Using the Tustin equivalence with the prewarping method fnd

Using the Tustin equivalence with the prewarping method, fnd the discrete equivalent of the following H(s): H s s s s s ( ) = + + + + 2 5 7 3 8 2 2 With h = 0.5, conduct a Bode plot comparison of continuous, Tustin equivalent, and backward differences.

Tustin Approximation

The Tustin or bilinear approximation yields the best frequency-domain match between the continuous-time and discretized systems. This method relates the s-domain and z-domain transfer functions using the approximation:

z=esTs1+sTs/21sTs/2.

In c2d conversions, the discretization H_d(z) of a continuous transfer function H(s) is:

Hd(z)=H(s),s=2Tsz1z+1

Similarly, the d2c conversion relies on the inverse correspondence

H(s)=Hd(z),z=1+sTs/21sTs/2

When you convert a state-space model using the Tustin method, the states are not preserved. The state transformation depends upon the state-space matrices and whether the system has time delays. For example, for an explicit (E = I) continuous-time model with no time delays, the state vector w[k] of the discretized model is related to the continuous-time state vector x(t) by:

w[kTs]=(IATs2)x(kTs)Ts2Bu(kTs)=x(kTs)Ts2(Ax(kTs)+Bu(kTs)).

T_s is the sample time of the discrete-time model. A and B are state-space matrices of the continuous-time model.

Tustin Approximation with Frequency Prewarping

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 discrete-time responses at the prewarp frequency.

The Tustin approximation with frequency prewarping uses the following transformation of variables:

Hd(z)=H(s),s=tan(Ts/2)z1z+1

This change of variable ensures the matching of the continuous- and discrete-time frequency responses at the prewarp frequency , because of the following correspondence:

H(j)=Hd(ejTs)

Tustin Approximation for Systems with Time Delays

You can use the Tustin approximation to discretize SISO or MIMO continuous-time 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 all-pass filter (Thiran filter) of specified order. To do so, use the FractDelayApproxOrder option ofc2dOptions.

To understand how the Tustin method handles systems with time delays, consider the following SISO state-space model G(s). The model has input delay _i, output delay _o, and internal delay .

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(_i /T_s), m_o = round(_o/T_s), and m = round(/T_s).. Also in this case, F_i(z) = F_o(z) = F(z) = 1.

If you set FractDelayApproxOrder to a non-zero 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 the input delay _i, the order of the Thiran filter F_i(z) is:

order(F_i(z)) = max(ceil(_i /T_s), FractDelayApproxOrder).

If ceil(_i /T_s) < FractDelayApproxOrder, the Thiran filter F_i(z) approximates the entire input delay _i. If ceil(_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^–m_i, where

m_i = ceil(_i /T_s) – FractDelayApproxOrder.

c2d uses Thiran filters and FractDelayApproxOrder in a similar way to approximate the output delay _o and the internal delay .

When you discretizetf and zpk models using the Tustin method, c2d first aggregates all input, output, and transfer delays into a single transfer delay _TOT for each channel. c2dthen approximates _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.

Zero-Pole Matching Equivalents

The method of conversion by computing zero-pole 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:

zi=esiTszzz

where:

Zero-Pole Matching for Systems with Time Delays

You can use zero-pole matching to discretize SISO continuous-time models with time delay, except that the method does not support ss models with internal delays. The zero-pole matching method handles time delays in the same way as the Tustin approximation.