Question 1

a) What is the Nyquist sampling theorem in discrete time systems? In practice, how should the sampling frequency be chosen in discrete time control systems?

b) Given the discrete-time system: y(k + 1) - y(k) = u(k + 1) find the system transfer function and its response to a sampled unit step.

c) A linear system is controlled using a discrete time controller shown in Figure 1

Figure 1: Block diagrarn of a discrete time control system

i) Write the equation of the discrete time open loop transfer function.

ii) Write the characteristic equation for this system.

iii) For a sampling time period T = 1sec. calculate the critical gain of this system The root locus leaves the unit circle at z = -1.