Convert model from continuous to discrete time
sysd = c2d(sys,Ts,method)
sysd = c2d(sys,Ts,opts)
[sysd,G] = c2d(sys,Ts,method)
[sysd,G] = c2d(sys,Ts,opts)
[sysd,G] = c2d(sys,Ts,method) returns a matrix, G that maps the continuous initial conditions x0 and u0 of the state-space model sys to the discrete-time initial state vector x . method is optional. To specify additional discretization options, use [sysd,G] = c2d(sys,Ts,opts).
Continuous-time dynamic system model (except frequency response data models). sys can represent a SISO or MIMO system, except that the 'matched' discretization method supports SISO systems only.
sys can have input/output or internal time delays; however, the 'matched' and 'impulse' methods do not support state-space models with internal time delays.
The following identified linear systems cannot be discretized directly:
For the syntax [sysd,G] = c2d(sys,Ts,opts), sys must be a state-space model.
String specifying a discretization method:
For more information about discretization methods, see Continuous-Discrete Conversion Methods.
Discretization options. Create opts using c2dOptions.
Discrete-time model of the same type as the input system sys.
When sys is an identified (IDLTI) model, sysd:
Matrix relating continuous-time initial conditions x0 and u0 of the state-space model sys to the discrete-time initial state vector x , 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.
Discretize the continuous-time transfer function:
with input delay Td = 0.35 second. To discretize this system using the triangle (first-order hold) approximation with sample time Ts = 0.1 second, type
H = tf([1 -1], [1 4 5], 'inputdelay', 0.35); Hd = c2d(H, 0.1, 'foh'); % discretize with FOH method and % 0.1 second sample time Transfer function: 0.0115 z^3 + 0.0456 z^2 - 0.0562 z - 0.009104 --------------------------------------------- z^6 - 1.629 z^5 + 0.6703 z^4 Sampling time: 0.1
The next command compares the continuous and discretized step responses.
Discretize the 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); % create transfer function hd = c2d(h, 0.1) % zoh is the default method
These commands produce the discrete-time transfer function
Transfer function: 0.01187 z^2 + 0.06408 z + 0.009721 z^(-3) * ---------------------------------- z^2 - 1.655 z + 0.7408 Sampling time: 0.1
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 and discretized models using
Discretize a state-space model with time delay, using a Thiran filter to model fractional delays:
sys = ss(tf([1, 2], [1, 4, 2])); % create a state-space model sys.InputDelay = 2.7 % add input delay
This command creates a continuous-time state-space model with two states, as the output shows:
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 (listed by channel): 2.7 Continuous-time model.
Use c2dOptions to create a set of discretization options, and discretize the model. This example uses the Tustin discretization method.
opt = c2dOptions('Method', 'tustin', 'FractDelayApproxOrder', 3); sysd1 = c2d(sys, 1, opt) % 1s sampling time
These commands yield the result
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 Sampling time: 1 Discrete-time 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 sampling time is 2.7, the third-order Thiran filter (FractDelayApproxOrder = 3) can approximate the entire time delay.
Discretize an identified, continuous-time transfer function and compare its performance against a directly estimated discrete-time 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 two models.
compare(z1, sys1d, sys2d)
The two systems are virtually identical.
Discretize an identified state-space model to build a one-step ahead predictor of its response.
load iddata2 sysc = ssest(z2, 4); sysd = c2d(sysc, 0.1, 'zoh'); [A,B,C,D,K] = idssdata(sysd); Predictor = ss(A-K*C, [K B-K*D], C, [0 D], 0.1);
The Predictor is a two input model which uses the measured output and input signals ([z1.y z1.u]) to compute the 1-steap predicted response of sysc.
Use the syntax sysd = c2d(sys,Ts,method) to discretize sys using the default options for method. To specify additional discretization options, use the syntax sysd = c2d(sys,Ts,opts).
To specify the tustin method with frequency prewarping (formerly known as the 'prewarp' method), use the PrewarpFrequency option of c2dOptions.
For information about the algorithms for each c2d conversion method, see Continuous-Discrete Conversion Methods.