In mathematics, discretization concerns the process of transferring continuous models and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. In order to be processed on a digital computer another process named quantization is essential.

• Euler discretization
• Zero order hold

Discretization is also somewhat connected to discrete mathematics.

Discretization of linear state space models

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Discretization is also concerned with the transformation of continuous differential equations into discrete difference equations, suitable for numerical computing.

The following continuous state space model

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

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

where v and w are continuous zero-mean white noise sources with covariances

$\backslash mathbf\{v\}(t)\; \backslash sim\; N(0,\backslash mathbf\; Q)$

$\backslash mathbf\{w\}(t)\; \backslash sim\; N(0,\backslash mathbf\; R)$

can be discretised, assuming zero-order hold for the input u and continuous integration for the noise v, to

with covariances

$\backslash mathbf\{v\}*\backslash sim\; N(0,\backslash mathbf\; Q\_d)$

$\backslash mathbf\{w\}*\backslash sim\; N(0,\backslash mathbf\; R\_d)$

where

$\backslash mathbf\; A\_d\; =\; e^\{\backslash mathbf\; A\; T\}\; =\; \backslash mathcal\{L\}^\{-1\}\backslash \{(s\backslash mathbf\; I\; -\; \backslash mathbf\; A)^\{-1\}\backslash \}\_\{t=T\}$

$\backslash mathbf\; B\_d\; =\; \backslash left(\; \backslash int\_\{\backslash tau=0\}^\{T\}e^\{\backslash mathbf\; A\; \backslash tau\}d\backslash tau\; \backslash right)\; \backslash mathbf\; B\; =\; \backslash mathbf\; A^\{-1\}(\backslash mathbf\; A\_d\; -\; I)\backslash mathbf\; B$, if $\backslash mathbf\; A$ is nonsingular

$\backslash mathbf\; C\_d\; =\; \backslash mathbf\; C$

$\backslash mathbf\; D\_d\; =\; \backslash mathbf\; D$

$\backslash mathbf\; Q\_d\; =\; \backslash int\_\{\backslash tau=0\}^\{T\}\; e^\{\backslash mathbf\; A\; \backslash tau\}\; \backslash mathbf\; Q\; e^\{\backslash mathbf\; A^T\; \backslash tau\}\; d\backslash tau$

$\backslash mathbf\; R\_d\; =\; \backslash mathbf\; R$

and $T$ is the sample time.

Derivation

Starting with the continuous model

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

we know that the matrix exponential is

$\backslash frac\{d\}\{dt\}e^\{\backslash mathbf\; At\}\; =\; \backslash mathbf\; A\; e^\{\backslash mathbf\; At\}\; =\; e^\{\backslash mathbf\; At\}\; \backslash mathbf\; A$

and by premultiplying the model we get

$e^\{-\backslash mathbf\; At\}\; \backslash mathbf\{\backslash dot\{x\}\}(t)\; =\; e^\{-\backslash mathbf\; At\}\; \backslash mathbf\; A\backslash mathbf\; x(t)\; +\; e^\{-\backslash mathbf\; At\}\; \backslash mathbf\; B\backslash mathbf\; u(t)$

which we recognize as

$\backslash frac\{d\}\{dt\}(e^\{-\backslash mathbf\; At\}\backslash mathbf\; x(t))\; =\; e^\{-\backslash mathbf\; At\}\; \backslash mathbf\; B\backslash mathbf\; u(t)$

and by integrating..

$e^\{-\backslash mathbf\; At\}\backslash mathbf\; x(t)\; -\; e^0\backslash mathbf\; x(0)\; =\; \backslash int\_0^t\; e^\{-\backslash mathbf\; A\backslash tau\}\backslash mathbf\; B\backslash mathbf\; u(\backslash tau)\; d\backslash tau$

$\backslash mathbf\; x(t)\; =\; e^\{\backslash mathbf\; At\}\backslash mathbf\; x(0)\; +\; \backslash int\_0^t\; e^\{\backslash mathbf\; A(t-\backslash tau)\}\; \backslash mathbf\; B\backslash mathbf\; u(\backslash tau)\; d\; \backslash tau$

which is an analytical solution to the continuous model.

Now we want to discretise the above expression. We assume that u is constant during each timestep.

$\backslash mathbf\; x*\backslash equiv\; \backslash mathbf\; x(kT)$

$\backslash mathbf\; x*=\; e^\{\backslash mathbf\; AkT\}\backslash mathbf\; x(0)\; +\; \backslash int\_0^\{kT\}\; e^\{\backslash mathbf\; A(kT-\backslash tau)\}\; \backslash mathbf\; B\backslash mathbf\; u(\backslash tau)\; d\; \backslash tau$

$\backslash mathbf\; x*=\; e^\{\backslash mathbf\; A(k+1)T\}\backslash mathbf\; x(0)\; +\; \backslash int\_0^\{(k+1)T\}\; e^\{\backslash mathbf\; A((k+1)T-\backslash tau)\}\; \backslash mathbf\; B\backslash mathbf\; u(\backslash tau)\; d\; \backslash tau$

$\backslash mathbf\; x=\; e^\{\backslash mathbf\; AT\}\; \backslash left[\; e^\{\backslash mathbf\; AkT\}\backslash mathbf\; x(0)\; +\; \backslash int\_0^\{kT\}\; e^\{\backslash mathbf\; A(kT-\backslash tau)\}\; \backslash mathbf\; B\backslash mathbf\; u(\backslash tau)\; d\; \backslash tau\; \backslash right+\; \backslash int\_\{kT\}^\{(k+1)T\}\; e^\{\backslash mathbf\; A(kT+T-\backslash tau)\}\; \backslash mathbf\; B\backslash mathbf\; u(\backslash tau)\; d\; \backslash tau$

We recognize the bracketed expression as $\backslash mathbf\; x*$, and the second term can be simplified by substituting $v\; =\; kT\; +\; T\; -\; \backslash tau$. We also assume that $\backslash mathbf\; u$ is constant during the integral, which in turn yields

which is an exact solution to the discretization problem.

Approximations

Exact discretization may sometimes be intractable due to the heavy matrix exponential and integral operations involved. It is much easier to calculate an approximate discrete model, based on that for small timesteps $e^\{\backslash mathbf\; AT\}\; \backslash approx\; \backslash mathbf\; I\; +\; \backslash mathbf\; A\; T$. The approximate solution then becomes:

which can further be approximated if $\backslash frac\{1\}\{2\}\; \backslash mathbf\; A\; T^2$ is small; yielding

Control theory | Numerical analysis

Diskretisierung