A new scheme for identification and control

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Zhu, Y., Driessen, M. H., & Damen, A. A. H. (1988). A new scheme for identification and control. (EUT report. E, Fac. of Electrical Engineering; Vol. 88-E-213). Technische Universiteit Eindhoven.

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I dentification and Control

by ZHU Yu-Cai M.H. Driessen A.A.H. Damen P. Eykhoff

EUT Report 88-E-213 ISBN 90-6144-213-3 December 1988

ISSN 0167- 9708

Faculty of Electrical Engineering

Eindhoven The Netherlands

A NEW SCHEME FOR IDENTIFICATION AND CONTROL

by

ZHU Yu-Cai

M.H. Driessen

A.A.H. Damen

P. Eykhoff

EUT Report 88-E-213

ISBN 90-6144-213-3

Eindhoven

December 1988

A.A

.H.*

t::]

I

P.

1.tf1Z

CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG

New

A new scheme for identification and control/by Zhu Yu-Cai,

M.H. Driessen, A.A.H. Damen and P. Eykhoff. - Eindhoven:

Eindhoven University of Technology, Faculty of Electrical

Engineering. - Fig. - (EUT report. ISSN 0167-9708; 88-E-213)

Met lit. opg., reg.

ISBN 90-6144-213-3

SISO 656.2 UDC 519.71.001.3 NUGI832

Trefw.: systeemidentificatie.

CONTENTS

Abstract 1. Introduction

2. The Problem and Old Solution

2.1. Manual Control 2.2. Identification of Processes in Closed Loop 1 2 3 3 3. The New Scheme for Identification and Control 6

3.1. A Two Step Scheme to Identification

and Control for Processes 7

3.2. Some Analysis of Identification Results 9

3.3. The Optimality of the Two-Loop System

4. 5. Structure A Case Study 4.1. Identification Results 4.2. Results of Control Conclusions Acknowledgement References 16 18 19 21 28 29 29

Zhu Yu-Cai, M.H. Driessen, A.A.H. Darnen and P. Eykhoff

A NEW SCHEME FOR IDENTIFICATION AND CONTROL.

Faculty of Electrical Engineering, Eindhoven University

of Technology, The Netherlands, 1988. EUT Report 88-E-2l3

Address of the Authors:

Measurement and Control Group, Faculty of Electrical Engineering, Eindhoven University of Technology,

P.O. Box 513,

5600 ME EINDHOVEN,

The Netherlands

The first author is on leave from the Xi'an Jiaotong University, Xi'an, China.

ABSTRACT

ZHU Yu-Cai, M.H. Driessen A.A.H. Damen and P. Eykhoff

A new scheme for identification and control is proposed. This method

can overcome such difficulties as: the process in open loop is

un-stable or nearly unun-stable, the process has some nonlinearity and some

time variation. In these cases open loop identification with a

lin-ear and time-invariant model set may be very difficult.

So the idea is: first use a primary controller for stabilizing the

loop, then identify this closed loop system; finally design a

con-troller in the second loop in order to optimize the system. It will

be shown that by this method one will obtain a better model during the identification procedure, and the two-loop system will maintain

the optimality under a condition that is easy to check. For

vali-dating the idea, a case study is given.

1 . INTRODUCTION

The class of processes considered here is assumed to be almost linear

and almost time-invariant. This implies that a linear time-invariant

dynamic model will give a reasonable approximation of the process be-haviour, so linear control theory and design techniques are

applic-able to this class of processes. A typical example of this class is

an industrial process operating around a fixed working point. The problem at hand is to control the process, in order to improve or

op-timize the performance of the whole system. The approach consists of

model-building by identification, and then controller design based on

the estimated model. Identification and controller design are two

aspects of the same problem, rather than two separate problems. The

"new scheme", discussed in this paper, means a new way of integrating identification and design procedures.

Often identification experiments are performed on systems operating

in closed loop, that is, under output feedback. For industrial

pro-cesses, in many cases, safety or production-economic reasons do not permit open loop experiments, especially when the process is unstable

or nearly unstable. We will study the problem mainly for such

situ-ations. In section 2, the conventional approaches to the problem

will be briefly reviewed. In section

3,

the new solution to the

pro-blem will be proposed and discussed. In order to validate this new

method, a case study will be given in section 4. Conclusions and

discussions will be given in section 5.

Although the identification and control problem will be treated form-ally by linear system theory, it is more realistic to allow the as-sumption that the true process has some nonlinearity and some time

variation. To explain the idea only single-input and single-output

(SISO) processes are used. But these theories have been proved for

multi-input mUlti-output (MIMO) processes too. This means that our

method is also suitable for MIMO processes.

2. THE PROBLEM AND OLD SOLUTIONS

We shall start with the situation that the open loop identification experiment is not allowed, due to the fact that the process is unsta-ble or nearly unstaunsta-ble, und/or the process output has too big a va-riance. We will review some of the conventional solutions to the

problem, and point out son~ of the difficulties when using these methods. Fig. 2.1

il'l'Ut

r - - - ,

noise

+

+

1---1>1 Process 1--i:>I

Experiment with mnnual COIltrol

2.1 Manual Control

When the process dynamics are slow, an experienced operator can con-trol the process while the testing signal is added at the process

in-put. The human operator can be seen as a feedback controller who

stabilizes the process and keeps the output variance small. This is

shown in Fig. 2.1.

The process input-output data are collected for calculating the pro-cess dynamics.

This method of experiment is unreliable or even impossible if the process has fast dynamics.

2.2 Identification of Processes in Closed Loop

This scheme is shown in Fig. 2.2, where

v

u

p

y

c

Fig. 2.2 Process identification in closed loop

P denotes the dynamics of the process, C denotes the controller.

The closed loop will be called 'system'. The signal u is called

'process input' and signal w is called 'system input'; y is at the same time the process and system output; and v is the process dis-turbance acting at the process output.

In practice, the system input w is the testing signal (an extra ex-citation) applied to the system in order to achieve a good

identifi-cation result. We will assume that w exists and is also measurable.

testing signal is absent, the excitation will be too low to guarantee the accuracy of the estimation, in most of the practical situations. Identification of processes when systems operate in closed loop is

called closed loop identification. This topic has attracted a lot

of interest and a great deal of literature has been dedicated to

this problem. This special area of identification has been studied

and surveyed thoroughly by Gustavsson, Ljung and Soderstrom (1976,

1981). The discussion here will be based on their results.

There are mainly three approaches to closed loop identification.

The first is called direct identification. This method treats the

process input-output data exactly as if they were obtained from the open loop experiment.

If the controller is linear and time-invariant, then an indirect way

of estimating the process parameters can be applied. The closed

loop system can be regarded as a whole and its parameters can be

de-termined by some method. In principle the open loop process

parame-ters may then be determined from the estimated closed loop system

pa-rameters using the knowledge of the controller. This second approach

is called indirect identification.

The third method treats the process input and output jointly as the

output of a system driven by noise only. This is called joint

input-output identification.

For each of these three approaches, different estimation methods can be used, such as correlation and spectral analysis, various paramet-ric estimation methods, etc.

The basic problems related to closed loop identification are process

identifiability and accuracy aspects. The question of

identifiabil-ity is whether or not the process characteristics can be obtained as the number of data tends to infinity; see Gustavsson et al. (1977,

1981) for the proper definition. When process identifiability is

as-sured, the next problem is the accuracy, i.e., the quality of the es-timates for a finite number of data.

Compared to an open loop experiment, the main problem caused by feed-back in the closed loop experiment for process identification is that the process input u is correlated with the disturbance v.

This correlation influences the process identifiability and the accu-racy of the estimation result. More precisely, it makes things

worse. For direct identification, ordinary correlation and spectral

analysis will not yield identifiability; the consistency of the transfer function estimate is not guaranteed if an independently pa-rametrized noise model is too simple, e.g., output error method. For

joint input-output identification, the same difficulties exist. For

indirect identification, the dynamics of the closed loop system is operating in "open loop", the system input w is independent of the disturbance v, see Fig. 2.2, therefore, no restriction regarding the estimation methods have to be introduced; for example, correlation

methods and the output error method will work in this case. In the

second phase of indirect identification the process dynamics is solv-ed from the clossolv-ed loop system dynamics and from the knowlsolv-edge of the

controller. This can be complex and it mayor may not be

success-ful.

Concerning identifiability and accuracy, the best way for closed loop identification, shown by Gustavsson et al. (1977, 1981), is the direct identification used with a prediction error estimation method,

that assumes a causal relationship between the process input and

out-put. "If they fail, no other approach will succeed" (Ljung, 1987).

In general, prediction error identification methods need a nonlinear minimization algorithm for computing the estimates, except in the

case of the equation error least-squares method. A necessary

condi-tion for the success of the algorithm is that the true process and

the intermediate models are all stable. This condition is violated

if the process is unstable or nearly unstable, which is often the

reason for closed loop experiments in practice. Also some

nonlinear-ity and/or time-variation may cause the algorithm to fail. We see that even the be:::;t methoci tOl: closed loop identification can easily fail in practice.

Some of the above discussions will be analysed in the next section. Summarizing, the following table provides a picture of the problems and difficulties of closed loop identification in practice.

Problems Status of the solutions

safety & economy

of experiment solved

model accuracy conflict with control*

divergency of the algorithms caused by

instability unsolved

nonlinearity for

Erocess identification unsolved

time-variance for

I2rocess identification unsolved

Table 2.1 Problems and difficulties of closed loop

identification in order to obtain a process model

*

Refer to equation (3.27)

This table shows that many difficult problems remain unsolved when closed loop identification is used for obtaining a process model.

This is the moment to ask the question: For control system design,

is it really necessary to identify the I2rocess dynamics when the

sys-tem is operating in closed loop? The idea of the "new structure" for

solving identification and control system design problem, is inspired by this question.

3 . THE NEW SCHEME FOR IDENTIFICATION AND CONTROL

It is well-known that feedback control can offer us the following be-nefits, cf. Ashworth (1982) and Klein and van Zelst (1967):

decreasing the effect of nonlinearity; decreasing the effect of time-variation; attenuation of the effect of disturbances;

stabilizing the unstable process or improving the (relative) stability;

achieving desired dynamic characteristics.

All these properties of feedback control are concerned with the

clos-ed loop system, not the original process. Consequently it is clear

that feedback will improve the result of closed loop system

identifi-cation, if linear identification techniques are used. These nice

properties of feedback control are not helpful for the process iden-tification, as explained in the last section.

Based on this observation, we propose the following solution to the problem:

3.1 A Two Step Scheme to Identification and Control for Processes

The procedures

1. closed loop experiment;

2. identification of the closed loop system dynamics;

3. designing the second loop controller for the closed loop system,

based on the closed loop system model, in order to improve or to optimize the performance of the system.

The procedures are shown in Fig. 3.1 and Fig. 3.2

Fig. 3.1

w

-+

p

C1

+

+

-I I I I I

!..---'=r---

J

y

Identification of the closed loop system

Fig. 3.2 Remarks

p

+

+

v

---,

I I I I

C1

:

L _ _ _ _ _ _ _ _ _ _ _ _ _ _

J

C2

y

Design the second loop controller for the closed loop

a. The first loop controller C_l is used to stabilize and/or to

re-duce the output variance and/or to rere-duce the effects of

non-linearity and the time variance. In practice, C

l is

deter-mined based on some rough a priori knowledge of the process dy-namics, which is often available.

b. The identification part is only half of the indirect

identifica-tion method menidentifica-tioned in the last secidentifica-tion; due to lack of a good name, we call it "blind identification" meaning that it ignores the dynamics of the original process and the controller,

al-though the latter should be known in practice; only the system

dynamics is taken into account.

c. The closed loop system is operating in "open loop", so that the

system input w is independent of the disturbance v. Therefore

good statistical properties of identification of the closed loop system is guaranteed, the estimation methods such as

spec-tral analysis method and output error method will work.

Remem-ber that they will usually fail for obtaining process models in

closed loop identification. It is clear that all the benefits

of feedback control are helpful to to closed loop system identi-fication.

d. Conditional to the proposed identification is a sufficiently rich system input signal w which is not always necessary for the process identification in the closed loop.

e. The implementation of the second loop controller C

2 is not

dif-ficult: if the two controllers are implemented in the same

com-puter, the new controller is C

1 + C2. If the first loop

con-troller C₁ already exists, giving reliable operation, one does

not need to remove it; the second loop controller can be

implem-ented in the computer. If the computer fails, the system goes

back to the old control, no disaster will occur.

3.2 Some Analysis of Identification Results

In this part, we assume that the process is stable and operating in a closed loop, so the closed loop identification for obtaining

pro-cess model can work. Now the accuracy aspect of the identification

results will be discussed.

The direct identification approach is used with a prediction error

method to obtain a process model. Also in "blind identification"

for identifying method is used.

w

closed loop system dynamics, the prediction The situation is shown in Fig. 3.3.

r1~~---'

I

e

I

I

+

_p

y

C1

Fig. 3.3 Process identification and closed loop system

identification.

Let a linear model of the discrete-time dynamic process be described by

y(t) G(q)u(t) + vItI

L

g(k)u(t-k) + vItI (3.1)

k~l

Here, y(t) and u(t) are the process output and input at time t,

res-- 1

pectively, q is the backward shift operator, vItI is an additive

disturbance. With this model we associate the transfer function

g(k)e- ikOl,

and i t is well-known that this function plays an important role in control system design.

(3.2)

With some abuse of the notation, we denote Go (eiOl) as the true

trans-fer function of the process P. The output disturbance {v(t)} is

as-sumed to be a zero-mean stationary stochastic process with spectral

density <l>v (00) •

Suppose that the impulse response coefficients g(k) have been deter-mined somehow using the data sequence

ZN ~ (y(l)u(l) . . . y(N)u(N».

Denote the corresponding transfer function by GN(eiOl) to emphasize

that it is an estimate based on N measurements.

(3.3)

The data sequence is assumed to be a realization of a random variable

which, in turn, means that the estimate GN(eiOl) is a random variable.

If the expected value E GN(eiOl) does not give a correct description

of the system, we say that the estimate is biased and denote the bias by

Then, the mean square error of the estimate can be expressed by

This error can be used to measure the accuracy of the estimate.

Prediction error method and asymptotic results for the estimates

Postulate a set of n-th order candidate models.

y(t)

=

G(q,S)u(t) + H(q,S)e(t)

S

~

DnGRd

(3.5)

(3.6)

(3.7)

where 8 is the parameter vector of the model set, Dn is the parameter

space, H(q,8) is the stable and invertible noise model, (e(t» is a

sequence of white, zero mean noise with variance

A.

For a fixed

va-lue of 8, G(q,S) and H(q,8) denote a particular model.

Let y(ti 8) denote the one-step ahead prediction according to the mo-del (3.7)

y(tIS)

=

H-1 (q,S) G(q,S)u(t) + ll-H- 1 (q,S)ly(t)

and let E(t,8) be the prediction error

E(t,8) =

y(t)-~(tIS)

= H-'(q,Sj[y(t) - G(q,S)u(tll

Then, the prediction error method is to calculate the estimate of 8 by

arg min

1.

I

E 2

N t=l (t,S)

The corresponding transfer function estimates are

GN(eiOl) HN(eiOl) iro ,.. G(e ,SN) 'Ol -H (e~ ,SN)

}

(3. L) (3.9) (3.10) (3.11)

In order to let the model set be large enough to contain the true process dynamics (if P is linear and time-invariant), or to give a

good approximation of the true process dynamics (if P has some

non-linearity and/or time variation), we will allow the model order n to

increase when the number of data N increases. This can be formally

expressed by the following assumptions

n ~ OQ as N ---t DO

2

n /N ~ 0 as N ~ =

where (3.11b) means that n is small compared to N.

Define the auto- and cross-spectral densities

<llu (00) ~ =

L

R ('t)e-i'too 't=-oo u

~

R ('t)e-i'too 't =_00 ue

where the following limits are assumed to exist:

1.

N lim E

L

u(t)u(t-'t) N~= N t~l

1.

N lim E

L

u (t ) e (t -'t ) N~= N t~l Rue('t) ~ 0 _{for 't} < 0

}

Assume also that the process input is persistently exciting:

C > <ll (00) > 0 u Theorem 3.1: (Ljung, 1987) (C is a constant) (3.lla) (3.llb) (3.12) (3.14) (3.14)

Given (3.1) - (3.14), and assuming that the process P is linear and

time-invariant. Then EG N (e ioo ) _{Go (e iffi)} (1 ) w.p. 1 as N~= (3.15 )

(2)

GN(eiOJ) EG_N (e iOJ)

(N

_{-->){( 0, P}

n (OJ)) HN (e iOJ) EHN (e iOJ)

as N --> ~

}

lim

i

P n (OJ) <l>v(OJ) <1>- , ( OJ) n--> ~

(3.16)

where H, (eiOJ) is the true noise filter, and

<I>(OJ) [ <I>u (OJ) <l>ue (OJ) (3.17)

o

The following discussion is based on this result.

Accuracy aspects of the process model and system model

Suppose that the closed loop experiment data is used with the predic-tion error method to identify both the process model and system mod-el, as shown in Fig. 3.3. And assume that the number of data N is large enough to let us use the asymptotic results in Theorem 3.1. Denote the closed loop system model by

and the real system dynamics by

(~ 1 + G, C, )

and correcting H, to H;: ~ SH" where H, is the true noise filter. The mean square error of the estimates will be used as the measure of accuracy.

Bias term. The first part of Theorem 3.1 says that the bias of the

estimate tends to zero when the number of data tends to infinity. In

turn this implies that if the identified dynamics is linear and time-invariant, the bias part of the model error can be neglected when the

condi-tion is often assured when the experiments are performed in closed

loop. In other words, for this class of processes, the bias of the

estimates are mainly caused by the nonlinearity and/or the time

vari-ation of the identified dynamics. Hence, if the process P has some

nonlinearity and time variation, they will cause bias if the linear time invariant model (3.1) is used to identify the process dynamics. But if we consider the closed loop system extended by the feedback,

i t will have less nonlinear - and less time-variant dynamics than the

original process has. Therefore, the process nonlinearity and time

variation will cause less trouble for the identification of closed

loop system dynamics where the bias term is concerned. This part of

the discussion is intuitive and heuristic.

Variance term. Heuristically, one could rewrite (3.16) as

If only the process model is considered, we have

var

_ 1<1> I'

ue

a) For closed loop system identification, ("blind

identifica-tion") .

The input-output data sequence used is

N

Zs ~ (y(l),w(l), , y(N),w(N»

and the equivalent system output disturbance is

Vs (t) ~ S (q)v_{o (t)} where

S(q) .~ 1

1 + Go (q)C 1 (q)

is called sensitivity function. Then

(3.18 ) (3.19) (3.20 ) (3.21) (3.22)

.

,

<l>vs(!O) ~ IS(e1!O)I <l>v(!O) (3.23)

It follows from (3.23) that I S(ei!O) I should be kept small for output

variance reduction; smaller I s(ei!O) I means better control. In

<l>we (OJ)

=

<l>ew (OJ)

o

Assume that the closed loop system model order is also taken to be

n. From Theorem 3.1, i t follows that

- I <I> _we I 151' [ • s iOJ

J

n var GN(e )

=-N <I> _w A (3.24 ) (3.25)

We see that better control (smaller 151) will reduce the variance of the system model, this improves the accuracy for closed loop system

identification. Therefore, when the closed loop system dynamics is

concerned, identification and control become mutually-supporting: good control of the first loop will decrease the variance of the sys-tem model, as well as the effects of nonlinearity and time variation; a better model of the closed loop system will further lead to a bet-ter second loop controller design (the problem of second loop con-troller will be discussed later).

b) For process identification in closed loop.

From the feedback relation

u(t) = w(t) - Cl(q)y(t) (3.26) we have u(t)

=

5(q)w(t) - 5(q)C_{l (q)H o (q)e(t)} Then (3.19) becomes 1

1;1'

(3.27)

From (3.27), we make the following observations:

(1) It is the system input (external excitation) that accounts for

the overall accuracy of the process transfer function estimate.

(2) Better control (smaller 151) will lead to poorer accuracy of the

process model. This is the well-known conflict (dilemma)

be-tween process identification and control.

Because different objects are concerned in closed loop system identi-fication and process identiidenti-fication, i t is hard to compare (3.25) with (3.27).

In the following table we see how the problems of identification are

solved for closed loop system identification. This should be

Problems Status of the solutions

safety & economy

of experiment solved

model accuracy mutually supporting with control*

divergency of the algorithms caused

by instability solved*

nonlinearity for

system identification difficulty reduced*

time-variation for difficulty reduced* system identification

Table 3.1 How the problems of identification are solved when

identifying the closed loop system dynamics. The

asterisks mean the benefits of the new scheme.

3.3 The Optimality of the Two-loop System Structure

After the model of the closed loop system has been identified, the

next step is to design the second loop controller C₂ based on the

system model (and, may be, also on the system noise model), see Fig.

3.2. Although it has been shown that the system model is usually

better than the process model, one might argue that the first loop controller C1 may introduce constraints for the second loop

control-ler. The question is: does the system in Fig. 3.2 loose the

opti-mality, in other words, can the optimal controller for the process be

realized by the two-loop structure? It will be shown here, under

what condition the two-loop system maintains optimality.

Let Go (q) denote the transfer function operator of the process, and

C1 (q) denote the one of the first loop controller Cl ; see (3.1) for

the definition. Then the transfer function operator of the first

s Go (q)

Go (q) - -:-l---'-+--=G~o --;(-=q7) """C'-l--;(-q ) (3.28 )

In practice, the controller design chooses a controller from the

class of controllers which stabilize the process. If the process

transfer function operator and controller operator are both real rat-ional, the set of all the stabilizing controllers can be parametrized by Youla's lemma, see Vidyasagar (1985).

Denote

S(Go ):- the set of all controllers which stabilize the process Go;

Similarly, denote

S (G~) .- the set of all controllers which stabilize the closed loop system G;

(3.29)

(3.30) Then i t is clear that the set of all controllers which stabilize the

process Go via the two-loop structure

C

l + S (G;)

We can now put the question in another Is i t true that

?

Theorem 3.2 (Vidayasagar, 1985).

Suppose Go (q) is real rational, C

1 (q) and + S (G;) _ S (Go) + S(G;) - _{S(G o )} is way:

S(Go ) (C 1 stabilizes Go)'

if and only if C

1 (q) is stable and proper.

(3.31 )

(3.32 )

(3.33) (3.34 )

Where a rational function is proper means that the degree of the de-nominator is greater than or equal to the degree of the numerator.

D

This theorem tells us that not all controllers in S(Go ) can be real-ized by the two-loop structure; the necessary and sufficient condi-tion to make such a decomposicondi-tion possible is that C

1 is proper and

practice. _{A process Go is said to be strongly stabilizable if S(Go )}

contains a stable controller. Go (q) is strongly stabilizable if and

only if the number of real poles (counting the multiplicity) of Go (q) between any pair of real zeros of Go (q) outside the unit circle is

even (where one also has to count the zero's at infinity); cf.

Vidyasagar (1985).

Summarizing, if the process Go is strongly stabilizable, by choosing C

1 to be proper and stable, the optimal controller for the process

can be realized by the two-loop structure.

In practice, the modelling error must be considered during the design step; then the two-loop system structure in our case is expected to

be superior to the one-loop structure. Concerning the robust

stabil-ity, when designing the second-loop controller, we are dealing with less restrictive gain and phase margins compared to the design of the single loop controller, because we have a better model for the closed loop system than for the process, this means that we gain more free-dom for improving the system performance.

4 . A CASE STUDY

Our case study is the identification and control of a ball balancing process which has been developed for research, educational and

demon-stration purposes; see Fig. 4.1.

~~

1 copper rail

---

2 metal ball I'~, 3 perspex tube

1

4 spindle 5 roll joining .~ 6 end switches ..?

C

7 servomotor 8 contraweight ; _•

.

< 9 frontside

10 turn axis (pivot)

11 flexible coupling

The metal ball rolls over a copper rail as a result of the change in

the angle of the rail. This change of angle is brought about by a

servomotor. A voltage is applied acroSS the rail from end to end and

the potential of the ball is detected as a measure for its position. With the aid of controllers implemented on a PC, the system can

dir-ect the ball to a desired position on the rail. The process input is

the voltage supplied to the amplifier of the servomotor, and the out-put is the ball position.

According to the physical modelling, the process has three poles close to the origin, and one real pole in the left half plane. The nonlinear part of the servomotor is modelled as a dead zone, which is measured roughly and compensated for by the PC.

Then, the physical model of the process is given by the following transfer function: 61 Go (s) ~ ~3--~~--s (s+8.35) 4.1 Identification Results (4.1)

All experiments are performed at the sample frequency 10 Hz.

Al-though this is too low to sample the model (4.1), experience has taught that the system is too sensitive to the disturbance if a

high-er sample frequency is used. The experiment time is 100 seconds,

hence the number of data N~lOOO. The later 500 samples are used for

identification, whereas the first 500 samples are used for model

va-lidation.

The identification calculation is done by using the System Identifi-cation Toolbox of PC-Matlab, written by L. Ljung.

Results from Manually Controlled Experiment Data

It is highly difficult to control the ball balancing process by hand, due to its triple integration behaviour. After some training,

manual control is possible. The white noise is used as the test

sig-nal, so the process input is the white noise plus the manual control action, see Fig. 4.2 (see the last part of this section).

After some experimentation, a 4th-order model with one unit

time-delay is chosen for the process. This is confirmed by the physical

model (4. 1) .

vali-dation by simulation; see Fig. 4.3. Prediction error methods fail to

converge. From Fig. 4.3 we see that the simulated output is quite

different from the measured output. The reason for this bad fit can

be:

(1) bad input signal due to the bad experiment condition, see Fig.

4.2;

(2) instability of the process;

(3) nonlinearity of the process; and

(4) numerical problem in simulation.

We know that (2) and (3) cause numerical problems for estimation al-gorithms.

When a simulation model has poles very close to the unit circle, as in our case, numerical problems can occur; see Astrom and Wittenmark

(1984) .

Results of Process Identification from Closed Loop Data

The experiment scheme is shown in Fig. 2.2. The process input-output

data is used for process identification. Fig. 4.4 shows the process

input and system input, where the system input is the white noise test signal with its variance being the same as the variance of the

test signal in the manual control experiment. The feedback

control-ler is a discrete PDD controlcontrol-ler and has the transfer function: 2

46.2q - 85.8q + 40

2

q

Again, the prediction error methods do not converge; the output error method seems to converges to a local minimum.

(4.2)

The process model has an order 4 with one delay. The output error

model is simulated, the simulated output is compared with measured

output, see Fig. 4.5. We see that the result is not satisfactory.

The reasons for this are (2), (3) and (4) as discussed for the manual control data.

Because the output error method and general prediction error method do not succeed for the process identification, a 4th-order ARX model

is used for equation error method identification. This model is

un-stable, and will be used for the single loop controller design.

Results of Closed Loop System Identification

The system input wand output yare collected for the system identi-fication, from the same closed loop experiment for process

identifi-cation. The feedback controller is the same PDD controller as in

If the process has an order 4, one might think that the system model

must be 6. Surprisingly, the results of identification show that

order 3 is high enough to model the system dynamics. For higher

orders zero/pole cancellation takes place. The reasons for this low

order can be (1) process nonlinearity has been reduced by feedback;

(2) fast dynamics of the process can not be recognized when sampling

at 10 Hz.

The system model, therefore, has order 3, with one delay; the

para-meters are estimated by an output error method. The simulated output

is compared with the measured output in Fig. 4.6 and the fit is

striking! This result should be compared with Fig. 4.5, where one

wants to identify the process. The crosscorrelation function between

the system input and output residuals is within the 95% confidence

interval; see Fig. 4.7. This means that the system input is

indepen-dent of output residuals, and the model quality is good.

4.2 Results of Control

The purpose of this subsection is to show that:

the new scheme of identification and control of processes works;

in the new scheme the theoretical design will be very close to reality, because the system model is of a good quality.

We are not trying to develop an optimal control system for the ball balancing process yet.

Single Loop System

Based on the 4th-order ARX model of the process (equation error

meth-od), a PDD controller is designed as in (4.3). The controller is in

the feedback path as shown in Fig. 4.8.

The setpoint is divided by the static gain of the closed loop

dynam-ics, denoted by k_st. The parameters of the PDD controller are

deter-mined such that the simulated closed loop system has a desired step response. 2 40q - 72q + 32.5 C (q) 2 (4.3) q

1

PQ

-

- -

_Process k 51 POO controller

Fig. 4.8 Single loop system with PDD controller

The sample frequency is 10 Hz, the set point input and system

res-ponse is shown in Fig. 4.9. The discrete time simulation is

perform-ed by using the ARX model and controller C(q). The system input is

the same as in the experiment.

The simulation is compared with the measured data, see Fig. 4.9.

We see that when the identified process models are used, the theoret-ical results about the system are different from the real measure-ments; in our case, they will give too optimistic conclusions about the system performance.

Two Loop System

In this case, the first loop controller C_l stays as part of the

sys-tem, and the second loop controller C₂ is designed based on the first

loop system model. Because the PDD controller C

1 (q) in (4.2) is

stable and proper, optimality is maintained in the two loop system.

The second controller is chosen to have the same structure as the first loop controller:

2

6.0299q - 10.8616 q + 5

2

q

and is placed parallel to the first controller.

(4.4)

The parameters of C₂ are determined such that the two loop system

give the desired stepresponse as in Fig. 4.11. The digital feedback

2

52.2299 q - 96.6616 q + 45

2 (4.5)

q

We see that controller implemented in the two loop structure will have the same order as the first loop controller, if C

1 (q) and C2(q)

have the same denominator.

The two loop system is shown in Fig. 4.10, where the set-point is divided by the static gain of the two-loop system, denoted by kst' which is measured from the step response.

1 --c

- -

_k

--V

Process

at -~

"'+

_C1

,/ +

C2

Fig. 4.10 Two loop control system

The experiment result of the two loop system is compared with the simulation result in Fig. 4.11, where the first loop system model

and second controller are used to simulate the two loop system. We

find that the simulation fits the real measurement very well. The

measured output has more overshoot than the simulated output; this is probably caused by (1) the centrifugal force of the rail which af-fects the ball position in a nonlinear fashion; (2) some slipperiness of the ball, which might happen when the ball moves very fast.

It is interesting to compare Fig. 4.9 and Fig. 4.11, where the single loop simulation shows better step response than the two loop simulation, but the real measurements show the opposite.

This case study shows that our proposal in Section 3 works well. We

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500

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I 1\\1

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I '

r

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rlrd'ldl

'

:

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1111

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il

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-10m L-lJ---'1_1LcO

l-::-.JL:JL--==---:c:::-'-~~_7;;:_-'l,

-;! o 50 1{)0 150 200 250 300 350 400 450 500

o

Samples

Fig. 4.2 _{Process input signal, manually controlled}

experiment ~(l()() lS00 ltlOO " ~ _5UO

-"

" .:::: E ₀ " 0 5 _, "- -500 6 ·\000 ·1500 -2000 0 Fig. 4.3

r..ka~ll!L'd output and simu\;lll-d output "snliJ" := nll'a~url'd OU1Pll!

, , .. ' ·'"1 ,. " '

,

' " _" !"',

n

)

\

\.. " rio:'

_,

"\ ,

..

: : :

\

: ... , .. ,

.

J "

" 50 100 150 200 250 300 350 400 450 500 Samples

Measured and simulated outputs, model from manually controlled data

': Fig. 4.4 -500 :1

I

i

50 100 150 200 " ;: "

'.

" 25U 30{) 350 400 Samples 450 500

Process input u and system input w, closed

loop experiment

:-kasun:d olltplll and simubtcd output

"solid·' := mca~ured output

"(bshcd" := simulated output

-1tI00'--~---:

o 50 10() !."iO 200 ~50 JOO 350 4()(] 450 50U SJlllples

Fig. 4.5 Measured and simulated process outputs,

Measured output and simulated output

1500,~~~~~~- ,----.--~---r----T--- --r---,-~~~-,

1000

500

o

·500 "solid" := measured output

"d:J.shed" := simulated omput

Samples

Fig. 4.6 _{Measured and simulated system outputs,}

model from closed loop system identification

CRUSS CORRELATION FUNCTION BETWEEN INPUT # I AND RESIDUALS

"uLl=-='

~-~

.

---~~-=.=---

._---'-1

", f---"..,,---T''---''--'---·0.05 ·01 -O.L~~:'i _-15 -]()

.,

_{u 5 10 15 20 25} lag

Fig. 4.7 Cross correlation function between

the system input wand output residuals

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D <J

30 20 _i I E ₁₀ ,( ~

l'

c .g _Of-'i' .~ 0-';l 10

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'--~~---' ·50,--· 0 5 --'~--:':c--_ ... -10 15 ~-'--'-~' Fig. 4.9 50 40 30 20

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10 ;.

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c , .9 ₀ J' ,~ 0 ~

,

·10

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_·20 ·30 ·4() .501 0 20 25 Time Ist'c.,

Measured and simulated responses, single-loop system

Input, measured output and simulated output "dashdot" := measured output

"dashed" := simulated output

5

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l

lU ;

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" 15 Time Isec.1 20 ! i :--. 25 30

1

30

Fig. 4.11 Measured and simulated responses, two-loop

difficult to separate the process from the system. The difficulties can be overcome when we view the system as a whole,and do not attempt to identify the process dynamics.

s.

CONCLUSIONS

In this work, a new way of integrating identification and control is

proposed. We have shown that it is possible to use the advantages

of feedback control to improve the quality of identification of ~­

tem dynamics, while it is well-known that there is a conflict between feedback control and closed loop identification of process dynamics. Hence we propose:

When the system operates in closed loop for identification experi-ments, one can identify the closed loop system dynamics, then design the second loop controller to meet the control requirement, and the

process dynamics can be ignored in the latter step. It has been

shown that any controller for the original process can be realized by the two loop system structure, provided that the first loop control-ler is proper and stable which is true for many practical situa-tions.

The two loop system is not new in control literature, but we believe that it is a new idea to combine identification and control system

design in such a scheme. The case study is a successful example of

the new method. Compared with conventional methods, the advantages

of our method are

solving the problem of identification/control for unstable pro-cesses;

reducing the difficulty caused by process nonlinearity, and time variation;

identification and control are mutually supporting.

These are specially beneficial for identification/control for

indus-trial processes. The nice thing about this approach is that we do

not have to pay for the benefits gained, because the method neither needs to develop a new identification theory/technique nor a control theory/technique, and the implementation of two loop system is not

difficult. The new method can be used not only for situations where

the closed loop experiment has to be performed, but also for

situ-ations where the open loop experiment is permitted. The advantages

for the latter situation are the reduction of the influences of non-linearity, time variation and disturbances on the identification, and the increase in experiment time (due to the better experiment condi-tion) .

ACKNOWLEDGEMENT

The first author would like to thank Dr. Petre Stoica for his

valu-able co~~ents and suggestions about this work.

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ANALOG AND DIGITAL SIMULATION OF LINE-ENERGIZING OVERVOLTAGES AND COMPARISON WITH MEASUREMENTS IN A 400 ,V NETWORK.

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MARTINUS VAN MARU~: A Dutch :...cientist in a revolutionary time. EUT Report 88-E-194. 1988. ISRN 90-6144-194-3

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MODEL BUILDING FOR AN INGOT HEffitJG PROCESS: Phy~ical modelling approach and identification approach.

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ANEW METHOD FOR DYNAfll C HUNT I NG EXTREMUll CONTROL, BASED ON COMPAR I SON OF MEASURED AND ESTII·IATED VALUE.

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THE FULL DECOilPOS I T I ON OF SEQUENT I AL I~ACH I NES WITH THE OUTPUT BEHAV lOUR REALIZATION.

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A FORMALISM [0 DESCRIBE CONCURRENT NON-DETERMINISTIC SYSTEMS AND Ahl APPLICATION OF IT BY ANALYSING SYSTE~1S FOR DANem

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DEADLOCK.

rUT Repor't. 86-[-200. 1981:1. l~t1N 9{)-6144-200-1

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HARDWARE SYNTHES I 5 WITH THE A I D OF DYNMll C PROGRMlt,ll NG. EUT Report 88-E-201. 1988. ISHN 90-6144-201-X

En~elshoven. R.J. van and R. van den Born

coT CALCULATION FOR I NCREt-tENTAL HARDWARE SYNTHES I S. EUT Repo,t 88-E-202. 1988. ISBN 90-6144-202-8 (203) Oelissen, J.G.M.

THE LINEAR REGRESSION !·iODEL: i10del structure selection and biased estimators. EUT Repo,t 88-E-203. 1988. ISBN 9Q-6144-203-6

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COMPARISON OF AN ANALYTICAL STUDY AND EMTP 11·IPlEI·IENTATION OF COMPLICATED THREE -PHASE SCHEflES FOR REACTOR I IHERRUPT I ON.

f~"""Wl')RDLENC1H EFFEC1S IN Illl,ITAL rJL l[RS:~it'VI.

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(206) B~llen, M.H.J. and G.A.P. Jacobs

TIITfiS) VE TEST) NG OF AN ALGOR) THI1 FOR TRAVELLI NG-WAVE-BASED D I RECTI ONAL DETECTION AND PHASE-SELECTION BY USING TWONFIL AND EMTP.

EUT Report 88-E-206. 1988. IS8N 90-6144-206-0

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STABILITY OF A TAYLOR-RELAXED CYLINDRICAL PLASMA SEPARATED FROM THE WALL BY A VACUUM LAYER.

EUT Report 88-E-207. 1988. ISBN 90-6144-207-9

Lucassen, F.H.R. and H.H. van de Ven

A NOTATION CONVENTION IN RIGID ROBOT MODELLING. EUT Report 8B-E-208. 1988. IS8N 90-6144-208-7

(209) J6iwiak, L.

M1Nl~tAL REALIZATION OF SEQUENTIAL f·1ACHINES: The method of maximal adjacencies.

EUT Report 88-E-209. 1988. ISBN 90-6144-209-5

(210) Lucassen, F.H.R. and H.H. van de Ven

OPTIMAL BODY FIXED COORDINATE SYSTEMS IN NEWTON/EULER MODELLING. EUT Report 88-E-210. 1988. ISBN 90-6144-210-9

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Hm-CONTROL: An exploratory ~>tudy.

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(212) Zhu Yu-Cai

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ONTHE R08UST STAB I I. 11 Y OF M !flO 1.1 NEAR FEEDBACK SYSTEMS.

EUT f~epor-t 88-E-212. 1988. ISBN 90-6144-212-5

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ANEW SCHEME FOR IDENTIFICATION AND CONTROL.

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