1. Chapter 2 Systems 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. 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

13. 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)

14. 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

15. can be obtained analytically in a closed form and expressed as
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
(total response = zero-input response + zero-state response)