1. Laplace Transforms Chapter 3 Standard notation in dynamics and control
2. Converts mathematics to algebraic operations
3. Advantageous for block diagram analysis
Chapter 3

2. Laplace Transforms and Transfer Functions
Provide valuable insight into process dynamics and the dynamics of feedback systems.
Provide a major portion of the terminology of the process control profession.
Are NOT directly used in the practice of process control.

3. Laplace Transforms Useful for solving linear differential equations.
Approach is to apply Laplace transform to differential equation. Then algebraically solve for Y(s). Finally, apply inverse Laplace transform to directly determine y(t).
Tables of Laplace transforms are available.

4. Chapter 3 Laplace Transform Usually define f(0) = 0 (e.g., the error)
Example 1:
Chapter 3
Usually define f(0) = 0 (e.g., the error)

5. Integration by parts i.e., Example Let: Then:
where C is an arbitrary constant of integration

6. Other Transforms
etc. for
Chapter 3
Euler identity
Note:

7. Chapter 3

8. Table 3.1 Laplace Transforms for Various Time-Domain Functionsa
f(t) F(s)
Chapter 3

9. Table 3.1 Laplace Transforms for Various Time-Domain Functionsa
f(t) F(s)
Chapter 3

10. Table 3.1 Laplace Transforms for Various Time-Domain
Functionsa (continued)
f(t) F(s)

11. Chapter 3 Example 3.1 Solve the ODE,
First, take L of both sides of (3-26), and use superposition rule
Chapter 3
Solve for Y(s)
Rearrange,
Take L-1,
Divide the numerator and Denominator by 5 and put all factor in the s+b form
From table it match (s+b3)/[s+b1)(s+b2) for b1=0.8, b2=0
From Table 3.1,

12. Partial Fraction Expansions
Expand into a term for each factor in the denominator.
Recombine RHS
Equate terms in s and constant terms. Solve.
Each term is in a form so that inverse Laplace transforms can be applied.

13. Example of Solution of an ODE
ODE w/initial conditions
Apply Laplace transform to each term
Solve for Y(s)
Apply partial fraction expansion
Apply inverse Laplace transform to each term

19. Chapter 3 Example: system at rest (s.s.)
To find transient response for y(t) = unit step at t > 0
1. Take Laplace Transform (L.T.)
2. Factor, use partial fraction decomposition
3. Take inverse L.T.
Step Take L.T. (note zero initial conditions)

20. Chapter 3 Rearranging, Step 2a. Factor denominator of Y(s)
Step 2b. Use partial fraction decomposition
Multiply by s, set s = 0

21. Chapter 3 For a2, multiply by (s+1), set s=-1 (same procedure
for a3, a4)
Step 3. Take inverse of L.T.
Chapter 3
(check original ODE)
You can use this method on any order of ODE,
limited only by factoring of denominator polynomial
(characteristic equation)
Must use modified procedure for repeated roots,
imaginary roots

22. Chapter 3 Laplace transforms can be used in process control for:
1. Solution of differential equations (linear)
2. Analysis of linear control systems
(frequency response)
3. Prediction of transient response for
different inputs
Chapter 3

23. Chapter 3 Factoring the denominator polynomial 1.
Transforms to e-t/3, e-t (real roots)
Chapter 3
(no oscillation)

24. Chapter 3 2. Transforms to (oscillation) From Table 3.1, line 17
(complete the square)

25. Chapter 3 Let h→0, f(t) = δ(t) (Dirac delta) L(δ) = 1
Use L’Hopital’s theorem
(h→0)
If h = 1, rectangular pulse input

26. Chapter 3 Difference of two step inputs S(t) – S(t-1)
(S(t-1) is step starting at t = h = 1)
By Laplace transform
Chapter 3
Can be generalized to steps of different magnitudes
(a1, a2).

27. Chapter 3 One other useful feature of the Laplace transform
is that one can analyze the denominator of the transform
to determine its dynamic behavior. For example, if
the denominator can be factored into (s+2)(s+1).
Using the partial fraction technique
Chapter 3
The step response of the process will have exponential terms
e-2t and e-t, which indicates y(t) approaches zero. However, if
We know that the system is unstable and has a transient
response involving e2t and e-t. e2t is unbounded for
large time. We shall use this concept later in the analysis
of feedback system stability.

28. Final Value Theorem
Allows one to use the Laplace transform of a function to determine the steady-state resting value of the function.
A good consistency check.

29. Initial-Value Theorem
Allows one to use the Laplace transform of a function to determine the initial conditions of the function.
A good consistency check

30. Apply Initial- and Final-Value Theorems to this Example
Laplace transform of the function.
Apply final-value theorem
Apply initial-value theorem

31. Chapter 3 Other applications of L( ): Example 3: step response
A. Final value theorem
“offset”
Example 3: step response
Chapter 3
offset (steady state error) is a.
Time-shift theorem
y(t)=0 t < to

32. Chapter 3 C. Initial value theorem by initial value theorem
by final value theorem

33. Transfer Functions Defined as G(s) = Y(s)/U(s)
Represents a normalized model of a process, i.e., can be used with any input.
Y(s) and U(s) are both written in deviation variable form.
The form of the transfer function indicates the dynamic behavior of the process.

34. Poles of the Transfer Function Indicate the Dynamic Response
For a, b, c, and d positive constants, transfer function indicates exponential decay, oscillatory response, and exponential growth, respectively.

35. Poles on a Complex Plane

36. Exponential Decay

37. Damped Sinusoidal

38. Exponentially Growing Sinusoidal Behavior (Unstable)

39. What Kind of Dynamic Behavior?

40. Unstable Behavior
If the output of a process grows without bound for a bounded input, the process is referred to a unstable.
If the real portion of any pole of a transfer function is positive, the process corresponding to the transfer function is unstable.
If any pole is located in the right half plane, the process is unstable.

41. An Example of Block Diagram Algebra

42. Solution of Example

43. What if the Process Model is Nonlinear
Before transforming to the deviation variables, linearize the nonlinear equation.
Transform to the deviation variables.
Apply Laplace transform to each term in the equation.
Collect terms and form the desired transfer functions.

44. Use Taylor Series Expansion to Linearize a Nonlinear Equation
This expression provides a linear approximation of y(x) about x=x0.
The closer x is to x0, the more accurate this equation will be.
The more nonlinear that the original equation is, the less accurate this approximation will be.

45. Linearize a Nonlinear Term
Linearization of the nonlinear term using a Taylor series approximation

46. Overview
The transfer function of a process shows the characteristics of its dynamic behavior assuming a linear representation of the process.
Transfer functions are not, in general, used in the industrial practice of process control because they require considerable effort to develop for industrial processes and they do not consider nonlinear behavior.