Laplace-domain input/output description of an LTI system , (extends the resolvent matrix to the forced problem).
Laplace Transform of the Complete System
- For the complete system, Laplace gives:
-
- (the same zero-input / zero-state split as in the time domain)
- with
- (using )
Transfer Function Matrix
- Output:
Transfer Function Matrix (TFM)
- transfer function matrix for the system (TFM)
Impulse Response Matrix
- We know
Impulse Response Matrix (LTI)
- The impulse response matrix is the inverse transform of the TFM
Remember
- The above is for LTI only! Laplace breaks down for LTV systems, since
- similarly
- and the same holds for , etc.
Modal (Partial-Fraction) Form
- From the modal decomposition of the response (with ), satisfies
- Substitute the modal expressions for ( rows of , columns of , entries of , / right / left eigenvectors):
TFM in Partial-Fraction Form
- Instantly available in partial fraction form!
- : how much of mode is visible in output
- : strength of the impact of input on mode
Invariance Under Coordinate Changes
- For LTI systems, a change of state coordinates does not change the TFM:
- i.e. ; proof in State Coordinate Transformations