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Transforming between continuous-time and discrete-time representations is useful, for example, if you have estimated a discrete-time linear model and require a continuous-time model instead for your application.

You can use `c2d`

and `d2c`

to transform any linear identified
model between continuous-time and discrete-time representations. `d2d`

is useful is you want to change the
sample time of a discrete-time model. All of these operations change
the sample time, which is called *resampling* the
model.

These commands do not transform the estimated model uncertainty.
If you want to translate the estimated parameter covariance during
the conversion, use `translatecov`

.

`c2d`

and `d2d`

correctly
approximate the transformation of the noise model only when the sample
time `T`

is small compared to the bandwidth of the
noise.

The following table summarizes the commands for transforming between continuous-time and discrete-time model representations.

Command | Description | Usage Example |
---|---|---|

`c2d` |
Converts continuous-time models to discrete-time models. You cannot use | To transform a continuous-time model mod_d = c2d(mod_c,T) where |

`d2c` |
Converts parametric discrete-time models to continuous-time models. You cannot use |
To transform a discrete-time model
mod_c = d2c(mod_d) |

`d2d` |
Resample a linear discrete-time model and produce an equivalent discrete-time model with a new sample time. You can use the resampled model to simulate or predict output with a specified time interval. | To resample a discrete-time model mod_d2 = d2d(mod_d1,Ts) |

The following commands compare estimated
model `m`

and its continuous-time counterpart `mc`

on
a Bode plot:

% Estimate discrete-time ARMAX model % from the data m = armax(data,[2 3 1 2]); % Convert to continuous-time form mc = d2c(m); % Plot bode plot for both models bode(m,mc)

A sampled signal is characterized only by its values at the
sampling instants. However, when you apply a continuous-time input
to a continuous-time system, the output values at the sampling instants
depend on the inputs at the sampling instants and on the inputs between
these points. Thus, the `InterSample`

data property
describes how the algorithms should handle the input between samples.
For example, you can specify the behavior between the samples to be
piece-wise constant (zero-order hold, `zoh`

) or linearly
interpolated between the samples (first order hold, `foh`

).
The transformation formulas for `c2d`

and `d2c`

are
affected by the intersample behavior of the input.

By default, `c2d`

and `d2c`

use
the intersample behavior you assigned to the estimation data. To override
this setting during transformation, add an extra argument in the syntax.
For example:

% Set first-order hold intersample behavior mod_d = c2d(mod_c,T,'foh')

`c2d`

, `d2c`

, and `d2d`

change
the sample time of both the dynamic model and the noise model. Resampling
a model affects the variance of its noise model.

A parametric noise model is a time-series model with the following mathematical description:

$$\begin{array}{l}y(t)=H(q)e(t)\\ E{e}^{2}=\lambda \end{array}$$

The noise spectrum is computed by the following discrete-time equation:

$${\Phi}_{v}(\omega )=\lambda T{\left|H\left({e}^{i\omega T}\right)\right|}^{2}$$

where $$\lambda $$ is the variance of the white
noise *e(t)*, and $$\lambda T$$ represents
the spectral density of *e(t)*. Resampling the
noise model preserves the spectral density $$\lambda $$*T *. The
spectral density $$\lambda $$*T * is
invariant up to the Nyquist frequency. For more information about
spectrum normalization, see Spectrum Normalization.

`d2d`

resampling of the noise model affects
simulations with noise using `sim`

. If you resample
a model to a faster sampling rate, simulating this model results in
higher noise level. This higher noise level results from the underlying
continuous-time model being subject to continuous-time white noise
disturbances, which have infinite, instantaneous variance. In this
case, the *underlying continuous-time model* is
the unique representation for discrete-time models. To maintain the
same level of noise after interpolating the noise signal, scale the
noise spectrum by $$\sqrt{{\scriptscriptstyle \raisebox{1ex}{${T}_{New}$}\!\left/ \!\raisebox{-1ex}{${T}_{Old}$}\right.}}$$, where *T _{new}* is
the new sample time and

`sim`

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