Taips Book Ch1

Preface

And having an equally venerable age theory or Processing Systems

Signals can be easily ascertained by studying its history the identification

Systems emerged and developed around with Automatica Therefore it

naturally become a fundamental subject of this area along with

disciplines mentioned above which is related When referring to

concepts such as process System regulator state or signal (first 4 being

Specific Automatic Control) discussion is based on the concept of model (mathematical)

Mathematical modeling entities on a suitable implementation means

Automatic calculation is a first and important objective of Automatic Control Default

mathematical modeling calls for construction of an assembly

consistent and rigorous ways entity considered to be identified with

the most appropriate mathematical model best adapted to the characteristics

its core In this type of problem has occurred and to

developed (and modeling) Identification Systems as interdisciplinary science

This book aims to describe the theoretical and practical aspects of modeling and

System Identification As a work of applied mathematics in technical (ie

in Automatic) which addresses the reader must have a knowledge base

located at least the first two years of training in a faculty of electrical profile

However the book was conceived in the spirit of the phrase self-contained ie

Presentation of gradual starting from the simplest terms by the complicated

without making numerous references to references nested and often difficult

available Whenever it was necessary the reader is reminded concepts

it needs to understand the arguments developed in chapters work

Of the four constituent chapters aims to introduce the first

discipline specific terminology is peppered with historical notes Identification

Systems is a discipline developed in a referential format of 3

major coordinates mathematical model associated entity to be identified

signal that it can be stimulated to provide data on behavior

its evolution and its intrinsic or the method that is determined model

especially math They constitute the nucleus of the 3 main chapters of the book

Each chapter concludes with a collection of exercises to be solved whose

purpose is to facilitate understanding

It is possible that some readers find too mathematized presentation especially

the end of the last chapter although the book contains numerous references to practical

discipline For those who prefer more practical purpose methods or

the algorithm rather than rigorous argumentation for construction and

their accuracy there is a paper alternative [SMS04]

Once learned the fundamentals of modeling and System Identification is expected

It can provide a basis for understanding other disciplines Automation

For example two of the subjects that has Systems Modelling and Identification

Setting a direct impact are automatic and automatic management of processes

We hope that the study material in the book to be as interesting and useful numbers

more readers

Chapter 1

Overview

The first chapter of this book is an opening

introduction to the terminology and problematic areas

Identification Systems (IS) Starting from the concepts of process

and system is specified fundamendală problem while being

highlighted the three coordinates that define the IS models

Mathematical they operate in practice the input signals

they stimulate the processes and methods of providing data

which is expected parameters selected models

11 The subject

Identification Systems (IS) is a discipline whose object of study is

process modeling dynamic systems using experimental data acquired

During their operation In this context modeling and construction means

determining a mathematical model associated entity usually progressive with a

momentum The entity is seen as a black box about whose structure

Internal no details A model is a mathematical relationship abstract

accurately describe certain characteristics and or dynamic operation of a

entities Mathematical models in the IS (also called identification models) are

used in a systemic manner More specifically the identification model reflects a relationship

of entry that stimulates a particular entity (called process or system) and

output encoding appropriate reaction to that entity Mathematical models

which operates in the IS are mainly based on the concepts of equation

Differential (for continuous-time systems development) and equation with differences

(for discrete-time systems development) However the models based on other

concepts are also used but more for the purpose of qualitative descriptions

behavior of the process to be identified Building models

Identification is based on experimental data provided by the black box (see

Figure 1)

The need to identify entities with unknown internal structure appears

numerous applications among which we mention only a few simulation in order

highlighting key characteristics and or behavior in various situations

forms recognition signal processing prediction forecast diagnosis of

defective design of automated systems management or control etc

There are two categories of identification techniques analytical and experimental In

Analytical identification aims to determine the physical parameters of a process

used for this purpose physicochemical laws at its base (ie the balance equations

mass energy equation of dynamic equilibrium etc) Experimental identification is

objective determination of general parameters without physical meaning but capable

describes the process behavior around a specific operating point

Compared with mathematical models obtained by writing equations balance

Results of expression laws of physics experimental models show identification

following characteristics

bull have a generality and validity limited to certain classes of processes signals

stimulus and even operating only at certain points of the same process

bull they have a physical interpretation difficult to give because in most cases

parameters have clear physical meanings parameters are rather used as

tools to facilitate description of the operation pocesului

bull Their determination is often made using algorithmic what they

gives efficiency and simplicity

In this book the discussion will focus on modeling and identification issues

Experimental trials A (short) refers to the identification of physical parameters

less complex processes can be found in [SMS04]

12 Historical Notes

IS Field was outlined in particular with the publications of KJ Aringstroumlm and

P Eykhoff in the 70s and 80s [AsEy71] [EyP74] [EyP81] In parallel there may be mentioned

important contributions to the development of research and outlining some directions

through his publications RL Kashyap and AR Rao [KaRa76] RK Mehra and D G Lainiois

[MeLa76] GC Goodwin and RL Payne [GoPa77] or T Soumlderstroumlm [SoT84]

Applications Identification and parameter estimation techniques (involving and modeling

mathematics) were not slow to appear They are described in a number of conferences

IFAC dedicated to IS and parameter estimation techniques such as those from Prague

(1967 1970) The Hague (1973) Tbilisi (1976) Darmstadt (1979) Washington DC (1982)

Numerous papers and overview synthesis were published especially

Automatic journals published by IFAC committee [IFAC80] [IFAC82] But one of the

more complete characterization of the domain was published in [SoSt89] - probably the

most cited reference in a decade They followed [LjGl94] and [LjL99] - two references

oriented identification algorithms

In Romania during the most prolific in terms of publications in the field of IS (years

70 -80) Does not go unnoticed Thus it can be said that the Romanian school

Identifications were initiated mainly through the works of C Penescu M and P Stoica Tertişco

[PITC71] [TeSt80] [TeSt85] [TSP87] An extremely practical vision related to IS (in

Automatic control systems context) was published by ID Landau

[LaID93] (in French) a book that was translated into Romanian [LaID97] A

Another paper dealing with practical aspects of the IS range is [SMS04]

Undoubtedly this brief history may not include vast panoply contributions

which led to diversification and enrichment IS domain Today the IS continues

in particular through the development of applications requiring openness to interdisciplinary approaches

The fast algorithms and techniques for identifying unconventional started

to appear since the early 90s through interaction with other fields

research especially with Signal Processing (PS) Artificial Intelligence (AI) and

Evolutionary Programming (EP)

13 Coordinates IS domain

Studying IS domain lies in the very concept of modeling

math Numerous applications Automation and or Computer Science

uses mathematical models In many cases the processes studied are so

complex that it is not possible to characterize them by describing phenomena

Physical underlying their behavior ie using the principles and laws of physics

expressed by balance equations Often it equations obtained in this way contain a

large number of unknown parameters In such cases the user is bound

circumstances to seek to identify models and experimental techniques

The specific work of IS is structured around three fundamental concepts

the mathematical model of the stimulus signal and the method of identification We refer on

In summary as to each of them They are described in detail in

The following three chapters of the book

Mathematical models are nonparametric and parametric The models

nonparametric are mainly used to obtain descriptions priori (preliminary)

more qualitative the process to be identified In this case the data

acquired are regarded as statistical data on the evolution dynamics of the process

Relatively simple statistical methods (generally based on technical (self-) correlation) are

applied to the data to obtain models both time domain and frequency as well

These models are described through charts or tables but without resorting to

Parameter concept They are useful in analyzing the process from different perspectives In

basically 4 types of analyzes can be performed transient analysis analysis

frequency analysis based on auto-correlation and spectral analysis

Parametric models most used applications in class ARMAX

(Auto-regressive moving average with exogenous control) As described in

Chapter 2 general class ARMAX model shows that the signal output

obtained as a result of the superposition of a useful signal obtained by filtering the signal

input and a signal obtained by filtering white noise parasite Particular cases

the most used applications are used ARX AR MA and ARMA The first model is

Typical applications optimal numerical control (or automatic control) while

last 3 are used particularly for modeling and predicting time series (less

specifically their stochastic component) A series of time (or a time stochastic process

discreet) is seen as an embodiment of a process driven by white noise

The choice of stimulus signals is based on a general principle if the

complex is integrated into a larger system - that works in closed loop -

The stimulus signal is then used during operation if the process can

function in open loop then a more accurate mathematical model is obtained by

stimulation of a persistent signal The concept of persistence is crucial

IS and will be described in detail in Chapter 3

The input signal is ideal white noise which has infinite persistence From

Unfortunately this signal can not be generated artificially More specifically the signals

artifacts (ie artificially produced) can not persist indefinitely order There are

artifacts signals with finite order of persistence approximate white noise in

auto-covariance purposes (and possibly the probability distribution) These

call pseudo-random binary signals (SPAB) or simply signed

Pseudo-Random (SPA) They are periodic as the algorithms used for

their generation using finite representational accuracy of the numerical values on a

Automatic calculation means Interestingly though their persistence is proportional order

period Moreover as the period increases they are closer to white noise

ie their values are becoming more correlated

Using SPAB or SPA IS is very frequent whenever the

identified open loop can be stimulated The models obtained using these

signals have high precision and are very versatile and can be used for a wide

range of operating points the stimulus signal and or configuration of the system

Finally identification methods aim to determine the parameters

naive model suggests either direct relationship computing or iterative procedures

However lack not only parameter values but their numbers

It entails adopting a strategy where the structural complexity iterative

model is gradually increased to the extent that its accuracy is not

significantly improved Specifically starting with the simplest model ie

parsimonios1 For each model given structure determine its parameters and

evaluate the error to the process (using a predefined criterion) If error

decreases significantly re-iterative process that is increase the number of

parameters then reassess them and error to the process Otherwise the process

iterative stalled and retain top model determined This model must be

finally validated using specific tests For example a model is valid if the error

between measured and simulated data has the characteristics of a Gaussian white noise

Determination of the unknown parameters of a mathematical model can be achieved in

mainly using methods drawn from optimization theory and or estimate Theory

(Statistics) A quick but discarded objects on these methods would put the

show their advantages and disadvantages Thus the methods of optimization algorithms provide

iterative (implemented) parameter estimation but estimates can not be

characterized statistically They provide the point of convergence

optimal but does not guarantee consistency in terms of statistical estimate (In this

context a consistent estimate of a parameter that tends to value

true that parameter as the number of data acquired from the process

tends to infinity whatever the data set used) From the other perspective Theory

Estimates consistency parameters can be tested but actual methods

neimplementabilitate assessment suffers generally rather a

theoretical support for other methods In addition these methods are based on assumptions

often restrictive in order to ensure consistency Fortunately the two theories

intersect The methods for identifying the most interesting and useful are the results

the combination of optimizing estimate They are implemented (possibly iterative) and

allow statistical characterization of the estimated parameters The prototype method you constiuie

Least squares (LSM) which will be presented in detail in Chapter 4

LSM is a kind of mother method which arose many

Other methods of identification adaptations inspired by the type of model used (Method

Instrumental Variables for ARX models Prediction Error Minimization Method

ARMAX models predictions Optimal Method for AR models etc) Although

However the IS is not limited to the family of methods generated by LSM

There are for example methods for the estimation of states in the Kalman filter or the method of

Robust identification The overwhelming majority of these methods are described in

[SoSt89] most of which are described in this volume

IS methodology but can not be a panacea but has its limits More

it should be used with caution and science The most important practical problems

appear in the identification process are listed below

bull Selection of sizes and can be measured There are situations where

sizes of paramount importance for the identification process can not be

measured directly inaccessible For example if it is desired

determining a model of the vibrations of a bearing integrated in a system

A mechanical fault detection its view it is very possible that sensors

vibration (accelerometers) can not be placed directly on the housing

bearing Their location to other locations may result in the combination and or

measuring the signal interfering with the vibration signals produced by other

mechanical system components so the possible mathematical models

inadequate To solve this difficult problem the user has

to formulate an alternative identification problem to extract information

studied the process of measured data in the context in which it is

Place this if known interactions between different subsystems

the global system Returning to the example of the bearing a method of extraction

desired vibration sensors is the use of directional orientation

bearing and use a method to mitigate interference (Such

The method has been patented in the US for example [CaDL96]) The cost of such

But solutions can be high so that the user will be restricted means

financial capabilities

bull Primary data acquisition and processing Trials identified

characterized by datasets acquired for the report semnalzgomot

(SNR - Signal-to-noise ratio) is reasonably high values Other

words the noise should not dominate useful data The more SNR is lower

so the model associated with risk be imprecise process and the process is more

less identifiable Increased SNR (signal dominance that is useful to the

Noise) can be achieved to some extent by processing

The primary data It consists mainly of a technical mitigation

noise (denoising) based filtering The user is faced here

Inadequate data distortion problem by choosing a filter or method

noise mitigation Unfortunately between useful and spurious data is not

can draw a clear line so that whatever method

Primary processing used some useful data likely to be eliminated while

What part of spurious data likely to be interpreted as useful data For

emphasizing the difference between useful data and parasitic methods are needed

Sophisticated processing undesirably complicating algorithm

Identification Consequently the user must carefully design the experiment

data acquisition so SNR equals big enough

bull Selecting a suitable mathematical model This can be a difficult problem

especially when the user is faced with a process having a strong

nonlinear behavior Identifying common models are linear One way

course is to address non-linearities in the selection of nonlinear models with

provided that non-linearities can be characterized in terms

mathematically Another approach would be based on the adaptation and implementation of a

Neural networks (The theory of neural networks [DuHa96] [TaI97] is

LSM is all that optimization technique during training network) Third

strategy closer to the IS is to use linear models

but with time-varying parameters that systematically adapts itself according

The acquired data Finally the process can be non-linear behavior

multi-shaped and the technique used Thus an appropriate model is chosen

a stored pre-defined depending on the current operating point

bull variability of processes over time This feature simply result in

that the true values of the parameters vary over time Thus it requires

use of mathematical models with time-varying parameters (as in the case

nonlinearities) The main problem now arises is related to

consistency estimates This time the parameter estimates should not

only tend statistically (ie with increasing horizon extent) to

their true values but to pursue precisely the time evolution

large enough The two requirements are obviously opposite so that

the main objective of the method of identification used (which can only be

Iterative) is to provide a good compromise between tracking ability

estimates and precision Another compromise to be made is that between

adaptability and robustness mathematical model as dynamic system It is

well-known that excessive lead to loss of robustness adaptability

systems (ie their ability to reject certain performance

containing shocks and disturbances to remain stable) In turn robustness

Excessive lead to poor tracking performance (ie adaptability)

Although brief presentation of this section of the discussion focused on

IS domain should be noted however that some of the techniques of identification may be

employed and to signal processing Especially if the process

I can not study highlights the input signals information about its evolution

is encoded in the output data set which is a time series Series models

frames are commonly used in spectral estimation [OpSc85] [OpWi85] [PrMa96]

prediction [TeSt85] [StD9603] [SMS04] or adaptive filtering [HaS86] - standard applications

PS rather than IS But between IS and PS can not draw a clear line

dividing at their intersection While undergoing some modern and effective methods and techniques

goals that serve both areas

14 Determinism and nondeterminism

The black box that we want to build a mathematical model must be

the ability to provide data and can potentially be stimulated for it In

Automation such entities operate with the input signals and provides

Output signals are generally referred systems (dynamic) A dynamical system

It is described by a set of differential equations (continuous-time) or the difference (in

discrete time) as

Also in the equations (1) I denoted by x any derivatives x1048581 (t) (continuous time

tisinR) or x [n + 1] (discrete time normalized nisinR) By convention brackets

Round indicates continuous time and the right - discrete time All sizes listed

in the equation (1) depends on time Figure 2 illustrates the usual representation of a system

dynamic Observe group separate from the internal quantities outside the system

Although most entities that can be represented using dynamic systems

are inherently nonlinear linear models are often preferred linear correspondent

a system of type (1) may be expressed as

where AisinN times n times m BisinN CisinRp times n times n DisinRq EisinRn times l times r FisinRp GisinRq times m

HisinRp times m matrix coefficients are constant or variable over time

The systems described by equations (2) is the subject of Theory

Linear Systems (TSL) [IoV85] where stability properties observability

controllability and robustness are extensively analyzed Within TSL faults v and w

play a secondary role being used primarily in the capacity of a system to study them

rejection offset or retain intrinsic stability whatever their nature Of

Therefore mathematical models of the TS (L) have a deterministic character despite

the presence of disturbances In other words at any point in time the quantities that describe

unique mathematical model values determined by that point of time whether

known or not

In reality however faults have a deterministic character and should be considered

as random variables (stochastic) are characterized by certain distributions

probability This means that at a certain time values

disturbances are not uniquely determined but varies in a certain range (range)

each having a certain probability of occurrence Therefore other

sizes that are trouble-deterministic inherit their character

becoming stochastic The study deterministic systems is affected by disturbances

conducted within the IS (though not the only area concerned about this

problematic) from a perspective different from that of TS (L) determining the parameters and

their characteristics In the case of linear systems (2) the parameters are coefficients

matrix H

To emphasize the distinction between behavior systems

deterministic and non-deterministic behavior we take the example of a

I continuous linear system of order whose ideal transfer function is

isinR where K is the gain and Tgt 0 is the time constant which is

system parameters In the absence of disturbances the two parameters can be easily determined

about graphics using step response of the system (ie the step response

unit) as shown in Figure 3 Thus the amplification of the asymptote is given by

horizontal and the time constant is found assesses the tangent at abscissa

origin of the graph intersects the horizontal asymptote

above is found rarely The curve may look more like Figure 4 Such

variation is called and development (system) Moreover repeating the experiment

system stimulation with the exact same type input stage and measurement unit

output can drive each time to a different deployment of previous achievements

as shown in Figure 5 This behavior is due to deterministic disturbances which

corrupt system output It can be seen from Figure 5 it is virtually impossible

determined pair of parameters that characterize the system operation more

namely (K T) using the same technique in the case of deterministic system

The set of all achievements system is called deterministic process

(stochastic) Given the nature of the data deterministically output image

Figure 5 should be complemented with information about the probability distribution of

perturbations that have produced this effect Often preliminary knowledge

the probability distribution associated with a stochastic process is however very difficult if

not impossible Fortunately a large class of stochastic processes are or may be

considered normally distributed ie with probability density

where y is the average and σ2 is the variance of the random variable y (t) This density

by Carl Gauss was introduced (1777-1855) its shape illustrated in Figure 6 the

like a bell (Gauss bell) with the opening of the σ measured

For having the probability distribution (4) further writes yisinN (y σ2)

(ie y belongs to the class of normal distributed processes mean and variance σ2 y)

In Class N (y σ2) falls often probability distribution processes

Unknown based on the Central Limit Theorem (TLC) of Statistică2

Distribution normal Gaussian will often adopted developments in this

book but applied systems discrete processes In this case definition (4) changes

only the argument temporal (t) will be replaced with (NTS) or more simply with

[n] where Ts is sampling period (default unit) Note that the argument

The main probability distribution is not time but the values of the random variable

that the output y in the example chosen

Obviously because p is a probability distribution check identity

int (()) () = 1

+ infin

-infin

p y t dy t foralltisinR (5)

what comes to property cert probability that the value of y to

It is included in R (ie the interval (-infin + infin)) is uniform at any time

(Area under the bell of Gauss is always uniform)

Another interesting property is as follows the probability that the value of y to

be included in the range (y - 3σ y + 3σ) (appointed ad hoc and 3σ interval) is at

less than 95 at any time

()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ

int p ge foralltisinR (6)

The area under the bell of Gauss is almost above the range 3σ

equal to the entire area This implies that the values of y will belong close

3σ safe range It is noted that this range is controlled by the values of σ

called standard deviation The range looks and how precisely is determined y located

The higher the standard deviation is higher much weaker y is located on the real axis

(Gauss bell is dilated) and therefore less precise Deviations

standard (or variant) small lead to high accuracy (Gauss bell being

tablet) Basically it is essential Gauss bell located above 3σ range

Finally a third interesting property which can be demonstrated as follows

(Y (t)) y (t) dy (t) y

+ infin

-infin

int p = foralltisinR (7)

indicating that the statistical mean value of y is even y at any time

time Of course media y may depend in turn time even if this

addiction does not appear explicitly in the definition (4)

Property (7) induces that t is most likely value or value

the expected y This means that a finite sequence of experiments

measurement of y y value at a certain time has the highest frequency

great appearance

In the case of systems processes discrete equations (5) (6) and (7) during continuous

It should be replaced with discrete time (normalized) But they show the same properties

We conclude this section by stressing again that the distinction between

System concepts deterministic (that is operating in the TS (L)) and the

deterministic or stochastic process system (that is operating within IS) Often

dynamic system is considered an idealization process data provider being

also it is used as generator of the data if necessary

Starting with the next section the discussion will focus on identifying patterns

having an input and an output (SISO - Single Input Single Output) Other types of

models (SIMO - Single Input Multiple Output MISO - Multiple Input Single Output

MIMO - Multiple Input Multiple Output) are not practical subject of this book

15 The general issues in IS

The main objective of the experiment IS is that starting from a process

P stochastic structure and behavior unknown to build and

determine a mathematical model M which matches well the process in a sense

defined Generally adequate trial model is tested using criteria

a priori defined The criteria can fall into 3 main categories

a empirical criteria (quasi-statistical)

b criteria optimization

c estimation criteria

Empirical criteria based on some elementary statistics and are

often used to describe non-parametric models These models are used to

The qualitative description of the often rough imprecise process In

next chapter will describe some identification techniques (called analysis) using

nonparametric empirical models and criteria on which to evaluate accuracy

thereof

More interesting however are appropriate criteria for parametric models In this

case the mathematical model is described by a number of unknown parameters

to be determined Note that not only parameter values are not known but

their number In the IS the unknown parameters are packed in a vector denoted by θ

size (still unknown) nθ (As in the TS the IS working with vectors

column) adopted fundamental assumption in this case is

H The process works like a mathematical model with real parameters

unknown values deterministic (possibly time-varying) whose

vector is denoted by θ and has nθ size

In this context there are two formulations of the fundamental problem identification

adequacy criterion used by nature the formulation of Optimization Theory

(TO) and the formulation of the Estimation Theory (Statistics) (TE)

151 The problem from the perspective IS TOO

Within TOO problem identification is a formulation that relies on image

in Figure 7

Thus both the P (θ ) and the M (θ) are stimulated with the same input

u which is the collection of data measured in a 1 N [] = U = As horizon

It is normally finished for example of length N isinN The process provides output data

measured in N y n 1 [] = Y = The mathematical model provides simulated data

n N yn 1 [] = θ YM M which depend on the parameters determined from the data vector

measured For the same set of measured data U Y one can obtain a collection of

vectors of unknown parameters (both as a value and as dimensions) and therefore sets

simulated data It was noted by ε [n θ] error between the measured data

Simulated (ie the process and model) at the current time

[n θ] y [n] y [n θ]

def

M ε = - forallnisin1 N (8)

All errors (8) is used to define the criteria for adequacy V (θ) which

It must be optimized to determine the unknown parameters Choosing

adequate mathematical model often involves an iterative process following which

set of possible models is reduced to a finite collection The model is then appropriate

especially in this set taking into account not only appropriate but also other considerations

implementabilitatea related to it This is the meaning of the arrow returned

the casings of the model in Figure 7

Efficiency and complexity of the optimization depends sensitive to the manner in

which was defined criterion of adequacy Note that adequate criterion value

evaluated for a particular model identification is often interpreted as a

an indicator of the accuracy of the model Next specified two criteria of adequacy

frequently used applications

V (θ) ε [ θ] (robust criterion) (9)

Σ =

=

N

n

f

n

1

V (θ) ε 2 [ θ] (quadratic criterion) (10)

Both criteria evaluates the total error between process and model but using applications

different The first criterion (the robust (9)) looks more natural because it sums

the absolute values of all errors The second criterion (the quadratic (10)) is both more

easy to use optimization as function module used in the first criterion

It is differentiable everywhere Optimization is performed within IS using primarily

gradient techniques (some of which are described in Chapter 4) Other methods

Optimization can also be used (for example AI-based techniques or

PE) Turning to criteria (9) and (10) appropriate parameter vector of measured data

Note N θ is obtained by minimizing nonlinear Troubleshooter

Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)

where S sube stability Rnθ is the model matematic3 and argmin

means the argument that minimizes (specified criterion function) Naturally

want the resulting model different as little of the data provider ie

leading to a minimal total error Typically applications to accurately determine the

the optimum point is not possible but there are techniques for approximating it with

precisely controlled (as are the techniques of gradient) These techniques are based on

iterative whose main quality characteristics are complexity

convergence and speed of convergence Of these the optimum convergence is

Foremost

The optimization problem (11) is solved for a finite size

vector parameters (ie nθisin1 Nθ) in order to provide finite collection

vectors which will subsequently choose suitable parameter vector It should be noted that it

Suitable parameters are always the most precise meaning of criterion V as

will see later

You can define other criteria than those listed above (for some

it is necessary to solve a problem of maximization and minimization not) The

The resulting data will be suitable but only for the purposes of general mathematical criterion

used It is unlikely that a suitable model by a certain criterion to continue

after changing the criteria to be suitable

152 The problem IS the light of TE

Theory estimate (TE) consists of a set of techniques for determining

unknown parameters using statistical concepts As the data provided by

are of stochastic process it is transferred and parameters which

They are called in this context and parameters estimated The method for producing

estimates of unknown parameters is called estimator The quality of a

estimator is given by the following main characteristics complexity consistency

(or statistical convergence) and efficiency (or convergence speed) Of these

features (described in the next chapter) consistency is of prime importance

Problem identification is in this context suggested forms of image

Figure 8

Figure 8 Identification parameters using criteria TE

This image and that of Figure 7 are very similar Consequently

There are many similarities between the two formulations of the problem of identification We

briefly only differences between them

So this time the output measured and simulated using

to build adequate criterion P (θ) called auto-covariance matrix

the estimation error As the name P (θ) is a matrix of order nθisin1 Nθ

It is defined using statistical average operator E (presented in chapter

next)

() () () T

def

P θ = E θ - θ θ - θ (12)

The definition (12) the difference θ - θ is the error of estimation parameters and bull T is

transposition operator It also notes the use of the external

two vectors (VVT) One can easily see that this criterion directly addresses

true parameters (which are unknown) so apparently it is impossible

rated However it can be shown (as in Chapter 4) that for certain

estimated matrix P (θ) can be estimated despite vector

True parameter is unknown

Identification problem in this case is reduced to a nonlinear problem

minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)

where the minimization is performed in the positive sense of ownership defining matrices

More specifically a square matrix P1 is less than or equal to a matrix P2

having the same size if the matrix P2 - P1 is positive semi-defined

2 1 0 P - P ge Minimization condition is not only natural but also required

Consistency property It can show the consistency that the property is

equivalent to the following condition

lim () N rarr infin N

P θ = 0 (14)

The problem (13) can turn equivalently or can relax by

formulating a problem of maximizing the probability of a particular estimate

parameter vector (as described in Chapter 4) However problems

Optimization of the TE are more difficult than those of the TO

To conclude this section will be highlighted advantages and

disadvantages of each type of translation solutions solve the problem identification

the two theories The TO-based solutions have the advantage of being produced

using relatively easy to implement algorithms on an automatic means of calculation but

the disadvantage that can not be characterized statistically In the TE

solutions provides the opportunity to evaluate a number of statistical properties but methods

Their often used to obtain a theoretical without leading to

Implemented algorithms However the IS aflicircndu is somewhat at the intersection of

two theories provide general problem solving methods that combine

advantages of each of them as can be noted in Chapter 4

16 Identification experiment

In general terms the issue raised in the previous section can be solved

(for models identification parameters) by organizing an experiment

Identification This section is devoted to description of the manner in which it can be

designed an experiment to identify and ends with an example (finding

heaters)

161 Diagram of an identification experiment

The experiment identifier means a series of sequential operations or

parallel is made according to the organizational chart in Figure 9 We will describe briefly

In continuation

A Initial data

Designing an experiment identification process begins by stating

Provider of data to be identified The efficiency and success of the experiment

depend on how much sensitive information is a priori known about

process This information should contain the following minimum

1048581 type of process strong non-linear non-linear almost linear linear Most

Nonlinear processes are nonlinear and strong but linearized around

rated operating points Therefore practically it should be laid

operating points around which it is desired to build a model

linear mathematical If I know the equations of continuous operation while

(by the application of physicochemical laws) they may be discretized

to obtain information on identifying the model structure would

You are chosen Otherwise they will be (eventually) take several experiments

Successive identification of a suitable model for the construction and valid

1048581 type of variation slow (over 5s) medium (between 1s and 5s) fast (sub 1s)

This information refers to the amount of stabilizing the output when

step process is stimulated with a certain amplitude accepted

(not leading to instability) and is useful in determining the period

sampling (Ts) to be chosen for the purchase of digital data

1048581 intrinsic dead time of the process While its value detection

and conversion to continuous (even imprecise) discrete-time (as a number

throughout the sampling period) lead to a simplification model

Identification choice In the absence of this information the mathematical model can

work with a number of parameters parasites which increase the complexity

thereof

In practice the determination of the dead time can be effected by stimulating the

preliminary stage of the process with a certain amplitude which does not

leading to instability

1048581 process variability while Many processes behave as

systems with time-varying parameters If the parameters vary slowly over time

it can be considered that they are constant provided that at certain intervals

time associated mathematical model to be reassessed If however the parameters

varies rapidly over time then the model must be adapted to the dynamics des

process The information on the temporal variability of the process is useful in the

choice of appropriate identification method non-recursive (off-line)

or recursive (on-line)

1048581 classes stimulus signals accepted by the process It is very possible

as though to obtain a more accurate model to be necessary to stimulate

process a certain type of signals which would lead to instability

In this case the mathematical model could be determined using only

signals of classes admitted by process (occurring in exploitation

The usual) The result however suffer from lack of generality Instability

processes is typically produced shocks with high amplitudes

entries Even if they can not determine the class of the stimulus signal

accepted the use of sufficiently small amplitude signals can

lead to general mathematical models and precise enough

Classes 1048581 interference posed to the process This information is

helpful in selecting a suitable model of noise that can affect

the measurements of the process output Without it experiment

Identification can be repeated several times until the establishment of a model

Noise appropriate

1048581 purpose of performing identification (simulation control command

numerical prediction generation of data etc) This information is useful in

choosing a mathematical model for the use of appropriate accuracy

For example disturbances affecting the output of the process model will be

namely where the model for predictive overall than if

its use in a numerical control application

B Organization econometric experiment

Econometrics is experimental measurement technique The result of the experiment

is a series econometric (generally finished) the data representing the point of

Physical size measured numerically The accuracy of this representation

It depends on the mathematical model constructed from the data measured Thats why

econometric experiment organization must be done carefully

In general the organization of an econometric experiment to identify

a process involves the following main stages which

Choose solution sampling method for quantifying signal stimulus

needed sensors and their locations

Choosing 1048581 sampling solution rests on the one hand the information

preliminary process and on the other hand the financial resources

available

The sampling period which is the core of the sample solution is

You may choose starting from the type of process variation Trials will slow

be sampled using sampling periods greater than the fast

A more precise information in choosing the sampling period (although

connected with the type of process variation but difficult to determine)

the process is bandwidth Common processes are vast

majority finite band filters This means according to his theorems

Joseph Fourier [StD9602a] [StD9602b] the output signal spectrum

is essentially located in a finite bandwidth pulse for example [0 Ωc]

where Ωcgt 0 is called pulsatile tăiere4

As is known the spectrum of a signal is determined by evaluation

Amplitude Fourier Transform (FT) of that signal In this section

Specific mathematical notation is shown TF definition signals

discrete A similar definition can be formulated for continuous signals

ie continuous time [OpSc85] [PrMa96]

Basically as suggested by Figure 10 the input signals spectrum

pulsation located beyond cutting the high frequencies are strongly

attenuated It also says that the process is insensitive to these signals

Sampling can be selected using sampling theoremss

Kotelnikov and Shannon Nyquist [StD9602a] These results led to the

the following rule-Nyquist sampling Shannon

bull minimum sampling frequency (or the critical frequency of

Nyquist) (FNYQ) is twice the cutoff frequency (Fc)

1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)

Under Nyquist frequency signal obtained by sampling meshed

the continuous distortion due mainly contains aliasing

high frequencies (aliasing) a phenomenon described for example in [OpSc85] and

[StD9602a] This implies that the process acquired data can

not reflect its actual behavior they are disturbed by noise

High-frequency sampling the more important by frequency

the sample is lower than the critical one

A very intuitive image sampling rule is given by

T meshing a sine period Three possible situations can be made

in evidence

a if Tsgt T 2 resulting discrete signal may be periodically period

at least 2TSgt T and non-routine (as the T Ts

or not a rational number)

b if Ts = T 2 the resulting discrete signal constant (non-scheduled)

c where Ts ltt 2 resulting periodic discrete signal and period

proportional ⎣T Ts ⎦sdotTs non-routine (as the T Ts


How in practice can not be obtained sampling periods with values

irrational virtually sampled sinewave is correct only if

is followed by Shannon-Nyquist (ie when c above)

So according to this rule sampling how time frequency

Sampling is lower higher much better rendering discrete signal

on the continuity that comes (in the sense of avoiding aliasing

ie minimal distortion of the original spectrum) One limitation however there

and it is time the cost of sampling solution chosen which may increase

lot for performances

Mark

bull The current technology allows operation with sampling frequencies of the order of several

MHz without excessive costs which increase with performance yet but linear The range

tens and hundreds of MHz up to hundreds of GHz the cost increases exponentially with value

sampling frequency But there is a theoretical upper limit The electron can

provide only a limited switching speed due to its inert mass (911sdot10-31 kg) Although

small it can not be neglected at high speed switching The ratio of

Plancks constant and the electron energy (6626sdot10-34 [J1048581s]) is the frequency of its own

oscillation - the order of 124 times 1011 GHz According to the principles of quantum limit [WiE71] report

energy and natural frequency of oscillation must be higher Plancks constant

This automatically implies the existence of a maximum sampling frequency which if

using electric field is situated below the threshold of 106 GHz The range

ultra-high frequencies a solution is foreseen is the use of photons rather than

electronically having masses of about 1010 times smaller than the electron

Considering the product of higher energy and a constant sampling frequency

sampling threshold photons moving around 1016 GHz

To avoid aliasing in practice the procedure of

Figure 11 above Thus before the actual sampling signal

It is filtered with a continual analog of low-pass (LPF) band

range It requires a certain cutoff frequency Fc depending on the type

The variation of process data provider (lower cutoff frequency

processes slow and fast processes bigger) Frequency

the sample is then chosen to be equal to twice the cutoff frequency

thus fixed Fs = 2FC On the one hand analog pre-filtering can result

the loss of high-frequency useful information On the other hand

treble attenuation positive effects of removing disturbances

all high frequency that could corrupt the measured data

Although aliasing was avoided by pre-filtration technique

analog data distortions are inevitably produced with

quantifying their operation By quantifying means the representation

numeric values using a finite number of bits each bit being awarded to

Δ a certain quantum of the ranges covered by the data set

Quantification can be uniform (most common) or irregular (if

that require greater precision in certain sub-ranges

Range) The effect on measured data quantification is widely

analyzed [PrMa96] or [JaNo84]

For example if the value range is [-5 + 7] and we want data representation

12-bit then quantification requires use of quantum uniform

Δ = 12 (211 -1) cong 00059 (taking into account the sign bit and that

the number of quanta is 211 -1) This means that the value is null

assigned to the entire range (-Δ 2 + Δ 2) and that the values are null

approximated to the nearest whole multiple of Δ Of course this

introduce errors that are much less important is how quantum Δ

smaller Quanta small built on the increasing number of bits

representation and therefore data storage capacity Nowadays

data storage capacity has increased significantly and is no longer

Practical problems in the 80s and 90s that However the quantification of not

It can be done using any number of bits Limiting the number of

bit quantization sampler is because all that is required to

record every bit numeric value at a time at most equal to a

sampling period Choosing an appropriate method for quantifying

(generally uniform) and the number of bits to represent the data

Numeric is therefore restricted the sample solution adopted

A uniform 12-bit quantization is often considered a very good

compromise Higher accuracies are obtained 16-bit but beyond

this number may force solution chosen quantification choosing

Unduly small sampling periods accompanied by a significant increase in the

costs

Note that the data transfer means operating automatic calculation

depictions 32 64 or 128 bits (ie more precise than those

adopted for quantification) does not increase accuracy On the contrary

calculation errors in the further processing of data are primarily

quantum of representation influenced by data values rather than accuracy

Once the word car (which would lead to much lower quanta)

Choosing 1048581 stimulus signals conform to the following principle

Overall

a If there is preliminary information about classes signals

process stimulus accepted by either selecting signal to beach

the richest frequencies or if it can not otherwise choose

signal used in the actual operation of the process

b If the information about allowable process entries are missing

then put it signals how much stimulation

Persistent

The concept of persistence will be presented in detail in Chapter 3 He refers

the ability of a stimulus signal to drive the calculation of

of the sequence desired number of weight values for a linear system or

equivalent to boost system (process) on a desired number of frequencies

Mathematical models obtained by the stimulation signals

Persistent are versatile and have a higher degree of generality than

the signal obtained with the low persistence

Effective stimulus process can be done either directly with signal

selected input (if the process is discrete) or more commonly through

a converted many analog (CNA) when the process evolves over time

continuously It is the instruments dual ADC Numeric

(CAN) responsible for sampling CNA performs interpolation

discrete sequences according to certain rules The most commonly used

order interpolation are at 0 and the order 1 Interpolatorul order 0

(Zero Order Holder - ZOH) Discontinued produce a signal in scale through

maintaining the sample at a constant value for a period of

sampling Interpolatorul of order 1 (First Order Holder - FOH) produce

a continuous signal but derived discontinuous (in general) using a

linear regulator (each a line is drawn between any two samples

adjacent) By interpolation the original input signal is more or

less distorted depending on the rule used FOH is more accurate

than ZOH but more difficult physically That is why ZOH is preferred

the majority of applications with sufficiently small sampling period

(fast processes or environments) Instead FOH should be preferred in case

slow processes (for medium and slow) to avoid shocks of

changing the rare stimulus signal values

1048581 sensors 5 by means of which one can collect process data should be elected

Depending on the nature of the physical quantity measured Their performance however

limited financial resources available

In general to not introduce significant distortions in the data sensors

must have characteristics that influence how much less the size of the

is measured How much linear conversion law as lower weight speed

The larger switching etc Linearity measured size conversion

size (usually electricity) sampled throughout the perception

Sensor is a primary requirement but whose satisfaction can

incur high costs Therefore often it is accepted choice of

Linear sensors with local characteristics around values specified a

expensive sensor can thus be replaced by a less expensive sensor network with

local linear features around different operating points The problem

a main sensor networks is the elimination of interference

of the signals measured by it Generally this problem

solves both by suitable location of the sensors (solution requires

but a fairly good knowledge of the process and location

its components) and through initial data processing methods

collected The location of the sensors is a difficult problem in general

In many applications the quantities to be measured are difficult to access or

the topological point of view the hostility of the environment in which it is to be

place the sensors For example when measuring the vibrations produced by

mechanical fault diagnosis view it is possible that just in

neighborhood defective part can not be placed accelerometers

that the measurements are performed In this case it is necessary

their location in an area as close as possible to the location of the component

damaged so that interference with the vibration induced by other components

mechanical (some of them also affected by possible defects) to

have amplitudes as small as possible Another example is the application of control

automatic chemical industry where often sensors must be placed in

corrosive environments which in time affect their performance

The more choice and placement problem is better solved sensors with

so the data collected is less affected by disturbances and models

mathematical results are more accurate Sensors with modest features

and or placed in locations affected by interference leading to sensitive

the need for primary data processing algorithms complexity

high and less accurate mathematical models

C Primary data acquisition and processing

After econometric experiment parameters have been set following

initiation to collect data which is called operation

data acquisition Note that the industry offers numerous solutions to collect

data from one process integrated form of direct acquisition boards

connected to an automatic means of calculation They differ in performance and

related costs

In principle a data acquisition board containing both analog filters as well as filters

digital They are used both in the choice of sampling rate and

solution quantization storage processing and the primary

thereof Schematic diagram of Figure 11 should continue with a block

digital filtering (ie using a digital filter) whose main objective is

to eliminate gross errors of measurement data and possibly alleviate

interference The design of these filters is by techniques described by

eg [PrMa96] Usually designing or LPF (most often) or type filters

Bandpass (FTB) or filters of high pass (FTS) depending on the application and

The purpose of identification

If interference affected data is generally required

primary processing more elaborate This involves using methods

complex than simple filtration methods aimed at extracting

Useful information of data acquired The problem of separation of the useful data

parasite is generally difficult to solve data having lower SNR More

regardless of the SNR no method of parasite data can not

ensure perfect extraction of useful information Always a part of

useful information is lost (when considered parasitic) while a part of

parasitic information is interpreted as useful Useful information can be encoded

the spectrum signals included in any frequency band Eg

voice intonation is encoded in the medium and high frequency spectrum

Speech while having significantly less power than the spectral

Lead voice formants A filter whose purpose is to remove

High frequency noise can alleviate parasite sensitive information about

intonation leaving unincorporated voice signal as would have been generated

artificially using a specialized system for voice synthesis

The result is the acquisition of datasets measured in N a 1 [] = U = and

N N y n 1 [] = Y = (in the case of off-line identification) or rows of the measured data

entry and exit (in the case of identification on-line) Note that entry can be

generated directly in numerical form or can be measured by the same method

as output

D Choosing class of models and model identification

Along with organizing and conducting econometric experiment acquisition

Data models can be specified class identification that works

(using preliminary information available) Class is most used in IS

ARMAX which has been mentioned in section 13 which will be described in

Chapter 2 It however is not the only set of patterns that may be

used as can be seen also in Chapter 2

Choosing a particular model type specified class is made taking

account several desirable characteristics It is good that the model be

bull sufficiently precise equations approached by the application of laws

describing physicochemical process operation (if these equations

They are available)

bull parsimonios with minimal complexity of the involved algorithms

methods needed to determine to

The two properties are obviously opposed It is well known that if

It wants greater precision of a mathematical model the complexity

it should be increased A compromise is always possible however In the IS

Parsimoniei resort to principle according to which is sacrificed accuracy

implementabilităţii model for his or method of identification

Chosen model type M (θ) is then used both in choosing the method of

identification as well as the determination of its parameters

The choice of method for identifying E

The mathematical model often determines the method for determining the parameters

Its as method by its complexity can force another election

mathematical model easier to determine For example for applications

NC highly used ARX model This is not because

ARX model would be the best in these applications but because it can be

2 simplest determined readily implemented (or LSM method

Instrumental Variables (MVI)) The models that respond better this

Applications are either the type Error Output (Output Error - OE) or the type

Filtered inputs Filtered disturbances (Filtered Input Filtered Noise - FIFN) more

Specifically than OE model However the methods by which one can determine parameters

these models ie Prediction Error Minimization Method (MMEP) (form

the LSM generalized) or Error Minimization Method Output (MMEI) are

much higher complexity than conventional LSM or MVI Some of

Identifying these models and methods are described in Chapters 2 and 4

F Determining the optimal models and model appropriate

The main loop identification experiment consists of

determining the actual parameters of the model chosen for the different indicators

nθ structural note in the chart of Figure 9 m (for ease) a

maximum structural index (Nθ = M) stop the process The models

They are optimal in the sense defined adequacy criteria chosen Each model

The determined optimal M (θm) and is assessed and precision using all criteria

appropriate Thus identifying models of the same type but different

structures can be compared with each other from the point of view of accuracy in

choosing the appropriate order

Choosing an appropriate model of the M available not performed

only based on its accuracy but also resorted to a series of tests

Suitably the most used are described in Chapter 4 These tests help

users to take a decision on the optimal structure that should have

adequate mathematical model Thus for example can be drawn chart

quadratic optimization criterion V (θm) as shown in Figure 12 The graph was

interpolated to provide a clearer picture It notes that while models

maximum structural index led to the highest precision (ie the value

Criterion less suitable) starting at a certain structural indicators

improve the accuracy of models no longer significant The principle

parsimoniei (mentioned earlier) it is unnecessary to operate with models

complex if they can not ensure a noticeably higher accuracy than models

simpler

The appropriate optimal model will have a structural indicator given the entry into

precision landing of the chart (denoted mo) and not its maximum From

Therefore it is appropriate that the pattern denoted by M (θo) was elected

using test criterion flattening accuracy This test is but one component

subjective which can be mitigated through other such criteria selection

optimal structural index

G test validity

For a model of proper identification data acquired to be adopted

it is necessary that he be tested for validity Only an appropriate model

It will be returned valid identification at the end of the experiment Validation is

testing the operation of the process model as compared to when the

starting a new session stimulating both entities with the same input

For the model to be valid the error between process and model must check

some properties will generally be determined by the method of identification used in

during the experiment The usual validation tests are described in detail in

Chapter 4

For example if the appropriate model has been determined using LSM

trial and error of the model defined by (8) must have characteristics

a normally distributed white noise (Gaussian) for the model to be validated

For this reason test validation test is called bleaching

If the model fails the test validation identification experiment

It should be resumed from its main stages Attempting be reviewing

econometric experiment (perhaps the data are too affected by disturbances)

be reconsidering class models and or type of model or a change

Method of identification

Note that it is extremely important that validation is performed in another set

different than the one used to determine the parameters of the model Therefore

practice the dataset acquired is always divided into two sub-sets

disjoint one for identification and one for validation

The following example illustrates the manner in which it conducted an experiment

having as an objective the identification of a mathematical model used in the construction of a loop

auto

One of the main features of a heating system such as that illustrated in

Figure 13 is to keep a constant temperature at the exit of the ventilation air in

Despite air temperatures absorbed This is achieved with a system of

temperature compensation based on a simple control loop If the heater is

placed in an environment having different temperatures of the air streams they can

they disrupt normal operation Therefore we assume desired

air heater in the open loop identification following the mathematical model obtained to be

used in the design of the controller to ensure disturbance rejection and maintaining

the temperature around a desired value

We will go through all the main stages of the experiment on identification related

building mathematical model desired

1 Informaţiipreliminare

Heating is an electro-mechanical system whose operation equations based

the laws of dynamics electricity and thermodynamics lead to the conclusion that the order

Maximum model identifier 2 The scheme is its functional principle

plotted in Figure 14

The cold air is ventilated by an electrical resistance powered by

means a power amplifier This may vary the voltage and or

intensity electric current going through Size Control

power amplifier is entering the system System output is

provided by the temperature sensor placed in the hot air stream produced by

resistance Both entry and exit as measurable

Basically the circuit from input to output of the process there are four components

power amplifier the electrical resistance hot air flow sensor

temperature The process is nonlinear but linearized around each of temperatures

Beach permissible that can provide electrical resistance (having power

The maximum) Also the process can be considered as having an average type of

variation because the settling time of a fixed temperature is about 2s

The process dead time is about 01s Its parameters can be considered

They are constants (Their variations become noticeable when the heater has a

given the high wear resistance aging electrical and or

power amplifier due to damaging the temperature sensor

its exposure to high temperatures etc) Heating can be controlled with a range

range of input signals including high persistence Their amplitude

however it will be limited by the capacity of the power amplifier

The disturbance comes from the cold air flow which can be both temperature as well as

Variable flow The variable temperature due to the environment in which it is

Located heater (that is an external noise) while variations in flux

rather they may be due to defects in the electric motor acting

fan (internal noise they produce) No distinction will be made between the two

types of disturbances and shall be deemed heater is affected by one

global6 noise Order the disturbance model is unknown but it can go

from simple designs to intricate as usual

The objective (or purpose) is to provide identification of a mathematical model needed

designing an automatic regulator that ensures both a good rejection

interference as well as maintaining the temperature of the hot air flow around an

specified The temperature is fixed manually via amplifier

power which is equipped with a thermostat An automatic adjustment scheme in which

compensator is designed based on the model identification can be seen in

Figure 15 The design of patterns starting from the numerical controllers Identification

It is described in detail in many works among which we mention only [LaID93]

[LaID97] [GePo97] or [DTFP04]

Process and data acquisition 2Stimularea

Based on preliminary information it is sufficient to select a frequency

tens of Hz sampling at most Covering may be fixed

Fs = 100 Hz ie Ts = 10ms This leads to an analog filter having

Fs = 50Hz cutoff frequency (frequency as forced convection air heaters power source)

and a normalized dead time nk = 10 It is yet to give pre-filtering

Digital because SNR is large enough Data collection can be done

be a general purpose acquisition board or a dedicated acquisition system a

Such a system is expensive but provides more flexibility in handling and

Primary data processing

The image in Figure 16 shows the portable system LMS Belgian acquisition

Roadrunner [LMS99] which has a micro-computer integrated (functionally

Windows operating system)

Among other things it provides pre filter data simultaneous data acquisition

at least 2 channels (4 channels features and can be expanded to 16 channels) a wide range

Sampling frequency (between 1 Hz and 100 kHz) compatibility with a large

number of sensors the possibility to trace spectra and even spectrogram (ie

Spectra variables over time) etc

To stimulate forced convection air heater choosing a high persistence signal generated

by means of an algorithm implemented on a computer One such signal is

the pseudo-random (binary) known under the abbreviation SPA (B) Some algorithms

generating SPA (B) are shown in Chapter 3 (As I already said

models obtained using high persistence signals are more specific

versatile than those obtained with low persistence signals) signal

Stimulus is interpolated using a CNA based ZOH for simplicity

Data acquisition horizon is fixed at N = 210 = 1024 Powers of 2

Signals favoring spectra evaluation using an algorithm type

Fast Fourier Transform (Fast Fourier Transform - FFT) [OpSc85] [PrMa96]

Data acquired (identification and validation) are quantized 12-bit and will

Computer representation in the form of floating-point numbers 32-bit

3Alegereaclaseidemodeleşiamodeluluispecific

Initially trying to determine the mathematical model of useful data It will return

subsequently adding a model of the disturbance Class of standard models

IS is ARMAX General equations describing ARMAX are presented in class

Chapter 2 For clarity we reproduce in this context

General Equation of class ARMAX [na nb nc nk] (a difference equation) is

as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where

bull u is the input signal or stimulus

bull y is the output signal or system response

bull e is ideal stochastic called white noise signal in statistical terms

White noise is the prototype total neautocorelate signals ie

2

0 E e [n] e [m] = λ δ [n - m] foralln misinZ (17)

where E is the statistical average operator 0 δ is the impulse unit

centered in origin (Kroneckers symbol) and λ2 is the variance of noise

unknown

bull Q-1 is a step delay operator (sampling) defined by

(q-1 f) [n] = f [n-1] forallnisinZ for any string of data f (scalar or vector)

bull A B C are finite polynomial⎢ ⎢ ⎢ ⎢⎣⎡= + + +

= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)

for which the coefficients ii na 1 isin ii nb b 1 isin ii nc c 1 isin (model parameters)

Grades na nb nc (structural indices of the model) (and sometimes nk - delay

intrinsic model) are unknown variables

Having a clear identification the usual pattern they operate

application control CNC is ARX [na nb] Using the information

preliminary order of this model is approximately equal to 2 This implies

that the appropriate model to be found among the ARX models [22 nk]

ARX [23 nk] ARX [32 nk] and ARX [33 nk]

4Alegereametodeideidentificare

Determination ARX model can be made using two methods LSM and MVI

(as revised version of LSM) Of these MVI is well adapted

model and will be taken further

5Determinareamodeluluiadecvat

After running the experiment for identifying the main loop you can choose

ARX appropriate type model [2310]

(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)

(intrinsic delay is already known and equal to 10 times of sampling)

Another suitable model possible but parsimonios is ARX [2210] Comparing

the two models it appears that the model parameter b3 (19) - (20) has a

much smaller than the absolute value of the absolute values of the other two parameters

polynomial B It follows that the term 2

3

b q may be neglected This leads to

reducing model size by one Note that this model parameters

parsimonios are identical to those that have more complex model such

that neglect of 2

3

bq- require reassessment of remaining in the competition parameters

6Validareamodeluluiadecvat

Especially suitable model (ARX [2210]) fails the validation test than the one

satisfactory probably because of interference pattern which is not

accurate Therefore identification should be resumed with a model experiment

mathematical model that includes a complex of disturbances

7Reconsiderareamodeluluimatematicşiametodeideidentificare

To work this time with more general models such ARMAX [22 nc 10]

The identification method is also changed from MVI in the extended version of the MMEP

MVI Maximum structural index of the MA is set to Nc = 10

8Redeterminareamodeluluiadecvat

This time the optimal model 10 produced by the main loop

Experimental identification model is suitable or ARMAX [22110] or

Again it appears that the absolute value of the model coefficient c2 second

It is negligible compared with the absolute value of the coefficient C1 Thats why

The model is preferred parsimonios ARMAX [22110] It should be noted once again that

model parameters are different from the corresponding models

ARX [2210] and ARMAX [22210]

9Validareanouluimodeladecvat

Assay validation is passed to the new model described by polynomials

the following⎢ ⎢ ⎢ ⎢⎣⎡= +

= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)

The model is sufficiently precise to play details of transitional arrangements

characterized heater Its step response but can be approximated

that of a system of order 1 (ie having only one pole) If the model (16) amp (21)

is too complex for automatic adjustment application then you can choose one

simple models of ARX [1210] or ARMAX [12110] slaughtering

the accuracy and validity area

1048581

The objective of this chapter was to briefly present the main coordinates of

IS domain Subsequent chapters will provide more details on these

although approaches will remain at the basic level introductory

17 Exercises proposed

Exercise 11 (mini-project)


deterministic and non-deterministic behavior take the example of a

second order linear continuous system and show how it can be determined about graphics

its parameters Write a program with which to generate MATLAB

various embodiments of the system particularly in the case where the output is corrupted by a

perturbation amplitude Gaussian additive chosen by the user (To

generate interference I can use MATLAB routines rand or randn To simulate

operation of a discrete linear dynamic system transfer function rational

MATLAB can be used for routine filter) What can be said about identifiable

system perturbation amplitude chose

Exercise 12

Demonstrate the properties (5) (6) and (7) of Gaussian distribution (4) using

Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13

Using Gaussian distribution (4) demonstrate the following property

(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin

int p - = σ foralltisinR


Write a program in MATLAB by which to regularly check sampling

Nyquist-Shannons where meshing a continuous sinusoid period

time Show graphs illustrating the operation of sampling accuracy It is

thus possible to obtain a sampled signal using a non-routine

Automatic calculation means If not show why If so build signal

non-routine and display graph or a long enough time horizon

This book aims to describe the theoretical and practical aspects of modeling and

System Identification As a work of applied mathematics in technical (ie

in Automatic) which addresses the reader must have a knowledge base

located at least the first two years of training in a faculty of electrical profile

However the book was conceived in the spirit of the phrase self-contained ie

Presentation of gradual starting from the simplest terms by the complicated

without making numerous references to references nested and often difficult

available Whenever it was necessary the reader is reminded concepts

it needs to understand the arguments developed in chapters work

Of the four constituent chapters aims to introduce the first

discipline specific terminology is peppered with historical notes Identification

Systems is a discipline developed in a referential format of 3

major coordinates mathematical model associated entity to be identified

signal that it can be stimulated to provide data on behavior

its evolution and its intrinsic or the method that is determined model

especially math They constitute the nucleus of the 3 main chapters of the book

Each chapter concludes with a collection of exercises to be solved whose

purpose is to facilitate understanding

It is possible that some readers find too mathematized presentation especially

the end of the last chapter although the book contains numerous references to practical

discipline For those who prefer more practical purpose methods or

the algorithm rather than rigorous argumentation for construction and

their accuracy there is a paper alternative [SMS04]






more readers

Chapter 1

Overview









11 The subject


















Figure 1)

























12 Historical Notes















































































































































































































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin














more readers

Chapter 1

Overview









11 The subject


















Figure 1)

























12 Historical Notes















































































































































































































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin









Chapter 1

Overview









11 The subject


















Figure 1)

























12 Historical Notes















































































































































































































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin














Figure 1)

























12 Historical Notes















































































































































































































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin

































12 Historical Notes















































































































































































































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin











12 Historical Notes















































































































































































































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin









12 Historical Notes















































































































































































































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin








































































































































































































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin


































































































































































































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin











































































































































































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin






















































































































































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin




















































































































































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin































































































































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin










































































































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin






































































































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin

















































































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin


























































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin






















































discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin






























discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin









discrete time) as



















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin



























known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin
















known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin













known or not











matrix H







































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin












































int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin






























int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin


















int (()) () = 1

+ infin

-infin








()

3

3

() () 095

y

y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin









y

dy t y t

+ Σ

- Σ











(Y (t)) y (t) dy (t) y

+ infin

-infin








great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin












great appearance


























thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin
















thereof















in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin











in Figure 7










[n θ] y [n] y [n θ]

def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin









def















Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin










Σ =

=

N

n

f

n

1











Argmin ()

n

N isin sube θ

=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin









=

θ

θ θ

And R

V (11)









Foremost





will see later
















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin






















Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin














Figure 8









next)

() () () T

def

P θ = E θ - θ θ - θ (12)









minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin












minimization

Argmin ()

n

N isin sube θ

=

θ

P θ θ

And R

(13)








P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin














P θ = 0 (14)




















heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin













heaters)




In continuation

A Initial data
























thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin



















thereof


























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin























Noise appropriate



















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin


















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin

















available
























1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin

















1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin














1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin











1 2 c

NYQ s c

s

F F F

T

Ω

= ge = =

π

(15)










in evidence

















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin






















Mark

























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin


























































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin


















































costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin


























costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin













costs








Overall







Persistent






































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin










































































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin




















































related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin




























related costs































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin





































as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin













as output












They are available)















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin



















































simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin





























simpler







G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin















G test validity









Chapter 4
















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin

















auto

























































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin

































































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin












































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin











































































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin






















































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin

































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin
































as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin



























as follows

A (Q 1) y [n] B (q -1) [not] C (Q 1) e [n]

AR X MA

- = - + - 10485811048581104858110485811048581 10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (16)

where





2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin













2




unknown




= +

= + + +

- - -

- - - -

- - -

nc

nc

nb nk

nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin









nb

NA

NA

C q c q c q

B q b q b q q

A q q q

1048581

1048581

1048581

1

1

1

1 1

1

1

1

1

1

() 1

() ()

() 1

(18)
















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin






















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin





















(1) [] (1) [] s [n]

X

B q u n

AR

A Q Y n = - +

10485811048581104858110485811048581 10485811048581104858110485811048581 forallnisinN (19)

with⎢ ⎢⎣⎡= +

= +

- - - -

- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin









- - -

() ()

() 1

2

3

1

1 2

1 October

2

2

1

1

1

B b b q b q q q

A q q q

(20)






3




that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin










that neglect of 2

3


















ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin













ARX [2210] and ARMAX [22210]




= +

= +

- -

- - -

- - -

1

1

1

1

1 2

1 October

2

2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin









2

1

1

1

() 1

() ()

() 1

C q c q

B b b q q q

A q q q

(21)







1048581
















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin


















Exercise 12


Poissons identity

2 e t dt

-infin

-

-infin

int = π

exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin









exercise 13


(Y (t)) (y (t) y) 2DY (t) 2

+ infin

-infin









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Taips Book Ch1