematical models of dynamic systems usingmeasurements of the system's input and output signals/data.

The system identification methodology requires that you:

1. Measure the input and output signals/data from yoursystem in time or frequency domain.

2. Select a model structure.

3. Apply an estimation method to estimate value for theadjustable parameters in the candidate model structure.

4. Evaluate the estimated model to see if the model isadequate for your application needs.

•System identification uses the input and output signals youmeasure from a system to estimate the values of adjustableparameters in a given model structure.

•Obtaining a good model of your system depends on how wellyour measured data reflect the behavior of the system.

Documentation center

Dynamical modelingFinal Statement

100

A way to distinguish the techniques usedwhen modeling dynamic processes is topicture them on a scale with two extremes atthe ends:

•at the one sideblack boxsystem identification

•at the other sidetheoretical modeling orfirst principles modeling

MATHEMATICAL MODELS5th CLASSIFICATION

1. ANALYTICAL SOLUTIONS DIFFERENTIAL CALCULUS

 ALGEBRA

2. NUMERICAL SOLUTIONS NUMERICAL CALCULUS

Ex.: numerical methods for PDEs

1.FINITE DIFFERENCE METHODS(FDM)

2.FINITE ELEMENT METHODS (FEM)

3.COLLOCATION METHODS

101

(based on the type of mathematical solution,

with exclusion of empirical models)

Modelli a “principi primi” (“First principleModels”)

Modelli basati sul “bilancio dipopolazione”

Modelli empirici o di “fitting”

Modelli dinamici

•Modelli dinamici “con rappresentazioneingresso-uscita”

•Modelli dinamici “con rappresentazionenello spazio di stato”

•Modelli dinamici a scatola nera (blackbox)

Time Series

Modelli statistici

102

MATHEMATICAL MODELS1st CLASSIFICATION

(on the base ofthe approach adopted for model development)

TIME SERIES:Definitions

A time series is a sequence of data points,measured typically at successive timesspaced at uniform time intervals.

Y1, Y2,…, a series of values measured orobserved for a variable Y at successivetimes t1, t2,… which may be reported as asequence of integers k = 0, 1, 2, . . .

Anonymous

Values taken by a variable over time (suchas daily sales revenue, weekly orders,monthly overheads, yearly income) andtabulated or plotted as chronologicallyordered numbers or data points.

http://www.businessdictionary.com/

 time series are NOT just data …

103

TIME SERIES:Examples

•the daily closing value of the DowJones index

•the annual flow volume of the NileRiver at Aswan

•the sampled pressure measurementsin a gas-phase chemical reactor

. . .

104