ABSTRACT
This can be seen by differentiating the solution, and substituting in Equation 2.19. The output y(t) is thus
y(t) = Cx(t)+Du(t) = CeAtξ+ ∫ t 0
CeA(t−τ)Bu(τ) dτ+Du(t) (2.22) Using the eigenstructure of A, we can write
y(t) = n∑
+ n∑
e−λiτu(τ) dτ+Du(t) (2.24)
Applying the Laplace Transform to the system Equations 2.19 and 2.20,
y(s) = C(sI −A)−1Bu(s)+Du(s) (2.25) Using the eigenstructure of A, we can substitute for (sI −A)−1 to get the Laplace Transform equation
y(s) = n∑
s−λi u(s)+Du(s) (2.26)
The matrix CviwHi B (s−λi) is called the residue matrix at the pole s = λi .
