Performance Specification and Analysis
Lecture 2: Performance
specification & analysis
Control system: Integration of several components such as sensors,
actuators, signal conditioning devices, and controllers etc.,
 In the design, selection, and prescription of these components their
performance requirements have to be specified within the functional
needs of overall control system
2.0 Performance Specification
and Analysis
 Engineering parameters for performance specification, may be given in
Time or Frequency domains
 Instrument ratings of commercial products are often developed on the
basis of these engineering parameters
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Parameters for Performance Specification
 In multicomponent system, the overall error depends on the component
(sensor & transducer) error. Component error degrades the
performance of a control system
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Lecture 2: Performance
specification & analysis
A great majority of instrument ratings provided by manufacturers are in the
form of static parameters
A perfect measuring device can have the following characteristics
 Output instantly reaches the measured value (fast response)
 Transducer output is sufficiently large (high gain, low output impedance,
high sensitivity)
 Output remains at the measured value (no drift, stability, and robustness)
unless the measurand itself changes (stability)
 Output signal level of the transducer varies in proportion to the
measurand (static linearity)
 Connection of measuring device does not distort the measurand itself
(loading effects are absent and impedances are matched)
 Power consumption is low (high input impedance)
Items 1 to 4 can be specified either in the time or frequency domains 3
Lecture 2: Performance
specification & analysis
Time Domain Analysis
Responses of 1st and 2nd order systems
The typical reference input (test) signals are impulse, step, ramp,
parabolic, and sinusoidal functions
Impulse function
Step function
is a constant
is a constant
Ramp function
0
Parabolic function
0
0
0
0
0
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Lecture 2: Performance
specification & analysis
Time response of First-order system
Lecture 2: Performance
specification & analysis
Example
Show that
Solution
Consider a first order system,
if
Response to a unit step:
Then,
;
The slope at
is given by
So,
/
is known as the time constant
The error signal is
The error is
/
/
The steady-state error,
The steady-state error is,
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Lecture 2: Performance
specification & analysis
Time response of a second-order system
Second-order system response subject to unit-step
Consider a standard second-order system,
Undamped natural frequency;
as
Damping ratio
 For ζ
0, poles at
,
 For ζ
1, poles at
,
1
For a unit-step response
 For ζ
⇒
sin
1, poles at
The poles of the transfer function are
 For ζ
1
⇒
where,
where
damped natural frequency 7
1
ζ
,
1, poles at
1
ζ
;
(Undamped)
ζ
cos
1
,
1
ζ
(Under damped)
ζ
(Critically damped)
ζ
1
1
ζ
cos
,
⇒
damping constant
Lecture 2: Performance
specification & analysis
(Over damped)
ζ
1,
ζ
ζ
1
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The unit-step responses for various values of
Lecture 2: Performance
specification & analysis
Lecture 2: Performance
specification & analysis
Time domain specifications
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Time domain specifications
Lecture 2: Performance
specification & analysis
Time response indices are as follows:
 Rise time ( )
- Time taken to pass the steady-state value for the first time or 90%
of the steady state value
- Indicates the speed of the system
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 Settling time ( )
- Time taken for the device response to settle down within a certain
) of the steady-state value
percentage (typically
 Percentage overshoot (P.O)
-
 Delay time ( )
- Time taken to reach 50% of the steady state value for the first time
Lecture 2: Performance
specification & analysis
Time domain specifications
;
is the peak value
- P. O is a measure of damping or relative stability in the device
 Steady-state error ( )
- Deviation from the actual steady-state value from the desired value
 Peak time ( )
- Time at the first peak of the device response
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Simple Oscillator Model
Lecture 2: Performance
specification & analysis
Simple oscillator is a versatile model, which can represent the
performance of a variety of devices.
Based on the level of damping, oscillatory and non-oscillatory behavior
can be represented
The model can be represented as
where, is the excitation; is the response,
frequency, is damping ratio
is undamped natural
Performance parameters using simple oscillator model
Performance parameter
The response of a system to a unit step excitation, with zero - initial
conditions is
Rise time
Peak time
Maximum overshoot
Time constant
Settling time (2%)
Example (1) Mechanical system
Expression
Percentage overshoot (P.O)
The damped natural frequency,
where,
Lecture 2: Performance
specification & analysis
.
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Lecture 2: Performance
specification & analysis
For the mechanical system shown in the figure,
is the mass, is the spring constant, is the
is the external force and
Friction coefficient,
is the displacement. From Newton’s 2nd law
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Example: Mechanical system
Lecture 2: Performance
specification & analysis
: the gain of the system; ζ: the damping ratio of the system
: the natural frequency of the system
The poles of the system are,
Unit-step response
General form of second-order system is
Transfer function:
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Example
Lecture 2: Performance
specification & analysis
An automobile weighs 1000 kg. The equivalent stiffness at each level,
including the suspension system, is approximately
N/m. If
the suspension is designed for a percentage overshoot of 1%, estimate
the damping constant that is needed at each wheel.
Solution:
Consider a simple oscillator model, which is of the form
where, equivalent mass,
Equivalent stiffness,
N/m
Equivalent damping constant,
Displacement excitation at wheel is
Lecture 2: Performance
specification & analysis
Solution: (Contd..)
By comparing the above equation with the oscillator model, we get,
Note: the equivalent mass at each wheel is taken as ¼ of the total mass
For a Percentage Overshoot of 1% from the table, we have
which gives
Substitute values in equation of . We get,
N/m/s
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Example
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Lecture 2: Performance
specification & analysis
Consider the system shown below, where
and
rad/sec.
and when the system is subjected to a unit-step input.
Find
Lecture 2: Performance
specification & analysis
Solution: (Contd..)
.
Rise time,
.
[sec]
.
Peak time,
Solution:
[sec]
.
.
Maximum overshoot,
The closed loop transfer function is
Settling time,
.
or
[sec]
This is a standard second-order system with damped response as
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Frequency Domain Analysis
Lecture 2: Performance
specification & analysis
Lecture 2: Performance
specification & analysis
In general, when the input phase is
Then, the steady-state output is
Advantages of frequency response method
i.e.
 Frequency response characteristics of elements having complex dynamics
can be obtained experimentally;
 Controller design can be carried out by shaping the frequency response of
open-loop systems;
 Control systems may be designed so that the effects of undesirable
noises/disturbances are negligible
The frequency response of a system is defined as the steady-state response
of the system to a sinusoidal input
Hence, the steady-state response is a sinusoidal signal with the same
frequency as the input. But, the amplitudes and phases are different and
are dependent on the input frequency
Observations
 The amplitude of the steady-state output is
 The output phase is
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Lecture 2: Performance
specification & analysis
and
are frequency dependent and are referred to as the
magnitude and phase responses of the system respectively.
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Example
Consider the stable system with transfer function
Notes:
. When the
Input sinusoidal signal can be represented by a phasor
signal passes through a system, its amplitude and phase will be changed. We
can think of the system itself as represented by a complex number
.
Find the steady-state output due to the input
Solution
Hence, the output is
Then,
Lecture 2: Performance
specification & analysis
Note that
Thus, at
rad/s,
The steady-state output of the system is
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Frequency response of systems
Lecture 2: Performance
specification & analysis
Frequency domain specifications (Bode diagram)
Lecture 2: Performance
specification & analysis
Steady-state response of systems to sinusoidal inputs
To decrease the rise time (or to increase the speed), the bandwidth must be
increased
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Useful parameters for performance specification
Lecture 2: Performance
specification & analysis
 Useful frequency range
- Corresponds to flat region in the gain curve
is several times smaller than
- The
 Instrument bandwidth
- Measure of the useful frequency range of an instrument
- Common definitions
* Frequency range where the transfer function is flat
* Half power bandwidth – Amplitude drops to 0.707
 Control bandwidth
- Bandwidth (speed) of the control signal
 Static gain (DC gain)
- Steady-state gain to a constant input and is equal to the system
transfer function evaluated at
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 Gain margin (GM) and Phase margin (PM)
Lecture 2: Performance
specification & analysis
GM and PM are measures of stability of a device
GM is defined as the amount of gain that can be added to the system where
the loop phase lag is
Similarly, at the frequency where the gain is unity, the amount (margin) of
phase lag that can be added to the system so as to make the loop phase lag
equal to is a measure of stability
Feedback system
Bode diagram with GM and PM
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Lecture 2: Performance
Example
specification & analysis
Consider a plant of transfer function,
. What is the
static gain of this plant? Show that the magnitude of the transfer
function reaches
of the static gain when the excitation frequency
rad/s. Note that the frequency,
rad/s, may be taken as
is
the operating bandwidth of the plant.
Solution
Consider the frequency transfer function,
in rad/s
Hence, static gain (from the definition)
Therefore,
half-power bandwidth.
, where
is
and when
Lecture 2: Performance
Summary
specification & analysis
Study of time responses to impulse, step, ramp, and parabolic functions
Time response of First-order system
Transfer function,
The response depends on time constant
A first-order system gives a ‘good’ response with respect to a step input,
but will yield a steady-state error of with respect to a unit-ramp input
Time response of Second-order system
The standard 2nd order system:
at this frequency. This corresponds to the
The poles are:
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Lecture 2: Performance
Summary (Contd..)
specification & analysis
The system response is determined by the damping ratio and undamped
natural frequency
un-damped response
under-damped response
critically damped response
over-damped response
Summary (Contd..)
Unit-step response of a 2nd order system (
 Instrument bandwidth: measure of the useful frequency range
)
Lecture 2: Performance
specification & analysis
Frequency domain specifications
 Useful frequency range: flat (static) region in the gain curve
 Control bandwidth: used to specify the maximum possible speed of
control
 Static gain: gain of measuring instrument within useful (flat) range
Rise time,
Maximum overshoot,
Settling time,
Peak time,
 Gain margin: margin of
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 Phase margin: margin of
when
when
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