Convert model from discrete to continuous time
Create the following discrete-time transfer function:
H = tf([1 -1],[1 1 0.3],0.1);
The sample time of the model is .
Derive a continuous-time, zero-order-hold equivalent model.
Hc = d2c(H)
Hc = 121.7 s - 8.407e-13 --------------------- s^2 + 12.04 s + 776.7 Continuous-time transfer function.
Discretize the resulting model,
Hc, with the default zero-order hold method and sample time 0.1s to return the original discrete model,
ans = z - 1 ------------- z^2 + z + 0.3 Sample time: 0.1 seconds Discrete-time transfer function.
Use the Tustin approximation method to convert
H to a continuous time model.
Hc2 = d2c(H,'tustin');
Discretize the resulting model, Hc2, to get back the original discrete-time model,
Estimate a discrete-time transfer function model, and convert it to a continuous-time model.
load iddata1 sys1d = tfest(z1,2,'Ts',0.1); sys1c = d2c(sys1d,'zoh');
Estimate a continuous-time transfer function model.
sys2c = tfest(z1,2);
Compare the response of
sys1c and the directly estimated continuous-time model,
The two systems are almost identical.
Convert an identified discrete-time transfer function model to continuous-time.
load iddata1 sysd = tfest(z1,2,'Ts',0.1); sysc = d2c(sysd,'zoh');
sys1c has no covariance information. The
d2c operation leads to loss of covariance data of identified models.
Regenerate the covariance information using a zero iteration update with the same estimation command and estimation data.
opt = tfestOptions; opt.SearchOptions.MaxIterations = 0; sys1c = tfest(z1,sysc,opt);
Analyze the effect on frequency-response uncertainty.
h = bodeplot(sysd,sys1c); showConfidence(h,3)
The uncertainties of
sysd are comparable up to the Nyquist frequency. However,
sys1c exhibits large uncertainty in the frequency range for which the estimation data does not provide any information.
If you do not have access to the estimation data, use the
translatecov command which is a Gauss-approximation formula based translation of covariance across model type conversion operations.
method— Discrete-to-continuous time conversion method
Discrete-to-continuous time conversion method, specified as one of the following values:
'zoh' — Zero-order hold on the inputs. Assumes
that the control inputs are piecewise constant over the sampling
'foh' — Linear interpolation of the inputs
(modified first-order hold). Assumes that the control inputs are
piecewise linear over the sampling period.
'tustin' — Bilinear (Tustin) approximation to
the derivative. To specify this method with frequency prewarping
(formerly known as the
'prewarp' method), use the
PrewarpFrequency option of
'matched' — Zero-pole matching method (for SISO
systems only). See
For information about the algorithms for each
conversion method, see Continuous-Discrete Conversion Methods.
opts— Discrete-to-continuous time conversion options
Discrete-to-continuous time conversion options, created using
d2cOptions. For example,
specify the prewarp frequency or the conversion method as an option.
sysc— Continuous-time model
Continuous-time model, returned as a dynamic system model of the same type
as the input system
sysd is an identified (IDLTI) model,
Includes both the measured and noise components of
sysd. If the noise variance is
sysd, then the
sysc has an indicated level
of noise spectral density equal to Ts*λ.
Does not include the estimated parameter covariance of
sysd. If you want to translate the covariance
while converting the model, use
translatecov (System Identification Toolbox).
G— Mapping of discrete-time states of state-space model to continuous-time states
Mapping of the states
xd[k] of the state-space model
sysd to the states
sysc, returned as a matrix. The mapping of the states
is as follows:
Given an initial condition
sysd and an initial input
u, the corresponding initial condition for
u[k] = 0 for
k < 0 is:
 Franklin, G.F., Powell,D.J., and Workman, M.L., Digital Control of Dynamic Systems (3rd Edition), Prentice Hall, 1997.
 Kollár, I., G.F. Franklin, and R. Pintelon, "On the Equivalence of z-domain and s-domain Models in System Identification," Proceedings of the IEEE® Instrumentation and Measurement Technology Conference, Brussels, Belgium, June, 1996, Vol. 1, pp. 14-19.