A) Consider the system represented in the figure

where

G_p(s) = 1/s(s+1) and D(z) = K

1. By using the Nyquist criterion, demonstrate that the system without sampler and ZOH is always stable for all values of K.

2. Let us consider the complete system with the sampler and the ZOH and T=1 s.

a. Determine the range of values of K for the system to be stable by using the Routh Hurwitz criterion and the Jury test. From both methods obtain the oscillation frequency at the boundary of stability.

b. Obtain the closed loop poles for K=2 and K=3. Check the result in (a).

c. Skecth the root locus of the system.

d. Obtain the system response at time instants t=kT to a unit step for K=2 and K=3.

B) Consider the scheme in Figure 1 and let

G_p(s) = e^-st/s(s+1) and D(z) = K

Skecth the root locus of the system and determine the stability range in terms of T and K using . Determine the value of K corresponding to equal closed loop poles.

C) Consider the loop gain L(z) = K [1/(z-1)(z-0.5)]

Obtain the root locus plot and the critical gain for stability without using the Jury test or the Routh Hurwitz criteria. Check the closed loop poles at the obtained value of K

D) Consider the scheme in Figure 1 and let

G_p(s) = 1/s² and D(z) = K

Sketch the root locus and demonstrate that the system is always unstable independently on the value of K. Check the result using the Jury test. What is the oscillation frequency at the border of stability?

E) Consider the scheme in Figure 1 and let

G_p(s) = 1/s² and D(z) = K.z-1/z-0.2

Sketch the root locus and find the range of values of K guaranteeing stability.

F) Consider the scheme in Figure 1 and let

G_p(s) = 1/s² and D(z) = K (z - 1/z + 0.98)

1. Sketch the root locus and find the range of values of K guaranteeing stability.

2. Obtain using a reasonable approximation the step response of the closed loop system.

G) Consider the same scheme as in the previous problem but with

G_p(s) = 1/(s + 1)

Using the jury test determine the stability range in terms of the paramter T and K. Plot the result in the (T,K) parameter space.

H) Consider the same scheme as in the previous problem but with

Let T=1 s. Using the Jury test obtain the stability range in the parameter space (K,τ).

1) Consider the system in Figure 1 with

G_p = 1/s(s+1)

Design a digital controller D(z) such thatthe damping ratio ζ of the dominant closed loop pole be 0.5 and the number of samples per cycle of damped sinusoidal oscillation is 8. Assume that the sampling period is 0.1 s. Determine the static velocity error constant. Also, determine the response of the designed system to a unit step input.

J. Design a proportional controller for the digital system with sampling period T=0.1 s to obtain:

a) ω_d =rad/s b) A time constant of 0.5 s, and a damping ratio ζ =0.7.

K. Consider the system in Figure 1 with

G_p = 1/s(s+5)

and sampling period T=0.1 s.

Design a proportional controller such that the steady state error due to ramp input es equal to 0.1 and a and a damping ratio ζ =0.7.

L. Consider the system in Figure 1 with

G_p = 1/(s + 2)(s + 1)

Using the bode diagram approach in the w plane, design a digital controller D(z) such that the phase margin is 60 degree, the gain margin is 12 dB or more, and the static velocity error constant is 5 s^-1. The sampling period is assumed to be 0.1 s.

M. Consider the system in Figure 1 with

G_p = 1/s(s+1) with T = 0.1 s.

Using the bode diagram approach in the w plane, design a digital controller D(z) such that the phase margin is larger than 40 degree at gain crossover frequency greater than 1 Hz, the gain margin is 10 dB or more at phase crossover frequency greater than 1 Hz, and the steady-state error for a ramp input is less than 5%.