d2c (Function Reference)

Convert discrete-time LTI models to continuous time

Syntax

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

Description

d2c converts LTI models from discrete to continuous time using one of the following conversion methods:

'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).

`'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).

The string method specifies the conversion method. If method is omitted then zero-order hold ('zoh') is assumed. 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)

MATLAB responds with

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 .

See Also
c2d Continuous- to discrete-time conversion

d2d Resampling of discrete models

logm Matrix logarithm

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.

ctrbf d2d