# Chapter 2 Systems Defined by Differential or Difference Equations - PowerPoint PPT Presentation

PPT – Chapter 2 Systems Defined by Differential or Difference Equations PowerPoint presentation | free to download - id: 6c4db5-ZDFmZ

The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
Title:

## Chapter 2 Systems Defined by Differential or Difference Equations

Description:

### Chapter 2 Systems Defined by Differential or Difference Equations Linear I/O Differential Equations with Constant Coefficients Consider the CT SISO system described ... – PowerPoint PPT presentation

Number of Views:51
Avg rating:3.0/5.0
Slides: 19
Provided by: Luca99
Category:
Tags:
Transcript and Presenter's Notes

Title: Chapter 2 Systems Defined by Differential or Difference Equations

1
Chapter 2Systems Defined by Differential or
Difference Equations
2
Linear I/O Differential Equations with Constant
Coefficients
• Consider the CT SISO system
• described by

System
3
Initial Conditions
• In order to solve the previous equation for
• , we have to know the N initial
conditions

4
Initial Conditions Contd
• If the M-th derivative of the input x(t) contains
an impulse or a derivative of an
impulse, the N initial conditions must be taken
at time , i.e.,

5
First-Order Case
• Consider the following differential equation
• Its solution is

or
if the initial time is taken to be
6
Generalization of the First-Order Case
• Consider the equation
• Define
• Differentiating this equation, we obtain

7
Generalization of the First-Order Case Contd
8
Generalization of the First-Order Case Contd
• Solving for
it is
• which, plugged into ,
• yields

9
Generalization of the First-Order Case Contd
If the solution of
is
then the solution of
is
10
System Modeling Electrical Circuits
resistor
capacitor
inductor
11
Example Bridged-T Circuit
Kirchhoffs voltage law
loop (or mesh) equations
12
Mechanical Systems
• Newtons second Law of Motion
• Viscous friction
• Elastic force

13
Example Automobile Suspension System
14
Rotational Mechanical Systems
• Inertia torque
• Damping torque
• Spring torque

15
Linear I/O Difference Equation With Constant
Coefficients
• Consider the DT SISO system
• described by

System
N is the order or dimension of the system
16
Solution by Recursion
• Unlike linear I/O differential equations, linear
I/O difference equations can be solved by direct
numerical procedure (N-th order recursion)

(recursive DT system or recursive digital filter)
17
Solution by Recursion Contd
• The solution by recursion for requires
the knowledge of the N initial conditions
• and of the M initial input values

18
Analytical Solution
• Like the solution of a constant-coefficient
differential equation, the solution of
• can be obtained analytically in a closed form
and expressed as
• Solution method presented in ECE 464/564

(total response zero-input response
zero-state response)