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 MY ANSWERS: 1) Yes. I think the typical characteristic expressions would be different for arbitrary inputs. If inputs were a constant like a step then I think that it would result in the typical characteristic expressions. 2) Yes. I think the typical characteristic expressions would be different for non-zero initial conditions and zero inputs. 3) Yes.