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# d2c

Convert model from discrete to continuous time

## Syntax

sysc = d2c(sysd)
sysc = d2c(sysd,method)
sysc = d2c(sysd,opts)
[sysc,G] = d2c(sysd,method,opts)

## 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 opts, specified using the d2cOptions command.

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

## Input Arguments

 sysd Discrete-time dynamic system model You cannot directly use an idgrey model with FcnType='d' with d2c. Convert the model into idss form first. method String specifying a discrete-to-continuous time conversion method: 'zoh' — Zero-order hold on the inputs. Assumes the control inputs are piecewise constant over the sampling period.'foh' — Linear interpolation of the inputs (modified first-order hold). Assumes the control inputs are piecewise linear over the sampling period.'tustin' — Bilinear (Tustin) approximation to the derivative.'matched' — Zero-pole matching method of [1] (for SISO systems only). Default: 'zoh' opts Discrete-to-continuous time conversion options, created using d2cOptions.

## Output Arguments

 sysc Continuous-time model of the same type as the input system sysd. When sysd is an identified (IDLTI) model, sysc: Includes both the measured and noise components of sysd. 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.Does not include the estimated parameter covariance of sysd. If you want to translate the covariance while converting the model, use translatecov. G Matrix mapping the states xd[k] of the state-space model sysd to the states xc(t) of sysc: ${x}_{c}\left(k{T}_{s}\right)=G\left[\begin{array}{c}{x}_{d}\left[k\right]\\ u\left[k\right]\end{array}\right].$ 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: ${x}_{c}\left(0\right)=G\left[\begin{array}{c}{x}_{0}\\ {u}_{0}\end{array}\right].$

## Examples

### Example 1

Consider the following discrete-time transfer function:

$H\left(z\right)=\frac{z-1}{{z}^{2}+z+0.3}$

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):

```c2d(Hc,0.1)
```
```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

```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.

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.

## 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

$H\left(z\right)=\frac{z+0.2}{\left(z+0.5\right)\left({z}^{2}+z+0.4\right)}$

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.

expand all

### 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.

### Algorithms

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

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

## 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.