I have a feedback loop as shown in figure. Broken link represents sampling and signal with asterisk as superscript represents discrete signal. ZOH is the abbreviation for zero order hold operation.
I tried solving for the closed loop transfer function but after following rules given in the books on Digital Control, I was able to come up with
\begin{equation} {C^*} = \frac{{Z(G{R})}}{{1 + Z(ZOHG)H(z)}}. \end{equation}
I have following questions
- Is it possible to get closed loop transfer function which can be written as $\frac{{{C^*}}}{{{R^*}}}$?
- Is it right if one converts $H(z)$ to continuous transfer function using tustin or other method, and writes a closed loop transfer function in continuous form?
Your post is somewhat confusing. First what is R*? I am assuming you mean R(s), the reference signal. Furthermore, you are interested in the control output C(s)?
Firstly, regarding your second question, it is far more customary to start of with a continuous time model and discretize it using Tustin or other methods then the other way around. So let start from that... Followingly
Herein, r is the reference signal, e is the error signal, u is the control output signal, y is the output of plant. Furthermore, C(s) is the transfer function describing the controller and P(s) is the transfer function describing the behavior of the plant.
Now we can easily derive the transfer functions from several signals to other signals. You requested the transfer function which describes the behavior from the reference signal to the control output.
$$\frac{U(s)}{R(s)} = CS(s) = \frac{C(s)}{1 + P(s)C(s)}$$
This is also called the controller sensitivity. When you would look at the bode diagram you would see were the controller is most effective.
This answers your first question because this is the same as in discrete time
$$\frac{U(z)}{R(z)} = CS(z) = \frac{C(z)}{1 + P(z)C(z)}$$
Because you will choose either to discretize $P(s)$ by a ZOH discretization scheme and $C(s)$ by, for instance, Tustin.