Control System Toolbox™

c2d

Convert from continuous- to discrete-time models

Syntax

sysd = c2d(sys,Ts) sysd = c2d(sys,Ts,method)

Description

sysd = c2d(sys,Ts) discretizes the continuous-time LTI model sys using zero-order hold on the inputs and a sample time of Ts seconds.

sysd = c2d(sys,Ts,method) gives access to alternative discretization schemes. The string method selects the discretization method among the following:

`'zoh'`	Zero-order hold. The control inputs are assumed piecewise constant over the sampling period `Ts`.
`'foh'`	Triangle approximation (modified first-order hold, see [1], p. 151). The control inputs are assumed piecewise linear over the sampling period `Ts`.
`'imp'`	Impulse-invariant discretization
`'tustin'`	Bilinear (Tustin) approximation
`'prewarp'`	Tustin approximation with frequency prewarping. You must specify the critical frequency `Wc` (in rad/s) as a fourth input as in sysd = c2d(sysc,ts,'prewarp',Wc)
`'matched'`	Matched pole-zero method. See [1], p. 147.

Refer to Continuous/Discrete Conversions of LTI Models for more detail on these discretization methods.

c2d supports MIMO systems (except for the 'matched' method) as well as LTI models with delays with some restrictions for 'matched' and 'tustin' methods.

For state-space systems,

[sysd,G] = c2d(sys,Ts,method)

returns a matrix G that maps the continuous initial conditions and to their discrete counterparts and according to

Example

Consider the system

with input delay second. To discretize this system using the triangle approximation with sample time second, type

H = tf([1 -1],[1 4 5],'inputdelay',0.35)
Transfer function:
                   s - 1
exp(-0.35*s) * -------------
               s^2 + 4 s + 5
Hd = c2d(H,0.1,'foh')
Transfer function:
0.0115 z^3 + 0.0456 z^2 - 0.0562 z - 0.009104
---------------------------------------------
        z^6 - 1.629 z^5 + 0.6703 z^4

Sampling time: 0.1

The next command compares the continuous and discretized step responses.

step(H,'-',Hd,'--')

Algorithm

ZOH, FOH, and IMP Methods

Exact Discretization

For most LTI systems, the ZOH, FOH, and IMP methods produce exact discretizations in the time domain. In this context, exact means that the time responses of the continuous and discretized models match exactly for the following classes of input signals:

Staircase inputs for ZOH
Piecewise linear inputs for FOH
Impulse trains for impulse IMP

This exact match makes these discretization methods well suited for time-domain simulations.

LTI models with exact discretizations include:

Linear systems without delays
Linear systems with delays at the inputs and outputs of the form

Approximate Discretization

For systems with delays in feedback loops, similar to the system in the following figure, the ZOH and FOH methods result in approximate discretizations.

For such systems, c2d uses the following steps to compute an approximate ZOH or FOH discretization:

The delay is decomposed as with .
The fractional delay is absorbed into H(s). Then, H(s) is discretized to H(z).
The discretized model is assembled as shown in the following figure:

For more information on time delays, see Time Delays in the Control System Toolbox™ documentation.

Tustin and Matched Methods

The Tustin and Matched methods typically perform better in the frequency domain because they introduce less gain and phase distortion near the Nyquist frequency. The Tustin method uses a bilinear transformation to compute the discretized model. This method rounds fractional delays to the nearest integer multiple of the sampling period.

References

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

[2] L.F. Shampine and P. Gahinet, "Software for Modeling and Analysis of Linear Systems with Delays," American Control Conference, 2004.

c2d