ME 413 Systems Dynamics & Control
Section 10-5: Steady State Errors and System Types
Chapter 10:
Time-Domain Analysis and Design of
Control Systems: Steady State Error and
System Types
A. Bazoune
10.5 STEADY STATE ERROR AND SYSTEM TYPES
Steady-state errors constitute an extremely important aspect of the system performance, for it would
be meaningless to design for dynamic accuracy if the steady output differed substantially from the
desired value for one reason or another.
The steady state error is a measure of system accuracy. These errors arise from the nature of
the inputs, system type and from nonlinearities of system components such as static friction,
backlash, etc. These are generally aggravated by amplifiers drifts, aging or deterioration. The steadystate performance of a stable control system is generally judged by its steady state error to step,
ramp and parabolic inputs.
Consider a unity feedback system as shown
in the Figure. The input is R ( s ) , the output
is C ( s ) , the feedback signal H ( s ) and the
difference between input and output is the
error signal E ( s ) .
E (s)
R (s)
G ( s)
C ( s)
H ( s)
From the above Figure
C (s)
G (s)
=
R (s) 1+ G (s)
(1)
C (s) = E (s) G (s)
(2)
On the other hand
Substitution of Equation (2) into (1) yields
E (s) =
1
R (s)
1+ G (s)
(3)
The steady-state error ess may be found by use of the Final Value Theorem (FVT) as follows:
ess = lim e ( t ) = lim SE ( s ) = lim
t →∞
s→0
1/15
s→0
sR ( s )
1+ G (s)
(4)
ME 413 Systems Dynamics & Control
Section 10-5: Steady State Errors and System Types
Equation (4) shows that the steady state error depends upon the input R ( s ) and the forward
transfer function G ( s ) . The expression for steady-state errors for various types of standard test
signals are derived next.
1.
Unit Step (Positional) Input
Input
r ( t ) = 1( t )
or
R ( s ) = L r ( t ) =
r ( t)
1
s
From Equation (4)
ess = lim
s→0
1
s (1 s )
sR ( s )
1
1
1
= lim
= lim
=
=
→
0
→
0
s
s
1+ G (s)
1+ G (s)
1 + G (s) 1 + G (0) 1 + Kp
t
where K p = G ( 0 ) is defined as the position error constant.
2.
Unit Ramp (Velocity) Input
r ( t ) = t or r ( t ) = 1
1
R ( s ) = L r ( t ) = 2
s
Input
or
From Equation (4)
r ( t)
1
s (1 s 2 )
sR ( s )
1
1
1
= lim
= lim
= lim
=
s→0 1 + G ( s )
s →0 1 + G ( s )
s →0 s + sG ( s )
s → 0 sG ( s )
Kv
ess = lim
1
t
where K v = lim sG ( s ) is defined as the velocity error constant.
s →0
3.
Unit Parabolic (Acceleration) Input
Input
1
r (t ) = t2
2
or
R ( s ) = L r ( t ) =
or r (t ) = 1
r ( t)
1
s3
From Equation (4)
s (1 s3 )
sR ( s )
1
1
1
= lim
= lim 2 2
= lim 2
=
s→ 0 1 + G ( s )
s→0 1 + G ( s )
s→0 s + s G ( s )
s →0 s G ( s )
Ka
ess = lim
where K a = lim s 2G ( s ) is defined as the acceleration error constant.
s→0
2/15
t
ME 413 Systems Dynamics & Control
Section 10-5: Steady State Errors and System Types
10.6 TYPES OF FEEDBACK CONTROL SYSTEMS
The open-loop transfer function of a unity feedback system can be written in two standard forms:
•
The time constant form
G (s) =
(
)
K (Tz1s + 1)(Tz 2 s + 1) Tzj s + 1
(
)
s Tp1s + 1 (Tz 2 s + 1) (Tzk s + 1)
n
(8)
where K and T are constants. The system type refers to the order of the pole of G ( s ) at s = 0 .
Equation (8) is of type n .
•
The pole-zero form
G (s) =
(
K ' ( s + z1 )( s + z2 ) s + z j
)
s ( s + p1 )( s + p2 ) ( s + pk )
n
(9)
The gains in the two forms are related by
∏z
K = K'
j
j
∏p
(10)
k
k
with the gain relation of Equation (10) for the two forms of G ( s ) , it is sufficient to obtain steady
state errors in terms of the gains of any one of the forms. We shall use the time constant form in the
discussion below.
Equation (8) involves the term sn in the denominator which corresponds to number of integrations in
the system. As s → 0 , this term dominates in determining the steady-state error. Control systems are
therefore classified in accordance with the number of integration in the open loop transfer function
G ( s ) as described below.
1.
Type-0 System
If n = 0, G ( s ) =
K
= K the steady-state errors to various standard inputs, obtained from Equations
s0
(5), (6), (7) and (8) are
1
1
1
=
=
1 + G (0) 1 + K 1 + Kp
ess ( Position )
=
ess ( Velocity )
= lim
s →0
ess ( Acceleration ) = lim
s →0
3/15
1
1
= lim
=∞
s
→
0
sG ( s )
sK
1
1
= lim 2 = ∞
s
→
0
s G (s)
sK
2
(11)
ME 413 Systems Dynamics & Control
Section 10-5: Steady State Errors and System Types
Thus a system with n = 0, or no integration in G ( s ) has
•
•
•
a constant position error,
infinite velocity error and
infinite acceleration error
2.
Type-1 System
If n = 1, G ( s ) =
K
, the steady-state errors to various standard inputs, obtained from Equations (5),
s1
(6), (7) and (8) are
1
1
1
= lim
=
=0
s
→
0
K 1+ ∞
1 + G (0)
1+
s
1
1
1
1
ess ( Velocity )
= lim
= lim
= =
s →0 sG ( s )
s→0
K K Kv
s
s
1
1
1
ess ( Acceleration ) = lim 2
= lim
= =∞
s →0 s G ( s )
s→0 2 K
0
s
s
ess ( Position )
=
(12)
Thus a system with n = 1, or with one integration in G ( s ) has
• a zero position error,
• a constant velocity error and
• infinite acceleration error
3.
Type-2 System
If n = 1, G ( s ) =
K
, the steady-state errors to various standard inputs, obtained from Equations (5),
s2
(6), (7) and (8) are
1
1
1
= lim
=
=0
1 + G ( 0 ) s→0 1 + K 1 + ∞
s2
1
1
1
ess ( Velocity )
= lim
= lim
= =0
s →0 sG ( s )
s →0
K ∞
s 2
s
1
1
1
1
ess ( Acceleration ) = lim 2
= lim
= =
s →0 s G ( s )
s →0 2 K
s 2 K Ka
s
ess ( Position )
=
Thus a system with n = 2, or with one integration in G ( s ) has
• a zero position error,
• a zero velocity error and
• a constant acceleration error
4/15
(13)
ME 413 Systems Dynamics & Control
Section 10-5: Steady State Errors and System Types
TABLE 1.
Steady-state errors in closed loop systems
r(t) = A
ess =
A
1 + Kp
ess = ∞
ess = ∞
r(t ) = At
ess =
ess = 0
A
Kv
ess = ∞
r (t ) =
ess = 0
ess = 0
K p = lim G(s) Kv = lim sG(s)
s→0
s →0
1 2
At
2
ess =
A
Ka
Ka = lim s2G(s)
s→0
where K p = G ( 0 ) is defined as the position error constant.
where K v = lim sG ( s ) is defined as the velocity error constant.
s →0
where K a = lim s 2G ( s ) is defined as the acceleration error constant.
s→0
█
Example 1 : Given the control system shown in the figure below, find the value of K so that
there is 10% error in the steady state.
R (s)
E( s)
K ( s + 5)
s ( s + 6 )( s + 7 )( s + 8 )
5/15
C( s)
ME 413 Systems Dynamics & Control
█
Section 10-5: Steady State Errors and System Types
Solution
Since the system type is 1, the error stated in the problem must apply to a ramp input; only a
ramp yields a finite error for in a type 1 system. Thus,
e (∞) =
1
= 0.1
kv
Therefore,
kv = 10 = lim sG ( s ) =
s →0
K ×5
5K
336 × 10
=
⇒ K=
= 672
6 × 7 × 8 336
5
10.7 STEADY STATE ERROR FOR NON-UNITY FEEDBACKSYSTEMS
R (s)
E(s)
C ( s)
G ( s)
H( s)
Add to the previous block two feedback blocks H1 ( s ) = −1 and H1 ( s ) = 1
R (s)
E(s)
G ( s)
C ( s)
H( s)
−1
R (s)
E(s)
G (s)
C ( s)
H ( s) −1
Ge ( s)
R (s)
E (s)
G( s)
1+ G( s) H( s) −G( s)
6/15
C ( s)
ME 413 Systems Dynamics & Control
Section 10-5: Steady State Errors and System Types
The equivalent forward transfer function is
Ge ( s ) =
█
G( s )
G( s )
=
1 + G ( s ) H ( s ) - G ( s ) 1 + G ( s ) H ( s ) - 1 
Example 2
For the system shown below, find
• The system type
• Appropriate error constant associated with the system type, and
• The steady state error for unit step input
R (s)
E (s)
C ( s)
100
s ( s + 10 )
1
s+5
█
Solution
The equivalent feed-forward transfer function is
Ge ( s ) =
G ( s)
G (s)
=
1 + G ( s ) H ( s ) − G ( s ) 1 + G ( s ) H ( s ) − 1
100
100
s ( s + 10 )
s ( s + 10 )
=
=

100  1
100  (1 − s − 5) 
1+
− 1  s ( s + 10 )( s + 5 ) +



s ( s + 10 )  ( s + 5 ) 
s ( s + 10 )  ( s + 5) 
=
100 ( s + 5)
100 ( s + 5 )
= 3
s ( s + 10 )( s + 5 ) + 100 (1 − ( s + 5) ) s + 15s2 − 50 s − 400
•
System Type: 0
•
Appropriate error constant K p = Ge ( 0 ) =
•
100 ( 0 + 5 )
5
=−
2
0 + 15 × 0 − 50 × 0 − 400
4
1
1
1
Steady State Error = ess =
=
=
= −4
1 + Kp 1 − 5 1 − 5
4
4
3
The negative value for steady state error implies that the output is larger than the input step.
7/15
ME 413 Systems Dynamics & Control
Section 10-5: Steady State Errors and System Types
8/15
ME 413 Systems Dynamics & Control
Section 10-5: Steady State Errors and System Types
9/15
ME 413 Systems Dynamics & Control
Section 10-5: Steady State Errors and System Types
10/15
ME 413 Systems Dynamics & Control
Section 10-5: Steady State Errors and System Types
11/15
ME 413 Systems Dynamics & Control
Section 10-5: Steady State Errors and System Types
12/15
ME 413 Systems Dynamics & Control
Section 10-5: Steady State Errors and System Types
13/15
ME 413 Systems Dynamics & Control
Section 10-5: Steady State Errors and System Types
14/15
ME 413 Systems Dynamics & Control
Section 10-5: Steady State Errors and System Types
15/15