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

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Chapter 2Systems Defined by Differential or

Difference Equations

Linear I/O Differential Equations with Constant

Coefficients

- Consider the CT SISO system
- described by

System

Initial Conditions

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

conditions

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.,

First-Order Case

- Consider the following differential equation
- Its solution is

or

if the initial time is taken to be

Generalization of the First-Order Case

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

Generalization of the First-Order Case Contd

Generalization of the First-Order Case Contd

- Solving for

it is - which, plugged into ,
- yields

Generalization of the First-Order Case Contd

If the solution of

is

then the solution of

is

System Modeling Electrical Circuits

resistor

capacitor

inductor

Example Bridged-T Circuit

Kirchhoffs voltage law

loop (or mesh) equations

Mechanical Systems

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

Example Automobile Suspension System

Rotational Mechanical Systems

- Inertia torque
- Damping torque
- Spring torque

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

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)

Solution by Recursion Contd

- The solution by recursion for requires

the knowledge of the N initial conditions - and of the M initial input values

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)