Discretization of LTI Systems

Let us consider a linear time-invariant system in the form $$\dot{x}(\tau )=Ax(\tau )+Bu(\tau ).$$

Integration of the system's dynamics over an iterval $\tau \in ⟨0,t+h⟩$ gives us $$x(t+h)=e^{A(t+h)}x(0)+{\int }_0^{t+h}{e}^{A(t+h-\tau )}Bu(\tau )d\tau .$$ To express its state at $x(t+h)$ with respect to the state $x(t)$ we may perform a series of manipulations $$\begin{array}{rl}x(t+h)& =e^{A(t+h)}x(0)+{\int }_0^{t+h}{e}^{A(t+h-\tau )}Bu(\tau )d\tau \\ & =e^{A(t+h)}x(0)+{\int }_0^t{e}^{A(t+h-\tau )}Bu(\tau )d\tau +{\int }_t^{t+h}{e}^{A(t+h-\tau )}Bu(\tau )d\tau \\ & =e^{Ah}\underset{x(t)}{\underbrace{\left(e^{At}x(0)+{\int }_0^t{e}^{A(t-\tau )}Bu(\tau )d\tau \right)}}+{\int }_t^{t+h}{e}^{A(t+h-\tau )}Bu(\tau )d\tau .\end{array}\text{\hspace{1em}(1)}$$ If we then set $u(\tau )=\hat{u}$, $\tau \in ⟨t,t+h)$ the integral $${\int }_t^{t+h}{e}^{A(t+h-\tau )}Bu(\tau )d\tau$$ can be manipulated into $$\begin{array}{rl}{\int }_t^{t+h}{e}^{A(t+h-\tau )}Bu(\tau )d\tau & =e^{A(t+h)}{\int }_t^{t+h}{e}^{-A\tau }d\tau B\hat{u}\\ & =e^{A(t+h)}{\left[-A^{-1}{e}^{-A\tau }\right]}_t^{t+h}B\hat{u}\\ & =e^{A(t+h)}\left(-A^{-1}{e}^{-A(t+h)}+A^{-1}{e}^{-At}\right)B\hat{u}\\ & =A^{-1}\left(e^{Ah}-I\right)B\hat{u}.\end{array}\text{\hspace{1em}(2)}$$ By substituting (2) into (1) we obtain $$x_{k+1}=e^{Ah}{x}_k+A^{-1}(e^{Ah}-I)B{u}_k\text{\hspace{1em}(3)}$$ where $x_{k+1}=x(t+h)$, $x_k=x(t)$, and $u_k=\hat{u}$.

If we use the first two terms of the taylor expansion $$e^{Ah}=I+Ah+\frac{(Ah{)}^2}{2!}+\dots$$ in (3) we get the system representation $$x_{k+1}=(I+Ah)x_k+Bh{u}_k.$$