Discretization

Discretization concerns the process of transferring continuous models and equations into discrete counterparts, suitable for numerical evaluation and implementation on computers.

Euler discretization
Zero order hold

Discretization is also somewhat connected to discrete mathematics.

Discretization of Linear State Space Models

The following continuous state space model

$\dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)$

$\mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t)$

may be discretized, assuming zero-order hold, to

$\mathbf{x}(k+1) = A_d \mathbf{x}(k) + B_d \mathbf{u}(k)$

$\mathbf{y}(k) = C_d \mathbf{x}(k) + D_d \mathbf{u}(k)$

where

$A_d = \mathcal{L}^{-1}\{(sI - A)^{-1}\}_{t=T_s} = e^{A T_s}$

$B_d = \int_{\tau=0}^{T_s}e^{A \tau}d\tau B = A^{-1}(A_d - I)B$ , if

A

is nonsingular

C d = C

D d = D

and $T s$ is the sample time.

Categories: Control theory

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Discretization

Discretization of Linear State Space Models

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Discretization

Discretization of Linear State Space Models