EE 448
Control Systems, Sensors and
Actuators
Instructor : Huỳnh Việt Thắng <[email protected]>
TA : Lại T. Kim Phụng
LA : Vũ Vân Thanh
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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