d2c - Convert model from discrete to continuous time

Syntax

Description

sysc = d2c(sysd) produces a continuous-time model sysc that is equivalent to the discrete-time dynamic system model sysd using zero-order hold on the inputs.

sysc = d2c(sysd,method) uses the specified conversion method method:

sysc = d2c(sysd,opts) converts sysd using the option set specified with d2cOptions object.

See Continuous-Discrete Conversion Methods for more details on the conversion methods.

[sysc,G] = d2c(sysd,...), for a state-space model sysd, also returns the matrix G that maps the states xd[k] of sysd to the states xc(t) of sysc:

Given an initial condition x0 for sysd and an initial input u0 = u[0], the corresponding initial condition for sysc (assuming u[k] = 0 for k < 0 is given by:

Tips

• 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.

• The D2C operation on an identified model transforms both its measured and noise components. If the noise variance is λ in sysd, then the continuous-time model sysc has an indicated level of noise spectral density equal to Ts*λ.

• You cannot directly use an idgrey with FcnType='d' with d2c. Convert the model into idss form first.

• The d2c operation does not translate the covariance information of the original model sysd. Consequently sysc has no parameter covariance information.

Examples

Example 1

Consider the discrete-time model with transfer function

and sample time T_s = 0.1 s. You can derive a continuous-time zero-order-hold equivalent model by typing

Hc = d2c(H)

Discretizing the resulting model Hc with the default zero-order hold method and sampling time T_s = 0.1s returns the original discrete model H(z):

c2d(Hc,0.1)

To use the Tustin approximation instead of zero-order hold, type

Hc = d2c(H,'tustin')

As with zero-order hold, the inverse discretization operation

c2d(Hc,0.1,'tustin')

gives back the original H(z).

Example 2

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.

Example 3

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.

Algorithms

d2c performs the 'zoh' conversion in state space, and relies on the matrix logarithm (see logm in the MATLAB documentation).

Limitations

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, [2]. 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

type:

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:

c2d(Hc,Ts)

yielding

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.

References

[1] Franklin, G.F., Powell,D.J., and Workman, M.L., Digital Control of Dynamic Systems (3rd Edition), Prentice Hall, 1997..

[2] 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.

See Also

c2d | d2cOptions | d2d | logm

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