Approximation for integral of Matrix Exponential

Question

I am trying to implement some algorithm in matlab. To this end, I need to discretise a system of differential equation as $\dot x = A x + B u$.

Starting with initial condition $x_0$ the system after a single time step $h$ will reach $$ x(h) = e^{Ah}x_0 + \int_0^h e^{A(h-\tau)}Bu(\tau)d\tau. $$

Assuming $u(\tau)$ to be constant in $[0,h]$, the previous expresion reduces to $$ x(h) = e^{Ah}x_0 + \left (\int_0^h e^{As}ds\right ) Bu, $$ with $s =h-\tau$.

The first exponential matrix $e^{Ah}$ is calculated with matlab command

expm(A*h).

As far as I know, there is no command which approximate the term $\int_0^h e^{As}ds$ so I thought to use the expansion series to approximate it as

$$ \int_0^h e^{As}ds = \int_0^h I + As + \frac{1}{2}A^2 s^2 + \cdots ds \\ = Ih + \frac{1}{2}Ah^2 + \frac{1}{6}A^2 h^3 + \cdots $$

I evaluate this expression in matlab up to some degree. The method works right but I'd like to ask you if there is a better approximation to it (maybe padé?).

Answer 1

This is EXACTLY from Wikipedia https://en.wikipedia.org/wiki/Matrix_exponential

"If $P$ and $Q_t$ are nonzero polynomials in one variable, such that $P(A)=0$, and if the memorphic function

$f(z)=\frac{e^{tz}-Q_t(z)}{P(z)}$

is entire, then $e^{tA}=Q_t(A)$"

Answer 2

Expanding on the answer of marshal craft, for the integral you might want to chose $Q_t$ such that $$ P(z)|\left(\frac{e^{tz}-1}z-Q_t(z)\right) $$

Answer 3

There is a very easy way to compute $\Phi(h)=\int_0^h e^{As}Bds$.

To this aim, define the matrix

$$M=\begin{bmatrix}A & B\\0 & 0\end{bmatrix}$$

and compute $e^{Mh}$, which is given by

$$e^{Mh}=\begin{bmatrix}e^{Ah} & \Phi(h)\\0 & I\end{bmatrix}.$$

This can be proven using the power series definition of the exponential.