Question

1+w Q. 2. (a) Show that the bilinear transform z = \frac{1+w}{1-w} maps points of z inside and outside unit circle onto points w that lie in the left and the right halves, respectively, of the complex plane. Similarly, points z on unit circle are mapped onto points w on imaginary axis. (2 Points) (b) A unity feedback sampled data system shown in Fig. 2 has an open- loop transfer function given by: 0.084z^2+0.17z+0.019 G_p(z) = \frac{0.084z^2+0.17z+0.019}{z^3-1.5z^2+0.553z-0.05} Using the bilinear transformation z = \frac{1+w}{1-w}, w being a complex variable, followed by Routh-Hurwitz stability test for charactristic polynomials in complex domain, determine the range of values of K for stability of the system. (4 Points) (c) Verify using Jury's stability test that the above system is stable for K_{ub} - 1, and unstable for K_{ub} + 1 where K_{ub} is the upper bound of K determined in 2. (b). (2 Points)

          1+w
Q. 2. (a) Show that the bilinear transform z = \frac{1+w}{1-w} maps points of z inside and outside unit circle onto
points w that lie in the left and the right halves, respectively, of the complex plane. Similarly, points z
on unit circle are mapped onto points w on imaginary axis.
(2 Points)
(b) A unity feedback sampled data
system shown in Fig. 2 has an open-
loop transfer function given by:
0.084z^2+0.17z+0.019
G_p(z) = \frac{0.084z^2+0.17z+0.019}{z^3-1.5z^2+0.553z-0.05}
Using the bilinear transformation z = \frac{1+w}{1-w}, w being a complex variable, followed by Routh-Hurwitz stability test for charactristic polynomials
in complex domain, determine the range of values of K for stability of the system.
(4 Points)
(c) Verify using Jury's stability test that the above system is stable for K_{ub} - 1, and unstable for K_{ub} + 1
where K_{ub} is the upper bound of K determined in 2. (b).
(2 Points)

1+w
Q. 2. (a) Show that the bilinear transform z = (1+w)/(1-w) maps points of z inside and outside unit circle onto
points w that lie in the left and the right halves, respectively, of the complex plane. Similarly, points z
on unit circle are mapped onto points w on imaginary axis.
(2 Points)
(b) A unity feedback sampled data
system shown in Fig. 2 has an open-
loop transfer function given by:
0.084z^2+0.17z+0.019
Gp(z) = (0.084z^2+0.17z+0.019)/(z^3-1.5z^2+0.553z-0.05)
Using the bilinear transformation z = (1+w)/(1-w), w being a complex variable, followed by Routh-Hurwitz stability test for charactristic polynomials
in complex domain, determine the range of values of K for stability of the system.
(4 Points)
(c) Verify using Jury's stability test that the above system is stable for Kub - 1, and unstable for Kub + 1
where Kub is the upper bound of K determined in 2. (b).
(2 Points)

Added by Robert F.

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