The document discusses the transfer function, a mathematical representation of linear time-invariant dynamical systems through the ratio of Laplace transforms of output and input, highlighting its formula, poles, and zeros. It outlines advantages such as simplifying analysis and aiding in controller design, alongside disadvantages, including limitations to linear systems and complexity in higher-order cases. Additionally, it details applications in system analysis, design, performance analysis, simulation, and feedback control, concluding that while transfer functions are fundamental, they are less effective for multiple-input multiple-output systems compared to state-space representations.

1. TRANSFER FUNCTION AND IT’S
APPLICATION
Submitted by
R.s

2. CONTENTS
 Introduction
 Transfer Function Definition
 Transfer Function Formula
 Poles and Zeros of Transfer Function
 Advantages and Disadvantages of Transfer Function
 Applications of Transfer Function
 Conclusion

3. Introduction: -
The transfer function is a convenient representation of a linear time invariant
dynamical system. Mathematically the transfer function is a function of complex
variables. For finite dimensional systems the transfer function is simply a rational
function of a complex variable. The transfer function can be obtained by inspection
or by simple algebraic manipulations of the differential equations that describe the
systems. Transfer functions can describe systems of very high order, even infinite
dimensional systems governed by partial differential equations. The transfer
function of a system can be determined from experiments on a system.

4. Transfer Function Definition: -
It is defined as the ratio of the Laplace transform of the output to the Laplace
transform of the input, assuming zero initial conditions.
G(s)=C(s)/R(s)
Transfer Function Formula: -
The transfer function formula in control system can be derived from the differential
equations governing the system. For a simple first-order system, the transfer
function is represented as:
T(s) = X(s)/Y(s)
In a Laplace transform, if the input is represented by R(s) and the output is
represented by C(s), then the transfer function will be:
G(s) = C(s)/R(s) ⇒ R(s).G(s) = C(s)

5. Poles and Zeros of Transfer Function: -
Factorizing the numerator and denominator polynomials of a transfer function
reveals its poles and zeros. Poles are values of s where the denominator becomes
infinite making the magnitude of transfer function infinity. Zeros are values where
the numerator becomes zero making the magnitude zero.
Poles determine system characteristics like stability, rise time, etc. while zeros affect
frequency response. Multiple poles/zeros occur when the factor is raised to a
power. A system is said to be stable if all its poles lie in the Left Half Plane on the s-
plane.
T(S) = /
The above equation can be written in the factorized form,
T(S) = /
The gain factor of the system is K

6. Poles of Transfer Function: -
Poles of the transfer function are defined as the values of the parameter ‘s’ that make the
denominator of the transfer function infinite when substituted.
In the above equation, if s is substituted with in the denominator, these values become the poles of
the transfer function.
When the denominator is set to zero, the resulting roots are known as the poles.
Assume we have a system, whose transfer function is
Put denominator = 0, to find the poles of the system
S= 0, -1
•These values are the poles of the above transfer function because substituting them into the
denominator results in an infinite transfer function.
•Poles of a transfer function generally fall into three categories: simple poles, repeated poles, and
conjugate poles.
•If the values are real and non-repetitive, they are known as simple poles.

7. Zeros of Transfer Function: -
We have already discussed that the poles are determined by the denominator of the transfer
function. However, the zeros of the transfer function are evaluated using the numerator.
The zeros of a transfer function are those values of s that, when substituted into the numerator,
result in the transfer function becoming zero.
Just like the poles, the zeros are the roots of the equation obtained when the numerator is set to
zero.
Zeros can also be categorized into three types: simple zeros (non-repetitive), repeated zeros, and
complex conjugate pairs.
Assume a system has a transfer function given below
To find zeros of the transfer function
(s+1) (s-2)= 0
s= -1, 2
These values are the zeros of the transfer function because substituting them into the numerator
makes the overall transfer function of the system equal to zero.

8. Advantages and Disadvantages of Transfer Function: -
Advantages: -
•Simplifies Analysis: Converts differential equations to algebraic equations, making analysis easier.
•Frequency Domain Analysis: Facilitates the study of system behavior in the frequency domain.
•System Design: Aids in designing controllers and compensators to achieve desired system
performance.
Disadvantages: -
•Linear Systems Only: Applicable only to linear time-invariant systems.
•No Initial Conditions: Assumes all initial conditions are zero, which may not be practical in all
cases.
•Complexity: For higher-order systems, deriving the transfer function can be complex and
cumbersome.

9. Applications of Transfer Function: -
1. System Analysis
•
Frequency Response Analysis: Transfer functions allow you to analyze how a system responds to different
frequencies. By examining the frequency response, you can determine the stability and performance
characteristics of the system.
•
Stability Analysis: Transfer functions help in assessing the stability of a system. By analyzing poles and
zeros in the transfer function, you can determine whether a system is stable, unstable, or marginally stable.
2. System Design
•
Controller Design: Transfer functions are used to design controllers such as Proportional-Derivative-
Integrative (PID) controllers. By modifying the transfer function with feedback, you can shape the closed-
loop response to meet desired performance criteria.
•
Compensator Design: Transfer functions help in designing compensators like lead, lag, and lead-lag
compensators to improve system performance. These compensators are used to modify the open-loop
transfer function to achieve better closed-loop characteristics.
3. Performance Analysis
•
Transient Response: Transfer functions can be used to determine the transient response of a system,
including rise time, settling time, overshoot, and damping. This helps in understanding how quickly and
effectively the system reaches its desired state.
•
Steady-State Response: By analyzing the transfer function, you can predict the steady-state behavior of the
system, such as the steady-state error and system output to a given input.

10. 4. System Simulation
•Simulation and Modeling: Transfer functions are used in simulations to model and analyze the
dynamic behavior of control systems. Software tools like MATLAB and Simulink use transfer functions to
simulate system responses and evaluate control strategies.
5. State Space Representation
•Conversion Between Representations: Transfer functions provide a way to convert between state-
space representations and frequency domain representations of systems. This is useful in analyzing and
designing systems using different approaches.
6. Feedback Control
•Design of Feedback Systems: Transfer functions are central to designing feedback control systems. By
analyzing the open-loop transfer function and designing appropriate feedback loops, you can ensure
that the closed-loop system meets specific performance criteria.
7. System Identification
•Modeling Real Systems: Transfer functions are used to identify and model real systems based on
observed input-output data. This helps in developing accurate models for control design and analysis.
8. Root Locus Analysis
•Root Locus Technique: Transfer functions are used in root locus analysis to study how the roots of the
closed-loop transfer function move in the s-plane as a system parameter (such as gain) varies. This
provides insight into the system’s stability and transient response.

11. Conclusion: -
The transfer function was the primary tool used in classical control
engineering. However, it has proven to be unwieldy for the analysis of
multiple-input multiple-output (MIMO) systems, and has been largely
supplanted by state space representations for such system. In spite of
this, a transfer matrix can be always obtained for any linear system.
Transfer function is a mathematical description that characterizes how a
dynamic system responds to any external input over time. It provides a
robust way to model and study control systems by transforming
differential equations into algebraic equations. Concepts like poles,
zeros, stability, and sensitivity can be analyzed from the transfer
function. Though limited to linear systems, transfer functions offer
many analytical and design advantages in a compact form. They form a
crucial part of control system analysis and remain important from
academic to industrial applications.

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YOU