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Chapter 2 Systems Defined by Differential or Difference Equations

Chapter 2 Systems 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

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Chapter 2 Systems Defined by Differential or Difference Equations

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  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 – Cont’d • 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 – Cont’d

  8. Generalization of the First-Order Case – Cont’d • Solving for it is which, plugged into , yields

  9. Generalization of the First-Order Case – Cont’d If the solution of is then the solution of is

  10. System Modeling – Electrical Circuits resistor capacitor inductor

  11. Example: Bridged-T Circuit Kirchhoff’s voltage law loop (or mesh) equations

  12. Mechanical Systems • Newton’s 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 – Cont’d • 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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