Mikles J., Fikar M. Process Modelling, Identification, and Control

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218 5 Discrete-Time Process Models

Y. Z. Cypkin. Foundations of Theory of Automatic Systems. Nauka, Moskva,

1977. (in Russian).

Y. Z. Cypkin. Grundlagen der Theorie automatischer Systeme. VEB Verlag

Technik, Berlin, 1981.

E. I. Jury. Digital Control Design. GIFML, Moskva, 1963. (in Russian).

B. Hanuˇs. Introduction to linear discrete control, volume I. ES VS

ST, Liberec,

1972. (in Czech).

Salamon. Control Theory.ESSV

ST, Bratislava, 1979. (in Slovak).

V. Strejc and kol. Control Design with Digital Computers.N

CSAV, 1965. (in

Czech).

H. Unbehauen. Regelungstechnik II. Vieweg, Braunschweig/Wiesbaden, 1987.

K. Ogata. Discrete-time control systems. Prentice Hall, Englewood Cliﬀs,

New Jersey, 1987.

R. Isermann. Digitale Regelsysteme. Springer Verlag, Berlin, 1977.

J. Mikleˇs and V. Hutla. Theory of Automatic Control. Alfa, Bratislava, 1986.

(in Slovak).

J. Lunze. Regelungstechnik 2. Springer Verlag, Berlin, 1997.

5.9 Exercises

Exercise 5.1:

Derive discrete-time state-space model of interacting tanks in series process

shown in Fig. 2.2 on page 17. Consider input variable q

and output variable

. Zero-order hold is assumed.

Exercise 5.2:

Derive discrete-time transfer functions of the process shown in Fig. 2.2. As-

sume zero-order hold.

Exercise 5.3:

Derive state-space model of the system

y(k +2)+3.1y(k +1)+2.2y(k)=u(k)

Exercise 5.4:

Find a general solution y(k) for an input u(k) and a system

−y

(k +2)−9y

(k +1)+y

(k)=u

(k)

(k +1)+3y

(k)=u

(k)

5.9 Exercises 219

Exercise 5.5:

Find the state transition matrix Φ(T

) for the system in Exercise 5.3.

Exercise 5.6:

Find the discrete-time transfer function matrix for the system in Exercise 5.4.

Process Identiﬁcation

Processes in chemical technology can be modelled in various ways. The most

natural representation is based on the state-space description resulting from

material and energy balances. The models mostly used for control purposes

are in form of linear diﬀerential or diﬀerence equations. We often assume that

their parameters are known and constant. However, in experimental conditions

it is often necessary to measure or estimate parameters from process input and

output signals with carefully chosen input signal types. In this case we will

speak about parameter estimation or system identiﬁcation.

The ﬁrst part of the chapter deals with classiﬁcation of the identiﬁcation

methods, the mostly used model structures, and guidelines for identiﬁcation.

The second part is devoted to identiﬁcation from step responses that is mainly

used to gain some basic information about the process. The third part will dis-

cuss least-squares estimation and its recursive version to identify parameters

of continuous and discrete time systems.

6.1 Introduction

While mathematical models based in mass and energy balances construct

model by the means of analysis, identiﬁcation strives to describe the process

according the information gained from relation between process inputs and

outputs. Also, analysis methods result mainly in state-space model descrip-

tions, whereas identiﬁcation produces usually input-output models.

Identiﬁcation Procedure

A general procedure for process estimation includes the following steps:

Determination of the model structure This often makes use of empirical ex-

perience about the process, or information from some basic experiments.

It is necessary to choose the model type (linear vs. nonlinear), as well

222 6 Process Identiﬁcation

as its complexity. There are various criteria that can be selected: quality,

ﬂexibility, or the model price. The choice of the structure still remains

more art than a systematic procedure.

Parameter estimation There are lots of procedures for parameter estimation.

It depends on the type and characteristics of the process input, as well as

the desired model structure.

Model veriﬁcation There are several important aspects at this stage. A suit-

able model should agree with the experimental data, it should describe the

process accurately, and it should meet the purpose it was obtained for.

Further, it can be veriﬁed whether the parameters obtained are within

physical limits. It is also possible to reduce the model and compare it

with the original model to see if a simpler model suﬃces.

Classiﬁcation of Identiﬁcation Methods

There are several possibilities to classify identiﬁcation methods:

A Passive or active experiment identiﬁcation. This is usually determined

by the given process technology and demands of the experiment. This

indicates whether it is permitted to generate special signals on inputs or

it is just possible to collect typical process inputs and outputs.

B From the point of view of the mathematics, it is possible to distinguish

the following methods:

• deterministic,

• stochastic.

Deterministic methods assume exact knowledge about the process inputs

and outputs and do not consider random sources and inﬂuences.

Stochastic methods include for example the least squares method and its

modiﬁcations, maximum likelihood method, the Bayesian approach, etc.

Any of these methods assumes some properties of random disturbances

and some knowledge about them (the least demanding from this point of

view are the least squares methods).

Of course, the choice of the method is not arbitrary. It depends on signal

to noise ration, disturbance properties, etc.

C From the processing point of view we can divide the methods into:

• single-shot (batch) methods,

• recursive methods.

Batch methods can further be divided into the manual and computer pro-

cessed. Batch methods are not suitable for computer processing because

it is diﬃcult to make algorithm from them, as they depend on some arbi-

trary performance evaluation (e. g. inﬂexion point at the step response).

Computer batch methods process some data not suitable for manual cal-

culation (least-squares, numerical integration, calculation of correlation

functions) and are sometimes called oﬀ-line methods.

6.1 Introduction 223

Recursive methods process the experimental data gradually and estimated

parameters are improved from each experiment. They can be used in on-

line control of processes and lay foundations for adaptive control.

6.1.1 Models of Linear Dynamic Systems

This section describes basic kinds of linear models employed in identiﬁcation.

We will distinguish between deterministic, stochastic and continuous, discrete

models.

Deterministic Models

Mostly used are continuous models described by a linear diﬀerential equation

with constant coeﬃcients of the form

y(t)

+ a

n−1

y(t)

n−1

+ ···+ y(t)=b

u(t)

+ ···+ b

u(t) (6.1)

A(p)y(t)=B(p)u(t) (6.2)

A(p)=1+a

p + ···+ a

n−1

+ a

(6.3)

B(p)=b

+ b

p + ···+ b

m−1

+ b

(6.4)

where p =d/dt is the diﬀerentiation operator. The identiﬁcation aim is to

estimate the coeﬃcients of the polynomials A, B for given degrees n, m. Very

often the time delay T

is also estimated, hence

A(p)y(t)=B(p)u(t −T

) (6.5)

Alternatively, the following equation will be considered in the case of linear

discrete-time deterministic models

y(k)=−a

y(k −1) −···−a

y(k −n)+b

u(k −1)+···+ b

u(k −m) (6.6)

A(q

−1

)y(k)=B(q

−1

)u(k) (6.7)

A(q

−1

)=1+a

−1

+ ···+ a

n−1

−n+1

+ a

−n

(6.8)

B(q

−1

)=b

−1

+ ···+ b

m−1

−m+1

+ b

−m

(6.9)

where q is the shift operator (q

−1

y(k)=y(k − 1)). The identiﬁcation aim is

again to estimate the coeﬃcients of the polynomials A, B.

Stochastic Models

When stochastic models are considered, we assume a random factor ξ(t)inthe

equation (6.1), or (6.6). Mostly, white noise ξ will be assumed, but also more

complex cases can be discussed. Stochastic models given below are described

in discrete-time.

224 6 Process Identiﬁcation

ARMA Process

Consider a stationary process v(k) that can be represented as white noise

ﬁltered by a linear process

v(k)=

C(q

−1

)

A(q

−1

)

ξ(k) (6.10)

where

C(q

−1

)=1+c

−1

+ ···+ c

−n

A(q

−1

)=1+a

−1

+ ···+ a

−n

Hence,

v(k)=−a

v(k − 1) −···−a

v(k − n

)

+ ξ(k)+c

ξ(k − 1) + ···+ c

ξ(k − n

) (6.11)

A(q

−1

)v(k)=C(q

−1

)ξ(k) (6.12)

Stochastic process described by this equation is called ARMA. It can be split

into two parts: AR (autoregressive), when n

v(k)+a

v(k − 1) + ···+ a

v(k − n

)=ξ(k),Av= ξ (6.13)

and MA (moving average), when n

v(k)=ξ(k)+c

ξ(k − 1) + ···+ c

ξ(k − n

),v= Cξ (6.14)

ARX Model

It is assumed that the white noise enters directly the process equation

Ay = Bu + ξ (6.15)

y(k)=−a

y(k − 1) −···−a

y(k − n

)

+ b

u(k − 1) + ···+ b

u(k − n

)+ξ(k) (6.16)

This model is denoted ARX, because AR describes the part A(q)y and X is

an extra input. The special case when n

= 0 is called FIR (ﬁnite impulse

response) model.

ARX model is due to its simplicity and linearity with respect to identiﬁed

parameters most often used in identiﬁcation.

ARMAX Model

In this model the random source enters the system as MA model, hence

Ay = Bu + Cξ (6.17)

y(k)=−a

y(k − 1) −···−a

y(k − n

)

+ b

u(k − 1) + ···+ b

u(k − n

)

+ ξ(k)+c

ξ(k − 1) + ···+ c

ξ(k − n

) (6.18)

6.2 Identiﬁcation from Step Responses 225

Frequently a modiﬁcation of ARMAX model is used, the so called ARI-

MAX (ARMAX Integrated) or CARIMA (Controlled ARIMA) model. In this

case a nonstationary disturbance is assumed when the white noise is ﬁltered

by an integrator

Ay = Bu +

e (6.19)

where Δ = 1 −q

−1

corresponds to the discrete integrator.

Output Error Model

The white noise is added to the output variable (measurement error)

y =

u + ξ (6.20)

w(k)=−a

w(k − 1) −···−a

w(k − n

)

+ b

u(k − 1) + ···+ b

u(k − n

) (6.21)

y(k)=w(k)+ξ(k) (6.22)

6.2 Identiﬁcation from Step Responses

The broad variety of methods based on step response identiﬁcation belong to

deterministic methods as the input signal is deterministic and random sources

are not considered in the process description. The results of the identiﬁcation

are coeﬃcients of the equation (6.5). Methods in this category aim at ﬁrst es-

timates of the process and provide information about an approximate process

gain, dominant time constant, and time delay.

The input signal used is a step change of one of the process inputs when

all other inputs are held constant. It is necessary that the controlled process

is in a steady state before the step change. The measured process response is

a real step response that needs to be further normalised for unit step change

and for zero initial conditions.

As the identiﬁed process can in general be nonlinear, it might be advan-

tageous to record several step responses with diﬀerent step changes and signs

of the input signals. To obtain the normalised step response, the following

formula can then be used

ˆy



k=1

Δu



k=1

(Δu

)

(6.23)

226 6 Process Identiﬁcation

where i is ith point of the step response,

k - kth measurement, k =1,...,N,

Δu

- step change of the input at the kth measurement,

- output value at the kth measurement in ith interval,

ˆy

- ﬁnal value of the step response in time i.

6.2.1 First Order System

Consider a ﬁrst order approximation of the identiﬁed process

G(s)=

Ts+1

−T

(6.24)

where Z is the process gain, T time constant, and T

time delay that need to

be determined. The step response corresponding to this transfer function can

be obtained via the inverse Laplace transform of the output as

y(t)=

⎧

⎪

⎨

⎪

⎩

0 t<T

⎛

⎝

1 −e

−

t −T

⎞

⎠

t ≥ T

(6.25)

Fig. 6.1. Step response of the ﬁrst order system

We assume the normalised step response. The process static gain is given

as the new steady-state output Z = y(∞) because the normalised input is the

unit step change (Fig. 6.1). If we assume that two points t

and t

from

the step response are known then it follows from (6.25)

6.2 Identiﬁcation from Step Responses 227

= Z

⎛

⎝

1 −e

−

− T

⎞

⎠

(6.26)

= Z

⎛

⎝

1 −e

−

− T

⎞

⎠

(6.27)

After some simple manipulations we get

T =

− t

Z−y

(6.28)

x −t

x −1

,x=

Z−y

(6.29)

The value of the time constant T can also be obtained from the step response

as it is shown in Fig. 6.1, or as the time when the output reaches 63% from

its new steady state.

Example 6.1: Approximation with the ﬁrst order system

www

Consider step response of dimensionless deviation output concentration

in a CSTR to step change of Δq

= 10 shown in Fig. 6.2. To obtain

the normalised step response, the original one was divided by the step

change value and shown in Fig. 6.2 (right) by a solid line. Two points were

chosen: [0.85; 0.0082] and [2.18; 0.0224]. The process gain was obtained as

Z =0.027 from y(∞). From the points t

and t

were calculated

the time constant T =0.94 and the time delay T

=0.51. The resulting

transfer function is then of the form

G(s)=

0.027

0.94s +1

−0.51s

The approximated step response is shown in the same ﬁgure as the vari-

able x

by a dashed line. Both curves coincide at the measured points.

However, there are signiﬁcant discrepancies elsewhere. We can see that

this procedure serves only for a crude estimate of process parameters.

6.2.2 Underdamped Second Order System

If the process step response exhibits underdamped behaviour (Fig. 6.3) it is

possible to approximate it by a second order system with a transfer function

of the form

G(s)=

+2Tζs+1

Zω

+2ω

ζs + ω

(6.30)

The identiﬁcation task is to determine the process gain Z, natural fre-

quency ω

, time constant T =1/ω

, and damping ζ. Given are points [t

] and the steady state output y(∞).

228 6 Process Identiﬁcation

0 2 4 6 8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2 4 6 8 10

0.005

0.01

0.015

0.02

0.025

0.03

Fig. 6.2. Measured step response of a chemical reactor using the input change

Δu = 10 (left), approximation of step response by the ﬁrst order step response

(right)

Fig. 6.3. Step response of the second order underdamped system

For the derivation of relations between the known and estimated variables

we will use the fact that the derivative of the step response with respect to

times is in the points t

(local extrema) zero.

The process static gain is as in the previous case given as the new steady-

state value of the process output Z = y(∞).

The step response is of the form (see the Laplace transform table)

y(t)=Z



1 −

−ζω

sin(ω

Pt+ ϕ)



,ϕ= arccos ζ, P =



1 −ζ

(6.31)

The derivative of y(t) with respect to time is given as