# Why Time Response Characteristics Derived from Zero State Equation

QUESTION:

1) Why are Time Response Characteristic's Expressions derived from only from Zero State Equations?
NOTE: Nise Control Systems Engineering 6ed uses step inputs to derive Time Response Characteristics for 1st
and 2nd order ordinary differential equations.
-What would the characteristic expressions be for arbitrary inputs?

2) Why is it not customary to derive general analytic characteristic expressions from Zero Input Equations?
-Would the characteristic expressions be different?

3) Why is it not customary to derive general analytic general analytic characteristic expressions from the Total Solution?
-Would the characteristic expressions be different due to arbitrary inputs and now the inclusion of non-zero
initial conditions?

4) Can Impulse Input Functions be considered as initial conditions?

ASSUMPTIONS:
-Time Response Characteristics. e.g.: settling time, rise time, percent overshoot, peak time
-All Time Response Characteristics are derived using a step input
-Zero State Equation is the solution to a differential equation with zero initial conditions and non-zero inputs
-Zero Input Equation is the solution to a differential equation with zero inputs and non-zero initial conditions

(SYSTEM) STABILITY & BOUNDED INPUTS:
-bounded inputs/forcing functions (i.e. no unbounded inputs, but can have harmonic, periodic, and constant input functions)
-aperiodic inputs such as ramp, parabola, or impulses are not considered, except of course the constant (step)
-system is stable (i.e. naturally decays to zero for infinite time)
-Not considering marginally stable systems: oscillatory nor constant. Therefore our inputs when operating at the system's natural frequency will not cause an unbounded total solution.

SOURCES:
-Nise Control Systems Engineering 6ed