# (b) A unity feedback system is characterized by the transfer function. Design a suitable compensator to meet the following specifications:

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1 1. (a) The open loop transfer function of a unity feedback control system is given by G(S) = K/S(1+0.1S)(1+S) (i) Determine the value of K so that the resonance peak M r of the system is equal to 1.4. (ii) Determine the value of K so that the Gain margin of the system is 20 db. (iii) Determine the value of K so that the phase margin of the system is 60 degrees. (b) A unity feedback system is characterized by the transfer function G(S) = Design a suitable compensator to meet the following specifications: (i) Settling time for 2% tolerance, band = 4 sec (ii) Steady state error for ramp input 10%.(1990) 2. Draw the root locus diagram for the following control system and calculate the breakin and breakaway points..(1992) 3. Sketch the desirable range of the location of the poles of the transfer function of a system s damping ratio is to lie between 0.3 and 0.7 and its natural frequency is to lie between 2 and 4rad/sec..(1993) 4. Draw a signal flow graph for the following equations: x 2 = t 12 x 1 + t 32 x 3 x 3 = t 23 x 2 + t 43 x 4 x 4 = t 24 x 2 + t 34 x 3 + t 44 x 4 x 5 = t 25 x 2 + t 45 x 4 5. Consider a feedback control system with the open loop transfer function G(s) = Design a series compensator to provide the following specification:.(1993)

2 (i) The phase margin of the system must be greater than (ii) When the input to the system is a ramp, the steady state error of the output in position should be less then 0.1 degree/deg/sec of the final output velocity..(1993) 6. The open loop transfer function of a system is given below: G(s) H(s) = For what values of K is the system stable?.(1994) 7. (a) The system shown below consists of a unity feedback loop containing a minor rate feedback loop. (i) (ii) Without any rate feedback (b = 0), determine the damping factor, natural resonant frequency, overshoot of the system to a unit step input, and the steady state error resulting from a unit ramp input. Determine the rate feedback constant which will increase the equivalent damping factor of the system to 0.8. Determine the overshoot of the system in this case to a unit step input and the steady state error resulting from a unit ramp input. (b) Using Bode plots, determine the gain margin and phase margin of a unit feedback system having an open loop transfer function: G(s) = By what constant factor should G(s) be multiplied for (i) a gain margin 20 db, and (ii) a phase margin of 24 0.(1994) 8. (a) Sketch the root-locus of a unity feedback system with an open-loop transfer function

3 G(s) = For what range of k will the system have damped oscillatory response? What is the highest value of k that can be used before continuous oscillations occur? (b) Consider the third order position control system with velocity feedback shown below. Determine the value of k so that the dominant poles of the transfer function of the closed loop system have a damping ratio of 0.5. What will be the response of the system to a unit step input for this value of k?.(1995) 9. Determine the value of k so that all the roots of the following polynomial are to the left of the line = F(s) = s 3 + 6s s + k.(1996) 10. Sketch the constant-m loci in the G-plane for a unity feedback system and derive the equations for the loci..(1996) 11. (a) Find the output in the following block diagram having three input: R, U 1 and U 2. (b) Enumerate the advantages of state space modeling Derive relations to find the poles and zeros of a system from the state space model. Determine the poles and zeros of the following system: X = X + C = [-17-5]X + [1]r.(1996)

4 12. Obtain the transfer function for the multi-loop control system shown..(1997) 13. A unity feedback control system has a forward transfer. Find the resonance peak and the corresponding frequency for the closed loop frequency response. Derive the formula you used..(1997) 14. For open loop transfer function A(s) = a negative feedback is applied with a feedback factor β. Find the value of βa, (i) corresponding to the breakaway point, (ii) for which the system becomes unstable..(1998) 15. (a) The open loop transfer function of a unity feedback control system is G(s) = Construct the root-locus diagram of the system and comment on the stability of the system..(1998) (b) For a proportional plus derivative (PD) controller plot the controller output and error vs time. Specify the equation for the controller..(1998) 16.

5 A dynamic vibration absorber is shown in the above figure. The system is seen in many situations involving machines containing several unbalanced components. The parameters M 2 and K 12 may be chosen such that the main Mass M 1 does not vibrate when F(t) = a sin ω 0 t. (a) Obtain the differential equation describing the system. (b) Draw the analogous electric circuit based on Force current analogy. (c) What is the condition for Mass M 1 not vibrating at frequency ω 0..(1999) 17. (a) Explain gain margin and phase margin from Nyquist diagram. (b) A closed loop transfer function of a unity feedback control system is = Determine the response of the system when the excitation applied to the input terminal is.(1999) (d) For the system shown below, determine the characteristic equation. Hence, find the following the excitation is a unit step: (i) Undamped natural frequency (ii) Damped frequency of oscillation (iii) Dampling ratio and damping factor (iv) Maximum overshoot (v) Setting time (vi) Number of cycles completed before the output is settled within 2%, 5% of its final value (vii) Time interval after which maximum and minimum will occur..(1999)

6 18. (i) Write the transfer function of a PID controller and state the effect of integral control on the performance of the system. (ii) A closed loop system has G(s) = and H(s) =1/s. Draw the Nyquist path for analyzing the stability of the system. 19. A second order control system with proportional derivative controller is shown in figure. Derive expression for its (i) steady state error to velocity input (ii) natural frequency of oscillation (iii) damping ratio in terms of the system parameters. 20. By analytical method calculate the gain margin in db of the unity feedback control system with transfer function..(2000) G(s) =.(2000) 21. For the mechanical system shown in Figure write the differential equation representing the system. Draw an integrator based electronic circuit to simulate this mechanical system to study the variations of x for different value of the parameters. Symbols used have their usual meaning..(2001) 22. (a) A control system is represented by the block diagram of figure. Find its characteristic equation using block diagram reduction technique. (b) Calculate its damping factor and undamped natural frequency for k V = 10. (c) What should be the value of k v for critical damping?

7 (d) For k V = 10, find the expression of C(t) and obtain the time at which first overshoot occurs. Also find the peak overshoot magnitude. 23. (a).(2001) Determine the value of k and velocity feedback constant k v so that the maximum overshoot in the unit step response is 0.2 and the peak time is 1 sec. With these values of k and k v, obtain the rise time and settling time. (b) Consider a closed loop system whose loop transfer function is G(s) H(s) = Determine the maximum value of the gain k for stability as a function of dead time T..(2002) 24. (a) Consider a control system with characteristic equation s(s + 4) (s 2 + 2s + 2) + K(s + 1) = 0 Draw the complete root loci labeling important values. Also find the angles of asymptotes and the intercept of asymptotes. (b) Consider a third order system with characteristic equation s s x 10 6 s+1.5 x 10 7 K = 0.

8 Find the critical value of K for stability using Routh-Hurwitz criterion. Also find the undamped frequency corresponding to the zero input response and the critical value of K..(2003) 25. For a single-loop feedback control system G(s) = and H(s) = Evaluate the steady state errors for three basic types of inputs. 26. (a) A three term controller is described by the equation (t) = 20 Where e(t) = system error (t) = controller output T r = reset time T d = derivative time This is used to control a process with transfer function G(s) = unity.(2003) feedback is used. (i) If integral action is not employed, find the derivative time required to make the closed-loop damping ratio unity. (ii) If this value of derivative time is maintained, determine the minimum value of reset time that can be used without instability arising. (b) Consider the following control system (i) Sketch the root locus diagram for 0 < K <. (ii) Determine the value of K that gives the system characteristic equation a damping ratio of 0.5..(2004) 27. A phase lead compensator has a transfer function G(s) = Determine the maximum value of the phase lead and the frequency at which it occurs. Sketch the Bode diagram for this network..(2004) 28. (a) Construct a signal flow graph for the following equations and evaluate y 5 /y 1 : y 2 = a 12 y 1 + a 32 y 3

9 y 3 = a 33 y 2 + a 43 y 4 y 4 = a 24 y 2 + a 34 y 3 + a 44 y 4 y 5 = a 25 y 2 + a 45 y 4 (b) The characteristic equation of a closed loop control system is s 3 + 3Ks 2 + (K+1)s +4 = 0 Find the range of K for which the system is stable. Shown all steps clearly..(2005) 29. (a) Consider a mechanical system as shown. Write the force equations and draw a fully labeled state diagram. (b) An amplifier with an open loop voltage gain of 500 delivers 10 W of output power at 5% second harmonic distortion when the input signal is 5 mv. If 20dB negative voltage is to remain 10 W, determine (i) the required input signal strength, and (ii) the percent second harmonic distortion..(2005) 30. State and explain the terms gain margin and phase margin. With neat sketches, explain how you can obtain gain margin and phase margin from Nyquist diagram and Bode plot..(2006) 31. (a) Find the value of gain k for the feedback control system shown in figure, such that the system will be underdamped and will respond with 16% overshoot. Then calculate the following parameters of the system: (i) Undamped natural frequency (ii) Damping ratio, (iii) Time required to reach the first maximum or peak, T p (iv) Time required for the transient to reach within 2% of the steady-state value, i.e., settling time, T s (v) Damped natural frequency, ω d.

10 (b) using the root-locus technique, discuss the stability of unity feedback first-order and second-order control system of gain k..(2006) 32. Draw the asymptotic Bode diagram for G(s) = and determine the value of G(j1000). 33. A unity feedback system has a forward has a forward loop transfer function:.(2006) G(s) = Determine: (i) the range of k for closed-loop system stability, (ii) the frequency of oscillations when the system is marginally stable..(2007) 34. (a) A closed-loop system is represented by = 144e where, e = r-0.5c is the actuating signal. Find the value of the damping ratio, damped and undamped frequency of oscillations. Draw the block diagram of the system. (b) A system employing a proportional and an error-rate control is hsown in the figure. Determine- (i) the error-rate factor k e, so that the damping ratio is 0.5; (ii) the settling time, maximum overshoot and steady-state error for unit ramp input..(2007) 35. (i) For the system shown in the figure, obtain the values of k and a, to satisfy, M r = 1.04 and ω r = rad/sec. (ii) A unity feedback system has an open-loop transfer function G(s) = Determine the steady-state error for r(t) = 3 +10t..(2007)

11 36. Determine the open loop transfer function, G(s) H(s), of a feedback control system whose Bode-Plot s magnitude characteristic is shown in the figure..(2008) 37. (a) If a unity feedback system having G(s) = is critically stable and oscillates with a frequency of 2.5 rad/sec, calculate the corresponding values of K and p. (b) For the block diagram of a unity feedback control system, define: (i) the steady error for K = 400 and unit ramp input. (ii) the value of K for which the steady state error for unit input will be (c) For the tachometer feedback control system shown in the figure. Determine the value of Kb to make the system s damping ratio equal to 0.8. Calculate the corresponding peak time, peak overshoot, damped frequency and settling time taking 2% of the steady state valued..(2008)

12 38. The open-loop transfer function of a unity feedback system is G(s) =, where K and T are positive constants. How many times the gain of the system, K s should be increased the peak overshoot from 40% to 60%?.(2008) 39. (a) Explain the difficulties involved in the application of Routh-Hurwitz criterion and also bring out limitations. Find the stability of the control system whose characteristic equation is given by: (s-1) 2 (s+2)(s+1) = 0. (b) Explain the effect of additional poles and zeros of G(s) H(s) on the shape of the Nyquist plot. Sketch the Nyquist diagram and determine stability of the transfer function: (i) G(s) H(s) = (ii) G(s) H(s) = (c) Obtain the overall transfer function C/R from the signal flow graph.(2009)

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