Lecture 12 ME 176 6 Steady State Error

1. ME 176
Control Systems Engineering
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Steady-State Errors

2. Background: Design Process


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3. Background: Analysis & Design Objectives

"Analysis is the process by which a system's performance is determined."

"Design is the process by which a systems performance is created or changed."

Transient Response
Steady State Response

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Steady State Error
Stability

4. Background: Steady-State Error

Definition : is the difference between the input and the output for a
prescribed test input as t approaches infinity.

Scope :

Linear - the relationship between the input and the output of the
system satisfies the superposition property. If the input to the
system is the sum of two component signals:





In general:

If, then,
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5. Background: Steady-State Error

Scope :

Time invariant systems - are systems that can be modeled with a
transfer function that is not a function of time except expressed
by the input and output.

"Meaning, that whether we apply an input to the system now or T
seconds from now, the output will be identical, except for a time delay
of the T seconds. If the output due to input x(t) is y(t), then the output
due to input x(t − T) is y(t − T). More specifically, an input affected by
a time delay should effect a corresponding time delay in the output,
hence time-invariant."


STABLE

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6. Background: Steady-State Error

Test Inputs :

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7. Evaluating: Steady-State Error


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1. Step Input:
Output 1 : No
Steady-State Error
Output 2 : Constant
Steady-State Error
of e2


2. Ramp Input
Output 1 : No
Steady-State Error
Output 2 : Constant
Steady-State Error
of e2
Output 3 : Infinite
Steady-State Error

8. Representation: Steady-State Error


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R(s) and C(s) : Input and Output Respectively
E(s) : Steady-State Error

a) General Representation:
T(s) : Closed loop transfer function


b) Unity Feedback Systems
G(s): Open loop transfer function

9. Sources: Steady-State Error


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Scope : Errors arising from configuration of the system itself and the
type of applied input.

a) Pure Gain : there will always be a
steady state error for a step input

b) Integrator : can have a zero steady
state error for a step input

10. Defining: Steady-State Error for Unity Feedback


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11. Example: Steady-State Error for Unity Feedback






Steady-state error for a unit step input:

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12. Defining: Steady-State Error for Unity Feedback


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13. Example: Steady-State Error for Unity Feedback






Find the steady-state errors for inputs
of 5u(t), 5tu(t), and 5t^2u(t). The function
u(t) is the step function.

Note Laplace transforms:
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14. Defining: Static Error Constants for Unity Feedback








Position Constant
Velocity Constant


Acceleration Constant
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15. Example: Static Error Constants for Unity Feedback

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16. Example: Static Error Constants for Unity Feedback

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17. Example: Static Error Constants for Unity Feedback
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18. System Types for
Unity Feedback:
Given the system shown, the
"system type" is defined as the
value of "n" in the denominator;
or, equivalently the number of pure
integrations in the feedforward path.

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19. Specifications: Steady-State Error

"Static error constants can be used to specificy the
steady-state error characteristics of a control system."


Knowing Kp = 1000 what can be learned of the system:

1.System is stable.
2.System is Type 0
3.Input Test signal is step.
4.Error per unit step:
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20. Example: Steady-State Error Specification






Find K so that there is a 10% error in steady state.
Since system is Type 1, error stated must apply to ramp function.

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21. Analysis: Steady-State Error for Disturbances




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"Steady-state error produced by a step
function can be reduced by increasing
the gain of G1(s) or decreasing the
gain of G2(s)."

22. Example: Steady-State Error for Disturbances









Find the steady-state error component due to a step disturbance.
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23. Definition: Steady-State Error for Nonunity Feedback





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Move R(s) to right
of summing
junction.

Compute resulting
G(s) and H(s).

Add and subtract
unity feedback
paths.

Combine negative
feedback path to H
(s).

Combine feedback
system consisting
of G(s) and [H(s)
-1].

24. Example: Steady-State Error for Nonunity Feedback

Find system type, appropriate
error constant, steady-state
error for unit step input.
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25. Definition: Steady-State Error for Nonunity Feedback
w/ Disturbances








General form: For step input and step distrubances:
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26. Definition: Steady-State Error for Nonunity Feedback
w/ Disturbances








For zero error:

1.System is stable
2.G1(s) is type 1.
3.G2(s) is type 0.
4.H(s) is type 0 with a dc gain of unity.
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27. Definition: Steady-State Error for Nonunity Feedback
w/ Disturbances









Steady-state value of the actuating signal Ea1(s)::


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28. Example: Steady-State Error for Nonunity Feedback
w/ Disturbances








Find the steady-state actuating signal for unity step input. Repeat for unit ramp
input:

Step: Ramp:
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29. Definition: Sensitivity

"The degree to which changes in system parameters affect
system transfer functions, and hence performance."

A system with zero sensitivity is ideal.
Greater the sensitivity, the less desirable.



"The ratio of the fractional change in the function to the fractional change
in parameter as the fractional change of parameters approaches zero"


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30. Example: Sensitivity





Calculate sensitivity of the closed-loop transfer function to changes in parameter a:

Closed-loop transfer function:
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31. Example: Sensitivity





Calculate sensitivity of the closed-loop transfer function to changes in parameter K
and a, with ramp inputs:
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