# c2d

Convert model from continuous to discrete time

## Description

discretizes the continuous-time dynamic system
model
`sysd`

= c2d(`sysc`

,`Ts`

)`sysc`

using zero-order hold on the inputs and a sample time of
`Ts`

.

## Examples

### Discretize a Transfer Function

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 Model with Fractional Delay Absorbed into Coefficients

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,'-')

### Discretize Model with Approximated Fractional Delay

*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.

### Discretize Identified Model

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

load iddata1 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.

### Build Predictor Model

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.

```
load iddata2
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

`sysc`

— Continuous-time dynamic system

dynamic system model

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.

`Ts`

— Sample time

positive scalar

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.

`method`

— Discretization method

`'zoh'`

(default) | `'foh'`

| `'impulse'`

| `'tustin'`

| `'matched'`

| `'least-squares'`

| `'damped'`

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.

`opts`

— Discretization options

`c2dOptions`

object

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

`sysd`

— Discrete-time model

dynamic system model

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`NoiseVariance`

property, 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`

.

`G`

— Mapping of continuous initial conditions of state-space model to discrete-time initial state vector

matrix

Mapping of continuous-time initial conditions *x*_{0} and *u*_{0} 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:

$$x\left[\text{\hspace{0.05em}}0\right]=G\cdot \left[\begin{array}{c}{x}_{0}\\ {u}_{0}\end{array}\right]$$

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**

### R2024a: Does not add extra states during conversion

The `c2d`

command no longer adds extra states when modeling
extra delays. By default, `c2d`

now uses internal delays and
returns the discretized model with the same number of states. This results in a more
predictable behavior and simplifies initial state mapping from continuous to
discrete time.

To revert to the old behavior before R2024a, create an option set using
`c2dOptions`

and set `DelayModeling`

to
`"state"`

.

### R2024a: Support for `mechss`

models

You can now discretize `mechss`

models using the bilinear Tustin
approximation method. For `mechss`

models, the
`'tustin'`

method computes the second-order form of the Tustin
discretization. This is equivalent to applying Tustin to the first-order
`sparss`

equivalent of the `mechss`

model.

### R2024a: Support for models with offsets and other improvements

You can now use `c2d`

to discretize models with offsets such as
gridded linear parameter varying models.

Additionally, `c2dOptions`

provides two new
options

`DelayModeling`

— Specify whether to model extra delays as internal delays (default) or additional states.`Consistency`

— Enforce state and delay consistency in state-space arrays. This option is helpful when discretizing gridded LPV or LTV models.

## See Also

`c2dOptions`

| `d2c`

| `d2d`

| `thiran`

| `translatecov`

(System Identification Toolbox) | Convert Model Rate

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