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2003 Course Control System - II

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Total No. of Questions : 12] [Total No. of Pages :5 P1274 [3864] - 224 B.E. (Electrical) CONTROL SYSTEM - II (2003 Course) Time : 3 Hours] [Max. Marks : 100 Instructions to the candidates: 1) Answers to the two sections should be written in separate books. 2) Neat diagrams must be drawn wherever necessary. 3) Figures to the right indicate full marks. 4) Your answers will be valued as a whole. 5) Use of logarithmic tables, slide rule, Mollier charts, electronic pocket calculator and steam tables is allowed. 6) Assume suitable data, if necessary. SECTION - I Q1) a) Compare the state space technique with the transfer function method for [8] the analysis of feedback control systems. b) Obtain state space representation in phase variable form for a system described by (Draw block diagram also) [10] Y(s) s+2 =3 U(s) s + 3s 2 + 4s + 5 OR Q2) A D.C. motor position control system is described by the following differential equations V - Eb = IaRa + L dI a dt T = KmIa; T = J dw + B w; Eb = Kb w dt d = angular velocity dt Ia = Armature current V = Armature voltage; Eb = Back emf Km = Motor torque constant; Kb = Back emf constant J = Rotor inertia; B = frictional constant. Where w = P.T.O. Obtain two different state models for the system using a) b) Q3) a) b) [x1, x2, x3 ]T = [ [x1, x2 , x3 ]T = [ , Ia , ] [18] T ] T Derive the solution of a nonhomogeneous state equation X = AX + BU .[8] Using the concept of similarity transformation obtain diagonal form for a system 3 1 1 1 X = 1 5 1 X + 2 U 1 1 3 3 [8] OR Q4) a) b) State Caley-Hamilton theorem and explain how it can be used for [8] computation of state transition matrix. Obtain state transition matrix using Laplace transform method for 1 0 A= . 3 4 Hence obtain solution for a homogeneous equation X = AX for the initial condition X (0) = [ 1 1]T. [8] Q5) a) b) Define controllability and observability of a system. Explain Kalman s test and Gilbert s test for controllability and observability. [10] Investigate controllability and observability of a system described by 1 2 1 1 0 1 0 X + 0 U X= 1 4 3 1 x1 Y = [ 1 0] x2 1 x3 [6] OR [3864]-224 -2- Q6) a) b) Explain Ackerman s method for pole placement using state feedback.[8] Consider a system defined by X = AX + BU where 1 0 0 0 A= 0 0 1 B = 0 1 5 6 1 Using state feedback control, it is desired to have the closed loop poles at s = 2 + j4; s = 10. Determine the state feedback gain matrix K. [8] SECTION - II Q7) a) Explain the method of describing function for analysis of nonlinear systems stating clearly the assumptions. What are the advantages and disadvantages of the method over phase plane method? [8] b) Obtain the describing function for the ON-OFF nonlinearity with dead zone shown in Figure 7b. [8] OR Q8) a) Write a short note on the following phenomena of nonlinear systems.[8] i) Jump resonance. ii) Subharmonic Oscillations. [3864]-224 -3- b) For the system shown in figure 8b, determine the amplitude and frequency of the limit cycle. [8] Q9) a) Find equilibrium points for the nonlinear systems described by the following state equations [6] i) x1 = x2 x2 = sin x1 ii) x1 = ax1 bx1 x2 x2 = cx1 x2 dx2 b) A linear second order system is described by the equation 2 e + 2 wn e + wn e = 0 where wn = 2 = 0.2 . Using the method of isocline, construct phase trajectory of the system for the initial conditions e (0) = 2 e (0) = 1 [10] OR Q10)a) b) Explain the direct method of Lyapunov to determine the stability of a [8] nonlinear control systems. Consider a system described by the state equations x1 = x2 x2 = x1 x2 Choosing Lyapunov function 2 V ( x) = x12 + x2 , comment on stability of the system [8] Q11)a) Explain the concept of performance index and different indices ISE, ITAE, IAE and ITSE. [9] b) Explain in brief different factors considered in designing optimal controller. [9] [3864]-224 -4- OR Q12)Write a note on : [18] a) Power transmitting techniques. b) c) Switches and relays. Hydraulic and pneumatic actuators. kbkb [3864]-224 -5-

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