Model-based Investigation of the Effect of Tuning Parameters o.docx

Model-based Investigation of the Effect of Tuning Parameters
on a
Servo-Motor Response and Mode Transition
1.0 Project Objective:
The objective of this project is to familiarize the students with
the use and limitations of
models in understanding the response of a practical servo-
system and evaluating its
usefulness. It introduces system modes as a tool of evaluating
the quality of a system’s
output. It also explores the ability of a controller’s tuning
parameters to affect the
system’s behavior and cause it to shift from one mode to
another.
2.0 Equipment: Matlab, Simulink and the EE380 textbook.
3.0 Background:
This section provides a brief background about the tools and
concepts needed to
understand the role of mathematical constructs in modeling,
predicting and tuning the
behavior of a system.
3.1: System Modes and the Quality of its Response
Control systems are enablers whose objective is to make a
servo-process (SP) yield to
the commands of an operator and provide him with useful work.

When the operator
issues a command, the SP can only respond by being in one of
the following modes:
o If the SP complies with the command of the operator it is in a
stable mode
o If it does not comply with the command, the mode is called
unstable
Figure-1: Stability does not imply useful outcome from the
servo-process
Obviously, the minimum expectation of the operator from the
servo-process is to be in a
stable mode. However, the system being in a stable mode need
not necessarily mean
that it is providing the operator with useful work. The quality of
the system response could
be too poor and practically useless. Take for example a car as a
servo-process (figure-1).
If the car fails to reach the destination, one may call the process
unstable. However, if the
process is stable and the car is able to reach the destination but
the path it took is too
long, rough, consumes too much fuel and contains many
detours, the effort derived from
the car cannot be called useful.
Measuring the usefulness of the outcome from a servo-process

is important since a
response that is not useful defeats the purpose of control. One
way to assess the quality
of a system’s response is though the use of performance
measures such as overshoot,
settling time, rise time and steady state error. A more general
way of describing the
quality of system behavior is through using system modes.
System modes may be used
to qualitatively describe the whole state of the response not
particular aspects of it, as in
performance measures. In industrial applications, six system
modes are used to
describe the response of a practical servo-process. They are
unstable, over-damped,
critically-damped, under-damped, oscillatory and chattering.
Their description and profile
are shown in table-1.
Mode Description Profile
1 Unstable Instability causes the position to diverge from the
reference position in either an oscillatory or an
exponential manner.
2 Over-
damped
A steady and slow motion towards the reference position
with no oscillation
3 Critically-
damped
Fast, steady and oscillation-free motion towards the

reference position.
4 Under-
damped
Oscillations whose strength decays with time are present
in motion as it approach the reference position.
5 Oscillatory Sustained position oscillations of equal magnitude
where
motion does not settle at the reference location.
6 Chattering Audible high-frequency, low-magnitude, sustained
oscillations around the reference position are present in
motion
Table-1: System modes
3.2: The Tuning Parameters of a System.
Assume that the quality of a system’s response was assessed. If
the quality is
satisfactory, the operator may use the system. On the other
hand, if the quality is not
satisfactory, the operator has to either replace the system or
adjust its performance to
meet the needed quality level. Being able to tune a servo-system
requires the system to
have a second input port besides the command port the operator
uses to drive the output
to the desired value (figure-2). Through the extra port, a set of
variables called the tuning
parameters set (β) affects the behavior of the system. In effect,
this makes the transfer
function of the system a function of both the complex frequency

S and the set of tuning
parameters (equation-1). Changing the tuning parameters will
simply change the transfer
function of the process.
)H(S,
X
Y
β= (1)
Such a situation is mostly encountered when feedback control is
used to adjust the
behavior of a servo-process. For example, in figure-3 position
and velocity feedback are
used with a servo-motor, β = {Kp, Kv}, where Kp is the forward
position gain and Kv is the
velocity feedback gain.
Figure-2: Servo-system with and without tuning parameters (β).
Figure-3: A servo-system with two tuning parameters Kp and
Kv.
3.3: Models and Virtual Experiments.
One can experiment with different values of the tuning
parameters and select the one’s
that provide the best output quality. This approach may be
undesirable since some

values of the tuning parameters could cause instability and
damage the servo-process. In
addition, physical experiments are usually costly and although a
must, they should be
used sparingly in tuning a system.
One way to lower the burden of physical experimentation is to
use simulation. The core of
a simulation experiment is a mathematical model of the servo-
process. A model is a
device or a process used by an engineer to predict the future
output of a system in
response to an action applied on the system in the present. A
Model is constructed by
studying the components of a system and mathematically
capturing their input-output
relation. An aggregator is then used to combine the individual
behaviors of the
components to yield the system behavior. For example, ohm’s
law, which is an empirical
law, is used to model circuit components (figure-4). Kirchhoff’s
laws are then used to
aggregate the individual relations to yield the equation that ties
the input of the electric
circuit to its output (figure-5).
Figure-4: Ohm’s law used to model individual circuit
components
Figure-5: Kirchhoff’s laws used to construct the overall circuit
(system) model.

The ability of a model to emulate a system’s response and
predict its behavior depends
on using full information about how the system works. This is
usually not possible; there
will always be missing information about the forces that act on
a system. As a result, the
ability of a model to predict the behavior of a system will
always be limited. This does not
mean that the model is useless; it only means that it cannot
predict everything about the
behavior of the system. In any case, some applications do not
require high accuracy and
full behavior spectrum prediction while other applications do.
The more information used
in constructing a model, the better is its ability to predict a
system’s behavior.
3.4: Analysis Tools: System Poles , Rootlocus and Routh-
Horowitz
Using a model of a physical system to assist in predicting its
behavior requires the
identification of the information bearing elements of the model
and the development of
tools that can extract the needed information from these
elements.
Figure-6: System behavior versus pole location.
Poles and zeros, especially poles, seem to be the most important
element that bears
information about the behavior of linear systems and the modes
it can assume (figure-6).

The most important behavior about a system is to know whether
it is stable or unstable.
In control system theory, the Routh–Hurwitz stability criterion
is a mathematical test that
is a necessary and sufficient condition for the stability of a
linear time invariant (LTI)
control system. The Routh test is an efficient recursive
algorithm to determine whether all
the roots of the characteristic polynomial of a linear system
have negative real parts (i.e.
the system is stable). It operates on the denominator of the
system transfer function (2). It
begins by constructing the Routh-Horowitz matrix (3) then
checking the signs of its first
column. If all the sign are the same, the poles of H(S) (zeros of
D(S)) are all in the right
hand side and the system is stable.
D(S)
N(S)
H(S)_ =
(2)
(3)
While the location of the system poles provides information
about system behavior, the
trajectories of the poles (figure-7) provide a good idea about
how the system modes

change.
Figure-7: Pole trajectory versus system response
Studying the trajectories the system poles as a function of the
tuning parameters used to
modify the behavior of the system is important in understanding
the effect of these
parameters on the system. It is also an important aid for the
designer that allows him to
determine the value or set of parameter that place the system in
a desired mode. In
control theory and stability theory, root locus analysis (figure-
8) is a graphical method for
examining how the roots of a system change with variation of a
certain system parameter.
The root locus plots the poles of the closed loop transfer
function in the complex s-plane
as a function of a gain parameter.
Figure-8: A Root-locus Plot.
In this project, the effect of changing the tuning parameters of a
controlled servo-motor
on its mode changes is investigated. Three different models of
the position servo with
different accuracies are used:
1- a second order linear model of position obtained by
neglecting the small

electric time constant of the motor and considering only the
large
mechanical time constant.
2- A third order linear model of position that includes both
electrical and
mechanical time constants of the motor.
3- A third order model of position that includes nonlinearities
such as servo-
amplifier saturation and deadzone nonlinearities induced by
phenomenon
such as static friction.
A single loop, position feedback control is used with a single
tuning parameter, the
forward error proportional gain Kp.
4.0 Second order motor model-based behavior prediction:
A motor is an electromechanical device with an electrical input
component connected to a
mechanical output component (figure-9). The system is assumed
linear and can be
modeled by an integrator and first order transfer function for
the mechanical and electrical
components with time constants τ m and τ e respectively.
Furthermore, the electric time
constant is assumed to be much lower than the mechanical one
and is ignored (τ m
>>>τ e). In this case, the position transfer function of the motor
reduces to the second
order transfer function in (4).

1S
K
S
K
1)S(S
K
S)(H
m
21m
+⋅
+=
+⋅
=
ττ m
(4)
Figure-9: An electromechanical machine in a motor mode.
A single loop position feedback is used to control the motor
(figure-10). The only tuning
parameter used to adjust the quality of system behavior is the
error proportional gain (Kp).

Figure-10: Servo-loop, 2nd order linear servo-motor
4.1 Mode transition for 2nd order system:
1- Choose τ m =0.5 sec and Km=5.
2- Derive the the tuning parameter (Kp) dependant transfer
function of the system in
figure-10.
3- Use matlab to draw the rootlocus of the closed loop system
with Kp as the free
variable.
4- From the root-locus plot, determine the modes of the servo-
motor which the second
order linear model can predict. Also, determine the region of
values of Kp that
would place the system in each one of these modes.
5- Select specific values for Kp that would place the feedback
system in each of these
modes and use matlab to plot on the same graph the step
response of the system
for all the values of the selected Kps.
5.0 Third order motor model-based behavior prediction:
To improve the accuracy of the servo-motor model, the electric
time (τ e) constant is not
neglected. In this case, the transfer function of the motor
becomes third order (5).
1S

K
1S
K
S
K
1)S(1)S(S
K
S)(H 3
e
21m
+⋅
+
+⋅
+=
+⋅ +⋅
=
mem ττττ
(5)
Similar to the second order model, the servo-motor is placed in
a position feedback
configuration with tuning parameter Kp (figure-11).
Figure-11: Servo-loop, 3rd order linear servomotor

5.1 Sensitivity to the electric time constant:
In this section, the effect of the electric time constant (τ e) on
the ability of the second
order model of the motor to predict the behavior of the third
order model (figure-11) is
examined.
1- Obtain the transfer function of the feedback system in figure-
11.
2- Select Km=5, τ m =0.5 and Kp as the critically damped value
you obtained
previously for the second order model (section 4.1).
3- Plot on the same graph the step response the following values
of ( τ e
=1, .5, .4, .3, .2, .1). Record your observations.
4- Plot the maximum overshoot from the step response versus
the electric time
constant for the following values of ( τ e =1, .5, .4, .3, .2, .1).
Record your
observations.
5- Use the transfer function to derive the rootlocus equation of
the system with the
electrical time constant (τ e) as the free variable (6)
0
D(S)
N(S)

1 e =⋅ +τ (6)
6- For the three values of Kp representing the other modes in
the 2nd order model
obtained in section 4.1,
7- Use the Routh-Horowtiz criterion to determine the range of τ
e for which the system
is stable,
8- use the root-locus plot to determine the range of values of τ e
where the third order
system may be approximated by a second order system with two
dominant poles.
This can be done by determining the range of τ e for which the
quickly fading pole
is about 10 times the real part of the two dominant complex
poles.
9- Is it always possible to find a value of τ e where
approximating a 3rd order system
with a 2nd order system is possible? If you find more than one
solution for τ e that
satisfies the quickly fading condition, which one to keep and
which one to reject?
10- Draw the 3 rootloci for each Kp on the same graph.
5.2 Mode transitions as a function of Kp:
1- Choose the electric time constant equal to τ e =0.05 and τ e
=.5.
2- What are the range of values for Kp for which the system is
stale?
3- Derive the root-locus equation of the system in figure-11

with Kp as the free
parameter.
4- Use the root-locus to determine the modes that the system
response can assume.
Also, determine the range of values of Kp that will place the
system in each one of
these modes.
5- Select a point from each range a mode can assume and plot
the step response on
the same graph for the 3rd order system.
6- Also, for the second order model, plot the step response for
the three values of Kp
on the same graph. Compare the two cases.
6.0 Third order motor, Nonlinear, model-based behavior:
Servo-systems, in particular motors, do experience a multitude
of forces that cannot be
expressed using linear models. For example, saturation in the
servo-amplifier, dead-zone
caused by static friction or semiconductor nonlinearity and
hysteresis caused by
mechanical couplings. This subsection will explore the effect of
saturation and dead zone
nonlinearities (figure-12) on the response of the servo-motor.
Saturation affects the high
magnitude of the motor actuating signal and causes a slowdown
in the system. Dead
zone affect the low magnitude of actuation and causes mainly

steady state error.
Figure-12: Saturation and dead zone nonlinearities affecting the
actuation signal
The presence of such nonlinearities can seriously alter the
behavior of the system and
disrupt the ability of linear models to predict the motor’s
behavior. For the model to better
predict the experiment, the effect of these nonlinearities is
usually introduced in series
with the motor (figure-13).
Figure-13: Servo-loop, 3rd order linear servomotor
Figure-14: Servo-loop, 3rd order linear servomotor – Simulink
realization
6.1 Effect of amplifier saturation on system response:
1- Use simulink (figure-14) to construct the system in figure-13
and choose the
saturation nonlinearity from the discontinuous components
menu.
2- Set Km=5, τ e =.05, τ m= .5
3- Select a step input from the sources block set of simulink and
the “to workspace’
block to export the output to matalb,

4- Repeat the following steps for the three values of Kp
representing each mode of the
third order linear model (section 5.2)
5- Assume that the saturation nonlinearity does not exist (you
may do so by setting
the saturation block level to a very high value), determine the
maximum absolute
value the input to the motor can assume when the system is
subjected to a unit
step input.
6- On the same graph, plot on the same graph the step response
for the following
saturation levels: 90%, 70%, 40%, 15% of the maximum
absolute value of the input
to the saturation-free case
7- Record your observation about the effect of saturation on the
step response of the
system
6.2 Mode transition as a function of the tuning parameters -
saturation:
1- Select a 70% saturation level of the maximum absolute value
2- Use simulink to explore the effect of Kp on the modes that
the system can assume.
3- Determine the range of values of Kp that will place the
system in each one of these
modes.

the step response,
5- Repeat the above for a 40% and 10% saturation levels,
6- Record your observations
6.3 Effect of dead zone nonlinearity on system response:
1- Use simulink (figure-14) to construct the system in figure-13
and choose the dead
zone nonlinearity from the discontinuous components menu.
2- Set Km=5, τ e =.05, τ m= .5
3- Select a step input from the sources and the “to workspace’
block to export the
output to matalb,
4- Repeat the following steps for the three values of Kp
representing each mode of the
third order linear model (section 5.2)
5- Assume that the dead zone nonlinearity does not exist (you
may do so by setting
the dead zone block level to a zero value), determine the
maximum absolute value
the input to the motor can assume when the system is subjected
to a unit step input.
6- On the same graph, plot the step response for the following
saturation levels: 30%,
20%, 10%, 5% of the maximum absolute value of the input to
the dead zone-free
case
7- Record your observation about the effect of dead zone on the

step response of the
system
6.4 Mode transition as a function of the tuning parameters –
Dead Zone:
1- Select a 30% Dead Zone level of the maximum absolute value
2- Use simulink to explore the effect of Kp on the modes that
the system can
assume.
3- Determine the range of values of Kp that will place the
system in each one of
these modes.
the step response,
5- Repeat the above for a 20% and 10% saturation levels,
6- Record your observations
7.0 Instructions for writing the report:
1- The report must be typed, well-organized and contain all
formal components of
objective, introduction, data section, data analysis section,
conclusions and
references
2- Details of the mathematical derivations has to be shown in
full and all used
equations must be numbered

3- All figures and tables must be numbered and provided with
small descriptive
captions
4- You have to comment on the results obtained. Figures,
equations etc. that are
merely stated with no comments will not be considered
5- You have to provide a meaningful conclusion section
6- You may explore the topic more. Any meaningful and well-
documented extra work
will be rewarded with extra credit.
7- Any constructive suggestion about how to enhance the
project will be taken into
consideration in marking the report.

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