Class 2 - Mathematical Modeling Using Transfer Function Approach

System Modeling Coursework Class 2: Mathematical Modeling of systems using transfer function approach P.R. VENKATESWARAN Faculty, Instrumentation and Control Engineering, Manipal Institute of Technology, Manipal Karnataka 576 104 INDIA Ph: 0820 2925154, 2925152 Fax: 0820 2571071 Email: [email protected], [email protected] Blog: www.godsfavouritechild.wordpress.com Web address: http://www.esnips.com/web/SystemModelingClassNotes

WARNING! • I claim no originality in all these notes. These are the compilation from various sources for the purpose of delivering lectures. I humbly acknowledge the wonderful help provided by the original sources in this compilation. • For best results, it is always suggested you read the source material. July – December 2008

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Why mathematical model is required? • It is required to understand, analyse and control the system • Fundamental physical laws of science and engineering are used for modeling. – Electrical Systems: Ohms, Kirchoffs and Lenz law – Mechanical Systems: Thermodynamic and Newton’s law

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Introduction to mathematical modeling • The mathematical model is usually in the form of differential equations. A differential equation can describe relationship between input and output • For a linear system, Laplace transform can be used to find the solutions of the differential equations • Using Laplace Transforms, we can represent the real system using transfer functions.

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Introduction • Deriving reasonable mathematical models is the most important part of the entire analysis. • The principle of causality is assumed throughout this course. • A trade-off exists between simplicity and accuracy. • Linear Systems. – Superposition applies.

• Linear Time-Invariant (vs. Time-Varying) Systems. • Differential equations with constant coefficients (vs. coefficients of functions of time). July – December 2008

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Transfer function approach • The physical system, the input and the output signals can be separated and easily visualized.

• Transfer function in the Laplace domain is that relation which algebraically relates the input and output of a control system, for the case when the initial conditions are zero. July – December 2008

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Transfer function approach Consider the linear time-invariant system defined by the differential equation: (n)

( n −1)

(n)

.

( n −1)

.

a0 y + a1 y + ⋅⋅⋅ + an −1 y + an y = b0 x + b1 x + ⋅⋅⋅ + bn −1 x + bn x

( n ≥ m)

where x = input and y = output

Transfer function = G ( s ) =

L[ y ] L[ x] zero initial conditions

Y ( s ) b0 s m + b1s m −1 + ⋅⋅⋅ + bm −1s + bm N ( s ) = = = n n −1 X ( s ) a0 s + a1s + ⋅⋅⋅ + an −1s + an D( s ) Order ( D( s )) = number of order of the system.

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Characteristics of transfer function • It is mathematical model to express the relationship between the output and the input variable. • It is independent of the magnitude and nature of the input or driving function. • It includes the units. However it does not provide any information concerning the physical structure. • If it is known, the output or response can be studied for various forms of inputs. • If it is unknown, it may be established experimentally by introducing known inputs and studying the output. Once established, it gives a full description of the dynamic characteristics.

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Example: Impulse response function Convolution Integral. Y (s) = G (s) X (s) t

t

0

0

where g (t ) = x(t ) = 0

for t < 0.

⇒ y (t ) = ∫ x(τ ) g (t − τ )dt = ∫ g (τ ) x(t − τ )dt

Let x(t ) = δ (t ). ⇒ X (s) = 1 ⇒ Y (s) = G (s) L−1[G ( s )] = g (t ): Impulse-Response Function. The transfer function and the impulse-response function contain the same complete information about the system dynamics.

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Laplace transform • Laplace transforms provide a method for representing and analyzing linear systems using algebraic methods. • In systems that begin undeflected and at rest the Laplace can directly replace the d/dt operator in differential equations. It is a superset of the phasor representation in that it has both a complex part, for the steady state response, but also a real part, representing the transient part. • As with the other representations the Laplace s is related to the rate of change in the system. D=s s = σ + jω (if the initial conditions/derivatives are all zero at t=0s) July – December 2008

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Procedure to apply Laplace transform • Convert the system transfer function, or differential equation, to the s-domain by replacing ’D’ with ’s’. (Note: If any of the initial conditions are non-zero these must be also be added.) • Convert the input function(s) to the s-domain using the transform tables. • Algebraically combine the input and transfer function to find an output function. • Use partial fractions to reduce the output function to simpler components. • Convert the output equation back to the time-domain using the tables.

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Summary • The understanding of the system is best obtained in linear systems is from deriving the transfer function of the system. • In order to make the analytical complexity easy to handle the transformation of time domain to a different domain is necessary. Laplace transformation helps in such transformation. T • he transfer function is obtained by identifying the input and output variables from the system and deducing using Laplace transforms July – December 2008

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References •

http://en.wikipedia.org/wiki/Laplace_transform

amongst

others…

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And, before we break… • The impossible is the untried

Thanks for listening… July – December 2008

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