1. Ch2 Mathematical models of systems
Introduction
Models in time-domain (Differential equations)
Models in frequency-domain(Transfer function)
Block diagram models
Signal-flow graph model
Simulation and Examples

2. 2.1 Introduction
Mathematical models can help us to understand and control complex system effectively. So it is necessary to analyze the relationships between the system variables and to obtain a mathematical model. Because the systems are dynamic in nature, the descriptive equations are usually differential equations, if they can be linearized, then Laplace transform can be used to simplify the method of solution.

3. Goals of modeling Modeling method Analysis Design or Control
Analytical method
Experimental method

4. Models under Consideration
Linear
Time-invariant (constant)
Parameter-lumped

5. 2.2 Differential equations of physical systems
Through-variable and across-variable
Analogous variables
Refer to (P34-37)
Voltage-Velocity analogy
Force-current analogy

6. How to obtain DE ? Newton’s laws for Mechanical system
Kirchhoff’s laws for electrical systems
Two examples
(Refer to script 2-2)

7. Solving the equation Analytical or classical method
Laplace transform method
Example of solving differential equation
( Refer to script 2-3,4)

8. Modes of dynamic system
Modes is determined by Characteristic roots:
real and distinct roots
real and repeated roots
complex conjugate roots

9. 2.3 Linear approximations of physical systems
Non-linearity is essential and pervasive
A system is defined as linear in terms of the
system excitation and response.
Principle of superposition
Homogeneity
A linear system satisfies the properties of superposition and homogeneity

10. Linear approximations method
Taylor series expansion
Example: pendulum oscillator model
Refer to P40

11. 2.4 The Laplace transform Review complex algebraic (refer to P41-47):
Inverse Laplace transform
residue evaluation
Final value theorem

12. Time response by Laplace transform
Time response solution is obtained by:
1. obtain the differential equations
2. obtain the Laplace transformation of
differential equations
3. solve the algebraic equation of the
variable of interest
4. obtain the response by the inverse
Laplace transform

13. 2.5 The transfer function of linear system
Transfer function is defined as the ratio of the Laplace transform of the output variable to the laplace transform of the input variable , with all initial conditions assumed to be zero.
Examples: (refer to P48)

14. Characteristics of TF （传递函数的特性）
Black-box model
rational real fractional function
transition between TF and DE
TF and impulse response
Transfer function is the Laplace transform of the impulse response and vice versa.

15. Implication of TF （传递函数的含义）
Zero initial condition
representation of poles and zeros
Inputs and outputs of system are zero at t=0-

16. TF of typical elements and dynamic systems
Basic factors of TF
Examples of TF
Refer to P50-57 and
Table 2.5 (P58-61)

17. 2.6 Block diagram models （结构图或方框图模型）
Block diagrams representation is prevalent in control engineering, and consist of unidirectional operational blocks that represent the transfer function of the variables of interests.
Representation of block diagram
Refer to P62-63

18. Transformation of Block Diagram
BD=Scheme+Equation
Four Components of BD (refer to 2-14,15)
Transition between TF and BD
Refer to 2-15,16,17

19. Characteristics of BD Unidirectional Principle Without load effect
Connection ways:
Series
parallel and
feedback connection

20. Block diagram reduction
Combination of blocks in cascade
Combination of blocks in parallel
Elimination of feedback loop
Moving pickoff points and summing points
Principle: Keep every signal to be equivalent

21. Examples of BD reduction
Refer to textbook ( P64-66)
Other examples : refer to (2-19,20,21)

22. Assignment (课后作业）
Review Ch2 (P32--)
E2.4
E2.6
E2.8
E2.18