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

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188 4 Dynamical Behaviour of Processes

1. the Nyquist diagram,

2. the Bode diagram.

Exercise 4.5:

Consider a system with the transfer function given as

T (s)=

G(s)R(s)

1+G(s)R(s)

where G(s) is in the ﬁrst exercise and R(s) is in the third exercise. Find:

1. response of the system to the unit step,

2. the Bode diagram.

Exercise 4.6:

Consider a system with the transfer function given as

S(s)=

1+G(s)R(s)

where G(s) is in the ﬁrst exercise and R(s) is in the third exercise. Find its

Bode diagram.

Discrete-Time Process Models

The aim of this chapter is to explain the reason why discrete-time process

models are used. It has a close relation to computer control of processes.

The chapter demonstrates how discrete-time processes and discrete transfer

functions are obtained. Basic properties of discrete-time systems are examined.

5.1 Computer Controlled and Sampled Data Systems

()

kTw

Computer

Discretised process

D/A converter

with S/H

()

Process

()

converter

with S/H

()

kTe

()

kTu

()

kTy

Fig. 5.1. A digital control system controlling a continuous-time process

Computer controlled process control indicates that the control law is cal-

culated by computer. A feedback scheme of such a control is shown in Fig. 5.1.

This is a simpliﬁed scheme of direct digital control. The scheme in Fig. 5.1

contains four basic blocks: computer, digital-to-analog (D/A) converter, con-

trolled process, and analog-to-digital (A/D) converter. Both converters con-

tain sample-and-hold (S/H) device (or simply sampler) that holds each sample

constant until a new information arrives. The control error e(kT

) is given as

the diﬀerence between the setpoint signal w(kT

) and controlled process out-

put y(kT

) in digital form in sampling times speciﬁed by the sampling period

. The computer interprets the signal e(kT

) as a sequence of numbers and

given the control law, it generates a new sequence of control signals u(kT

190 5 Discrete-Time Process Models

The discretised process represents a system with input being the sequence

[u(kT

)] and output being the sequence [y(kT

)]. This system is characterised

by a discrete-time model that deﬁnes relations between sequences [u(kT

)]

and [y(kT

)]. A/D converter implements transformation of a continuous-time

signal to a sequence of numbers (Fig. 5.2). D/A converter with a sampler

implements transformation of a digital signal to a continuous-time signal that

is constant within one sampling period (Fig. 5.3). A/D and D/A converters

operate synchronously.

0 t

converter

()

kTy

Sampler

continuous-time

signal

discrete-time

signal

∗

Fig. 5.2. Transformation of a continuous-time to a discrete-time signal

A possible realisation of a sampler is zero-order hold with the transfer

function of the form

G(s)=

1 −e

−T

(5.1)

When processes are controlled digitally, it is important to choose a suitable

sampling period T

that will capture the process dynamics correctly. There

are two contradictory requirements on its choice. When computational load

is considered, the sampling period as large as possible should be chosen. On

the other hand, large sampling times imply loss of information.

In any case, the sampling period cannot be larger than some critical value

at which important information about the continuous-time signal can be lost.

5.1 Computer Controlled and Sampled Data Systems 191

D/A converter

with S/H

()

kTu

Zero-order hold

continuous-time

signal

discrete-time

signal

Fig. 5.3. Continuous-time signal step reconstruction from its discrete-time coun-

terpart

If a continuous-time sine signal is considered, loss of information occurs

when the following inequality does not hold

sin

(5.2)

where T

sin

is the oscillation period of the sine signal. This can be seen in

Fig. 5.4.

If (5.2) is not valid, the phenomenon occurring with the sine signal can

also occur with other sampled continuous-time signals.

If this phenomenon is studied using magnitude frequency spectra then it

will be shown by aliasing of individual spectral densities.

Let us now investigate properties of an ideal sampler shown in Fig. 5.5. Its

output variable y

∗

can be represented as a sequence of modulated δ functions.

A periodical sequence of δ functions can be represented as

∗

(t)=

∞



k=−∞

δ(t −kT

) (5.3)

Let us deﬁne the sampling frequency

2π

(5.4)

192 5 Discrete-Time Process Models

sin

Fig. 5.4. A possible loss of information caused by sampling

()

∗

Fig. 5.5. An ideal sampler

Then this sequence can be expressed using the exponential form of the Fourier

series

∗

(t)=

2π

∞



n=−∞

−jnω

(5.5)

The output variable of the ideal sampler can then be written as

∗

(t)=y(t)δ

∗

(t) y(t) ≡ 0,t <0 (5.6)

Substituting δ

∗

(t) from (5.5) into equation (5.6) yields

∗

(t)=

2π

y(t)

∞



n=−∞

−jnω

(5.7)

5.1 Computer Controlled and Sampled Data Systems 193

∗

(t)=

y(t)

∞



n=−∞

−jnω

(5.8)

The Fourier transform of this function if y(0) = 0 is given as



∞

∗

(t)e

−jωt

dt =



∞

y(t)e

−jωt



∞



n=−∞

−jnω



dt (5.9)

Next, the following holds

∗

(jω)=

∞



n=−∞



∞

−j(ω+nω

y (t)dt (5.10)

∗

(jω)=

∞



n=−∞

Y [j(ω + nω

)] (5.11)

as for the Fourier integral holds



∞

y(t)e

−j(ω+nω

dt = Y [j(ω + nω

)] (5.12)

Substituting s for jω in equation (5.11) gives

∗

(s)=

∞



n=−∞

Y (s +jnω

) (5.13)

It is clear from (5.11) that the transform Y (jω) is only nω

shifted form of

the function y(t). If the spectral density function of the variable y(t)is|Y (jω)|

(see Fig. 5.7) then the spectral density of the sampled signal y

∗

(t) is given as

∗

(jω)| =



∞



n=−∞

Y [j(ω + nω

)]



(5.14)

-ω

|Y(jω)|

Fig. 5.6. Spectral density of a signal y(t)

194 5 Discrete-Time Process Models

|Y*(j

Fig. 5.7. Spectral density of a sampled signal y

∗

(t)

Two cases can occur. In the ﬁrst one the frequency ω

is smaller than or

equal to half of the sampling period, hence

≤

(5.15)

In this case the spectral density of |Y

∗

(jω)| is composed of spectra of

|Y (jω)| shifted to the right of nω

and that are non-overlapping.

The second case occurs if the frequency ω

is larger than half of the sam-

pling period

Here the spectral density of |Y

∗

(jω)| consists of spectra |Y (jω)| shifted to the

right of nω

and overlapping. Hence, the spectral density of the signal |Y

∗

(jω)|

is distorted.

The previous analysis has shown that it is imperative for non-overlapping

spectral densities that the sampling period obeys the relation

≥ 2ω

(5.16)

Overlapping of the spectra can be removed when a suitable anti-aliasing ﬁlter

is used for the original continuous-time signal before sampling.

The sampling period choice is rather a problem of experience than some

exact procedure. In general is has a strong inﬂuence on dynamic properties of

controlled system, as well as on whole closed-loop system.

Consider a dynamical system of the ﬁrst order of the form

dy(t)

+ y(t)=u(t) (5.17)

where y(t) is the output variable, u(t) is the input variable, and T is the process

time constant. The sampling period can be chosen based on the relation

5.2 Z – Transform 195

Consider a dynamical system of the second order of the form

y(t)

+2ζT

dy(t)

+ y(t)=u(t) (5.18)

where T

is its time constant andζ the damping coeﬃcient. The sampling

period is usually chosen within the interval

0.25 ≤

< 1.50.7 ≤ ζ ≤ 1 (5.19)

If ζ>1 then an approximate model of the form (5.17) can be used. If

a closed-loop system is considered, the overall damping coeﬃcient should be

larger than 0.7.

5.2 Z – Transform

0 t

()

∗

()

∗

Fig. 5.8. An ideal sampler and an impulse modulated signal

Let us again consider an ideal sampler shown in Fig 5.8. This sampler

implements transformation of a continuous-time signal f(t) to an impulse

modulated signal. Individual impulses appear on the sampler output in the

sampling times kT

, k =0, 1, 2,... and are equal to functions f (kT

), k =

0, 1, 2,.... This impulse modulated signal containing a sequence of impulses

is denoted by f

∗

(t). The signal f

∗

(t) can be expressed as

∗

(t)=

∞



k=0

f(kT

)δ(t −kT

) (5.20)

196 5 Discrete-Time Process Models

The Laplace transform of this function is

L{f

∗

(t)} = F

∗

(s)=

∞



k=0

f(kT

−kT

(5.21)

Let us introduce a new variable in equation (5.21)

z =e

(5.22)

Then we can write

L{f

∗

(t)} =

∞



k=0

f(kT

−k

(5.23)

Z-transform of the signal f(t)orf

∗

(t) is deﬁned by the expression

F (z)=Z{f (t)} =

∞



k=0

f(kT

−k

(5.24)

Z{f(t)} = F

∗

(s)|

z=e

(5.25)

Z-transform is mathematically equivalent to the Laplace transform and diﬀers

only in the argument. Z-transform exists only if some z exists such that the

series in (5.24) converges.

Consider the unit step function 1(t) deﬁned as

1(t)=

1,t≥ 0

0,t<0

(5.26)

Sampled unit step function is given as

f(kT

)=1,k≥ 0 (5.27)

and thus

Z{1(t)} =

∞



k=0

−k

(5.28)

If |z| > 1 then the series in (5.28) is convergent and Z{1(t)} is

Z{1(t)} =

1 −z

−1

(5.29)

Z{1(t)} =

z −1

(5.30)

5.2 Z – Transform 197

Consider now an exponential function

f(t)=e

−at

1(t) (5.31)

where a =1/T and T is a time constant. Then

L{f

∗

(t)} =

∞



k=0

−akT

−kT

(5.32)

Its Z-transform is given as



−at

1(t)



∞



k=0



−aT

−1



(5.33)



−1

−aT



< 1 then



−at

1(t)



1 −z

−1

−aT

z − e

−aT

(5.34)

Z-transforms of other functions can be calculated in a similar way. Ta-

ble 5.1 lists some functions and their Laplace and Z transformations.

Z-transform is a linear operation.

Z-transform of a delayed function is given as

Z{f(t −kT

)} = z

−k

F (z) (5.35)

where k is a positive integer and f(t)=0fort<0.

For an initial function value holds

lim

k→0

f(kT

) = lim

z→∞

z −1

F (z) (5.36)

For a ﬁnal function value holds

lim

k→∞

f(kT

) = lim

z→1

(1 −z

−1

)F (z) (5.37)

Reciprocally, given the Z-transform of a function, we can ﬁnd the function

values in sampling times. The inverse Z-transform is symbolically written as

−1

{F (z)} =[f (0),f(T

),f(2T

),...] (5.38)

−1

{F (z)} =[f (kT

)] = f

∗

(t) (5.39)

However, using the inverse Z-transform, only values of f in sampling times

are found. It is not possible to ﬁnd the sampling time. Also, a unique function

F (z) can be found for several continuous-time functions f(t).

Discrete functions can practically be found from Z-transforms using the

polynomial division or the expansion as a sum of partial fractions.