Deriving Discrete Equivalent Control Algorithms Deriving Dis

Deriving Discrete Equivalent Control Algorithms

Deriving Discrete Equivalent Control Algorithms Derive a causal algorithm that can be implemented in a processor that will implement a lead-lag controller (ie Gc = K(s+a)/(s+b)) using a sample rate Fs = 1/Ts using the tustin transform approximation. (you can check your results by choosing a representativeTs, a & b values and using the Matlab function \'c2d\').

Solution

Answer :

Syntax

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

Description

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

sysd = c2d(sys,Ts,method) discretizes sys using the specified discretization method method.

sysd = c2d(sys,Ts,opts) discretizes sys using the option set opts, specified using the c2dOptions command.

[sysd,G] = c2d(sys,Ts,method) returns a matrix, G that maps the continuous initial conditions x0 and u0 of the state-space model sys to the discrete-time initial state vector x[0]. method is optional. To specify additional discretization options, use [sysd,G] = c2d(sys,Ts,opts).

Input Arguments

sys

Continuous-time dynamic system model (except frequency response data models). sys can represent a SISO or MIMO system, except that the \'matched\'discretization method supports SISO systems only.

sys can have input/output or internal time delays; however, the \'matched\' and \'impulse\' methods do not support state-space models with internal time delays.

The following identified linear systems cannot be discretized directly:

idgrey models whose FunctionType is \'c\'. Convert to idss model first.

idproc models. Convert to idtf or idpoly model first.

For the syntax [sysd,G] = c2d(sys,Ts,opts), sys must be a state-space model.

Ts

Sample time.

method

String specifying a discretization method:

\'zoh\' — Zero-order hold (default). Assumes the control inputs are piecewise constant over the sample time Ts.

\'foh\' — Triangle approximation (modified first-order hold). Assumes the control inputs are piecewise linear over the sample time Ts.

\'impulse\' — Impulse invariant discretization.

\'tustin\' — Bilinear (Tustin) method.

\'matched\' — Zero-pole matching method.

For more information about discretization methods, see Continuous-Discrete Conversion Methods.

opts

Discretization options. Create opts using c2dOptions.

Output Arguments

sysd

Discrete-time model of the same type as the input system sys.

When sys is an identified (IDLTI) model, sysd:

Includes both measured and noise components of sys. The innovations variance of the continuous-time identified model sys, stored in itsNoiseVarianceproperty, is interpreted as the intensity of the spectral density of the noise spectrum. The noise variance in sysd is thus /Ts.

Does not include the estimated parameter covariance of sys. If you want to translate the covariance while discretizing the model, use translatecov.

G

Matrix relating continuous-time initial conditions x0 and u0 of the state-space model sys to the discrete-time initial state vector x [0], as follows:

x[0]=G[x0u0]

For state-space models with time delays, c2d pads the matrix G with zeroes to account for additional states introduced by discretizing those delays. See Continuous-Discrete Conversion Methods for a discussion of modeling time delays in discretized systems.

sys	Continuous-time dynamic system model (except frequency response data models). sys can represent a SISO or MIMO system, except that the \'matched\'discretization method supports SISO systems only. sys can have input/output or internal time delays; however, the \'matched\' and \'impulse\' methods do not support state-space models with internal time delays. The following identified linear systems cannot be discretized directly: idgrey models whose FunctionType is \'c\'. Convert to idss model first. idproc models. Convert to idtf or idpoly model first. For the syntax [sysd,G] = c2d(sys,Ts,opts), sys must be a state-space model.
Ts	Sample time.
method	String specifying a discretization method: \'zoh\' — Zero-order hold (default). Assumes the control inputs are piecewise constant over the sample time Ts. \'foh\' — Triangle approximation (modified first-order hold). Assumes the control inputs are piecewise linear over the sample time Ts. \'impulse\' — Impulse invariant discretization. \'tustin\' — Bilinear (Tustin) method. \'matched\' — Zero-pole matching method. For more information about discretization methods, see Continuous-Discrete Conversion Methods.
opts	Discretization options. Create opts using c2dOptions.