solution of odes using laplace transforms...alternatively we can use partial fraction expansion to...
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Solution of ODEs using Laplace Transforms
Process Dynamics and Control
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Linear ODEs
■ For linear ODEs, we can solve without integrating by using Laplace transforms
■ Integrate out time and transform to Laplace domain
Multiplication
Integration
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Common Transforms
Useful Laplace Transforms 1. Exponential 2. Cosine
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Common Transforms
Useful Laplace Transforms 3. Sine
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Common Transforms
Operators 1. Derivative of a function, , 2. Integral of a function
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Common Transforms
Operators 3. Delayed function
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Common Transforms
Input Signals 1. Constant 2. Step 3. Ramp function
L [f(t)] =
Z 1
0ate�stdt = � ate�st
s
�1
0
+
Z 1
0
ae�st
sdt =
a
s2
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Common Transforms
Input Signals 4. Rectangular Pulse 5. Unit impulse
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Laplace Transforms
Final Value Theorem Limitations: Initial Value Theorem
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Solution of ODEs
We can continue taking Laplace transforms and generate a catalogue of Laplace domain functions.
The final aim is the solution of ordinary differential
equations.
Example Using Laplace Transform, solve Result
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Solution of ODEs Cruise Control Example
➤ Taking the Laplace transform of the ODE yields (recalling the Laplace transform is a linear operator)
Force of Engine (u)
Friction
Speed (v)
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Solution of ODEs
■ Isolate
and solve
➤ If the input is kept constant
its Laplace transform ➤ Leading to
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Solution of ODEs
■ Solve by inverse Laplace transform: (tables)
➤ Solution is obtained by a getting the inverse Laplace transform from a table ➤ Alternatively we can use partial fraction expansion to compute the solution using simple inverse transforms
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Solution of Linear ODEs
■ DC Motor
■ System dynamics describes (negligible inductance)
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Laplace Transform
■ Expressing in terms of angular velocity
➤ Taking Laplace Transforms
➤ Solving
➤ Note that this function can be written as
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Laplace Transform
Assume then the transfer function gives directly Cannot invert explicitly, but if we can find such that
we can invert using tables. Need Partial Fraction Expansion to deal with
such functions
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Linear ODEs
We deal with rational functions of the form where degree of > degree of
is called the characteristic polynomial of the function The roots of are the poles of the function Theorem:
Every polynomial with real coefficients can be factored into the product of only two types of factors ➤ powers of linear terms and/or ➤ powers of irreducible quadratic terms,
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Partial fraction Expansions
1. has real and distinct factors expand as 2. has real but repeated factor expanded
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Partial Fraction Expansion
Heaviside expansion For a rational function of the form Constants are given by
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Partial Fraction Expansion
Example
The polynomial
has roots
It can be factored as
By partial fraction expansion
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Partial Fraction Expansion
By Heaviside
➤ becomes
➤ By inverse laplace
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Partial Fraction Expansion
Heaviside expansion For a rational function of the form Constants are given by
↵n�1 =d
ds
✓(s+ b)n
p(s)
q(s)
◆�
s=�b
↵1 =1
n!
dn
dsn
✓(s+ b)n
p(s)
q(s)
◆�
s=�b
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Partial Fraction Expansion
Example
The polynomial
has roots
It can be factored as
By partial fraction expansion
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Partial Fraction Expansion
By Heaviside
➤ becomes
➤ By inverse laplace
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Partial Fraction Expansion
3. has an irreducible quadratic factor
➤ Gives a pair of complex conjugates if
➤ Can be factored in two ways a) is factored as
b) or as
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Partial Fraction Expansion
Heaviside expansion For a rational function of the form Constants are given by
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Partial Fraction Expansion
Example The polynomial
has roots
It can be factored as
By partial fraction expansion
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Partial Fraction Expansion
By Heaviside, which yields Taking the inverse laplace
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Partial Fraction Expansion
The inverse laplace Can be re-arranged to
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Partial Fraction Expansion
Example The polynomial
has roots
It can be factored as ( )
Solving for A and B,
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Partial Fraction Expansion
Equating similar powers of s in, yields hence Giving Taking the inverse laplace
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Partial Fraction Expansions
Algorithm for Solution of ODEs
➤ Take Laplace Transform of both sides of ODE ➤ Solve for
➤ Factor the characteristic polynomial ➨ Find the roots (roots or poles function in Matlab) ➨ Identify factors and multiplicities
➤ Perform partial fraction expansion ➤ Inverse Laplace using Tables of Laplace Transforms
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Partial Fraction Expansion
■ For a given function
■ The polynomial has three distinct types of roots ➤ Real roots
➨ yields exponential terms ➨ yields constant terms
➤ Complex roots ➨ yields exponentially weighted sinusoidal
signals ➨ yields pure sinusoidal signal
■ A lot of information is obtained from the roots of