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:

`'zoh'`	Zero-order hold on the inputs. The control inputs are assumed piecewise constant over the sampling period.
`'tustin'`	Bilinear (Tustin) approximation to the derivative.
`'prewarp'`	Tustin approximation with frequency prewarping.
`'matched'`	Matched pole-zero method of [1] (for SISO systems only).

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.