ME-314 Control Systems
Lecture 5
• Modeling in the Frequency Domain
– The Transfer Function
• The transfer function algebraically relates a
system’s output to its input.
• Transfer function allows to separate the input,
system, and output into three distinct parts.
• It enables us to algebraically combine
mathematical representations of subsystems to
yield a total system representation.
– Consider a general nth-order, linear, time-invariant
differential equation:
– Where c(t) is the output, r(t) is the input, and the ɑi’s,
bi’s, and the form of the differential equation represent
the system. Taking Laplace transform of both sides of
the above equation;
– Assuming all initial conditions to be zero and reducing
above equation;
– Forming the ratio of the output transform C(s), divided
by the input transform, R(s);
– The ratio G(s) is called the transfer function of the
system. (We will develop the so called characteristic
equation from the denominator of the above transfer
function after some simplification later.)
– Transfer function can be represented using block
diagram as:
– To further understand, we will develop transfer
functions for the three representative engineering
systems studied during the previous two lectures.
Writing Simultaneous Loop Equations and Developing Transfer Function for an
Electrical Network:
(a) Electrical Network for Loop Analysis
(b) Network with Element Voltages Expressed in Terms of the Loop Currents
By applying Kirchhoff’s voltage law around the i1(t) and i2(t) loops and simplifying:
These equations can be Laplace transformed, assuming zero initial conditions:
Above equations can be written in a standard form as follows:
– As the above system has two independent output
variables i.e. i1(t) and i2(t), we will develop the system
transfer function using the output variable associated
with the loop in which the voltage source is
connected. (The number of independent output
variables is termed as degrees of freedom (DOF) of
any system).
– The ‘system transfer function’ can be developed by
solving the above two equations for I1(s) using
Cramer’s rule.
– The total/overall or system transfer function is
determined between the main input to the system
(voltage, force, torque etc.) and the main or observed
output of the system (i.e. response of the element to
which the input is directly applied). Outputs of all
other elements within a single-input, single-output
system may be linked to the main output through the
mathematical relationships that appear in the
system’s dynamic equations.
Writing Simultaneous Mechanical Network Equations and Developing Transfer
Function:
(a) A Translational Mechanical Network
(b) Free-Body Diagrams for the Masses
The simultaneous differential equations in x1 and x2 for the translational system are:
These equations can be Laplace transformed, assuming zero initial conditions:
Above equations can be written in a standard form as follows:
– As the above system has two independent output
variables i.e. x1(t) and x2(t), we will develop the
system transfer function using the output variable
associated with the mass to which the force is directly
applied.
– The ‘system transfer function’ can be developed by
solving the above two equations for X2(s) using
Cramer’s rule.
Writing Simultaneous Rotational Mechanical Network Equations and Developing
Transfer Function:
(a) A Rotational Mechanical Network
(b) Free-Body Diagrams for the Rotary Masses with Numerical Values
The simultaneous differential equations in θ1 and θ2 for the rotational system are:
These equations can be Laplace transformed, assuming zero initial conditions:
Above equations can be written in a standard form as follows:
– As the above system has two independent output
variables i.e. θ1(t) and θ2(t), we will develop the
system transfer function using the output variable
associated with the rotational mass to which the
torque is directly applied.
– The ‘system transfer function’ can be developed by
solving the above two equations for Θ2(s) using
Cramer’s rule.
• Reading Assignment
– Text Book / Chapter 2 / Topic 2.3.