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Control SystemsEE-4751Part 3 Steady State Error and Rouths
Stability CriteriaSteady State Errors2Steady State Error & Test
SignalsSteady State Error is the difference between input and
output of system for some specific test input signal as time
reaches to infinity.Step test inputs represents constant position
input and helpful in determining the ability of system to position
itself w.r.t some stationary target.Ramp input signals represents
the constant velocity and used to determine the ability of system
towards constantly increasing amplitude and to track a constant
velocity.Parabolic input signal represents the constant
acceleration and used to track the acceleration.
3Table of Test Waveforms for Steady State Error4
Cont.5
Output 1 has zero steady state error.Output 2 has finite steady
state error.Output 1 has zero error.Output 2 has finite
error.Output 3 has infinite error.
Source of Steady State ErrorNon linarites are source of steady
state error like Back Lash in gears or motors.Due to pure Gain in
Forward path error always exist some finite and non-zero value.
By increasing the gain value error can be reduced to zero for a
step input case.
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Steady State Error for Unity Feedback SystemsFor above system
mathematical expression for steady state error will be.
In steady state our interest is in the final value which can be
found by using final value theorem. Stated as follows.7
Example(Steady State Error)8
Steady State Error for unity feedbackNow we will see the error
for the system in above diagram.
By applying final value theorem we will get the steady state
error.9
ContNow by substituting different input signals we will conclude
the relation between open loop system and nature of steady state
response.For Step input
For ramp input
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ContFor Parabolic input
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Static Error ConstantsSteady state error performance
specifications are called static error constants.We already
discussed the relations of steady state error for different input
signals. Terms in denominator with limit determine the steady state
error and called as static error constants.12Cont13
ContFollowings are three static error constants.14
ContRelation between Steady state error and other factors.15
System Stability & Routh Stability Criteria16Stability of
SystemThere are following definitions for the LTI system
stability.
BIBO Stable System is defined as 17
ContDefinitions on the bases of pole location.Stable system has
all poles in LHP.Unstable system has at least one pole in
RHP.Marginally stable system has poles on imaginary axis.(boundary
of LHP and RHP)18ContStable and unstable system responses are
shown19
Cont20
Routh-Hurwitz Stability CriteriaYields stability information
without solving closed loop system poles.This method yields the
location of poles.(either in LHP, RHP or imaginary axis. No
coordinate just position according to plane )There are two basic
stepsGenerate Data Table(Routh Table)Interpret table to get
location of poles)
21Step 1:Generating Routh TableLet consider the following
transfer function.
Label the table using Xtic equation
22
ContNow complete table as
23
Example(Routh Table)24
Step 2: Interpretation of Routh tableNumber of poles in the
right hand plane are equal to number of sign changes in first
column of routh table.System will be stable if there are no sign
changes in first column of routh table other wise unstable.
25Special Case 1: Zero Only in First ColoumnIf the first element
of row is 0 than all the elements of next row will be 0. to avoid
this epsilon is assigned to zero.Following example will explain the
procedure in more details.Let suppose for the following transfer
function first element of 3rd row will be zero.26
ContRouth table is shown below in which zero is replaced by
epsilon.27
ContIn next step we will take epsilon as positive and negative
both to see the sign changes in first column as shown.
From the table it can be seen that this system is unstable.
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Alternate Method (Reverse Coefficients)Alternate method to avoid
the special case 1 is to reverse the coefficients of characteristic
equation and than generate Routh table and see the sign changes to
determine the stability.For example for the system.29
ContCharacteristic equation is
After reversing the coefficients
Routh Table
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Special Case 2: Entire Row is ZeroIn some cases entire row
becomes zero in that case we will follow these steps.Form auxiliary
equation from the coefficients of row right above zero row.Take
derivative of that row and replace 0 row by the coefficients of
that.Construct remaining table normally as discussed
earlier.31Example (Special case 2)Transfer function.
Third row will be entire zero in that case.
Auxiliary equation will be
Its derivative
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Cont.Final table will be of the form.33
Practice ExersciseQ.134
ContQ.2
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ContQ.3
36
ContQ.4
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