Transient, or start up, analysis pretty much requires a time domain simulation. For Steady-State analysis there are a couple of tools.
I do a frequency domain analysis that pretty simple in something like MATLAB. If you input can be represented with a PSD then you multiply the PSD by the square of the magnitude of your system transfer function. (This is done at discrete frequencies for both magnitude and transfer function.) For a typical closed loop system the transfer function of interest is the disturbance rejection. You would multiply the input PSD by the magnitude squared of the disturbance rejection. The result is the PSD of your system residual error.
Caveats:
1) This only account for linear behavior. Start up, transient, behavior is often nonlinear as filters, etc. are impulsed at system start up and need time to settle before they behave linearly. Systems with nonlinearities need to be simulated in the time domain or certain linearizing assumptions must be made and understood.
2) The system must be stationary and ergodic. (I will eventually add a section on stochastics to my wiki -
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if your system has an integrator applied to a random, stationary process - such as random noise - then the output of the integrator needs to be checked. It is fairly likely that the output of the integrator is not stationary.
3) This can be a pretty miserable analysis to do without the aid of something like MATLAB or MATHCAD.
Gabe I've started a new wikipedia style site for control systems
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for controls - without the spam
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