Write the characteristic equation of the closed-loop system

Question

Question 1: Use the Routh-Hurwitz Stability Criterion to determine the stability of the control systems whose characteristic equations are given by:

a) q(s) = s³ + s² + 2s + 24 = 0
b) q(s) = s⁵ + 2s⁴ + 2s³ + 4s² + 11s + 10 = 0
c) q(s) = s⁵ + 2s⁴ + 24s³ + 48s² - 25s - 50 = 0
d) q(s) = s⁴ + 3s³ + 3s² + 2s + K = 0

Question 2: Use the Jury Stability test to determine the stability of the control systems whose characteristic equations are given by:

a) P(z) = z⁴ - 1.2z³ + 0.7z² + 0.4z - 0.2 = 0
b) P(z) = z³ - 0.8z² - 0.3z + 0.2 = 0
c) P(z) = z³ - 1.1z² - 0.1z + 0.3 = 0

Question 3: The characteristic equations of 4 DT control systems are as follows:

i) z² -1.2z + 0.4 = 0                   ii) z² -1.45z +1 = 0

iii) z³ -1.7z² +1.2z - 0.45 = 0       iv) z³ +0.6z² -1.5z + 0.24 = 0

a) Use the Routh-Hurwitz test to determine the stability of the system.

b) Use the Jury test to determine the stability of the system.

c) Check your results by finding the roots of the characteristic equations.

Question4: Consider the discrete-time unity feedback control system whose open-loop pulse transfer function is given by:

G(z) = (K (0.37 z + 0.26)) / (z - 0.37 ) (z - 1)

a) Find the closed-loop pulse transfer function of the system.

b) Using the Jury stability test, determine the range of the gain K for stability.

c) Using the Routh-Hurwitz Stability Criterion, determine the range of gain K for stability.

Question 5: Consider a continuous-time unity feedback control system whose open-loop transfer function is given by:

G(s) = 1 - e^-s/s .1/(s(s + 2))

a) Determine the Z-transform of G(s) when T = 0.5 sec.

b) Find the closed-loop pulse transfer function for the system.

c) Determine the stability of the closed-loop control system.

Question 6: Consider the control system described by the following block diagram:

The digital filter D(z) is described by D(z)=K.

a) Write the characteristic equation of the closed-loop system.

b) Use the Routh-Hurwitz test to determine the range of K for which the system is stable.

c) Use the Jury test to determine the range of K for which the system is stable.

d) Find the frequency at which the system will oscillate when K > 0.

e) Consider the system with all sampling removed (i.e. remove the sampler and the ZOH).

Find the range of K for which the system is stable.

f) By comparing the results of c) and e), discuss the effects of sampling.

g) If T=0.1, use the Jury test to determine the range of K for which the system is stable.

h) By comparing the results of c) and g), discuss the effects of reducing the sample time T on the stability of the system.