Solution of ODEs using Laplace Transforms Process Dynamics and Control 1
Linear ODEs For linear ODEs, we can solve without integrating by using Laplace transforms Integrate out time and transform to Laplace domain Integration Multiplication 2
Common Transforms Useful Laplace Transforms 1. Exponential 2. Cosine 3
Common Transforms Useful Laplace Transforms 3. Sine 4
Common Transforms Operators 1. Derivative of a function,, 2. Integral of a function 5
Common Transforms Operators 3. Delayed function 6
Common Transforms Input Signals 1. Constant 2. Step 3. Ramp function 7
Common Transforms Input Signals 4. Rectangular Pulse 5. Unit impulse 8
Laplace Transforms Final Value Theorem Limitations: Initial Value Theorem 9
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 10
Solution of ODEs Cruise Control Example Speed (v) Friction Force of Engine (u) Taking the Laplace transform of the ODE yields (recalling the Laplace transform is a linear operator) 11
Solution of ODEs Isolate and solve If the input is kept constant its Laplace transform Leading to 12
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 13
Solution of Linear ODEs DC Motor System dynamics describes (negligible inductance) 14
Laplace Transform Expressing in terms of angular velocity Taking Laplace Transforms Solving Note that this function can be written as 15
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 16
Linear ODEs We deal with rational functions of the form > degree of where 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, 17
1. has real and distinct factors Partial fraction Expansions expand as 2. has real but repeated factor expanded 18
Partial Fraction Expansion Heaviside expansion For a rational function of the form Constants are given by 19
Partial Fraction Expansion Example The polynomial has roots It can be factored as By partial fraction expansion 20
Partial Fraction Expansion By Heaviside becomes By inverse laplace 21
Partial Fraction Expansion Heaviside expansion For a rational function of the form Constants are given by 22
Partial Fraction Expansion Example The polynomial has roots It can be factored as By partial fraction expansion 23
Partial Fraction Expansion By Heaviside becomes By inverse laplace 24
3. has an irreducible quadratic factor Partial Fraction Expansion Gives a pair of complex conjugates if Can be factored in two ways a) is factored as b) or as 25
Partial Fraction Expansion Heaviside expansion For a rational function of the form Constants are given by 26
Partial Fraction Expansion Example The polynomial has roots It can be factored as By partial fraction expansion 27
Partial Fraction Expansion By Heaviside, which yields Taking the inverse laplace 28
Partial Fraction Expansion The inverse laplace Can be re-arranged to 29
Partial Fraction Expansion Example The polynomial has roots It can be factored as ( ) Solving for A and B, 30
Equating similar powers of s in, Partial Fraction Expansion yields hence Giving Taking the inverse laplace 31
Algorithm for Solution of ODEs Partial Fraction Expansions 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 32
Partial Fraction Expansion For a given function The polynomial Real roots has three distinct types of 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 33