SYSTEM IDENTIFICTION OF SERVO RIG

Prof Ian French MSc Control & Electronics

TEESSIDE UNIVERSITY

SCHOOL OF SCIENCE AND TECHNOLOGY

MSc CONTROL AND ELECTRONICS

System Identification and Adaptive Control

Karthik Kumar Naddi

J9026070

11 April 2023

1Karthik Kumar Naddi(J9026070)


PART 1-SYSTEM IDENTIFICATION:

1) Describe with the aid of diagrams why it is important to select an appropriate

sampling time when performing system identification. Describe a sampling method

that may be used to select an appropriate sampling time for the lab servo.

2) Why it is important to remove outliers, bias and trends prior to parameter estimation.

Illustrate your answer using results based on the s_datagen_b.mdl simulink model.

3) Why will the basic Least Squares method result in biased estimated when used to

estimate the parameters of an ARMA type model. Show that IV will result in

unbiased estimates.

4) Using the results obtained during the practical sessions, outline the steps needed to

obtain discrete transfer function models of the lab servo, for both the velocity and the

position.

PART 2-ADAPTIVE CONTROL:

5) Describe the Recursive Least Squares algorithm. Why is the RLS algorithm more

suited to the adaptive control problem than the basic LS algorithm?

6) Outline the theoretical basis of the phase margin design algorithm used in the self-

tuning PI control tutorial.

7) Using the results obtained during the in the practical sessions, outline the steps needed

to design and apply a PI controller for the velocity of the Lab Servo.

8) Using the results obtained during the in the practical sessions, outline the steps needed

to design and apply a PIP controller for the position of the Lab Servo.



Abstract:

In this assignment we are discussing system identification and adaptive or model based

control. The system identification scope includes all aspects of system modelling and

identification, from theoretical and methodological developments to practical applications. It

also includes the topics of model selection, identification methods, fault detection,

experiment design etc. In this we are using a servo-rig system to obtain mathematical model

which corresponds to the working of the servo-rig without using known laws, equation or

system behaviour which is associated with the device. We are also using Matlab and

Simulink programs to eliminate time consuming calculations which are associated in the

Automatic PIP controller for this servo-rig.

In this we use identification tool box to obtain appropriate discrete parametric models such as

position, velocity for the laboratory servo rig by using matlab programs like data collection,

structure selection and data conditioning. Following we are using the models to obtained

design and implement in pip controllers from laboratory servo-rig and then compare the

simulink response that are obtained with the real system response.



CONTENTS

1) INTRODUCTION:........................................................................................................................5

2) SAMPLING TIME:.......................................................................................................................5

3) Data Preprocessing:...................................................................................................................8

4) Bias of the Least Squares Estimator:..........................................................................................9

4.1) Instrumental variable methods:...........................................................................................11

5) Discrete transfer functions for velocity and position of servo rig:..............................................12

5.1)Young’s Information Criteria (YIC):........................................................................................14

6) Recursive Least Squares algorithm:.........................................................................................18

7) Phase Margin Algorithm:.........................................................................................................19

8) PI CONTROLLER FOR SERVO:....................................................................................................20

8.1) STPI controller for Position of servo rig:..............................................................................23

9) PIP CONTROLLER FOR SERVO:..................................................................................................24

10) Conclusion:..........................................................................................................................28

11) Reference:..........................................................................................................................29



1) INTRODUCTION:

System Identification is defined as the name given to the process of identifying models from

experimental data. This process is in a compact way and most useful to summarise the

knowledge about a process. The system response modelled with a mathematical relationship

with the given system input and obtained system output by the classification of the systems in

the sense of the system response is also known as system identification. For communication

and education, this model is very useful tool.

This model is designed and used for a number of reasons but in the field of engineering it is

used for three different tasks.

Used for the design of control system,

For forecasting and prediction and

Used for fault analysis and condition monitoring.

By one mathematical model this process is not characterised. The hierarchy of models should

depend on the models ranging from complex and detailed simulation to simple and easily

manipulated models. The gross features of the system behaviour are obtained by the simple

models which are used for exploratory purposes and to develop complicated models it takes

long time. The models can be available by two various principles from the prior in terms of

physical laws or by the experimentation of the process. Here the experimental process is

known as the system identification.

2) SAMPLING TIME:The selection of the sampling time is very important when using least squares system

identification techniques to estimate the parameters of dynamic discrete time systems. If the

sampling time is taken too long then the important information is lost and the process will

appear to be simple gain element. And if the sampling time is taken too short then changes in

the observed output will approach the noise levels of the system and all the meaningful



information is lost. In selecting sampling time, some conditions are usually provided to

accommodate processes with different output responses. Intuitively, sampling time has to be

smaller than the settling time for discrete control to be effective in order to allow the system

act without contradicting interference.

The choice of sampling time is a key issue in automatic digital control. Selection of sampling

time has major influences on many properties of the system under observation such as

rejection of load disturbances and measurement noise, and sensitivity to un-modelled

dynamics.

The selection of an appropriate sampling interval can be obtained by statistical analysis of the

data, based on ‘information’ content. It is, however, somewhat easier to use an iterative

approach to the model building process and use simple test to give the information required

to select the sampling time.

If we define the rise time of the process as the time taken for the process to move from 5% to

95% of its final steady state value. Then a good thumb rule suggests that the sampling period

should lie in the range Trise/15 < Ts < Trise/5 this rule is consistent with the sampling theorem.

Fig1: Data collect servo model



Fig2: Velocity of the servo

Data collect servo model is used to collect the data from the servo rig. In the above graph we

can see the data collected from the servo rig for the velocity. To get the reasonable sampling

time, we have to calculate the rise time (Tr). By using the below graph we can clearly say that

the rise time (Tr) is 1sec.

Fig3: velocity of the servo

From the above graph we calculate the sampling time

Here rise time =1sec

The thumb rule is that the sampling time must be in the range of Tr/15 < Ts < Tr/5.

Here, sampling time(Ts) = Tr/10

Tr= rise time

Sampling time(Ts)= 1/10

=0.1s



3) Data Preprocessing:

Fig4: s_datagen_b model

In the above model we can see, some constants were added. These can be adjusted to get the approximate result. Run the above model.

Load the Ident GUI by typing ident in the Matlab command window. Next import the data to

be analysed (which will be u & z ) by clicking on Time domain data in the Import data drop

down menu. Next perform the linear parametric models operation.

Fig5: identification toolbox before removing means



Now pre-process the data click on Remove means from the Pre-process drop down menu. An

additional data set will appear with the last letter d. Drag the new data set to the Working

data and Validation data positions.

Fig6: identification toolbox for removing means

In the above diagram, when we observe the time plot dialog box it is clearly evident that the

starting point is changed after performing the remove means operation. Data conditioning is

very important aspect in the system identification.

4) Bias of the Least Squares Estimator:The question to be addressed is whether the estimates obtained from similar experiments will

cluster about the true value.

Defining the bias as

b = E(θ)- θ

now for the least squares algorithm

θ = [XTX]-1XTY

and for the case the observations of the output, y, are subject to measurement errors we may

write;

Y=X θ + e

Therefore we can write:

b=E{[XTX]-1XT[Xθ+e]}- θ

b= E{[XTX]-1XTXθ-Iθ}+ E{[XTX]-1XTe}

b= E{[XTX]-1XTe}

from the above it is clear that the bias will be zero if the expected or mean value of e is zero.



Consider the output of the following dynamic system sampled at point r, Yr

The output of the process is dependent upon the current state of the input and what that input

was doing for some time in the past. This leads to defining a model with the following

structure:

yr=b0ur+b1ur-1+b2ur-2........+bru0 known as moving average(MA)

The above model can also be expressed in the following structure

yr(z)=b0ur(z)+b1ur(z)z-1+b2ur(z)z-2........+brur(z)z-r

Where z-1 represents a time shift of one sample into the past and is defined by

z-1 = e-sT

T, is the sampling period and is usually fixed.

Alternative logic to above, however, may argue that the value of the output at the current

sample must be in some way related to its value at the previous sample therefore the best

model structure may be defined as:-

yr= -a1yr(z)z-1 –a2yr(z)z-2......-aryr(z)z-r known as Auto Regressive AR

which may be expressed as:

yr(z) = -a1yr(z)z-1 –a2yr(z)z-2........-aryr(z)z-r

The obvious solution to this dilemma is to recognise that the present value of the output is

dependent upon the present and past values of the input together with the past values of the

output. The most probable model structure is therefore

yr = -a1yr-1-a2yr-2.....-apyr-p AR

b0ur+b1ur-2.....+bqur-q MA+

ARMA



If we rewrite the above structure in matrix form:

Yr=xrT θ

Where

θT = [a1,a2.....ap,b0,b1,b2.....bq]

xrT = [-yr-1,-yr-2....-yr-p,ur,ur-1,ur-2....ur-q]

The equations can be written as

Y = [ y1

.

.yr

] X=[ x1T

.

.xr

T ]Which suggest the least squares procedure can be applied to find θ̂

It is clear from above that the data vector Xr is dependent on past values of the output, y-r to

yr-p. Since the observations of y are in general subject to measurement noise then the estimate

of the parameters, θ, obtained using the standard least squares algorithm must be biased. We

must therefore look for some method for removing this bias.

4.1) Instrumental variable methods:

The bias due to correlation between the data vector and the error can be avoided by

modifying the least squares estimator into the variable estimator:

θ = [ZTX]-1ZTY

Where Z is a matrix closely related to X in which the error correlated repressor are replaced

by other variable (the instrumental variables or just instruments) not correlate with the error.

An additional requirement is that

Det ZTX ≠ 0

Bias of the IV estimator:

Writing the system model, when both X and Y are subject to measurement noise, as

Y = Xθ + (V- Wθ)

Where

M = Y-V

= X-W

And V and W are mutually uncorrelated, zero mean noise sequences.



The bias of the IV estimator is given by:

b=E (θ ¿ – θ = E {[ZTX]-1ZTe}

=E {[ZT (U+W)]-1ZTV} – E {[ZT (U+W)]-1ZTW}θ

Since Z and e are uncorrelated it follows that θ is an unbiased estimator of θ.

4) Using the results obtained during the practical sessions, outline the steps needed to obtain

discrete transfer function models of the lab servo, for both the velocity and the position.

5) Discrete transfer functions for velocity and position of servo rig:In the identification algorithms, the data collected from the final test is always not good.

Fig7: data collect siac model



Fig8: graph for the position of servo

In the above graph we can clearly observe, the dc level of the servo is different when

compared to the input. The dc level can be changed by dataconditioning.

Fig9: Graph for velocity of the servo

The above graph is for the velocity of servo. Even this graph is conditioned to get the output

with less noise.

Now, the steps to find the discrete transfer function of the servo for both velocity and

position.

Data conditioning:

For data conditioning we have to download an m file called datacond.m.

Datacond.m : this file is used to bias the output signal and filter the noise.

This is a matlab program which is used to condition the data. We have to call this file

whenever we required this function.

For Position,

uu = datacond(u,0.5);

By using the above command the input is conditioned.

pp = datacond(pos,0.5);

now, the position of the servo is conditioned.



To get the conditioned output, we use the following command

Plot(t,uu,t,pp)

Fig10: Data conditioned output for position

For Velocity,

uu=datacond(u,0.5);

By using the above command the input is conditioned.

vv=datacond(vel,0.5);

now, the velocity of the servo is conditioned.

To get the conditioned output, we use the following command

Plot(t,uu,t,vv)

Fig11: data conditioned output for velocity

5.1)Young’s Information Criteria (YIC):

Different fits and parameter estimates are produced by the particular data sets, different types

of model structures. Generally the parameters numbers are increased with improve in the fit.

The improvements in the parameter numbers are dramatic, until the point of threshold is

reached. The threshold point represents the optimal number of parameters which are needs

for the particular set models. The schemes are proposed to perform this model structure



selection automatically by searching through a range of structures and assigning to each a

figure of merit denoting efficiency. One suitable figure of merit is known as Young’s

information criteria (YIC).

To get the structure of the servo rig we use YIC model for position.

[nn score fit]=yic(pp,uu,5,5,5)

nn = score = fit =

2 2 1 58.5087 97.5303

1 3 1 58.1364 96.9660

3 1 1 57.9475 95.4774

These fit values denotes us to find the strength of the transfer function. For this position

transfer function these are the fit values. After the procedure follows we will get the best fit

value for the transfer function of the position.

By entering the ident command on the command page, we get the ident toolbox.



Fig12: system identification tool box for Position

After opening the ident toolbox, we have to select the time domain data at the import data

button as our results are time domain based. Now, we have to give the input and output

variables. Then the graph is visible. And then the linear parameter model dialog box we

should give the order and the method is IV. By pressing the estimate button we get the

transfer function.

The transfer function for the position of the servo is,

posu

=0.07133 z−1+0.04348 z−2

1−1.23 z−1+0.225 z−2



To get the structure of the servo rig we use YIC model for velocity.

[nn score fit]=yic(vv,uu,5,5,5)

nn = score = fit =

1 1 1 58.5397 93.9686

2 4 1 54.3793 94.0302

3 5 1 54.2231 94.0030

By entering the ident command on the command page, we get the ident toolbox.

Fig13: system identification tool box for Velocity

The transfer function for velocity of servo is,



velu

= 0.4872 z−1

1−0.3018 z−1

6) Recursive Least Squares algorithm:The least squares method is inefficient, in that, if an estimate of the parameters, θ̂ , is made

from r observations then it is not possible to make use of this prior knowledge if subsequently

it is required to update the estimates having obtained one more observation making a total or

r+1. It is again required to fill out the matrices X and Y and then perform the computational

time consuming matrix inversion, [XTX]-1.

Consider the least squares estimates biased on r observations:

θ̂r=( X rT X r)-1X r

T Y r

Where

Y = [ y1

.

.yr

] X=[ x1T

.

.xr

T ]To establish the recursive least squares algorithm, we consider the situation in which we

make one more observation:YY+1

Defining

Pr=[X rT X ¿-1

Then

Pr+1=[[ X r

X r+1T ]

T

[ X r

X r+1T ]]

−1

= [ X rT X r+xr+1 xr+1

T ]

The reculsive least squares (RLS) is used for on-line identification of parameters that vary

quickly with time. In many instances, the identification problem is extremely ill-conditioned.

Consequently, when designing algorithms for such problems, it is essential to exercise care,

otherwise there may be no precision in a computed solution.



Why is the RLS algorithm more suited to the adaptive control problem than the basic LS

algorithm?

By using least squares, the calculations have to be done for every new sample and it is not

efficient. Using RLS the calculation part is very simple and the parameters are varied for

every new sample instantly.

Outline the theoretical basis of the phase margin design algorithm used in the self-tuning PI

control tutorial.

7) Phase Margin Algorithm:For a phase margin design we must ensure that the gain of the FR of the open loop transfer

function (TF).

L02= -Π + ϕm rads

To calculate the phase angle of L02(z) we need to remember that

Z=est

(z-1=e-st)

Now for a frequency response we substitute s=jw.

Hence:

z=ejwT

(z-1=e-jwT)

Thus,

L02(jw)=Kc(1+KiT)b1 e− jwT+b2 e− j 2wT+b3 e− j 3 wT +… ..+b6e− j 6wT

1−e− jwT

Now we know

e-jθ = cosθ – jsinθ

Hence,

∟L02= -tan-1 sin wT¿¿



Using the above formula (for ϕm design) we need to find the frequency ω(which we will call

ωgc)

This is usually done using a simple ‘line search’ algorithm since ωgc must lie in the ramge

0 ≤ ωgc ≤ πT

8) PI CONTROLLER FOR SERVO:A PI Controller (proportional-integral controller) is a feedback controller which drives the plant to be controlled with a weighted sum of the error (difference between the output and desired set-point) and the integral of that value. This STPI describes a variation of Iterative Feedback technique, which aids the PI controller compensating at a fixed level, which is also used for online monitoring which is the case in this assignment. The advantage in using STPI is that user will not be required to distract the system severally, which as a result tuning will not be observed if status of the system does not change.

Technically the STPI uses superimpositions principles, which over reacts to a small injected test signal to levels on the top of fixed set points. As a matter of fact any change in set point will stop the tuning process; this means that STPI cannot be used in variable set point environment.

The self tuning PI controller is designed to control the servo rig, the velocity and position models designed.

The self tuning PI controller selftu1.mdl is loaded into the Matlab current directory.



Fig14: Model for finding the velocity

The above model is the simulink model for the velocity. After running the model, we get the values of control parameters Kc and Ki. In this model we add two displays for finding the valoues of Ki and Kc .We give one dispaly to the product which gives the value of Kc and other one to the mux gives the value of Ki.

Next the transfer function of velocity is given to the model ,then the model is simulated and the values for Kc and Ki are displayed in the model.

Then the ouput at the scope is obtained as shown in the figure below.

Fig15: output from rls.m file

The above graph is the output for the reculsive least squares.



Fig16: output graph for the simulink model for velocity

Fig17: comparing Real-time and simulink model

Fig18: Output for real-time



Fig19: Output for the both simulink and real time model

Above graph is the output for the comparison of the simulink and real time to control the

velocity of the servo rig. In the above graph yellow graph is for real time servo, purple graph

is the set point and the green graph is for the simulink model. We can observe by seeing the

graph is that the real time servo rig is following the simulink graph.

8.1) STPI controller for Position of servo rig:

Fig20: comparing simulink and real-time model

By comparing the real-time and simulink model we can clearly observe the controllability of

the controller.



Fig21: output for the model

In the above graph we can clearly observe that the PI controller is not performing according

to the requirement. For the first few cycles it controlled but when we observe at the last part

of the wave, it is very clear that the waveform of the real time servo is not following the

waveform of the simulink model.

So, to control the position of the servo rig PIP controller is designed.

9) PIP CONTROLLER FOR SERVO:To design the pip controller, we have to solve the transfer function of the position and make

in ‘n’ notation. From that we get a matrix form.

PIP controller for position:

Calculating the gain values for position.

Posn- 1.23 Posn-1 +0.225 Posn-2=0.07135un-1+0.0435un-2

Posn = 1.23 Posn-1 -0.225 Posn-2+0.07135un-1-0.0435un-2

Posn+1 1.23 -0.2250 0.0435 0 Posn 0.0713

Posn 1 0 0 0 Posn-1 0



= + un

Un 0 0 0 0 Un-1 1

Mn+1 -1.23 0.2250 -0.0435 1 Mn -0.0713

a =

1.2300 -0.2250 0.0435 0

1.0000 0 0 0

0 0 0 0

-1.2300 0.2250 -0.0435 1.0000

b =

0.0713

0

1.0000

-0.0713

q =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

r=1

By using the command,

k=optpip(a,b,q,r)

We get the values of K

k = 3.7875 -0.8084 0.1563 -0.5575

By using the above gain values, implement the PIP controller.



The simulink description below is obtained when this gain values were inserted appropriately

into the default PIP controller block diagram. The figure below shows the scope of the

position response of the PIP controller, which is a first order response curve, attaining a

steady state in less than 2 seconds of start up with a step function used as the input signal.

Fig22: Simulink model for position

In case of position we are having four gain values. In this case we are taking three discrete

filters. By using this plant of PIP controller we are getting the output for position.

The output for position of this PIP controller is shown below.

Fig23: output for the simulink model



Fig24: comparison of simulink and real time servo

The above model is the comparison of the real time servo and the simulink model. The transfer function for the position is given in the discrete filter.

Fig25: o/p from real model

The above graph is the output of the real time servo rig.



Fig26: comparing the outputs of real time and the simulink model

In the above graph, pink colour wave is the setpoint or the input. The green line is the output

for the simulink model and the yellow colour wave is the output for the real time servo rig.

We can clearly see how the position of the servo is following the simulink model. We can say

that, by designing the pip controller the position of the servo rig is controlled.

10) Conclusion:

The mathematical models of the servo rig demonstrate how physical systems can be

represented and how the representation can be used to design controller to regulate the

operation of such system. This report, along with the prior knowledge of servo system, helps

one to understand the basic dynamics of the servo rig.



11) Reference:

1) System identification notes by Mr Ian French

2) AstrÖm, K.J and Wittenmark, (1995) B, Adaptive Control, Second Edition, Addison-Wesley Publishing Company.

3) Zhu, Y (2001), Multivariable System Identification for Process Control, Elsevier Science and Technology Books.


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SYSTEM IDENTIFICTION OF SERVO RIG