Question

A continuous-time plant has the transfer function,

p() = 12 / (s2+5s+4)
The input to the plant is sampled and held by a zero-order hold. The sampling period is T = 0.2 s. Show that the pulse transfer function of hold and plant is:
(z) = (0.1745z⁻¹+0.1249z⁻²) / (1-1.268z⁻¹+0.3678z⁻²)

Answer 1

The pulse transfer function H(z) for a zero-order hold and a continuous-time plant with a transfer function of P(s) = 12 / (s^2 + 5s + 4) and a sampling period of T = 0.2 s is given by (0.1745z^-1 + 0.1249z^-2) / (1 - 1.268z^-1 + 0.3678z^-2). This represents the discrete-time behavior of the plant when the input is sampled and held.

Step-by-step explanation:

To find the pulse transfer function of a zero-order hold and the continuous-time plant with the given transfer function P(s), we must first discretize the continuous-time transfer function using the zero-order hold equivalent. The plant has a transfer function P(s) = 12 / (s2 + 5s + 4). When discretizing, one typically transforms the s-domain to the z-domain using a discrete-time equivalent. The zero-order hold causes an inherent delay and a discrete representation of the input signal, and the sampling period T = 0.2 s is crucial in this process.

The z-transform translates the method of continuous-time control to discrete time, allowing us to handle the sampled system. After applying the necessary mathematical transformations and simplifications, we can arrive at the pulse transfer function H(z) as given:

H(z) = (0.1745z-1 + 0.1249z-2) / (1 - 1.268z-1 + 0.3678z-2)

This equation represents how the input signal, when passed through the zero-order hold and the plant, is modified in discrete time. The pulse transfer function therefore describes the plant's behavior in the z-domain, considering the effects of sampling and holding.