The c2d
command discretizes continuoustime models. Conversely,
d2c
converts discretetime models to continuous time. Both
commands support several discretization and interpolation methods, as shown in the
following table.
Discretization Method  Use When 

ZeroOrder Hold  You want an exact discretization in the time domain for staircase inputs. 
FirstOrder Hold  You want an exact discretization in the time domain for piecewise linear inputs. 
ImpulseInvariant Mapping
(c2d only)  You want an exact discretization in the time domain for impulse train inputs. 
Tustin Approximation 

ZeroPole Matching Equivalents 

Least Squares (Control System Toolbox)
(c2d only) 

The ZeroOrder Hold (ZOH) method provides an exact match between the continuous and discretetime systems in the time domain for staircase inputs.
The following block diagram illustrates the zeroorderhold discretization H_{d}(z) of a continuoustime linear model H(s).
The ZOH block generates the continuoustime input signal u(t) by holding each sample value u(k) constant over one sample period:
$$u\left(t\right)=u\left[k\right],\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}k{T}_{s}\le t\le \left(k+1\right){T}_{s}$$
The signal u(t) is the input to the continuous system H(s). The output y[k] results from sampling y(t) every T_{s} seconds.
Conversely, given a discrete system H_{d}(z), d2c
produces a continuous system H(s). The ZOH discretization of H(s) coincides with H_{d}(z).
The ZOH discretetocontinuous conversion has the following limitations:
d2c
cannot convert LTI models with poles at z = 0.
For discretetime LTI models having negative real poles, ZOH
d2c
conversion produces a continuous system with higher
order. The model order increases because a negative real pole in the
z domain maps to a pure imaginary value in the
s domain. Such mapping results in a continuoustime model
with complex data. To avoid this issue, the software instead introduces a
conjugate pair of complex poles in the s domain.
You can use the ZOH method to discretize SISO or MIMO continuoustime models with time delays. The ZOH method yields an exact discretization for systems with input delays, output delays, or transport delays.
For systems with internal delays (delays in feedback loops), the ZOH method results in approximate discretizations. The following figure illustrates a system with an internal delay.
For such systems, c2d
performs the following actions to
compute an approximate ZOH discretization:
Decomposes the delay τ as $$\tau =k{T}_{s}+\rho $$ with $$0\le \rho <{T}_{s}$$.
Absorbs the fractional delay $$\rho $$ into H(s).
Discretizes H(s) to H(z).
Represents the integer portion of the delay kT_{s} as an internal discretetime delay z^{–k}. The final discretized model appears in the following figure:
The FirstOrder Hold (FOH) method provides an exact match between the continuous and discretetime systems in the time domain for piecewise linear inputs.
FOH differs from ZOH by the underlying hold mechanism. To turn the input samples u[k] into a continuous input u(t), FOH uses linear interpolation between samples:
$$u\left(t\right)=u\left[k\right]+\frac{tk{T}_{s}}{{T}_{s}}\left(u\left[k+1\right]u\left[k\right]\right),\text{\hspace{1em}}k{T}_{s}\le t\le \left(k+1\right){T}_{s}$$
In general, this method is more accurate than ZOH for systems driven by smooth inputs.
This FOH method differs from standard causal FOH and is more appropriately called triangle approximation (see [2], p. 228). The method is also known as rampinvariant approximation.
You can use the FOH method to discretize SISO or MIMO continuoustime models with time delays. The FOH method handles time delays in the same way as the ZOH method. See ZOH Method for Systems with Time Delays.
The impulseinvariant mapping produces a discretetime model with the same impulse response as the continuous time system. For example, compare the impulse response of a firstorder continuous system with the impulseinvariant discretization:
G = tf(1,[1,1]);
Gd1 = c2d(G,0.01,'impulse');
impulse(G,Gd1)
The impulse response plot shows that the impulse responses of the continuous and discretized systems match.
You can use impulseinvariant mapping to discretize SISO or MIMO continuoustime
models with time delays, except that the method does not support
ss
models with internal delays. For supported models,
impulseinvariant mapping yields an exact discretization of the time delay.
The Tustin or bilinear approximation yields the best frequencydomain match between the continuoustime and discretized systems. This method relates the sdomain and zdomain transfer functions using the approximation:
$$z={e}^{s{T}_{s}}\approx \frac{1+s{T}_{s}/2}{1s{T}_{s}/2}.$$
In c2d
conversions, the discretization H_{d}(z) of a continuous transfer function H(s) is:
$${H}_{d}\left(z\right)=H\left({s}^{\prime}\right),\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}{s}^{\prime}=\frac{2}{{T}_{s}}\frac{z1}{z+1}$$
Similarly, the d2c
conversion relies on the inverse
correspondence
$$H\left(s\right)={H}_{d}\left({z}^{\prime}\right),\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}{z}^{\prime}=\frac{1+s{T}_{s}/2}{1s{T}_{s}/2}$$
When you convert a statespace model using the Tustin method, the states are not preserved. The state transformation depends upon the statespace matrices and whether the system has time delays. For example, for an explicit (E = I) continuoustime model with no time delays, the state vector w[k] of the discretized model is related to the continuoustime state vector x(t) by:
$$w\left[k{T}_{s}\right]=\left(IA\frac{{T}_{s}}{2}\right)x\left(k{T}_{s}\right)\frac{{T}_{s}}{2}Bu\left(k{T}_{s}\right)=x\left(k{T}_{s}\right)\frac{{T}_{s}}{2}\left(Ax\left(k{T}_{s}\right)+Bu\left(k{T}_{s}\right)\right).$$
T_{s} is the sample time of the discretetime model. A and B are statespace matrices of the continuoustime model.
If your system has important dynamics at a particular frequency that you want the transformation to preserve, you can use the Tustin method with frequency prewarping. This method ensures a match between the continuous and discretetime responses at the prewarp frequency.
The Tustin approximation with frequency prewarping uses the following transformation of variables:
$${H}_{d}\left(z\right)=H\left({s}^{\prime}\right),\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}{s}^{\prime}=\frac{\omega}{\mathrm{tan}\left(\omega {T}_{s}/2\right)}\frac{z1}{z+1}$$
This change of variable ensures the matching of the continuous and discretetime frequency responses at the prewarp frequency ω, because of the following correspondence:
$$H\left(j\omega \right)={H}_{d}\left({e}^{j\omega {T}_{s}}\right)$$
You can use the Tustin approximation to discretize SISO or MIMO continuoustime models with time delays.
By default, the Tustin method rounds any time delay to the nearest multiple of the
sample time. Therefore, for any time delay tau
, the integer
portion of the delay, k*Ts
, maps to a delay of
k
sampling periods in the discretized model. This approach
ignores the residual fractional delay, tau

k*Ts
.
You can to approximate the fractional portion of the delay by a discrete allpass
filter (Thiran filter) of specified order. To do so, use the
FractDelayApproxOrder
option of
c2dOptions
.
To understand how the Tustin method handles systems with time delays, consider the following SISO statespace model G(s). The model has input delay τ_{i}, output delay τ_{o}, and internal delay τ.
The following figure shows the general result of discretizing G(s) using the Tustin method.
By default, c2d
converts the time delays to pure integer time
delays. The c2d
command computes the integer delays by rounding
each time delay to the nearest multiple of the sample time
T_{s}. Thus, in the default case, m_{i} =
round(τ_{i}/T_{s}),
m_{o} =
round
(τ_{o}/T_{s}),
and m =
round
(τ/T_{s}).. Also in this case,
F_{i}(z) =
F_{o}(z) =
F(z) = 1.
If you set FractDelayApproxOrder
to a nonzero value,
c2d
approximates the fractional portion of the time delays
by Thiran filters
F_{i}(z),
F_{o}(z), and
F(z).
The Thiran filters add additional states to the model. The maximum number of
additional states for each delay is FractDelayApproxOrder
.
For example, for the input delay τ_{i}, the order of the Thiran filter F_{i}(z) is:
order
(F_{i}(z))
=
max
(ceil
(τ_{i}/T_{s}),
FractDelayApproxOrder
).
If
ceil
(τ_{i}/T_{s})
< FractDelayApproxOrder
, the Thiran filter
F_{i}(z)
approximates the entire input delay τ_{i}. If
ceil
(τ_{i}/T_{s})
> FractDelayApproxOrder
, the Thiran filter only approximates a
portion of the input delay. In that case, c2d
represents the
remainder of the input delay as a chain of unit delays
z^{–mi},
where
m_{i} =
ceil
(τ_{i}/T_{s})
– FractDelayApproxOrder
c2d
uses Thiran filters and
FractDelayApproxOrder
in a similar way to approximate the
output delay τ_{o} and the internal delay
τ.
When you discretizetf
and zpk
models
using the Tustin method, c2d
first aggregates all input,
output, and transport delays into a single transport delay
τ_{TOT} for each channel.
c2d
then approximates
τ_{TOT} as a Thiran filter and a chain of
unit delays in the same way as described for each of the time delays in
ss
models.
For more information about Thiran filters, see the thiran
reference page and [4].
This method of conversion, which computes zeropole matching equivalents, applies only to SISO systems. The continuous and discretized systems have matching DC gains. Their poles and zeros are related by the transformation:
$${z}_{i}={e}^{{s}_{i}{T}_{s}}$$
where:
z_{i} is the ith pole or zero of the discretetime system.
s_{i} is the ith pole or zero of the continuoustime system.
T_{s} is the sample time.
See [2] for more information.
You can use zeropole matching to discretize SISO continuoustime models with time
delay, except that the method does not support ss
models with
internal delays. The zeropole matching method handles time delays in the same way
as the Tustin approximation. See Tustin Approximation for Systems with Time Delays.
The least squares method minimizes the error between the frequency responses of the continuoustime and discretetime systems up to the Nyquist frequency using a vectorfitting optimization approach. This method is useful when you want to capture fast system dynamics but must use a larger sample time, for example, when computational resources are limited.
This method is supported only by the c2d
function and only for SISO
systems.
As with Tustin approximation and zeropole matching, the least squares method provides a good match between the frequency responses of the original continuoustime system and the converted discretetime system. However, when using the least squares method with:
The same sample time as Tustin approximation or zeropole matching, you get a smaller difference between the continuoustime and discretetime frequency responses.
A lower sample time than what you would use with Tustin approximation or zeropole matching, you can still get a result that meets your requirements. Doing so is useful if computational resources are limited, since the slower sample time means that the processor must do less work.
[1] Åström, K.J. and B. Wittenmark, ComputerControlled Systems: Theory and Design, PrenticeHall, 1990, pp. 4852.
[2] Franklin, G.F., Powell, D.J., and Workman, M.L., Digital Control of Dynamic Systems (3rd Edition), Prentice Hall, 1997.
[3] Smith, J.O. III, "Impulse Invariant Method",
Physical Audio Signal Processing, August 2007. https://www.dsprelated.com/dspbooks/pasp/Impulse_Invariant_Method.html
.
[4] T. Laakso, V. Valimaki, "Splitting the Unit Delay", IEEE Signal Processing Magazine, Vol. 13, No. 1, p.3060, 1996.
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