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EE 350 Control Systems Assignments

Assignment 3: Root locus of a physical system (cont'd...)

3.16 The block diagram of a control system is shown in fig. 3.5. Draw the root locus of the system with $\alpha$ as varying parameter.
a) Determine the steady state error of the unit ramp input, damping ratio and settling time for the system without derivative feedback.
b) Discuss the effect of derivative feedback on the transient as well as steady state behavior of the system assuming $\alpha$ = 0.2.
c) Determine the value of $\alpha$ for the system to be critically damped.

Fig 3.5 Feedback control system

3.17 Transfer function of an Helicopter of pitch attitude is determined as $G(s) = \dfrac{10(s+0.04)}{(s+0.5)(s^{2}-0.4s+0.2)}$. In order to overcome the inherent instability of a helicopter (w.r.t. pitch attitude), a stabilizing feedback loop is introduced in its autopilot system as shown in fig. 3.6. The gain $K_{h}$ is adjustable.
a) Plot the root locus of the system with variable $K_{h}$.
b) Find the value of $K_h$ for the close loop system to have a dominant pole damping of $1/\sqrt{2}$.
c) With this value of $K_{h}$, find the steady state error $(e = r-c)$ caused by wind gust disturbance of $T_{d} = 1/s$.

Fig 3.6 Helicopter pitch attitude control

3.18 An autonomous guided vehicle is used to carry payload to various destinations by an on-board computer. The vehicle having four wheels of which front wheels are powered by two dc servomotors of identical ratings, the back wheel are free. Each of the motors is independently controlled by a controller. The basic block diagram of the control scheme is shown in fig. 3.7, wherein the motor is provided with an internal feedback loop. The controller parameters are $\alpha = 100$, $\beta = 275$. Torque constant $K_T$, and back emf constant $K_b$ are = $0.066 Nm/A$, armature resistance $R_{a} = 2.32 \Omega$, and $J_{m} = 0.0002 kgm^{2}$. Motor inductance and viscous friction can be neglected. Load inertia $J_{L} = 0.002 kgm^{2}$; directly coupled to motor. Tachometer constant $K_{t} = 11 V/rad(s)$.
a) Calculate the value of amplifier gain $K_{A}$ through the root locus method, such that the dominant pole pair has a damping coefficient of $\xi = 1$.
b) Also calculate and plot the unit step response of the system. What cause the response to have an overshoot and what is its magnitude. comment.

Fig 3.7 DC motor control

Reference:

(i) Katsuhiko Ogata, "Modern Control Engineering", Prentice Hall of India Pvt Ltd, 4th edition, 2007.
(ii) John Van de Vegte, "Feedback Control systems", Printice Hall, Upper Saddle River, New Jersey 07458, Third Edition, 1994.
(iii) I.J.Nagrath, and M. Gopal, "Control Systems Engineering", New Age International Publishers, New Delhi, 5th edition, reprint, 2014.