# c2d

Convert model from continuous to discrete time

## Description

example

sysd = c2d(sysc,Ts) discretizes the continuous-time dynamic system model sysc using zero-order hold on the inputs and a sample time of Ts.

example

sysd = c2d(sysc,Ts,method) specifies the discretization method.

example

sysd = c2d(sysc,Ts,opts) specifies additional options for the discretization.

[sysd,G] = c2d(___), where sysc is a state-space model, returns a matrix, G that maps the continuous initial conditions x0 and u0 of the state-space model to the discrete-time initial state vector x[0].

## Examples

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Discretize the following continuous-time transfer function:

$H\left(s\right)={e}^{-0.3s}\frac{s-1}{{s}^{2}+4s+5}.$

This system has an input delay of 0.3 s. Discretize the system using the triangle (first-order-hold) approximation with sample time Ts = 0.1 s.

H = tf([1 -1],[1 4 5],'InputDelay', 0.3);
Hd = c2d(H,0.1,'foh');

Compare the step responses of the continuous-time and discretized systems.

step(H,'-',Hd,'--')

Discretize the following delayed transfer function using zero-order hold on the input, and a 10-Hz sampling rate.

$H\left(s\right)={e}^{-0.25s}\frac{10}{{s}^{2}+3s+10}.$

h = tf(10,[1 3 10],'IODelay',0.25);
hd = c2d(h,0.1)
hd =

0.01187 z^2 + 0.06408 z + 0.009721
z^(-3) * ----------------------------------
z^2 - 1.655 z + 0.7408

Sample time: 0.1 seconds
Discrete-time transfer function.

In this example, the discretized model hd has a delay of three sampling periods. The discretization algorithm absorbs the residual half-period delay into the coefficients of hd.

Compare the step responses of the continuous-time and discretized models.

step(h,'--',hd,'-')

Since R2024a

Create a continuous-time state-space model with two states and an input delay.

sys = ss(tf([1,2],[1,4,2]));
sys.InputDelay = 2.7
sys =

A =
x1  x2
x1  -4  -2
x2   1   0

B =
u1
x1   2
x2   0

C =
x1   x2
y1  0.5    1

D =
u1
y1   0

Input delays (seconds): 2.7

Continuous-time state-space model.

Discretize the model using the Tustin discretization method and a Thiran filter to model fractional delays. The sample time Ts = 1 second.

opt = c2dOptions('Method','tustin','ThiranOrder',3);
sysd1 = c2d(sys,1,opt)
sysd1 =

A =
x1       x2
x1  -0.4286  -0.5714
x2   0.2857   0.7143

B =
u1
x1  0.5714
x2  0.2857

C =
x1      x2
y1  0.2857  0.7143

D =
u1
y1  0.2857

(values computed with all internal delays set to zero)

Internal delays (sampling periods): 1  1  1

Sample time: 1 seconds
Discrete-time state-space model.

By default, the function models extra delays as internal delays in the discretized model. To model extra delays as states in the discretized model, set the DelayModeling option of c2dOptions to 'state'.

opt2 = c2dOptions('Method','tustin','ThiranOrder',3,'DelayModeling','state');
sysd2 = c2d(sys,1,opt2)
sysd2 =

A =
x1         x2         x3         x4         x5
x1    -0.4286    -0.5714   -0.00265    0.06954      2.286
x2     0.2857     0.7143  -0.001325    0.03477      1.143
x3          0          0    -0.2432     0.1449    -0.1153
x4          0          0       0.25          0          0
x5          0          0          0      0.125          0

B =
u1
x1  0.002058
x2  0.001029
x3         8
x4         0
x5         0

C =
x1         x2         x3         x4         x5
y1     0.2857     0.7143  -0.001325    0.03477      1.143

D =
u1
y1  0.001029

Sample time: 1 seconds
Discrete-time state-space model.

The discretized model now contains three additional states x3, x4, and x5 corresponding to a third-order Thiran filter. Since the time delay divided by the sample time is 2.7, the third-order Thiran filter ('ThiranOrder' = 3) can approximate the entire time delay.

Estimate a continuous-time transfer function, and discretize it.

sys1c = tfest(z1,2);
sys1d = c2d(sys1c,0.1,'zoh');

Estimate a second order discrete-time transfer function.

sys2d = tfest(z1,2,'Ts',0.1);

Compare the response of the discretized continuous-time transfer function model, sys1d, and the directly estimated discrete-time model, sys2d.

compare(z1,sys1d,sys2d)

The two systems are almost identical.

Discretize an identified state-space model to build a one-step ahead predictor of its response.

Create a continuous-time identified state-space model using estimation data.

sysc = ssest(z2,4);

Predict the 1-step ahead predicted response of sysc.

predict(sysc,z2)

Discretize the model.

sysd = c2d(sysc,0.1,'zoh');

Build a predictor model from the discretized model, sysd.

[A,B,C,D,K] = idssdata(sysd);
Predictor = ss(A-K*C,[K B-K*D],C,[0 D],0.1);

Predictor is a two-input model which uses the measured output and input signals ([z1.y z1.u]) to compute the 1-step predicted response of sysc.

Simulate the predictor model to get the same response as the predict command.

lsim(Predictor,[z2.y,z2.u])

The simulation of the predictor model gives the same response as predict(sysc,z2).

## Input Arguments

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Continuous-time model, specified as a dynamic system model such as tf, ss, or zpk. sysc cannot be a frequency response data model. sysc can be a SISO or MIMO system, except that the 'matched' discretization method supports SISO systems only.

sysc can have input/output or internal time delays; however, the 'matched', 'impulse', and 'least-squares' methods do not support state-space models with internal time delays.

The following identified linear systems cannot be discretized directly:

• idgrey models whose FunctionType is 'c'. Convert to idss model first.

• idproc models. Convert to idtf or idpoly model first.

Sample time, specified as a positive scalar that represents the sampling period of the resulting discrete-time system. Ts is in TimeUnit, which is the sysc.TimeUnit property.

Discretization method, specified as one of the following values:

• 'zoh' — Zero-order hold (default). Assumes the control inputs are piecewise constant over the sample time Ts.

• 'foh' — Triangle approximation (modified first-order hold). Assumes the control inputs are piecewise linear over the sample time Ts.

• 'impulse' — Impulse invariant discretization

• 'tustin' — Bilinear (Tustin) method. To specify this method with frequency prewarping (formerly known as the 'prewarp' method), use the PrewarpFrequency option of c2dOptions.

• 'matched' — Zero-pole matching method

• 'least-squares' — Least-squares method

• 'damped' — Damped Tustin approximation based on the TRBDF2 formula for sparss models only.

For information about the algorithms for each conversion method, see Continuous-Discrete Conversion Methods.

Discretization options, specified as a c2dOptions object. For example, specify the prewarp frequency, order of the Thiran filter or discretization method as an option.

## Output Arguments

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Discrete-time model, returned as a dynamic system model of the same type as the input system sysc.

When sysc is an identified (IDLTI) model, sysd:

• Includes both measured and noise components of sysc. The innovations variance λ of the continuous-time identified model sysc, stored in its NoiseVarianceproperty, is interpreted as the intensity of the spectral density of the noise spectrum. The noise variance in sysd is thus λ/Ts.

• Does not include the estimated parameter covariance of sysc. If you want to translate the covariance while discretizing the model, use translatecov.

Mapping of continuous-time initial conditions x0 and u0 of the state-space model sysc to the discrete-time initial state vector x[0], returned as a matrix. The mapping of initial conditions to the initial state vector is as follows:

$x\left[\text{ }0\right]=G\cdot \left[\begin{array}{c}{x}_{0}\\ {u}_{0}\end{array}\right]$

For state-space models with time delays, c2d pads the matrix G with zeroes to account for additional states introduced by discretizing those delays. See Continuous-Discrete Conversion Methods for a discussion of modeling time delays in discretized systems.

## Version History

Introduced before R2006a

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