Transforming Between Discrete-Time and Continuous-Time Representations
Why Transform Between Continuous and Discrete Time?
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
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
These commands do not transform the estimated model uncertainty. If you want to
translate the estimated parameter covariance during the conversion, use
approximate the transformation of the noise model only when the sample time
T is small compared to the bandwidth of the
Using the c2d, d2c, and d2d Commands
The following table summarizes the commands for transforming between continuous-time and discrete-time model representations.
Converts continuous-time models to discrete-time models.
You cannot use
To transform a continuous-time model
mod_d = c2d(mod_c,T)
Converts parametric discrete-time models to continuous-time models.
You cannot use
To transform a discrete-time model
mod_c = d2c(mod_d)
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
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)
Specifying Intersample Behavior
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
foh). The transformation formulas for
d2c are affected by the
intersample behavior of the input.
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
% Set first-order hold intersample behavior mod_d = c2d(mod_c,T,'foh')
Effects on the Noise Model
d2d change the sample time of both the dynamic model
and the noise model. Resampling a model affects the variance of its noise
A parametric noise model is a time-series model with the following mathematical description:
The noise spectrum is computed by the following discrete-time equation:
where is the variance of the white noise e(t), and represents the spectral density of e(t). Resampling the noise model preserves the spectral density T . The spectral density 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
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 , where Tnew is the
new sample time and Told is the
original sample time. before applying