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:
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 = 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 idtf
, idss
, or idpoly
.
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 whoseFunctionType
is'c'
. Convert toidss
model first.idproc
models. Convert toidtf
oridpoly
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 timeTs
.'foh'
— Triangle approximation (modified first-order hold). Assumes the control inputs are piecewise linear over the sample timeTs
.'impulse'
— Impulse invariant discretization'tustin'
— Bilinear (Tustin) method. To specify this method with frequency prewarping (formerly known as the'prewarp'
method), use thePrewarpFrequency
option ofc2dOptions
.'matched'
— Zero-pole matching method'least-squares'
— Least-squares method'damped'
— Damped Tustin approximation based on theTRBDF2
formula forsparss
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 modelsysc
, stored in itsNoiseVariance
property, is interpreted as the intensity of the spectral density of the noise spectrum. The noise variance insysd
is thus λ/Ts.Does not include the estimated parameter covariance of
sysc
. If you want to translate the covariance while discretizing the model, usetranslatecov
.
G
— Mapping of continuous initial conditions of state-space model to discrete-time initial state vector
matrix
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:
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 R2006aR2024a: 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
(Control System Toolbox) | translatecov
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