# discretization of continuous systems

Consider a continuous-time system with the following state space representation

 $P:\ \ \ \ \left\{\begin{array}[]{ccl}\dot{x}(t)&=&Ax(t)+Bu(t),\\ y(t)&=&Cx(t)+Du(t),\end{array}\right.$ (1)

where $x(t)\in{\mathbb{R}}^{n}$, $u(t)\in{\mathbb{R}}^{r}$ and $y(t)\in{\mathbb{R}}^{m}$ are the state vector, input vector and output vector of the system, respectively; $A\in{\mathbb{R}}^{n\times n}$, $B\in{\mathbb{R}}^{n\times r}$, $C\in{\mathbb{R}}^{m\times n}$ and $D\in{\mathbb{R}}^{m\times r}$ are the constant real or complex matrices.

Suppose that the sampling interval is $\tau$. By using the step invariance transform or the zero-order hold (ZOH), i.e., $u(t)=u(k\tau),\ k\tau\leq t<(k+1)\tau$, discretizing the system in (1) gives a discrete-time model,

 $P_{\tau}:\ \ \ \ \left\{\begin{array}[]{ccl}x(k\tau+\tau)&=&G_{\tau}x(k\tau)+F% _{\tau}u(k\tau),\\ y(k\tau)&=&Cx(k\tau)+Du(k\tau),\ k=0,1,2,\cdots\end{array}\right.$ (2)

where $x(k\tau)=x(t)\left|{}_{t=k\tau}\right.$, $y(k\tau)=y(t)\left|{}_{t=k\tau}\right.$, and

 $G_{\tau}:={\rm e}^{A\tau},\ F_{\tau}:=\int^{\tau}_{0}{\rm e}^{At}{\rm d}t\ B.$ (3)
Title discretization of continuous systems DiscretizationOfContinuousSystems 2013-03-22 15:50:45 2013-03-22 15:50:45 mathcam (2727) mathcam (2727) 7 mathcam (2727) Topic msc 93C55 Transform