Department of Electrical Engineering,
Department of Electrical Engineering,
College of Engineering, University of North Texas, Denton
Control Systems Design (EENG 4310-001/EENG 5310-001)
Semester-End Examination (Fall 2023)
Facilitator: Dr. Parthasarathy Guturu
Learner:
Time: 1 week. Total Points: 20
Q. 1. An analog system such as one in Fig. 1, but without the sampler and zero order hold, has Gps= 4s(s+2) . For this system, you designed an analog compensator Gcs so that the compensated system would have a static velocity error constant Kv of 20 sec-1, a phase margin of at least 50o, and gain margin of at least 10 dB (Refer to Miguel Riveras class notes).The Gcs you designed is given by Gcs=41.7s+4.36s+18.6 . Now, you plan to convert the analog system into a digital system by replacing Gcs by an equivalent digital compensator and GosGcs by Gpz. You may assume a sampling time interval T of 0.001 sec. throughout. The following part questions address this conversion process.
(a) Show that an analog compensator Gcs=K.s+as+b can be converted into an equivalent digital compensator Gcz=K.z-Az-B by equating the values of the two functions at three critical points s=-a, s=-b, and s=0. Using this result, convert the analog compensator above into digital form. (2 Points)
(b) Derive the Trustins approximation for mapping s onto z and use this to solve the problem in 1(a). Show that this result tallies with that in 1(a). (2 Points)
65700829310Zero Order Hold
+
Plant
Fig. 1
r(t)
y(t)
Compensator
Gcs00Zero Order Hold
+
Plant
Fig. 1
r(t)
y(t)
Compensator
Gcs(c) The zero-order hold circuit in the equivalent digital system of Fig. 1 is given by G0s=1-e-sTs (where T is the sampling time interval). Find the closed loop transfer function Gclosed_loopz=Gforward(z)1+Gforward(z) for the compensated system. (4 Points)
(d) Find the first 4 sample values of the step response of the system, that is, output for input Rz=zz-1 by expressing Gclosedloopz.Rz=N(z)D(z) as a series in z-1 by dividing the numerator polynomial N(z) by the denominator polynomial D(z). (4 Points)
Q. 2. (a) Show that the bilinear transform z=1+w1-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)
237744078740 +
Plant with zero order hold
Gp(z)Fig. 2
r(t)
y(t)
K
00 +
Plant with zero order hold
Gp(z)Fig. 2
r(t)
y(t)
K
(b) A unity feedback sampled data system shown in Fig. 2 has an open-loop transfer function given by:
Gpz=0.084z2+0.17z+0.019z3-1.5z2+0.553z-0.05
Using the bilinear transformation z=1+w1-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 Jurys 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)
