Convert model from discrete to continuous time
sysc = d2c(sysd)
sysc = d2c(sysd,method)
sysc = d2c(sysd,opts)
[sysc,G] = d2c(sysd,method,opts)
Discrete-time dynamic system model
You cannot directly use an idgrey model with FcnType='d' with d2c. Convert the model into idss form first.
String specifying a discrete-to-continuous time conversion method:
Discrete-to-continuous time conversion options, created using d2cOptions.
Continuous-time model of the same type as the input system sysd.
When sysd is an identified (IDLTI) model, sysc:
Matrix mapping the states xd[k] of the state-space model sysd to the states xc(t) of sysc:
Given an initial condition x0 for sysd and an initial input u0 = u, the corresponding initial condition for sysc (assuming u[k] = 0 for k < 0 is given by:
Consider the following discrete-time transfer function:
Suppose the model has sample time Ts = 0.1 s. You can derive a continuous-time zero-order-hold equivalent model with the following commands:
H = tf([1 -1], [1 1 0.3], 0.1); Hc = d2c(H)
Hc = 121.7 s + 3.026e-12 --------------------- s^2 + 12.04 s + 776.7 Continuous-time transfer function.
Discretizing the resulting model Hc with the default zero-order hold method and sampling time Ts = 0.1s returns the original discrete model H(z):
ans = z - 1 ------------- z^2 + z + 0.3 Sample time: 0.1 seconds Discrete-time transfer function.
To use the Tustin approximation instead of zero-order hold, type
Hc = d2c(H,'tustin');
As with zero-order hold, the inverse discretization operation
gives back the original H(z).
Convert an identified transfer function and compare its performance against a directly estimated continuous-time model.
load iddata1 sys1d = tfest(z1,2,'Ts',0.1); sys1c = d2c(sys1d,'zoh'); sys2c = tfest(z1,2); compare(z1,sys1c,sys2c)
The two systems are virtually identical.
Analyze the effect of parameter uncertainty on frequency response across d2c operation on an identified model.
load iddata1 sysd = tfest(z1, 2, 'Ts', 0.1); sysc = d2c(sysd, 'zoh');
sys1c has no covariance information. Regenerate it using a zero iteration update with the same estimation command and estimation data:
opt = tfestOptions; opt.SearchOption.MaxIter = 0; sys1c = tfest(z1, sysc, opt); h = bodeplot(sysd, sysc); showConfidence(h)
The uncertainties of sysc and sysd are comparable up to the Nyquist frequency. However, sysc 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 translatecov which is a Gauss-approximation formula based translation of covariance across model type conversion operations.
The Tustin approximation is not defined for systems with poles at z = –1 and is ill-conditioned for systems with poles near z = –1.
The zero-order hold method cannot handle systems with poles at z = 0. In addition, the 'zoh' conversion increases the model order for systems with negative real poles, . The model order increases because the matrix logarithm maps real negative poles to complex poles. Single complex poles are not physically meaningful because of their complex time response.
Instead, to ensure that all complex poles of the continuous model come in conjugate pairs, d2c replaces negative real poles z = –α with a pair of complex conjugate poles near –α. The conversion then yields a continuous model with higher order. For example, to convert the discrete-time transfer function
Ts = 0.1 % sample time 0.1 s H = zpk(-0.2,-0.5,1,Ts) * tf(1,[1 1 0.4],Ts) Hc = d2c(H)
These commands produce the following result.
Warning: System order was increased to handle real negative poles. Zero/pole/gain: -33.6556 (s-6.273) (s^2 + 28.29s + 1041) -------------------------------------------- (s^2 + 9.163s + 637.3) (s^2 + 13.86s + 1035)
To convert Hc back to discrete time, type:
Zero/pole/gain: (z+0.5) (z+0.2) ------------------------- (z+0.5)^2 (z^2 + z + 0.4) Sampling time: 0.1
This discrete model coincides with H(z) after canceling the pole/zero pair at z = –0.5.
Use the syntax sysc = d2c(sysd,'method') to convert sysd using the default options for'method'. To specify tustin conversion with a frequency prewarp (formerly the 'prewarp' method), use the syntax sysc = d2c(sysd,opts). See the d2cOptions reference page for more information.
 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.