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

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6.3 Least Squares Methods 249

s−1

s +3s+2

Transfer Fcn

data

To Workspace

x’ = Ax+Bu

y = Cx+Du

State−Space1

x’ = Ax+Bu

y = Cx+Du

State−Space

rls

S−Function

Parameters

Band−Limited

White Noise

yf’’, −yf, −yf’

uf, uf’

Fig. 6.11. Simulink schema for estimation of parameters of the second order

continuous-time system

0 2 4 6 8 10

−1

Param

Fig. 6.12. Estimated parameter trajectories for the second order continuous-time

system

s−1

s +3s+2

Transfer Fcn

Terminator1

Terminator

Numerator

Denominator

C−identification

SISO

Continuous identification

Band−Limited

White Noise

Fig. 6.13. Alternate Simulink schema for parameter estimation of the second order

continuous-time system using blocks from library IDTOOL

250 6 Process Identiﬁcation

6.4 References

There are many books dealing with system identiﬁcation. Fundamental are:

P. Eykhoﬀ. Trends and Progress in System Identiﬁcation. Pergamon Press,

Oxford, 1981.

L. Ljung. System Identiﬁcation: Theory for the User. MIT Press, Cambridge,

Mass., 1987.

L. Ljung and T. S

oderstr

om. Theory and Practice of Recursive Identiﬁcation.

MIT Press, Cambridge, Mass., 1983.

T. S

oderstr

om and P. Stoica. System Identiﬁcation. Prentice Hall, Inc., 1989.

H. Unbehauen and G. P. Rao. Identiﬁcation of Continuous Systems. North-

Holland, Amsterdam, 1987.

The least squares method was derived by Gauss two centuries ago:

K. F. Gauss. Theoria motus corporum celestium. English translation: Theory

of the Motion of the Heavenly Bodies, Dover, New York, 1963, 1809.

For further reading we also recommend:

V. Strejc. Approximation aperiodischer

Ubertragungscharakteristiken. Regelung-

stechnik, 7:124 – 128, 1959.

K. J.

Astr

om. Introduction to Stochastic Control Theory. Academic Press,

New York, 1970.

G. J. Bierman. Factorization Methods for Discrete Sequential Estimation.

Academic Press, New York, 1977.

V. Bob´al, J. B

ohm, and J. Fessl. Digital Self-tuning Controllers. Springer

Verlag, London, 2005.

T. R. Fortescue, L. S. Kershenbaum, and B. E. Ydsie. Implementation of

self tuning regulators with variable forgetting factors. Automatica, 17:

831–835, 1981.

G. C. Goodwin and K. S. Sin. Adaptive Filtering Prediction and Control.

Prentice Hall, Inc., Englewood Cliﬀs, New Jersey, 1984.

P. Hudzoviˇc. Identifcation and Modelling.ESSV

ST, Bratislava, 1986. (in

Slovak).

B. Roh´al’-Il’kiv. System Identiﬁcation.SjFSV

ST, Bratislava, 1987. (in Slo-

vak).

M. Fikar and J. Mikleˇs. System Identiﬁcation. STU Press, 1999. (in Slovak).

The method LDDIF used for identiﬁcation in this book is in more detail

described in:

R. Kulhav´y and M. K´arn´y. Tracking of slowly varying parameters by di-

rectional forgetting. In Proc. 9th IFAC World Congr., Budapest, pages

79–83, 1984.

6.5 Exercises 251

S. Bittanti, P. Bolzern, and M. Campi. Exponential convergence of a modiﬁed

directional forgetting identiﬁcation algorithm. Systems & Control Letters,

14:131–137, 1990.

L’ .

Cirka and M. Fikar. Identiﬁcation tool for Simulink. Technical report

KAMF9803, Department of Process Control, FCT STU, Bratislava, Slo-

vakia, 1998.

Identiﬁcation Toolbox for Simulink IDTOOL that contains blocks simplifying

identiﬁcation of continuous and discrete time singlevariable and multivariable

systems is freely available from the authors web site http://www.kirp.chtf.

stuba.sk/~fikar/.

6.5 Exercises

Exercise 6.1:

Explain why it is possible to approximate time constant T for the ﬁrst order

system using the tangent line from Fig. 6.1.

Exercise 6.2:

Approximate the periodic system step response in Fig. 6.3 using the Strejc

and Broida methods.

Exercise 6.3:

Derive equations (6.40), (6.46) by the Strejc method.

Exercise 6.4:

Derive the Broida method of the step response approximation.

Exercise 6.5:

Derive relations (6.95) a (6.96) in recursive least squares method.

Exercise 6.6:

Design an identiﬁcation scheme for processes with transfer functions of the

form:

−1

1+a

−1

−2

(6.120)

(s)=

(Ts+1)

(6.121)

The Control Problem and Design

of Simple Controllers

Generally speaking control system design involves a series of steps that pro-

duce a control system with a satisfactory behaviour of a controlled process. If

a fairly comprehensive list of steps is to be considered, a step by step design

is as follows:

1. Study of the controlled process and deﬁnition of control objectives.

2. Process modelling.

3. Determination of process properties and behaviour.

4. Selection (pairing) of manipulated and controlled variables.

5. Determination of structure of the control system.

6. Determination of the regulator type.

7. Design of performance speciﬁcations based on control objectives.

8. Controller design.

9. Analysis of the closed-loop system.

10. Simulations of the closed-loop system, repeat from 2 if necessary.

11. Implementation of the controller on chosen hardware and software.

12. Testing and validation on-line.

The aim of this chapter is to explain steps 7 and 8, why control is used,

and what are its speciﬁcations and demands. Next, structure and design of

simple controllers are introduced, performance speciﬁcations are discussed. It

is supposed that the controlled system is suﬃciently precisely described by a

linear model and the controller is linear as well.

7.1 Closed-Loop System

A general scheme of a closed-loop system is shown in Fig. 7.1a and consists of

four parts: controller, actuator, controlled process, and measurement device.

To simplify the scheme, the actuator and the measurement device are usually

included into the controlled process. In that case, the scheme in Fig. 7.1b is

obtained.

254 7 The Control Problem and Design of Simple Controllers

Controller Actuator Process Measurement

−

- - - - e-

- -

R(s) G(s)

e- - e-

−

we u

Fig. 7.1. Detailed and simpliﬁed closed-loop system scheme

The signals in this scheme are w – setpoint, e – control error, u – ma-

nipulated variable, d – disturbance, and y – controlled output. The controller

is described by the transfer function R(s) and the controlled process by the

transfer function G(s).

The study of dynamical properties of closed-loop systems involves re-

sponses to setpoint and disturbance variables. Applying block diagram algebra

to ﬁg. 7.1b yields for the output variable

Y (s)=D(s)+R(s)G(s)[W (s) − Y (s)] (7.1)

Hence,

Y (s)=

1+R(s)G(s)

D(s)+

R(s)G(s)

1+R(s)G(s)

W (s) (7.2)

From this expression follows for regulation and tracking:

1. For D(s) = 0 is the transfer function between w and y given as

(s)=

R(s)G(s)

1+R(s)G(s)

(7.3)

2. For W (s) = 0 is the transfer function between d and y given as

(s)=

1+R(s)G(s)

(7.4)

Both transfer functions contain common partial transfer function S(s)ofthe

form

S(s)=

1+G

(7.5)

7.1 Closed-Loop System 255

that is called the closed-loop sensitivity and G

= RG.IfwesetW (s)=

0,D(s) = 0, disconnect the loop at any place, and deﬁne new output and

input signals y

, y

(Fig. 7.2) we get transfer function

= −GR = −G

(7.6)

We can see that the transfer function G

represents dynamics of the open-loop

system.

R(s) G(s)

e- - e-

−



Fig. 7.2. Open-loop system

7.1.1 Feedback Control Problem Deﬁnition

The control problem design constitutes one of the most important tasks in the

process control. If we put aside the problem of controller structure selection

and technical realisation of the closed-loop system, the problem reduces to

determination of the controller for the given controlled process.

The requirements on closed-loop system are as follows:

1. Stability.

2. Disturbance rejection.

3. Controlled variable should follow the setpoint as fast and exact as possible.

4. Robustness to parameter changes.

The items 2 and 3 can mathematically be described as

(s) = 0 (7.7)

(s) = 1 (7.8)

Example 7.1: Non-realisability of an ideal controller

Consider for simplicity a controlled process described by a ﬁrst order dif-

ferential equation and with the transfer function of the form

G(s)=

Ts+1

and a controller as another dynamical system with the transfer function

R(s). An ideal controller according to (7.7), (7.8) is then given from

256 7 The Control Problem and Design of Simple Controllers

Ts+1

R(s)

Ts+1

R(s)

=1,

Ts+1

R(s)

We can see that both conditions cannot be fulﬁlled for any controller.

Their validity is only for R(s) →∞. However, such a controller would be

unsuitable because:

• an arbitrarily small control error would cause a very high value of the

control signal. However, minimum and maximum values of the control

signal always exist in practice. This would mean a bang-bang type of

controller.

• the same gain would be applied to disturbances.

• the controller gain would be too high and instability of the closed-loop

system could occur for higher order systems.

7.2 Steady-State Behaviour

The steady-state behaviour of the closed-loop system describes its properties

after all transient eﬀects have ﬁnished. To analyse the control accuracy, we

analyse the value of the steady-state control error. For purposes of the analysis

assume the open-loop transfer function of the form

(s)=

(s)

−T

(7.9)

where

(s)=b

+ b

s + ···+ b

(7.10)

(s)=1+a

s + ···+ a

−k

≥ m

(7.11)

and k is a nonnegative integer deﬁning the system type. Its increased value

adds pure integrators to the system. Based on the stability analysis it can be

shown that it is desirable to have the smallest gain Z

and the smallest k in

the open-loop system.

For the steady-state behaviour examination assume that the A

(s) is sta-

ble.

If we set

E(s)=W (s) −Y (s) (7.12)

then from (7.2) follows

E(s)=

1+G

(s)

(W (s) −D(s)) (7.13)

This relation shows that impacts of disturbances and setpoints are (up to sign)

the same and that it is not necessary to study them both.

7.3 Control Performance Indices 257

Let us assume that the variable y

acting as the open-loop system input

(see Fig. 7.2) is given as y

(t)=y

1(t), i =0, 1, 2 (step,ramp,...) with the

Laplace transform Y

(s)=y

i!/s

i+1

. The control error is then given as

E(s)=

1+G

(s)

(s) (7.14)

Its steady-state value is in the time domain given by the Final value theorem

lim

t→∞

e(t) = lim

s→0

sE(s) = lim

s→0

k−i

+ Z

(7.15)

It is then clear that steady-state control error is a function of the input sig-

nal and the open-loop system properties: gain Z

and the coeﬃcient k.For

the zero steady-state control error holds k>i, i. e. the higher order of the

input variable, the more integrators the open-loop system should contain. If

k = i, non-zero steady-state control error results and its value is inversely

proportional to the gain Z

. As the gain is given as the product of gains of

the controlled system and the controller, the lowest value of the steady-state

control error is attained with the highest controller gain. Hence, the control

design problem is in opposition to requirements on stability. The least desired

state is k<iwhen e(∞)=∞.

7.3 Control Performance Indices

The third requirement on closed-loop system in Chapter 7.1.1 dictates that the

controlled output follows the reference value as precisely as possible. Several

performance indices can describe this issue more quantitatively.

7.3.1 Time Domain

For performance speciﬁcations in time domain it is suitable to choose a stan-

dard input variable behaviour and to evaluate the controlled output. Most

often a step response is chosen. A typical closed-loop output trajectory is

shown in Fig. 7.3.

Performance quantities characterising the transient response of the closed-

loop system are then as follows:

• Maximum overshoot e

max

deﬁnes (in percent) the normalised maximum

control error after the process output reaches setpoint for the ﬁrst time. It

can mathematically be written as

max

− y

∞

100 (7.16)

If the transient response is aperiodic and without overshoot then e

max

=0.

The recommended value for majority of cases is smaller than 25%.

258 7 The Control Problem and Design of Simple Controllers

max

100

e(∞)

Fig. 7.3. A typical response of a controlled process to a step change of a setpoint

variable

• Settling time T



is given as the time after which the absolute value of

the control error remains in some prescribed region. Usually the region

is characterised in percent of the maximum control error with the mean

value being the process output new steady state. The recommended value

is about 1% − 5%. The lower value of T



, the better control performance

results.

• Damping ratio – ratio of the ﬁrst and the second maximum value of the

controlled output. The value of 0.3 or less can be suitable.

• Time T

determines the point obtained as the intersection of the time axis

and tangent of the response at the inﬂexion point.

• Time T

determines the point obtained as the intersection of the settling

value of the controlled output with the tangent of the response at the in-

ﬂexion point. Both times T

are used in identiﬁcation from the process

step response.

• Rise time is the time required for the response to rise from 0% to n%of

its ﬁnal value. For underdamped responses time T

100

is considered (time

when the output reaches its ﬁnal value for the ﬁrst time. For overdamped

systems time T

is commonly used as time between 10% and 90% of

the new steady state output. However, a too small value can cause high

overshoot and settling time.

• Dominant time constant T

DOM

can be thought of as an alternative to

the settling time and can be deﬁned as the largest time constant of the

process. If for example the ﬁrst order system contains a term e

−γt

then the

dominant time constant is T

DOM

=1/γ. For higher order systems T

DOM

can be approximated from a similar second order system with a damping

7.3 Control Performance Indices 259

coeﬃcient ζ and the natural undamped frequency ω

.Inthiscaseitcan

be shown that

DOM

≥

ζω

(7.17)

• Steady-state control error e(∞) deﬁnes deviation between the setpoint

w and the controlled output in the new steady state. It is non-zero for

controllers without integral action.

Time indices deﬁne speed of control, other characterise control quality.

Similar characteristics can be deﬁned for disturbance rejection (Fig. 7.4).

max

Fig. 7.4. A typical response of the controlled process to a step change of disturbance

7.3.2 Integral Criteria

A large number of control quality speciﬁcations can lead to a diﬃcult control

design. Usually, it is easier to deﬁne a single performance index. One possibility

is for example to deﬁne a weighted mean value of several quantities, e. g.

I = a

+ a

max

+ a



(7.18)

and then to minimise the index I. An issue then remains how to choose weight-

ing coeﬃcients a

,....

Another possibility is to use integral criteria when an area (or some func-

tion of it) of control error is minimised. The advantage of using integral criteria

is that the entire transient response is evaluated, not only some of its charac-

teristics. From the mathematical point of view a general cost function can be

deﬁned as



∞

[e(t)]dt (7.19)

where the function f

can contain various terms. Suitable candidates should

lead to a higher value of I

when the control quality deteriorates.

Most often, the following functions are used: