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System Identification, Estimation and

Filteringy(t)^

Dr Martin Brown

Room: E1k, Control Systems CentreEmail: martin.brown@manchester.ac.uk

Telephone: 0161 306 4672

EEE Intranet

ControllerPlant

Model

u(t)

u(t)

y(t)

d(t)

^

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Course Resources

Core texts

L Ljung, System Identification: Theory for the User, Prentice Hall,1998

Notes by P. Wellstead, which provide useful background &supplementary information, see course Intranet

Other resourcesJP Norton, An Introduction to Identification, Academic Press, 1986

S Haykin, Adaptive Filter Theory, Prentice Hall, 2003

B Widrow, SD Stearns, Adaptive Signal Processing, Prentice Hall, 1985

CR Johnson Lectures in Adaptive Parameter Estimation, Prentice Hall,

1989

In addition, a lot of relevant material is contained on the Mathworks web-site and have a look at http://ocw.mit.edu/ for open course ware,especially the system identification course.

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Course Structure

Each 3 hourlecture will be organised:

2*50 minute lectures 50 minute (optional?) laboratory session A19

I expect that you will complete the laboratory sessions in your owntime, as they reinforce the concepts that we develop in thelectures and will gradually build up to the assignment. Theydevelop your Matlab & Simulink programming skills (W1&2) thatwill be used on the assignments.

Two assignments EF: set ~W6, submit W9, 17% of total marks

IS: set ~W8, submit W11, 17% of total marks

Exam: 3 hours, 4 questions from 6, 66% course marks

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Course Topic Areas

Matlab &

Simulink

Recursive

parameter

estimation

Linear

modelling

Self-tuning

control

Ordinary differential

/difference equation

Modelselection

Linear algebra

Statistics

See http://ocw.mit.edu/ for some refresher courses (or any good bookshop)

Least squaresestimation

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Elements of the Course

The course has five main elements to it:

1. Introduction to system identification

2. Non-parametric and parametric modelling

3. Least squares and recursive parameter estimation

4. Model selection

5. Self-tuning control

These concepts build on each other in the sense that anything that can

be done on-line, can be performed in an off-line design scenario.

However, off-line design typically involves a greater range of

validation and human involvement, whereas on-line is completelyautomatic.

Well be largely concerned with discrete-time system identification,

but will dip into continuous-time representation, as necessary.

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Lecture 1: Introduction to System

Identification

Introduction to system identification

Input and output signals

Examples

Identifying a system using data ARX and least squares parametric modelling

System identification process

The aim of this lecture is to give you an introduction andoverview of the main topics in system identification

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In order to design a controller, you must have a model

A model is a system that transforms an input signal into an

output signal.Typically, it is represented by differential ordifference

equations.

A model will be used, either explicitly or implicitly, in control

design

Introduction to System Identification

Controller Plant

Model

u(t)

y(t)

y(t)^

u(t)

d(t)

r(t)

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Signals & Systems Definitions

Typically, a system receives an input signal,x(t) (generally a vector)and transforms it into the output signal, y(t). In modelling andcontrol, this is further broken down:

u(t) is an input signal that can be manipulated (control signal)

w(t) is an input signal that can be measured (measurabledisturbance)

v(t) is an input signal that cannot be measured (unmeasurabledisturbance)

y(t) is the output signal

Typically, the system may have hidden statesx(t)

u(t)

w(t)

v(t)

y(t)x(t)

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Example 1: Solar-Heated House (Ljung)

The sun heats the air in the solar panels

The air is pumped into a heat storage (box filled with pebbles)

The stored energy can be later transferred to the house

Were interested in how solar radiation, w(t), and pump velocity, u(t),affect heat storage temperature, y(t).

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Example 2: Military Aircraft (Ljung)

Aim is to construct a mathematical model of dynamic behaviour to

develop simulators, synthesis of autopilots and analysis of itsproperties

Data below used to build a model of pitch channel, i.e. how pitch rate,

y(t), is affected by three separate control signals: elevator (aileron

combinations at the back of wings), canard (separate set of rudders

at front of wings) and leading flap edge

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How are Models Used?

Mathematical models are abstractions/simplifications of reality, whichare good enough for the purpose for which they were developed

In scientific modelling we aim to increase our understanding aboutcause-effect relationships. The models predictive ability can beused to test the model (e.g. Newton, Halley)

Models can be used forprediction and control. Here the predictive

ability is a key aspect, but this can be influenced by the modelssimplicity if it has to be estimated from exemplar data (e.g. modelpredictive control).

Models can be used forstate estimation. Here the aim is to trackvariables which characterize some dynamical behaviour byprocessing observations afflicted errors (e.g. estimating position

and velocity of Apollo moon landing).Models can be used forfault detection. Here predicted behaviour is

assessed against the actual behaviour to determine whether theplant is operating normally or not.

Models can be also be used forsimulation and operator training.

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Identifying a System

A system is constructed from observed/empirical data

Models of car behaviour (acceleration/steering) are built fromexperience/observational data

Generally, there may exist some prior knowledge (often formedfrom earlier observational data) that can be used with theexisting data to build a model. This can be combined in

several ways: Use past experience to express the equations

(ODEs/difference equations) of the system/sub-systems anduse observed data to estimate the parameters

Use past experience to specify prior distributions for theparameters

The term modelling generally refers to the case whensubstantial prior knowledge exists, the term systemidentification refers to the case when the process is largelybased on observed input-output data.

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Introduction: ARX Model

Consider a discrete-time ARX model (no disturbances)

(well fully define these terms later). This can be used for

prediction using:

and introducing the vectors:

the model can be written as

Note, that the prediction is a function of the estimated

parameters which is sometimes written as

)()1()()1()( 11 mtubtubntyatyaty mn ++=+++

)()1()()1()( 11 mtubtubntyatyaty mn +++=

[ ]

[ ]T

T

mn

mtutuntytyt

bbaa

)()1()()1()(

11

=

=

x

x )()( tty T=

x )()|( tty T=

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Introduction: Least Squares

EstimationBy manipulating the control signal u(t) over a period of time 1tN, we

can collect the data set:

Lets assume that the data is generated by:

Where is the true parameter vector and (0,2) generates zero mean,normally distributed measurement noise with standard deviation .

We want to find the estimated parameter vector, , that best fits this data

set.

Note that because of the random noise, we cant fit the data exactly, but

we can minimise the prediction errors squared using

Where X is the matrix formed from input vectors (one row per observation,

one column per input/parameter) and y is the measured output vector

(one row per observation

)}(),(,),1(),1({ NyNuyuZN =

),0()()( 2+= x tty T

yXXXTT 1)( =

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Example: First Order ARX model

Consider the simple, first order, linear difference equation, ARX

model:

where 10 data points Z10 = {u(1),y(1), ,u(10),y(10)} are collected.

This produces the (9*2) input matrix and (9*1) output vector:

Therefore the parameters =[ab]T can be estimated from

Using Matlab

thetaHat = inv(X*X)*X*y

)1()1()( =+ tbutayty

=

=

)10(

)3()2(

)9()9(

)2()2()1()1(

y

yy

uy

uyuy

yX

yXXX TT 1)( =

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Model Quality and Experimental Design

The variance/covariance matrix to be inverted

Strongly determines the quality of the parameter estimates.This is turn is determined by the distribution of the measured

input data.

Control signal should be chosen to make the matrix as well-

conditioned as possible (similar eigenvalues) Number of training data and discrete sample time both affect

the accuracy and condition of the matrix

Experimental design is the concept of choosing the

experiments to optimally estimate the models parameters

=

t nt nt n

t ntt

t ntt

T

txtxtxtxtx

txtxtxtxtx

txtxtxtxtx

)()()()()(

)()()()()(

)()()()()(

2

21

2

2

212

121

2

1

XX

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System Identification Process

In building a model, the designer has

control over three parts of theprocess

1. Generating the data setZN

2. Selecting a (set of) model structure

(ARX for instance)3. Selecting the criteria (least squares

for instance), used to specify the

optimal parameter estimates

There are many other factors that

influence the final model, however,

this course will focus on these three

factors and the method for

(recursive) parameter estimationValidate

Model

Calculate Model

Choose

Criterion

of Fit

Choose

Model Set

Data

Experiment

Design

Prior

knowledge

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Lecture 2: Signal and System Representation

1. Signal representation Continuous and discrete time signals

Impulse and step signals

Exponential signals

Time shifting signals

2. Systems representation

Differential equations

Difference equations Linear/non-linear systems

Non-parametric system representations

Signal

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Continuous & Discrete-Time SignalsContinuous-Time (CT) Signals

Most signals in the real world are

continuous time, as the scale isinfinitesimally fine.

Eg voltage, velocity,

Denote byx(t), where the time intervalmay be bounded (finite) or infinite

Discrete-Time (DT) Signals

Some real world and many digitalsignals are discrete time, as they aresampled

E.g. pixels, daily stock price (anythingthat a digital computer processes)

Denote byx(k), where kis an integervalue that varies discretely

Sampled continuous signal

x(tk) =x(kT) Tis sample time

Often the notationx(t) is just used,

and ttakes integer values.

x(t)

t

x(tk)

tk

Signal

Signal

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Discrete Unit Impulse and Step Signals

The discrete unit impulse signal is defined:

Critical in convolution as a basis foranalyzing other DT signals

The discrete unit step signal is defined:

Note that the unit impulse is the first

difference (derivative) of the step signal

Similarly, the unit step is the running sum(integral) of the unit impulse.

=

==0100)()(

ttttx

>

C

a

0

0

>

1, stretching if 0

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ODE System Representation

The signals y(t)=vc(t) (voltage across capacitor) andx(t)=v

s(t) (source

voltage) are patterns of variation over time

Note, we could also have considered the voltage across the resistor orthe current as signals and the corresponding system would be

different.

+

-i(t) vc(t)vs(t)

R

C)()(

)(

)()(

)()()(

tvtvdt

tdvRC

dt

tdvCti

tvtvtRi

scc

c

cs

=+

=

=

Step (signal) vs(t) at t=0

RC= 1 First order (exponential)

response forvc(t)

vs,v

c

System

System

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General LTI ODE SystemsA general Nth-order LTI differential equation is:

This system has the following properties:

Ordinary differential equation: the system only involves derivatives

involving a single variable: time.

Linear: the system is a sum of (weighted) input and outputderivatives only. So ax1(t)+bx2(t)ay1(t)+by2(t)

Time invariant: the coefficients {ak,bk} are constant and do not

depend on time. Sox(t)y(t) x(t-T)y(t-T)

Causal: the system does not respond before an input is applied. Sothe RHS does not include terms likex(t+T), where T>0.

These LTI ODE systems can represent many electrical and physical

systems

Obvious Laplace transform of the CT system

== =M

kk

k

k

N

kk

k

kdt

txd

bdt

tyd

a00

)()(

System

System

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General LTI ODE Solution

Consider a signal y(t) = Cert (C, rare Complex) and substitute it

into the left hand side of the (unforced) LTI ODE:

When r=rk, a root of the above polynomial, this is a solution.

There are Nsuch solutions, and the overall solution y(t) is a

linear combination of these solutions. As {ak} are real, then the

roots rk must be complex or imaginary conjugate pairs, or real

If all the solutions are exponential decays, Re(rk)0, the system is unstable

0

0

0

0

=

=

=

=

N

k

k

k

N

k

k

k

rt

ra

raCe

System

System

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Discrete Time Difference Equation Systems

Consider a simple (Euler)

discretisation of thecontinuous time system

Discrete-Time Systems

Discrete time systems

represent how discrete

signals are transformed

via difference

equationsE.g. discrete electrical

circuit, bank account,

( )( ) ( )

( 1) ( )( ) ( )

( ) (1 ) ( 1) ( 1)

cc s

c cc s

T Tc c sRC RC

dv tRC v t v t

dtv t v t

RC v t v tT

v t v t v t

+ =

+ + =

= +

First order, linear, time invariant

difference equation

y(t)

t

System

System

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General LTI Difference Systems

A general Nth-order LTI difference equation is

This system has the following properties:

Involves a single index (discrete time) t.

Linear: the system is a sum of (weighted) input and outputderivatives only. So ax

1(t)+bx

2(t)ay

1(t)+by

2(t)

Time invariant: the coefficients {ak,b

k} are constant and do not

depend on time. Sox(t)y(t) x(t-T)y(t-T)

Causal: the system does not respond before an input is applied.So the RHS does not include terms likex(t+k), where k>0.

These LTI difference equation systems can represent many

electrical and physical systems

Obvious z-transform representation of the DT system

== =M

k

k

N

k

k ktxbktya10

)()( a0=1

System

System

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General LTI Difference Solution

Consider a powersignal y(t) = Czt (C, zare Complex) and

substitute it into the left hand side of the (unforced) diff eqn

When z=zk, a root of the above polynomial, this is a solution.

There are Nsuch solutions, and the overall solution y(t) is a

linear combination of these solutions. As {ak} are real, then

the roots zkmust be complex or imaginary conjugate pairs, or

real

If all the solutions are exponential decays, |zk|1, the system is unstable

0

0

0

0

)(

=

=

=

=

N

k

kN

k

N

k

kN

k

Nt

za

zaCz

y

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L1&2 Conclusions

Basic introduction to system identification and parametric identification

Reviewed some basic results about signals

simple impulse and step basis signals

linear time transformations of signals

complex exponential signals

Reviewed some basic results about systems

differential and difference equation representation

basic properties, LTI, stable, causal,

how to solve the ODE/difference equations

Much of the course will assume a familiarity with such basic theory.

Next lectures, have a look at classical system identification

(impulse/frequency response) and calculating a prediction

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L1&2 Matlab Exercises

Remember, when starting Matlab, change to the P: drive and turn the

diary on/save Simulink files.

1. Create the CT first-order system described on Slide 24 into Simulink

and check the response

2. Enter the DT first-order system described on Slide 27 into a Matlab

function and verify its behaviour (note the stem() function may be

useful). Try RC=3, with a sample time of=0.1s.

3. Investigate how you can use the roots() function in Matlab to

solve (unforced) LTI differential and difference equations

4. Use the information on slide 16 to identify the parameters of the

discrete time model described in question (2). Use the eigs()

function to check the condition/structure of the XTX matrix

5. Complete any of the programming exercises from the first 4

Matlab/Simulink labs. It is very important that you are a competent

Matlab programme and Simulink model developer.

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L1&2 Theory Exercises

****

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Appendix A: General Complex Exponential Signals

So far, considered the real and periodic complex exponential

Now consider when Ccan be complex. Let us express Cis polar formand a in rectangular form:

So

Using Eulers relation

These are damped sinusoids

0

jra

eCC j

+=

=

tjrttjrjat eeCeeCCe )()( 00 ++ ==

))sin(())cos(( 00)( 0 teCjteCeeCCe rtrttjrjat +++== +

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Appendix B: Periodic Complex

Exponential & Sinusoidal SignalsConsider when a is purely imaginary:

By Eulers relationship, this can be expressed

as:

This is a periodic signals because:

when T=2/0

A closely related signal is the sinusoidal

signal:

We can always use:

tjCetx 0)( =

tjtetj

00 sincos0 +=

tj

Ttj

etjt

TtjTte

0

0

00

00)(

sincos

)(sin)(cos

=+=+++=+

( ) += ttx 0cos)( 00 2 f =

( ) ( )( ) ( ))(0

)(

0

0

0

sin

cos

+

+

=+

=+tj

tj

eAtA

eAtA

T0 = 2/0=

cos(1)

T0 is the fundamental

time period

0 is the fundamental

frequency