Control System Toolbox™

d2c

Convert from discrete- to continuous-time models

Syntax

sysc = d2c(sysd)
sysc = d2c(sysd,method)

Description

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

sysc = d2c(sysd,method) gives access to alternative conversion schemes. The string method selects the conversion method among the following:

See Continuous/Discrete Conversions of LTI Models for more details on the conversion methods.

Example

Consider the discrete-time model with transfer function

and sample time second. You can derive a continuous-time zero-order-hold equivalent model by typing

Hc = d2c(H)

Discretizing the resulting model Hc with the zero-order hold method (this is the default method) and sampling period gives back the original discrete model . To see this, type

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 .

Algorithm

The 'zoh' conversion is performed 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 and is ill-conditioned for systems with poles near .

The zero-order hold method cannot handle systems with poles at . In addition, the 'zoh' conversion increases the model order for systems with negative real poles, [2]. This is necessary because the matrix logarithm maps real negative poles to complex poles. As a result, a discrete model with a single pole at would be transformed to a continuous model with a single complex pole at . Such a model is not meaningful because of its complex time response.

To ensure that all complex poles of the continuous model come in conjugate pairs, d2c replaces negative real poles with a pair of complex conjugate poles near . The conversion then yields a continuous model with higher order. For example, the discrete model with transfer function

and sample time 0.1 second is converted by typing

Ts = 0.1
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)

Convert Hc back to discrete time by typing

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 after canceling the pole/zero pair at .

References

[1] Franklin, G.F., J.D. Powell, and M.L. Workman, Digital Control of Dynamic Systems, Second Edition, Addison-Wesley, 1990.

[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, d2d, logm

ctrlpref

d2d