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2003 Course Control Systems - II

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Total No. of Questions : 9] P1340 [Total No. of Pages : 3 [3764]-204 B.E. (Electrical) CONTROL SYSTEMS - II (2003 Course) Time : 3 Hours] [Max. Marks : 100 Instructions to the candidate: 1) Answer any one question from each pair of questions Q.No.1 and Q.No.2, Q.No.3 and Q.No.4, Q.No.5 and Q.No.6 from Section I. 2) Questions No.7, 8, 9 from Section II are compulsory. 3) Answers to the two sections should be written in separate answer books. 4) Neat diagrams must be drawn wherever necessary. 5) Figures to the right indicate full marks. 6) Use of logarithmic tables slide rule, Mollier charts, electronic pocket calculator and steam tables is allowed. 7) Assume suitable data, if necessary. SECTION - I Q1) a) b) Define and explain the terms: i) State iii) State space [8] ii) State variables iv) State equation Obtain state model by direct decomposition method of a system whose Y (s ) 5s 2 + 6 s + 8 =3 transfer function is ( ) . U s s + 3s 2 + 7 s + 9 [8] OR Q2) a) b) Derive the relationship between transfer function and the state variable model representation. [8] Find the Transfer function of the system having state model, 1 1 & 0 X= X + 0 U and Y = [1 0]X . 2 3 Q3) a) Define and explain the terms i) Eigen values iii) Modal Matrix [8] [8] ii) Eigen vector iv) Vander Monde Matrix P.T.O. 1 b) Obtain the eigen values, eigen vectors and modal matrix for 1 0 0 A= 3 0 2 12 7 6 Q4) a) b) and prove that M 1AM = = Diagonal matrix. OR Define STM and explain various methods to obtain it. Using Cayley Hamilton method, find eAt for 1 0 A= . 6 5 Q5) a) b) b) [8] [8] Define controllability and observability applied to state space represented system. Discuss the methods of determining these values. [9] Evaluate controllability and observability of the following models, 0 0 0 40 A = 1 0 3 B = 10 C = [0 0 1] 0 1 4 0 OR Q6) a) [8] [9] Derive Ackermann's formula for determination of the state feedback gain matrix k. [9] & Consider the system defined by X = AX + BU where 1 0 0 0 0 B = 0 A= 0 1 1 5 6 1 By using the state feedback control u = - kx, it is desired to have the closed loop poles at S = 2 j 4 and S = -10. Determine the state feedback gain matrix k. [9] [3764]-204 2 -2- SECTION - II Q7) Solve any two : a) Derive an expression for describing function for "Dead zone with saturation Nonlinearity". [9] b) For the system as shown in Fig. 4(a), the nonlinear element is relay with dead-zone. For a step input r = 3 and zero initial condition, draw phase portrait using method of isoclines. [9] c) Find solution for following Nonlinear Systems. i) ii) [9] & x = 3x 3 . 2 Q8) Solve any two: a) Find equilibrium point and analyze the Liapunov's Stability, for the 2 & & = c, a > [8] system given by v + 2a v v + bv x =1+ x 0, b > 0, c > 0 . b) State Stability theorem. Also analyze the stability of the system given by &+ y + (2 + sin t ) y = 0 & [8] y& c) Using Lyapunov function approach, analyze the stability of the system given by & x = A (t ) x V (t , x ) = xT P (t ) x where c1 I P(t ) c2 I , t , ci > 0 [8] Q9) Solve any two : a) With suitable example, explain the integration of following control system component : Sensors, actuators, relays and switches. [8] b) Detail out the mathematical procedure for optimal control design by calculus of variation method. [8] c) Enumerate qualitative analysis of performance indices used in optimal control theory. [8] [3764]-204 3 -3-

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