Basic Concepts and Terminology

Block Diagrams and Transfer Functions - The block diagram sơ đồ khối is a method of representing a control system that retains only this important feature of each component - Signal line

1 BASIC CONCEPTS AND

TERMINOLOGY

- Example: Wall’s fly-ball governor - 1769

§2 Block Diagrams and Transfer Functions

- The block diagram (sơ đồ khối) is a method of representing a control system that retains only this important feature of each component

- Signal lines (đường tín hiệu) indicate the input and output signals of the component

- The energy source is not shown on block diagrams However, many components

§2 Block Diagrams and Transfer Functions

2.1 Block diagrams

- Example: Block representations of various components

• Thermocouple temperature sensor: a thermocouple is a junction between two

different metals that produces a voltage related to a temperature difference

2.1 Block diagrams

• Amplifier: an amplifier or simply amp, is a device for increasing the power of a

signal

§2 Block Diagrams and Transfer Functions

2.1 Block diagrams

• DC Motor

2.1 Block diagrams

• Valve: valve is used to control flow rate by opening/closing the gate position

response to signals received from controllers

§2 Block Diagrams and Transfer Functions

2.1 Block diagrams

• Liquid flow transmitter: a device used to measure the flow of liquids in pipelines and convert the results into proportional electric signals

2.1 Block diagrams

• Driver of automobile

§2 Block Diagrams and Transfer Functions

2.1 Block diagrams

• Automobile power steering unit: steering is the term applied to the collection of

linkages which will allow a vehicle to follow the desired course

2.1 Block diagrams

• Automobile

§2 Block Diagrams and Transfer Functions

2.2 Transfer functions

- The most important characteristic of a component is the relationship between the input signal and the output signal This relationship is expressed by the transfer function of the component

- A transfer function (𝑇𝐹) is defined as the output divided by the input

𝑇𝐹 = 𝑜𝑢𝑡𝑝𝑢𝑡

𝑖𝑛𝑝𝑢𝑡 =

𝑦(𝑡)𝑥(𝑡)

- Steady state values for the 𝑇𝐹, which is sometimes called simply the gain

𝑔𝑎𝑖𝑛 = 𝑇𝐹𝑠𝑡𝑒𝑎𝑑𝑦 𝑠𝑡𝑎𝑡𝑒 = 𝑠𝑡𝑒𝑎𝑑𝑦 𝑠𝑡𝑎𝑡𝑒 𝑜𝑢𝑡𝑝𝑢𝑡

𝑠𝑡𝑒𝑎𝑑𝑦 𝑠𝑡𝑎𝑡𝑒 𝑖𝑛𝑝𝑢𝑡

2.2 Transfer functions

- Example: A potentiometer is used as a position sensor The pot is configured in such a way that 00 of rotation yields 0𝑉 and 3000 yields 10𝑉 Find the transfer function of the pot

Solution The transfer function is output divided by input In this case, the input to the pot is “position in degrees” and output is volts

§2 Block Diagrams and Transfer Functions

2.2 Transfer functions

- Example: For a temperature-measuring sensor, the input is temperature, and the output is voltage The sensor transfer function is given as 0.01𝑉/deg Find the sensor output voltage if the temperature is 6000𝐹

- An calibrated setting, previously determined by some sort of calibration procedure

or calculation, is need to obtain the desired result

- Example: A simple robot arm open-loop position system

§3 Open-Loop Control

- Example: The system consists of an electric motor driving a gear train, which is driving a winch The motor turns at 100𝑟𝑝𝑚𝑚 for each volt (𝑉𝑚) supplied The output shaft of the gear train rotates at 1/2 of the motor speed The winch (with a

3 𝑖𝑛𝑐ℎ shaft circumference) converts the rotary motion (𝑟𝑝𝑚𝑤) to linear speed

3𝑖𝑛/𝑚𝑖𝑛1𝑟𝑝𝑚𝑤 = 150

𝑖𝑛/𝑚𝑖𝑛

𝑉𝑚

Knowing this value, we can calculate the system output for any system input For

example, if the input to the this system is 12𝑉 (to the motor), the output speed of the winch is calculated as follows

𝑖𝑛/𝑚𝑖𝑛

- Example: A simple robot arm closed-loop position system

Open-loop control

Closed-loop control

§4 Closed-Loop Control

- Example: position control of machine tool table

Machine tool table moves from left to right with 𝑥0 = 400𝑚𝑚 and 𝑥𝑓 = 640𝑚 The sampling time of the system is 𝑑𝑡 = 0.1𝑠

§4 Closed-Loop Control

- Compare between open-loop control and close-loop control

Open-loop Control Close-loop Control

- shows a open-loop action: simple,

low cost

- can only counteract against

disturbances, for which it has been

designed; other disturbances cannot

be removed

- cannot become unstable - as long as

the controlled object is stable

- need to calibrate the system regularly

- shows a open-loop action: complex, high cost

- can counteract against disturbances

- can become unstable

- no need to calibrate the system

§5 Linear vs Nonlinear

- Linear Element

• Sinusoidal input waveform ⟹ Linear Element ⟹ Output waveform ?

• Use the input/output graph to determine the value of the output for each value of the input:

at time 𝑇𝑖, input value 𝑥𝑖 yields output value 𝑦𝑖

- Nonlinear Element

• Sinusoidal input waveform ⟹ Nonlinear Element ⟹ Output waveform ?

• Notice how the nonlinear element distorts the shape of the input waveform, resulting in a non-sinusoidal output waveform

§5 Linear vs Nonlinear

- Linear Element with Dead Band Nonlinearity

• Dead band is range of values through which an input can be changed without producing an observable change in the output

• The dead band effect occurs whenever the input change direction, example: backlash in mechanical gear trains

• Sinusoidal input waveform ⟹ Dead Band Nonlinearity ⟹ Output waveform ?

- Nonlinear Element with Hysteresis

• Hysteresis occurs when the I/O graph follows different curved paths when the input increase and decrease

• The result in I/O graph that forms a loop, and the value of the output for any given input depends on the history of the previous inputs

• Notice the distortion in the output waveform caused by the hysteresis

§5 Linear vs Nonlinear

- Linear Element with a Saturation Nonlinearity

• Saturation refers to the limitations on the range of values for the output of a component

• All real components reach a saturation limit when the input is increased or decreased beyond its limit values

- The gain of the controller determines a very important characteristic of a control system’s response: the type of damping or stability that the system displays in response to a disturbance

§7 Objectives of a Control System

- Control Objectives: after a load or set-point change, the control system should

• Minimize the maximum value of the error: min (𝑐𝑚𝑎𝑥 − 𝑐𝑓𝑖𝑛𝑎𝑙)

• Minimize the setting time: min (𝑡𝑠)

• Minimize the residual error (steady-state): min (𝑐𝑓𝑖𝑛𝑎𝑙 − 1)

- To evaluate a control system effectively, two decisions must be made

• the test must be specified

• the criteria of good control must be selected

- A step change in set-point or load is the most common test

- The three most common criteria of good control

• Quarter amplitude decay (suy giảm một phần tư biên độ)

• Minimum integral of absolute error (tích phân sai lệch tối thiểu)

• Critical damping (giảm chấn tới hạn)

§8 Control Performance Criteria

1 Quarter amplitude decay

+ This criterion specifies a damped oscillation in which each successive positive peak value is ¼ of the preceding positive peak value

+ Quarter amplitude decay is a popular criterion because it easy to apply and

2 Minimum integral of absolute error

+ This criterion specifies that the total area under the error curve should be minimum

+ The error is the distance between 𝐶2 and the controlled variable curve

+ This criterion is easy to used when a mathematical model is used to evaluate

a control system

§8 Control Performance Criteria

3 Critical damping

+ This criterion is used when overshoot above the set-point is undesirable

+ Critical damping is the least amount of damping that will produce a response with no overshoot an no oscillation

- Block reduction can be used to simplify more elaborate block diagrams containing multiple closed loop The method involves the reduction of portions of the block diagram until the desired simplification is obtained

- Block diagrams reduction procedure

Example: Reduce the block diagram shown in the figure into a single block

1 Assign variable names to all signal lines in the original diagram

Assign the variable names as shown in the figure

2 Select the blocks to reduce to a single block

§9 Block Diagram Simplification

3 Use the block transfer functions to obtain the input/output equation for each block selected in step 2

+ The output of a block = the input to the block × the block transfer function + The output of the summing junction = the algebraic sum of its inputs

Write the I/O equations for each block

5 Solve the equation from step 4 for the ratio of the output signal over the input signal The right hand side of the resulting equation is the single block that replaces the blocks selected in step 2

Solve Eq.(5) for 𝑉/𝑆 𝑉 = 𝑇𝐹1 × 𝑆 − 𝑇𝐹1 × 𝑇𝐹2 × 𝑉)

𝑇𝐹𝑡𝑜𝑡 = 𝑉

𝑆 =

𝑇𝐹1

1 + 𝑇𝐹1 × 𝑇𝐹2

§10 Classification of Control Systems

1 Classification of control systems

- Control systems are classified in a number of different ways

- Classification of control systems

a Feedback

• with feedback: close-loop control

• without feedback: open-loop control

b Types of signal

• continuous: analog control

• discrete: digital control

c Set-point

• seldom changed: regulator systems

• frequently changed: follow-up systems

• programmable controllers

§10 Classification of Control Systems

2 Analog and Digital Control

2 Analog and Digital Control

- Analog controlled system

• consists of traditional analog devices and circuits, that is, linear amplifiers

• the feedback signal is updated continuously

- Digital controlled system

• the controller uses a digital circuit In most cases, this circuit is actually a computer, usually microprocessor/microcontroller-based

• the feedback signal is updated after every sampling time

§10 Classification of Control Systems

3 Process Control

- Process control system maintains a variable in a process at its set-point

- Example: A closed-loop oven-heating system

3 Process Control

- Example: Process control in mixing paint

Automatic flow control

Manual control

§10 Classification of Control Systems

3 Process Control

- Example: Multi-process control

Individual local controllers

Direct computer control system

Distributed control system

4 Sequential Control

- A sequential control system performs a set

of operations in a prescribed manner

- Example: washing machine

§10 Classification of Control Systems

5 Servomechanism

- Servomechanisms are feedback control systems in which the controlled variable

is physical position or motion

- Example: the positioning system for a radar antenna

- Example: Basics of a numerical

control milling machine

§10 Classification of Control Systems

7 Robotics

- Example: Industrial Robots in car assembly lines

§10 Classification of Control Systems

8 Work-Cell

- A work-cell is an arrangement of

resources in a manufacturing

environment to improve the quality,

speed and cost of the process

Work-cells are designed to improve

these by improving process flow

and eliminating waste

9 Control System Examples

- Mechanical Speed

Control System

§10 Classification of Control Systems

9 Control System Examples

- DC Motor Speed Control System

9 Control System Examples

- Water Level Control System

§10 Classification of Control Systems

9 Control System Examples

- Temperature Control System