EE 448

Control Systems, Sensors and

Actuators

Instructor : Huỳnh Việt Thắng <thang143@gmail.com>

TA : Lại T. Kim Phụng

LA : Vũ Vân Thanh

1

Lecture 4. Transfer functions of some

typical systems

Transfer Functions of Physical Systems

• Electrical networks

• Translational mechanical systems

• Rotational mechanical systems

• Gearing Systems

• Electromechanical systems

• Electric Circuit Analogs

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Electrical Networks

• Please review Operational Amp. in Examples 2.14 and 2.15.

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Ex. 2.14: Inverting OP-AMP Circuit

• PID controller

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Exercise

• Find the transfer function

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Rotational Mechanical System

Transfer Functions

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K: spring constant

D: coefficient of viscous friction

J: moment of inertia

T: Torque θθθθ: angular displacement

ωωωω: angular velocity

Mô-men lực [Nm](mô men xoắn)

Góc quay [rad] Hằng số lò xo [Nm/rad]

Vận tốc góc [rad/s]

Mô-men quán tính [kgm2]

Transfer Functions for Systems with Gears• Rotational systems (especially those driven by motors) are

often associated with gear trains driving the load

• Gears provide mechanical advantage to rotational systems

• Example: riding a bicycle with gears

– Going uphill, you shift to provide more torque and less speed

– On the straightaway, you shift to obtain more speed and less torque

• Thus, gears allow you to match the drive system and the load

a trade-off between speed and torque!

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Transfer Functions for Systems with Gears

(cont.)

• An input gear with radius r1 and N1 teeth is rotated through

angle θθθθ1(t) due to a torque T1(t).

• An output gear with radius r2 and N2 teeth responds by

rotating through angle θθθθ2(t) and delivering a torque T2(t).

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Electromechanical System

Transfer Functions

• Electromechanical systems: systems that are hybrids of

electrical and mechanical variables

– DC motor, DC servo motor

• Applications: motor controller, robot controls, sun and star

trackers, computer tape and disk-drive position controls

• DC Motor

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DC motor

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Electromechanical System

Transfer Functions (cont.)

Review question

• What are the component parts of the mechanical

constants of a motor's transfer function?

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DC Motor Transfer function

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• Transfer function

• Simplified form:

DC Motor parameters

• Ea: applied armature voltage [V]

• θθθθm: angular displacement [rad]

• Tm: torque [N-m]

• ωωωωm: angular velocity [rad/s]

• Kt: motor torque constant [N-m/A]

• Kb: back emf constant [V-s/rad]

• Jm: equivalent inertia at the armature [kg-m2]

• Dm: equivalent viscous damping [N-m-s/rad]

• Ra: resistance of armature [Ω]

• La: inductance of armature [H]13

Torque – Speed relation

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Torque – Speed relation (cont.)

• Laplace domain

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• Time domain

Torque – Speed relation (cont.)

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Meaning of Equation (2.159)?

Torque – Speed relation (cont.)

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Example 2.23 & Exercise

• Please read example 2.23 in page 83 (NISE) (~30 mins.)

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Example 2.23 & Exercise (cont.)

• Please read example 2.23 in page 83 (NISE) (~30 mins.)

• Re-do example 2.23 with new parameters for the system:

– Ja = 3 kg-m2

– Da = 2 N-m s/rad

– N1 = 50

– N2 = 800

– JL = 1000 kg-m2

– DL = 600 N-m s/rad

Note: Parameters that are not mentioned hold the old values as in

example 2.23.

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Homework

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Case study: Antenna Control

Transfer Function

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Pre-amplifier & Power Amplifier

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Motor and Load

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System responses with LTI Viewer

• Your task: use Matlab and LTIviewer to plot the step response

and impluse response of some subsystems in the antenna

position control system:

– Power amplifier subsystem

– Motor and Load subsystem

• How?

– define the transfer function of corresponding subsystem in Matlab

workspace (use “tf”)

– use LTIviewer to show and analyze the responses (you should also give

your own comments to the receiving figures)

– document your work electronically in MS Word or MS power-point

slides

– in groups of 2 students

– Due time: 10h30 TODAY25