Process Identification and PID Control

11

Model Conversion from Discrete-Time to Continuous-Time Linear Models

11.1 Transfer Function of Discrete-Time Processes

In Part One, the Laplace transform was used to derive the transfer function of a continuous-time process. For a discrete-time process, the z-tranform is used. Consider the following discrete-time process:

The z-transform of y(kΔt) is defined as

One of the notable properties of the z-transform is y(z)z⁻¹ = Z{y(k − 1)Δt)} if y(kΔt) = 0, k < 0. The property is derived straightforwardly by comparing (11.2) with (11.3):

From (11.2) and (11.3), it is clear that y(z)z⁻¹ = Z{y(k − 1)Δt)} if y(kΔt) = 0, k < 0. Equivalently, y(z)z^−d = Z{y(k − d)Δt)} if y(kΔt) = 0, k < 0. Then, (11.4) is obtained by applying the z-transform to (11.1):

Rearranging (11.4), the following transfer function for the discrete-time process is obtained:

or

11.2 Frequency Responses of Discrete-Time Processes ...