Digital filters transfer function

The digital PID controller is implemented as a recursive filter with a biquadratic transfer function. Controller wordlength, set-point input, controller output, and controller parameters are 10-bit fixed-point values in the range [0,1). The controller parameters also have an additional sign-bit. 

Rearranging gives the transfer function for the digital filter using trapezoidal approximation... 

This is the transfer function for a first-order digital filter. Thus the precompensator slows down the input to the controller so that it does not see a step change in the error signal (see (Fig. 20.3h). 

Equation (29.5) yields the discrete transfer function of a first-order digital filter. The noise-free signal (the output of the filter) is given by... 

What is a discrete transfer function, and what is it needed for Develop the discrete transfer function for (a) a proportional control algorithm, (b) the velocity form of a PI control algorithm, and (c) a second-order digital filter. 

A powerful relationship used for analyzing digital filters is the tranter function, which is found by solving for the ratio of output (7) to input (X) in the Z-transformed filter expression. The transfer function for Equation 3.4 can be solved by using simple algebra ... 

This analysis can be extended to any digital LTI filter. As the general form transfer function, H z) is defined as the ratio of two polynomials... 

---