# Nonlinear rational model identification and control

Τεχνίτη Νοημοσύνη και Ρομποτική

19 Οκτ 2013 (πριν από 4 χρόνια και 7 μήνες)

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Nonlinear rational model
identification and control

Professor Quan M. Zhu

Bristol Institute of Technology

University of the West of England

Frenchay Campus

Coldharbour Lane, Bristol BS16 1QY, UK

q
uan.zhu@uwe.ac.uk

Contents

1) Background knowledge

2) Rational models and representations

3) Structure detection and parameter
estimation

4) Correlation based validation

5) Controller design

6) Conclusions

Model identification

Input/output data from
instrument measurements and
expert perceptions

Parametric model structure

Parameter estimation

Validity tests

model

training/

identification

plant

Model validation

Examine residuals

Correlation tests

A valid model’s
residuals should be
reduced to uncorrelated
sequence with zero
mean and finite
variance

model

examination/

diagnosis

plant

_

residual

A general modelling and control structure

plant

model

training

control

input

output

residual

Target

Rational models (1)
---

Expression

Rational models (2)
---

Example

Rational models (3)
---

Characteristics

1) The model can be much more concise than a polynomial
expansion, for example

2) The model can produce large deviations in the output,
for example

Rational models (4)
---

Errors

Rational models (5.1)
---

Representations

1)
The polynomial NARMAX models is a special case of
RM by setting denominator polynomial b(t) = 1.

2
)

The

model

is

non
-
linear

in

both

the

parameters

and

the

regression

terms,

this

is

induced

by

the

denominator

polynomial
.

3
)

Modelling

of

chemical

kinetics,

bio

dynamics,

brain

image
.

Rational models (5.2)
---

Representations

4)
Fuzzy systems with centre defuzzifier, product inference
rule, singleton fuzzifier, and Gaussian membership function.

5
)

The

normalised

basis

function

network

is

also

a

type

of

rational

model
.

When

the

centres

and

widths

need

to

be

estimated

this

becomes

a

rational

model

parameter

estimation

problem
.

6
)

Difference

in

time

domain

and

frequency

domain

Structure detection and
parameter estimation (1)

Prediction error method

Extended least squares method

Orthogonal structure detection procedure

Recursive least squares method

Back propagation method

Implicit leas squares method

Correlation based validation (1)

A basic concept for correlation based model

validity tests:

that if a model structure is correct and its

parameter estimation is unbiased, its residuals

should form a random (in theory) / uncorrelated (in practice)

sequence with zero mean and finite variance.

Correlation based validation (2)

Controller design (1)

1) Indirect (transformation) method: neural
network based design approach (Kumpati
Narendra )

using neural network to approach rational
models and then design control systems

2) Direct (analytical) method: U
-
model
based design approach

there is nothing lost to use U
-
model to
express ration models.

Controller design (2)

K. Narendra’s work can be referred from his publications below

K.S. Narendra and K. Parthasarathy, Identification and control of
dynamic systems using neural networks, IEEE Trans., on
Neural Networks, Vol. 1, No. 1, pp. 4
-
27, 1990.

J.B.D. Cabrera and K.S. Narendra, Issues in the application of
neural networks for tracking based on inverse control, IEEE
Trans., on Automatic Control, Vol. 44, No. 11, 1999.

L.G. Chen and K.S. Narendra, Nonlinear adaptive control using
neural networks and multiple models, Automatica, Vol. 37,
pp. 1245
-
1255, 2001.

Controller design (3)

U
-
model based NL control system design

1) Concise and efficient in structure

2) Wider representations

Challenges

1) Model structure detection and parameter estimation

2) State space realisation

3) Model reduction

4) Control system design

5) Stability analysis

Conclusions

QM Zhu’s relevant publications (1)

S
.
A
.

Billings

and

Q
.
M
.

Zhu,

Rational

model

identification

using

an

extended

least

squares

algorithm,

Int
.

J
.

Control

(International

Journal

of

Control),

Vol
.

54
,

No
.

3
,

pp
.

529
-
546
,

1991
.

Q
.
M
.

Zhu

and

S
.
A
.

Billings,

Recursive

parameter

estimation

for

nonlinear

rational

models,

Journal

of

Systems

Engineering,

No
.

1
,

pp
.

63
-
76
,

1991
.

Q
.
M
.

Zhu

and

S
.
A
.

Billings,

Parameter

estimation

for

stochastic

nonlinear

rational

models,

Int
.

J
.

Control,

Vol
.

57
,

No
.

2
,

pp
.

309
-
333
,

1993
.

QM Zhu’s relevant publications (2)

S
.
A
.

Billings

and

Q
.
M
.

Zhu,

Structure

detection

algorithm

for

nonlinear

rational

models,

Int
.

J
.

Control,

Vol
.

59
,

No
.

6
,

pp
.

1439
-
1463
,

1994
.

S
.
A
.

Billings

and

Q
.
M
.

Zhu,

Nonlinear

model

validation

using

correlation

tests,

Int
.

J
.

Control,

Vol
.

60
,

No
.

6
,

pp
.

1107
-
1120
,

1994
.

H
.
Q
.

Zhang,

S
.
A
.

Billings,

and

Q
.
M
.

Zhu,

Frequency

response

function

for

nonlinear

rational

model,

Int
.

J
.

Control,

Vol
.

61
,

No
.

5
,

pp
.

1073
-
1097
,

1995
.

QM Zhu’s relevant publications (3)

S.A. Billings and Q.M. Zhu, Model validity tests for
multivariable nonlinear models including neural networks, Int. J.
Control, Vol. 62, No. 4, pp. 749
-
766, 1995.

Q.M. Zhu and S.A. Billings, Fast orthogonal identification of
nonlinear stochastic models and radial basis function neural
networks, Int. J. Control, Vol. 64, No. 5, pp. 871
-
886, 1996.

QM Zhu’s relevant publications (4)

Q
.
M
.

Zhu

and

L
.
Z
.

Guo,

A

pole

placement

controller

for

nonlinear

dynamic

plants,

Proc
.

Instn
.

Mech
.

Enger,

Part

I
:

Journal

of

Systems

and

Control

Engineering,

Vol
.

216
,

No
.

6
,

2002
.

Q
.
M
.

Zhu,

A

back

propagation

algorithm

to

estimate

the

parameters

of

nonlinear

dynamic

rational

models,

Applied

Mathematical

Modelling,

Vol
.

27
,

pp
.

169
-
187
,

2003
.

Q
.
M
.

Zhu,

An

implicit

least

squares

algorithm

for

nonlinear

rational

model

parameter

estimation,

Applied

Mathematical

Modelling,

Vol
.

29

pp
.

673
-
689
,

2005
.

QM Zhu’s relevant publications (5)

L
.
F
.

Zhang,

Q
.
M
.

Zhu,

and

A
.

Longden,

A

set

of

novel

correlation

tests

for

nonlinear

system

variables,

Int
.

J
.

Systems

Science,

Vol
.

38
,

pp
.

47
-
60
,

2007
.

Q
.
M
.

Zhu,

L
.
F
.

Zhang,

and

A
.

Longden,

Development

of

omni
-
directional

correlation

functions

for

nonlinear

model

validation,

Vol
.

43
,

pp
.

1519
-
1531
,

Automatica,

2007
.

L
.
F
.

Zhang,

Q
.
M
.

Zhu

and

A
.

Longden,

A

correlation

tests

based

validation

procedure

for

identified

neural

networks,

Vol
.

20
,

pp
.

1
-
13
,

IEEE

TNN,

2009
.

QM Zhu’s relevant publications (6)

Q
.
M
.

Zhu,

L
.
F
.

Zhang,

and

A
.

Longden,

A

correlation

test

based

validity

monitoring

procedure

for

online

detecting

the

quality

of

nonlinear

noise

cancellation,

Int
.

J
.

Systems

Science,

in

print
.

Q
.
M
.

Zhu,

An

analytical

design

procedure

for

control

of

nonlinear

dynamic

rational

model

based

systems,

(under

preparation),

2010
.