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2003 Course Digital Control Systems

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Total No. of Questions : 12] P942 [Total No. of Pages : 4 [3664]-189 B.E. (Electrical) DIGITAL CONTROL SYSTEMS (403149) (2003 Course) Time : 3 Hours] [Max. Marks : 100 Instructions to the candidates : 1) Answer any one question from each pair of questions Q.1 & Q.2, Q.3 & Q.4, Q.5 & Q.6, Q.7 & Q. 8, Q.9 & Q.10, and Q. 11 & Q. 12. 2) Answers to the two sections should be written in separate books. 3) Neat diagrams must be drawn wherever necessary. 4) Figures to the right indicate full marks. 5) Use of logarithmic tables, electronic unprogrammable pocket calculator is allowed. 6) Assume suitable data, if necessary. SECTION - I Q1) a) b) Explain the sampling & reconstruction process, state sampling theorem. [6] Draw graphical representation of: [6] (i) Unit impulse or Delta function; (ii) Unit step sequence; (iii) Unit ramp sequence. c) Sketch a D.T. signal x(n) = z n for 2 n 2 and obtain y(n) = 2x(n) + (n).[5] OR Q2) a) Determine whether the following systems are shift (time) invariant or not? [9] i) y(n) = x(n) x(n 1). ii) y(n) = n x(n). iii) y(n) = x( n). b) For the given sequence x(n) = {4, 1, 5, 2, 2}. i) Delay the sequence by 2 samples, [8] ii) Compress the sequence by time scale 2, iii) Attenuate the sequence by amplitude scale 2, iv) Fold the sequence and advance by one sample. P.T.O. Q3) a) Explain with neat diagrams the Direct form - I and Direct form - II structure representations of a discrete time system. [8] b) Obtain Linear convolution of following sequences by multiplication method and then verify the result by Tabulation method: [8] X(n) = {1, 2, 1, 2} and h(n) {2, 2, 1, 1} OR Q4) a) Draw a neat block diagram of digital measurement of speed and explain function of each block. [8] b) Prove that LTI system is completely characterised by Unit impulse response h(n). [8] Q5) a) Define Z - transform and state important properties of Z - transform.[5] b) Obtain Z - transform of the following functions: i) f(t) = e at cos t . ii) [12] f(t) = sin t . iii) Find initial and final values of function 1 + z 1 . X(z) = 1 0.25 z 2 OR Q6) a) b) Explain any two methods of obtaining the inverse Z-transform. Evaluate the inverse - Z transform of the, i) 10 z . x(z) = ( z 1) ( z 2 ) ii) [7] [10] 3z 2 + 2 z + 1 x(z) = 2 . z + 3z + 2 SECTION - II Q7) a) Explain precisely Schurcohn Stability Criterion , as applied to the discrete time systems. [8] b) The characteristic polynomial of a certain discrete time control system is given by F(z) = z4 + 3z3 + 5z2 + 4z + 0.8 By applying Jury s Stability Test, Find whether the system is stable or not. [8] OR [3664] - 189 2 Q8) a) Define Pulse Transfer Function , obtain the pulse transfer function for impulse sampler located at the input of ZOH in cascade with G(s). [8] b) Show how a mapping of left half of the S-plane is done into Z-plane. Mark the stable and unstable regions in both the planes. [8] Q9) a) Discuss the various methods used for the computation of State Transition Matrix (STM) from given state difference equation: [8] x(k + 1) = G x(k) + H u(k) b) Obtain STM ( k ) of the following difference equation. x(k + 1) = G x(k) + H u(k) where 1 0 ; G= 0.2 1 1 H = . 1 [9] OR Q10)a) b) Derive the solution of a non-homogeneous state equation of a discrete time system from the first principles. [8] Consider the system x (k + 1) = G x(k) + H u(k); 1 0 ; Where G = 0.16 1 0 H = . 1 Determine a suitable state feed back Gain Matrix K such that the system will have closed loop poles at z = 0.5 + j 0.5, z = 0.5 j 0.5. [9] Q11)a) Explain the terms, characteristic equation, eigen values, eigen vectors, and vander monde matrix. [8] b) Obtain the state - space representation of a DT system by Direct Decomposition method, from the pulse transfer function given by; [9] ( z ) 4 z 3 12 z 2 + 13 z 7 . =3 U( z ) z 4 z 2 + 5z 2 OR [3664] - 189 3 Q12)a) Explain clearly the concept of controllability and observability of a Discrete Time (DT) system. How these are determined? [8] b) Determine the controllability and observability of the following DT system; [9] 1 0 x(k + 1) = x (k ) + 0 2 2 3 u(k); y(k) = [1 5] x(k). Y [3664] - 189 4

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