This document discusses steady-state error in control systems. It defines steady-state error and describes how it arises from system configuration and input type. Examples are provided to illustrate calculating steady-state error for various system types and inputs, including step, ramp, and disturbances. Sensitivity analysis is also introduced to analyze how changes in system parameters affect steady-state error.
state space representation,State Space Model Controllability and Observabilit...Waqas Afzal
State Variables of a Dynamical System
State Variable Equation
Why State space approach
Block Diagram Representation Of State Space Model
Controllability and Observability
Derive Transfer Function from State Space Equation
Time Response and State Transition Matrix
Eigen Value
Ch5 transient and steady state response analyses(control)Elaf A.Saeed
Chapter 5 Transient and steady-state response analyses. From the book (Ogata Modern Control Engineering 5th).
5-1 introduction.
5-2 First-Order System.
5-3 second-order system.
5-6 Routh’s stability criterion.
5-8 Steady-state errors in unity-feedback control systems.
state space representation,State Space Model Controllability and Observabilit...Waqas Afzal
State Variables of a Dynamical System
State Variable Equation
Why State space approach
Block Diagram Representation Of State Space Model
Controllability and Observability
Derive Transfer Function from State Space Equation
Time Response and State Transition Matrix
Eigen Value
Ch5 transient and steady state response analyses(control)Elaf A.Saeed
Chapter 5 Transient and steady-state response analyses. From the book (Ogata Modern Control Engineering 5th).
5-1 introduction.
5-2 First-Order System.
5-3 second-order system.
5-6 Routh’s stability criterion.
5-8 Steady-state errors in unity-feedback control systems.
Root locus is a graphical representation of the closed-loop poles as a system parameter is varied.
It can be used to describe qualitatively the performance of a system as various parameters are changed.
It gives graphic representation of a system’s transient response and also stability.
We can see the range of stability, instability, and the conditions that cause a system to break into oscillation.
This Presentation explains about the introduction of Frequency Response Analysis. This video clearly shows advantages and disadvantages of Frequency Response Analysis and also explains frequency domain specifications and derivations of Resonant Peak, Resonant Frequency and Bandwidth.
It gives how states are representing in various canonical forms and how it it is different from transfer function approach. and finally test the system controllability and observability by kalman's test
Mathematical Modelling of Control SystemsDivyanshu Rai
Different types of mathematical modeling in control systems [which include Mathematical Modeling of Mechanical and Electrical System (which further includes, Force-Voltage and Force-Current Analogies)]
State space analysis, eign values and eign vectorsShilpa Shukla
State space analysis concept, state space model to transfer function model in first and second companion forms jordan canonical forms, Concept of eign values eign vector and its physical meaning,characteristic equation derivation is presented from the control system subject area.
Systems Analysis & Control: Steady State ErrorsJARossiter
In the context of control engineering feedback loops, these slides describe how to find the steady-state error between a target and the system.
Links to more slides at
http://controleducation.group.shef.ac.uk/OER_index.htm
Root locus is a graphical representation of the closed-loop poles as a system parameter is varied.
It can be used to describe qualitatively the performance of a system as various parameters are changed.
It gives graphic representation of a system’s transient response and also stability.
We can see the range of stability, instability, and the conditions that cause a system to break into oscillation.
This Presentation explains about the introduction of Frequency Response Analysis. This video clearly shows advantages and disadvantages of Frequency Response Analysis and also explains frequency domain specifications and derivations of Resonant Peak, Resonant Frequency and Bandwidth.
It gives how states are representing in various canonical forms and how it it is different from transfer function approach. and finally test the system controllability and observability by kalman's test
Mathematical Modelling of Control SystemsDivyanshu Rai
Different types of mathematical modeling in control systems [which include Mathematical Modeling of Mechanical and Electrical System (which further includes, Force-Voltage and Force-Current Analogies)]
State space analysis, eign values and eign vectorsShilpa Shukla
State space analysis concept, state space model to transfer function model in first and second companion forms jordan canonical forms, Concept of eign values eign vector and its physical meaning,characteristic equation derivation is presented from the control system subject area.
Systems Analysis & Control: Steady State ErrorsJARossiter
In the context of control engineering feedback loops, these slides describe how to find the steady-state error between a target and the system.
Links to more slides at
http://controleducation.group.shef.ac.uk/OER_index.htm
Time response of first order systems and second order systemsNANDHAKUMARA10
It is the time required for the response to reach half of its final value from the zero instant. It is denoted by tdtd. Consider the step response of the second order system for t ≥ 0, when 'δ' lies between zero and one. It is the time required for the response to rise from 0% to 100% of its final value.
Analysis and Design of Conventional Controller for Speed Control of DC Motor ...IJERA Editor
The objective of this paper is to control the speed of the motor using conventional controller; compensator is used to improve the steady state error. To evaluate the performance of the controller, time response analysis is carried out. The time response analysis consists of two type of analysis. One is unit step response analysis and other is performance indices analysis. The paper describes the designing of a closed loop model of the dc motor drive for controlling speed. Accuracy and the dynamic responses are better in a closed loop system. The compensator is used to compensate the parameter of the system in such a way to meet the specification, so that it improves the steady state response of the system and get desired response.
Control systems Unit-I (Dr.D.Lenine, RGMCET, Nandyal)Lenine D
INTRODUCTION: Concepts of control systems – Open loop and closed loop control systems and their differences, examples – Types of feedback control systems
Mathematical modeling of Electrical & Mechanical(translational & rotational) systems, differential equations- Electrical analogous (F-V,F-I) of mechanical system- use of Laplace transforms in control systems-Transfer function: concepts, features-Transfer functions of above systems
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 Error
Steady State Response Stability
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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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7. Evaluating: Steady-State Error
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
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8. Representation: Steady-State Error
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
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9. Sources: Steady-State Error
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
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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
"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)."
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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
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].
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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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