1. Chapter 4 Transfer Function and Block Diagram Operations
§ Linear Time-Invariant Systems
§ Transfer Function and Dynamic Systems
§ Transfer Function and System Response
§ Block Diagram Operations for Complex Systems

2. § 4.1 Linear Time-Invariant Systems (1)
LTI Systems:
Differential Equation Formulation

3. § 4.1 Linear Time-Invariant Systems (2)
Solution Decomposition:
y(t)=y(I.C., system)+y(system, input)
y(I.C., system)=yh(t)
I.C.-dependent solution
Homogeneous solution
Natural response
Zero-input response
y(system, input)=yp(t)
Forcing term dependent solution
Particular solution
Forced response
Zero-state response

4. § 4.1 Linear Time-Invariant Systems (3)
Solution Modes:
Characteristic equation
Eigen value
Solution modes

5. § 4.1 Linear Time-Invariant Systems (4)

6. § 4.1 Linear Time-Invariant Systems (5)
Output Response:
(1) y(t)=yh(t)+yp(t)
yh(t): Linear combination of solution modes
yp(t): Same pattern and character as the forcing function
The RH side of LTI model affects only the coefficients of solution modes.
The LH side of LTI model dominates the solution modes of the response.
(2) y(t)=ys(t)+yt(t)
yt(t): Transient solution
ys(t): Steady state solution
Transient solution is contributed by initial condition and forcing function.

7. § 4.2 Transfer Function and Dynamic Systems (1)
Input is transfered through system G to output.
Definition:
Key points: Linear, Time-Invariant, Zero initial condition

8. § 4.2 Transfer Function and Dynamic Systems (2)
Pierre-Simon Laplace (1749 ~ 1827)
Monumental work “ Traite de mécanique céleste ”

9. § 4.2 Transfer Function and Dynamic Systems (3)
Laplace Transform
Definition:
Time function
Existence Condition
Inverse Laplace Transform
Signals that are physically realizable (causal) always has a Laplace transform.

10. § 4.2 Transfer Function and Dynamic Systems (4)
Important Properties:
t – Domain s – Domain
Linearity
Time shift
Scaling
Final value theorem
Initial value theorem
Convolution
Differentiation
Integration

11. § 4.2 Transfer Function and Dynamic Systems (5)
Unit impulse
Unit step
Ramp
Exponential decay
Sine wave
Cosine wave

12. § 4.2 Transfer Function and Dynamic Systems (6)
Fundamental Transfer Function of Mechanical System:
Elements Function Block Diagram T.F Example
Static element
(Proportional element)
Integral element
Differential element
Transportation lag

13. § 4.2 Transfer Function and Dynamic Systems (7)
States and Constitutive Law of Physical Systems:

14. § 4.2 Transfer Function and Dynamic Systems (8)
Inverse Laplace Transform and Partial Fraction Expansion:
Roots of D(s)=0:
(1) Real and distinct roots
From Laplace transform pairs

15. § 4.2 Transfer Function and Dynamic Systems (9)
(2) Real repeated roots
From Laplace transform pairs

16. § 4.2 Transfer Function and Dynamic Systems (10)
(3) Complex conjugate pairs
From Laplace transform properties and pairs

17. § 4.2 Transfer Function and Dynamic Systems (11)
Dynamic System Equation and Transfer Function:
Differential Equation and Transfer Function
Differential Equation: Transfer Function:
Problems associated with differentiation of noncontinuous functions, ex. step function, impulse function.

18. § 4.2 Transfer Function and Dynamic Systems (12)
Integral Equation and Transfer Function
The transfer function of a system is the Laplace transform of its
impulse response

19. § 4.3 Transfer Function and System Response (1)
Transfer Function G(s):
Rational T.F.
Irrational T.F.
Proper T.F.

20. § 4.3 Transfer Function and System Response (2)
Response by T.F.:
Partial fraction expansion is employed to find y(t).

21. § 4.3 Transfer Function and System Response (3)
Ex:

22. § 4.3 Transfer Function and System Response (4)
Poles, Zeros, and Pole-zero Diagram:
For an irreducible proper rational transfer function G(s),
a number (real or complex) is said to be
Pole-zero diagram
Representation of poles and zeros distribution by using “x” and “o”,
respectively in complex plane.
Ex:
Characteristic Equation
i.e.
characteristic roots: The roots of characteristic equation
i.e. The poles of G(s).

23. § 4.3 Transfer Function and System Response (5)
Impulse Response of Poles Distribution

24. § 4.3 Transfer Function and System Response (6)
Effects of Poles and Zeros
A pole of the input function generates the form of the forced response.
A pole of the transfer function generates the form of the natural response.
The zeros and poles generate the amplitude for both the forced and natural responses.
The growth, decay, oscillation, and their modulations determined by the impulse response of the poles distribution.

25. § 4.3 Transfer Function and System Response (7)

26. § 4.4 Block Diagram Operations for Complex Systems(1)
Fundamental Operations:
Signal operation
Summer Y(s)=X1(s)+X2(s)
Comparator Y(s)=X1(s)-X2(s)
Take-off point Y(s)=X1(s)
Component combinations
Serial
Parallel
Feedback

27. § 4.4 Block Diagram Operations for Complex Systems(2)
Moving junction / sequence
Ahead of a block
Past a block
Exchange sequence

28. § 4.4 Block Diagram Operations for Complex Systems(3)
Negative Feedback System:

29. § 4.4 Block Diagram Operations for Complex Systems(4)
Loading Effect:
Cascade
Realization Isolated Amp by 741OP

30. § 4.4 Block Diagram Operations for Complex Systems(5)
Network 1:
Network 2:
Loading effect
Loading effect

31. § 4.4 Block Diagram Operations for Complex Systems(6)
Note: For MIMO System
Output
Vector
Transfer
Matrix
Input
Vector

32. § 4.4 Block Diagram Operations for Complex Systems(7)
Example: Armature control DC servomotor
Static characteristics (Ideal)

33. § 4.4 Block Diagram Operations for Complex Systems(8)
Dynamic characteristics
I/O Block Diagram Reduction
Total
Response
Command
Response
Disturbance
Response

34. § 4.4 Block Diagram Operations for Complex Systems(9)
Model Reduction

35. § 4.4 Block Diagram Operations for Complex Systems(10)
Static gain is dominated by feedback gain Kb=1 / Km in system dynamics.
Key points: Linear time-invariant motor
No load
No delay
No damping
No inertia
No resistance
No inductance