Chapter 2 Systems Defined by Differential or Difference Equations

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Chapter 2Systems Defined by Differential or
Difference Equations
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Linear I/O Differential Equations with Constant
Coefficients

• Consider the CT SISO system
• described by

System
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Initial Conditions

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

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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.,

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First-Order Case

• Consider the following differential equation
• Its solution is

or
if the initial time is taken to be
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Generalization of the First-Order Case

• Consider the equation
• Define
• Differentiating this equation, we obtain

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Generalization of the First-Order Case Contd
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Generalization of the First-Order Case Contd

• Solving for
  it is
• which, plugged into ,
• yields

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Generalization of the First-Order Case Contd
If the solution of
is
then the solution of
is
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System Modeling Electrical Circuits
resistor
capacitor
inductor
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Example Bridged-T Circuit
Kirchhoffs voltage law
loop (or mesh) equations
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Mechanical Systems

• Newtons second Law of Motion
• Viscous friction
• Elastic force

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Example Automobile Suspension System
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Rotational Mechanical Systems

• Inertia torque
• Damping torque
• Spring torque

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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
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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)
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Solution by Recursion Contd

• The solution by recursion for requires
  the knowledge of the N initial conditions
• and of the M initial input values

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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)