Chapter 5 Dynamics and Regulation of Low-order Systems

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Chapter 5 Dynamics and Regulation of Low-order
Systems

• 5.1 General Effects of Feedback
• 5.2 Dynamics and Regulation of 1st-order
  System
• 5.3 Dynamics and Regulation of 2nd-order
  System
• 5.4 Time-domain Specifications of System
  Performance

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5.1 General Effects of Feedback (1)

• Closed-loop System

• Negative Feedback Control

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5.1 General Effects of Feedback (2)

• Proportional Regulator

Under Regulation
Out of Regulation
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5.1 General Effects of Feedback (3)

• Proportional Controller

• Mechanical and Hydraulic Controller

Electronic Controller
Governing of mechanical plant is liberated by
using OP to realize electronic controller.
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5.1 General Effects of Feedback (4)

• Effectiveness of Feedback

(1) Set point regulation

• Error Response

• Following the first law of Newtonian Mechanics
• At rest Zero error in position regulation
• At constant velocity Zero error in speed
  regulation

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5.1 General Effects of Feedback (5)

• Dynamic Response

Poles of Closed-loop Transfer Function
The dynamic response is governed by new poles as
D1(s)D2(s)N1(s)N2(s)0 For unity feedback
system, feedback has no effect on system zeros.
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5.1 General Effects of Feedback (6)
(2) Parameter Variations

I/O transmission is dominated by sensor for high
loop gain.

• System variations

The effect of large variation is investigated by
using robust analysis.
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5.1 General Effects of Feedback (7)

• Sensitivity analysis (small variation)
• Variation of G
• 
  

Loop gain
The Sensitivity of I/O variation is reduced by
employing high loop gain.
Variation of H
Feedback sensor directly and constantly affect
the I/O transmission. In general, is
higher than . High loop gain reduces
the I/O variation due to system parameters.
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5.1 General Effects of Feedback (8)
(3) Disturbance Rejection

• Disturbance Transmission

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5.1 General Effects of Feedback (9)

• Disturbance Error

Note steady state set point - offset error -
disturbance error
parameter uncertain error

• Overall Effects of Feedback

Advantages Command following
Disturbance rejection
Improve system robustness Disadvantages Reduce
gain Increase system
complexity Introduce
instability
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5.2 Dynamics and Regulation of 1st-order
Systems (1)

• Definition A system that can store energy in
  only one form and location. Physical
  Examples

System pole-zero diagram
t0 , ON
Input signal
Output signal
Input signal
Output signal
t0
Open gate
Input signal
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5.2 Dynamics and Regulation of 1st-order
Systems (2)

• System and Input Model
• Differential Eq. Model

Transfer Function Model
Standard Form of Pure Dynamics
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5.2 Dynamics and Regulation of 1st-order
Systems (3)

• System Dynamics
• (1) Step response from differential equation

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5.2 Dynamics and Regulation of 1st-order
Systems (4)

• Response of initial relaxed system

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5.2 Dynamics and Regulation of 1st-order
Systems (5)

• (2) Step response from pole-zero diagram

Steady state (Pure static gain)
Transient (Pure dynamics)
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5.2 Dynamics and Regulation of 1st-order
Systems (6)

• Overall response

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5.2 Dynamics and Regulation of 1st-order
Systems (7)

• Closed-loop Regulation
• Ex Liquid-level regulation

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5.2 Dynamics and Regulation of 1st-order
Systems (8)

• Unity Feedback Regulator

• Set Point Regulation

Equivalent I/O transmission
Proportional regulator (Kp) changes the static
amplification (K) and response speed ( ).
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5.2 Dynamics and Regulation of 1st-order
Systems (9)
(1) Effect of Kp on Pure Dynamics
Pole-zero diagram
Response is faster when the proportional gain is
increased.
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5.2 Dynamics and Regulation of 1st-order
Systems (10)
(2) Effect of Kp on Steady State Response
Steady state error (offset) is reduced when the
proportional gain is increased.
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5.2 Dynamics and Regulation of 1st-order
Systems (11)

• Set Point and Response Regulation

Response speed is a pure dynamic behavior,
therefore the comparisons of response speeds are
referred to the steady state responses at
different Kp.
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5.2 Dynamics and Regulation of 1st-order
Systems (12)

• Disturbance Rejection

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5.2 Dynamics and Regulation of 1st-order
Systems (13)

• High Gain Regulation

For infinite high gain Kp, the unity feedback
regulator approaches ideal static system.
Finite gain regulation will improve the speed of
dynamic response but it has offset error in
steady state.
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5.3 Dynamics and Regulation of 2nd-order
Systems (1)

• Definition A system having two separate
  energy-storage elements.
• Physical Examples

System pole-zero distribution
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5.3 Dynamics and Regulation of 2nd-order
Systems (2)

• System and Input Model
• Differential Eq. Model

Transfer Function Model
Standard Form of Pure Dynamics
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5.3 Dynamics and Regulation of 2nd-order
Systems (3)

• Step Response From Differential Equation
• For underdamping and initial relaxation system
  

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5.3 Dynamics and Regulation of 2nd-order
Systems (4)
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5.3 Dynamics and Regulation of 2nd-order
Systems (5)

• Step Response From Pole-zero distribution

Pole-zero distribution
Poles distribution
Response
Steady state
Transient
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5.3 Dynamics and Regulation of 2nd-order
Systems (6)

• Overall response

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5.3 Dynamics and Regulation of 2nd-order
Systems (7)

• Closed-loop Regulation
• Ex DC Servomechanism

Servo motor
Motion and power transmission by reduction gear
(Gear Ratio N)
Position sensor by potentiometer
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5.3 Dynamics and Regulation of 2nd-order
Systems (8)
Command response
Disturbance response
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5.3 Dynamics and Regulation of 2nd-order
Systems (9)

• Unity Feedback Regulator

• Set Point Regulation

Equivalent I/O transmission
Proportional regulator (Kp) changes the static
amplification (K) and dynamic response(
) through increasing the stiffness of a
system.
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5.3 Dynamics and Regulation of 2nd-order
Systems (10)
(1) Regulation of Pure Dynamics
Pole-zero distribution
(2) Steady-state response The same result as
that of 1st-order system with offset error
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5.3 Dynamics and Regulation of 2nd-order
Systems (11)

• Disturbance Rejection

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5.4 Time-domain Specifications of System
Performance (1)

• Step Testing of Black-Box System

(2) Dynamic test (Within linear range)
(1) Static test
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5.4 Time-domain Specifications of System
Performance (2)

• Step Testing and Identification of Low-order
  Systems

(1) Performance testing
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5.4 Time-domain Specifications of System
Performance (3)
(2) System Identification
1st-order system
2nd-order system
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5.4 Time-domain Specifications of System
Performance (4)

• Effects of Additional Poles an Zeros

Effect of real pole
The effect of the real pole is to make the
response more sluggish.
Effect of real LHP zero
The effect of the real zero is to make the
response more oscillatory.
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5.4 Time-domain Specifications of System
Performance (5)
Effect of real RHP zero RHP zero
Nonminimum-phase zero
The effect of nonminimum-phase zero is to cause
initial reversal motion in step response.
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5.4 Time-domain Specifications of System
Performance (6)

• Dynamic Model Simplification

Order reduction High order system Low
order model Ignore
real pole (zero) in oscillatory system modes
Pole-zero cancellation
Ignorance of far away
poles and zeros Poles
and zeros only have s.s. effects