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

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Chapter 2 Systems Defined by Differential or Difference Equations

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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

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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)
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