PlanetMath: discretization of continuous systems

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discretization of continuous systems

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

$\displaystyle P:\ \ \ \ \left\{\begin{array}{ccl} \dot{x}(t)&=&A x(t)+Bu(t),\\ y(t) &=& Cx(t)+D u(t),\end{array}\right.$

(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,

$\displaystyle P_\tau:\ \ \ \ \left\{\begin{array}{ccl}x(k\tau+\tau)&=&G_\tau x(... ...tau),\\ y(k\tau) &=& C x(k\tau)+Du(k\tau),\ k=0, 1, 2, \cdots\end{array}\right.$

(2)

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

$\displaystyle G_\tau:={\rm e}^{A \tau}, \ F_\tau:=\int^{\tau}_{0}{\rm e}^{A t}{\rm d}t\ B.$

(3)

"discretization of continuous systems" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Cross-references: interval, matrices, complex, real, vector, representation, state space
There are 81 references to this entry.

This is version 4 of discretization of continuous systems, born on 2006-04-13, modified 2006-10-04.
Object id is 7827, canonical name is DiscretizationOfContinuousSystems.
Accessed 4256 times total.

Classification:

AMS MSC:

93C55 (Systems theory; control :: Control systems, guided systems :: Discrete-time systems)

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