# Stability of impulse response

I'm doing a past paper exam question and I've gotten stuck. The system can be seen in the picture below.

Using initial conditions given in the paper, I've calculated the zero input response Yb(t) to be 10cos(2t-60), it could also be calculated to be Ae^(2jt) + Be^(-2jt) with A and B as arbitrary constants. The impulse response would then have characteristic roots of -2j and 2j. Am i correct in thinking that? The problem comes in with the impulse response of system c. The characterisitc root is -2j. Thus there is a repeated root of -2j. How will this affect the stability of the system? Is the system unstable because of the repeated root on the imaginary axis? Should I consider it this way or should i just leave the impulse response of system B in terms of a trigonometric function - resulting in marginal stability of system B and marginal stability of system C? Any resources pointing in the direcction of how to approach this would be greatly appreciated.

In this system, the total response is the sum of System A to D.

If you interpret this simply -- A, B and C are stable or marginally stable. D is exponential growth, so it is unstable. (Given D is not exactly canceled by another system) The sum is unstable.

System B and C responses share same roots, the same roots remain when adding them together, just the amplitudes change, regardless how you represent the mathematics (trigonometric or exponential). Notice you are adding and not chaining the two systems together.

• +1: this is one of those try to get an overview before you calculate anything question types. Hint: always when someone asks for stability in an exam. The actual calculation takes too much time for the most exams, given there are other questions, too. Commented Nov 6, 2016 at 20:55