Sharanabasava C Pilli Principal, KLE Societys

1
ME55 CONTROL ENGINEERING
Sharanabasava C PilliPrincipal, KLE Societys
College of Engineering and Technology,
Udyambag, Belgaum-590008Email
scpilli_at_yahoo.co.in
2
CONTENT
Steady state errors in unity feedback control
system
3
STEADY STATE ERRORS IN UNITY FEEDBACK CONTROL
SYSTEMS

• Causes of errors
• Changes in reference input cause transient and
  steady state errors.
• Imperfections such as friction, backlash, drift,
  aging cause steady state errors.
• Inability to follow the particular type of input.

4
Typical Input Signals
step , ramp , parabolic or acceleration
5
FINAL VALUE THEOREM
Final value theorem relates the steady state
behavior of f(t) to behavior of sF(s) in the
neighborhood of s0. Applies only when lim t??
f(t) exists f(t) settles down to a definite value
for t??. If all poles of sF(s) lie to the left
half s plane, lim t?? f(t) exists. If sF(s0 has
poles on the imaginary axis or in right half of s
plane, f(t) will contain oscillating or
exponentially increasing time functions and lim
t?? f(t) does not exist.e.g. f(t) sin(?t), has
roots on imaginary axis as s?j?.
6
POLES

• Points in the s-plane at which the
• function G(s) is not analytic are called
  singular points.
• Singular points at which the function G(s) or its
  derivatives approach infinity are called poles.
• The points at which the function G(s) equals zero
  are called zeros.

has zero at s -2, poles at s 0, -1,
-3,-3,-1?j3
x
7
Classification of Control Systems
Consider a unity feedback system with open loop
transfer function G(s)
8
Steady State Error
9
Error Constants

• Error constants Kp, Kv, Ka describe the ability
  of the unity feedback system to reduce or
  eliminate steady state error.
• Velocity error is used to express steady state
  error for a ramp input. It is error in position
  due to a ramp input and not an error in velocity.
• Acceleration error is used to express steady
  state error for a parabolic input. It is error in
  position due to a parabolic input and not an
  error in acceleration.

10
Static Position Error Constant Kp
11
Static Position Error Constant Kp
For type 0 system
For type 1 or higer system

• Response of a feed back system
• involves a steady state error if there is no
  integration in the feed forward path.
• steady state error can be reduced by a large gain
  K.
• with large K , stability problems are
  encountered.
• for zero ess, system of Type 1 or higher be used.

12
Static Velocity Error Constant Kv
13
Static Velocity Error Constant Kv

• Response of a feed back system
• Type 0 system is incapable of following a ramp
  input in steady state.
• Type 1 system with unity feedback can follow ramp
  input with finite error i.e. input and output
  velocities are the same with finite positional
  error.
• Error can be reduced with large gain.
• Type 2 and higher systems can follow ramp input
  with zero steady state error.

14
Static Acceleration Error Constant Ka
15
Static Acceleration Error Constant Ka

• Response of a feed back system
• Type 0, 1 incapable of following a parabolic
  input in steady state.
• Type 2 can follow parabolic input with
  finite error.
• Type 3 and higher can follow ramp input with
  zero ess.

16
Summary of Steady State Errors
17
Summary of Steady State Errors

• To improve the steady state performance we can
• increase the type of the system by adding an
  
• integrator(s) to the feed forward path.
• Addition of integrators introduces the stability
• problems.
• The step, ramp, and parabolic error constants are
  significant for the error analysis only when the
  input signal is a step, ramp and parabolic
  function respectively.

18
Summary of Steady State Errors

• Since the error constants are defined wrt forward
  path transfer function G(s), the method is
  applicable to unity feedback system only.
• Error relies on final value theorem, first check
  to see if sE(s) has any poles on imaginary axis
  or in the right half of s-plane.
• If configuration differs, we can either simplify
  to unity feedback system or establish error
  signal and apply final value theorem.
• Principle of superposition holds good in
  combination of these inputs.

19
Comparison of ess in Open and Closed Loop Systems
20
ess in Open / Closed Loop System
For a unit step input
Setting Kp very large ess can be reduced
Aging with time will introduce error.
21
Example 1
For a unity negative feedback system determine
the steady state error due to unit step, unit
ramp, and unit parabolic input of the following
systems.
poles at s -10, -5
poles at s 0, -2?j28
poles at s 0,0,-2?j3 zeros at s -0.25, -0.50

22
Example 1
Type 0 hence can only follow unit step input with
finite error at steady state and can not follow
ramp and parabolic inputs at steady state.
23
Example 1
Type 1 system hence can only follow unit step
input with zero error at steady state, can follow
unit ramp input at steady state With finite error
at steady state, and can not follow parabolic
input at steady state.
Large K can lead to unstable state
24
Example 1
Large K can lead to unstable state
Type 2 system hence can only follow unit step,
and ramp input with zero error at steady state,
can follow unit parabolic input with finite error
at steady state
25
Example 2
The closed loop system is unstable for all values
of K, and error analysis is meaningless.
26
RECAP

• Causes of error
• Typical input signals
• Classification of control systems
• Steady state error
• Error constants
• static position, velocity and
  acceleration constants
• Comparison of errors in open loop and closed loop
• Examples