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Prof. Dr.-Ing. Dirk Nissing
4 Time-Domain Characteristics of Control Systems
154
Controller Design Process
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4 Time-Domain Characteristics of
Control Systems
classification
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4.1 Basic System Types
Proportional transfer behavior
P
Ideal proportional behavior
Proportional behavior with delay (PTn)
Integral transfer behavior
I
Ideal integral behavior
Integral behavior with delay (ITn)
Derivative transfer behavior
D
Ideal derivative behavior (differentiator)
derivative behavior with delay (DTn)
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4.2 Proportional Behavior
(ideal) proportional behavior (P-system)
• (algebraic) equation: 𝑞 𝑡 = 𝐾𝑢(𝑡)
• Step response: 𝑞 𝑡 = 𝐾ℎ(𝑡)
• Ideal P-system is a steady-state system
• 𝐺 𝑠 =
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4.2 Proportional Behavior
Example: Systems with proportional behavior
spring
Ohms resistance
lever
gear
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4.3 Proportional Behavior with Delay
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4.3 Proportional Behavior with Delay
Proportional behavior with delay 1st-order (PT1)
• Differential equation 𝑇1 𝑦(𝑡) + 𝑦(𝑡) = 𝐾𝑢(𝑡)
• Step response 𝑞 𝑡 = 𝐾 1 − 𝑒
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−1
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𝑇1 𝑡
4.3 Proportional Behavior with Delay
Proportional behavior with delay 1st-order (PT1)
𝐺 𝑠 =
Im
s
Re
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4.3 Proportional Behavior with Delay
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4.3 Proportional Behavior with Delay
Proportional behavior with delay 2nd-order (PT2)
• Differential equation 𝑇2 ²𝑦 + 𝑇1 𝑦 + 𝑦 = 𝐾𝑢 or
𝑦 + 2𝐷𝜔0 𝑦 + 𝜔0 ²𝑦 = 𝐾 𝑢
𝑇1
1
with 𝜔0 = 𝑇 , 𝐷 =
2𝑇2
2
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4.3 Proportional Behavior with Delay
Proportional behavior with delay 2nd-order (PT2)
𝐺 𝑠 =
Im
Poles for
a) Overdamped
b) Critically damped
c) Damped
d) Undamped
s
Re
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4.3 Proportional Behavior with Delay
Proportional behavior with delay 2nd-order (PT2)
Poles for
a) Overdamped
b) Critically damped
c) Damped
d) Undamped
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4.3 Proportional Behavior with Delay
Proportional behavior with delay nth-order (PTn)
• Differential equation
𝑇𝑛 𝑛 𝑦 (𝑛) + 𝑇𝑛−1 𝑛−1 𝑦 (𝑛−1) + ⋯ + 𝑇1 𝑦 + 𝑦 = 𝐾𝑢
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4.4 Integral Behavior
Example: System with Integral Behavior
1
ℎ 𝑡 =
𝐴
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𝑄e 𝑡 − 𝑄a (𝑡) d𝑡
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4.4 Integral Behavior
(ideal) integral behavior (I-system)
• Differential equation 𝑦 =
𝑡
𝐾𝐼 0 𝑢
• Step response 𝑞 𝑡 = 𝐾𝐼 𝑡
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𝜏 𝑑𝜏 or 𝑦 = 𝐾𝐼 𝑢
4.4 Integral Behavior
(ideal) integral behavior (I-system)
𝐺 𝑠 =
Im
s
Re
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4.4 Integral Behavior
Example: System with integral behavior and delay
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4.4 Integral Behavior
Integral behavior with delay 1st-order (IT1-system)
• Differential equation
𝑡
𝑇1 𝑦 + 𝑦 = 𝐾𝐼 0 𝑢 𝜏 𝑑𝜏 or 𝑇1 𝑦 + 𝑦 = 𝐾𝐼 𝑢
• Step response 𝑞 𝑡 = 𝐾𝐼
𝑡 − 𝑇1 + 𝑇1 𝑒
−𝑡
𝑇1
Series connection of
I- and PT1-system
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4.4 Integral Behavior
Integral behavior with delay 1st-order (IT1-system)
𝐺 𝑠 =
Im
s
Re
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4.5 Derivative Behavior
Example: System with derivative behavior  shock
absorber
d𝑥(𝑡)
𝐹 𝑡 =𝑑
d𝑡
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4.5 Derivative Behavior
Ideal derivative behavior, differentiator (D-system)
• Differential equation 𝑦 = 𝐾𝐷 𝑢
• Step response 𝑞 𝑡 = 𝐾𝐷 ℎ 𝑡 = 𝐾𝐷 𝛿(𝑡)
• Ideal D-system is not causal  in reality only
with delay nth-order
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4.5 Derivative Behavior
Ideal derivative behavior, differentiator (D-system)
𝐺 𝑠 =
Im
s
Re
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4.5 Derivative Behavior
Derivative behavior with delay 1st-order (DT1-system)
• Differential equation 𝑇1 𝑦 + 𝑦 = 𝐾𝐷 𝑢
• Step response
−𝑡
−𝑡
𝑑
𝐾
𝐷
𝑞 𝑡 = 𝑞𝑃𝑇1 𝑡 = 𝐾𝐷 1 − 𝑒 𝑇1 =
𝑒 𝑇1
𝑑𝑡
𝑇1
Series connection of
D- and PT1-system
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4.5 Derivative Behavior
Derivative behavior with delay 1st-order (DT1-system)
𝐺 𝑠 =
Im
s
Re
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4.7 Combined System Types
• Standard form
𝑇𝑛 𝑛 𝑦 𝑛 𝑡 + ⋯ + 𝑇1 𝑦 𝑡 + 𝑦 𝑡
𝑡
= 𝐾𝐼
•
•
•
•
𝑢 𝜏 𝑑𝜏 + 𝐾𝑃 𝑢 𝑡 + 𝐾𝐷 𝑢(𝑡)
0
P I D Tn
Parallel connection of P-, I-, and D-type system
Tn characterized through highest order on left side
PID characterized through right side (input)
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Check your Understanding
1. How is a system with proportional transfer behavior
characterized?
2. Formulate the differential equation of a PT1-system!
3. Draw qualitatively the impulse response of a PT1-system.
4. How many parameters are required defining a PT2-system?
5. Draw qualitatively the step response of a PT2-system when
a) The damping ratio is 𝐷 ≥ 1
b) The damping ratio is 0 < 𝐷 < 1, and
c) The damping ratio is 𝐷 < 0
6. Characterize the system type of the following differential
equation: 𝑎𝑦 + 𝑐𝑦 + 𝑑𝑦 = 𝐾𝑢 when 𝑢 is the input and 𝑦 is the
output variable. Explain your answer.
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4.7 Combined System Types
Basic system types
• Here: Basic system characteristics describe plant
behavior
• But: Controller can be described by basic system
types as well!
• Example: P-Controller
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4.7 Combined System Types
Plant
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4.7 Combined System Types
Important combined system types
• PI-system (with delay)
• PD-system (with delay)
• PID-system (with delay)
These system types represent important controller
types!
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4.7 Combined System Types
PI
-System
𝑡
𝑦 𝑡 = 𝐾𝑃 𝑢 𝑡 + 𝐾𝐼
𝑢 𝜏 d𝜏 = 𝐾𝑃
0
1
𝑢 𝑡 +
𝑇𝐼
𝑦 𝑡 = 𝐾𝑃 𝑢 𝑡 + 𝐾𝐼 𝑢 𝑡 = 𝐾𝑃
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𝑡
𝑢 𝜏 d𝜏
0
𝐾𝑃
with 𝑇𝐼 =
𝐾𝐼
1
𝑢 𝑡 + 𝑢(𝑡)
𝑇𝐼
4.7 Combined System Types
PI-System
Differential equation: 𝑦 𝑡 = 𝐾𝑃 𝑢 𝑡 + 𝐾𝐼 𝑢 𝑡
Im
TF:
s
Re
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4.7 Combined System Types
PD -System
𝑦 𝑡 = 𝐾𝑃 𝑢 𝑡 + 𝐾𝐷 𝑢 𝑡 = 𝐾𝑃 𝑢 𝑡 + 𝑇𝐷 𝑢(𝑡)
𝐾𝐷
with 𝑇𝐷 =
𝐾𝑃
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4.7 Combined System Types
PD-System
Differential equation: 𝑦 𝑡 = 𝐾𝑃 𝑢 𝑡 + 𝐾𝐷 𝑢 𝑡
Im
TF:
s
Re
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4.7 Combined System Types
PID -System
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4.7 Combined System Types
PID-System
Differential equation: 𝑦 𝑡 = 𝐾𝐼 𝑢 𝑡 + 𝐾𝑃 𝑢 𝑡 + 𝐾𝐷 𝑢(𝑡)
Im
TF:
s
Re
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Example: RC-Network
𝑉𝑜𝑢𝑡 (𝑡) + 𝑅𝐶 𝑉𝑜𝑢𝑡 (𝑡) = 𝑉𝑖𝑛 (𝑡)
a) What is the TF of the
process when 𝑉𝑖𝑛 is the
input and 𝑉𝑜𝑢𝑡 is the
output?
b) What is the standard system type of the process?
c) Determine the parameters of the standard system.
d) Draw the step response.
e) Calculate the impulse response.
f) Draw the impulse response.
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Example: RC-Network
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Check your Understanding
1. How is a system with integral transfer behavior
characterized?
2. A step response of a system with integral behavior is
given. How can you determine the gain KI?
3. Formulate the differential equation of a DT2-system.
4. Sketch qualitatively the step response of a DT2-system.
5. Characterize the system type of the following differential
equation: 𝑎1 𝑦 + 𝑎0 𝑦 = 𝑏0 𝑢 + 𝑏1 𝑢 when 𝑢 is the input and 𝑦
is the output variable. Explain your answer.
6. An ideal PID-controller is characterized through how many
parameters?
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