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2003 Course Digital Control Systems

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Total No. of Questions : 12] [Total No. of Pages : 4 [3864] - 232 P 1064 B.E. (Electrical) DIGITAL CONTROL SYSTEMS (2003 Course) (403149) 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 indicates full marks. 5) Use of logarithmic tables, slide rules, electronic unprogrammable pocket calculator is allowed. 6) Assume suitable data, if necessary. SECTION - I Q1) a) b) Explain the various standard discrete input test signals with neat diagrams. [6] [10] Consider the discrete time sequence as given below : { x ( n ) = 1, 1, 4,1,2 } Explain in detail and draw graphs for : i) Compress the sequence by 2 samples. ii) Delay the sequence by 2 samples. iii) Fold the sequence and advance by one sample. iv) Attenuate the sequence by amplitude scale 2. v) Prove that folding and time delay operations are not commutative. OR Q2) a) b) Explain sampling and reconstruction process. State Sampling theorem and give its importance. [6] Classify the following sequences as static / dynamic, time-invariant / time-variant, linear / non-linear, Causal / non-causal, stable / unstable giving detail justification. n +1 x(k ) i) y( n ) = ii) y(n) = x(n).cos(w0n). k = [10] P.T.O. Q3) a) b) Explain what do you understand by Direct Form - I and Direct Form - II structure representations of a discrete-time system; in detail. [8] Obtain Linear convolution of following sequences by Tabulation and Multiplication method. [8] { } x ( n ) = 1,1, 0,1,1 { h ( n ) = 1, 2, 3, 4 } OR Q4) a) b) Draw a neat sketch of speed control scheme and explain its working.[8] Discuss the various methods of obtaining the convolution of discrete time systems. [8] Q5) a) State and prove Initial value Theorem and Final value Theorem in discrete-time systems. [8] Find the one sided Z-transform of the following discrete sequences : i) F(k) = Ka(k 1) ii) F(k) = K2 [10] b) OR Q6) a) b) State the properties and theorems of the Z-transform. Give the proof of the Final Value Theorem . [8] Determine the Inverse Z-transform of : [10] i) 4z2 2z F( z ) = 3 z 5z 2 + 8z 4 ii) F( z ) = z 0.4 z +z+2 2 SECTION - II Q7) a) b) Define Pulse Transfer Function . Obtain the pulse transfer function for error sampled unity feedback control system with forward path linear transfer function G(s). [6] A certain unity feedback error sampled data control system has ; ST 1 e 1 ; Z 0H .G h ( s ) = and sampling period T = 1 sec. G( s ) = s s ( s + 1) For unit step input, determine i) The Z-transform of the output. ii) The output response at sampling instants. [10] [3864] - 232 -2- OR Q8) a) b) c) Q9) a) Explain the concept of stability of discrete-data control system. [4] What is Bilinear transformation? How it is used for testing the stability of discrete-time control system? [4] By applying Jury s test examine the stability of the discrete-data system represented by characteristic polynomial, [8] F(z) = z4 1.368z3 + 0.4z2 + 0.08z + 0.002 Define state Transition Matrix for the discrete data system. For the system represented by the discrete state model, 1 1 x1 ( k ) 0 x ( k + 1) = + u( k ) 0 1 x 2 ( k ) 1 b) Find the state transition matrix ( t ) . [9] Find the Z-transfer function for the following discrete-time system. x1 x 2 ( k + 1) 0 2 x1 ( k ) 2 = + u( k ) ( k + 1) 1 3 x 2 ( k ) 1 x1 ( k ) y ( k ) = [3 2 ] x2 ( k ) [9] OR Q10)a) b) Explain any one method for finding the state transition matrix ( t ) for a discrete time system of the form with usual notations x(k + 1) = Fx(k) + Gu(k) y(k) = Hx(k) [6] Draw the state variable diagram in phase variable canonical form and determine the discrete time state variable model for a system having Y( z ) z2 + z = pulse transfer function : U( z ) z 3 0.8 z 2 0.21z + 0.01 Q11)a) b) [12] Discuss the following methods of the realization of the pulse transfer function into a discrete state model form, i) Parallel decomposition. ii) Cascade decomposition. [8] Explain the PID controller for a digital control system. Draw the relevant block diagram. [8] [3864] - 232 -3- OR Q12)a) Define eigen values and obtain the same for the discrete-data system represented in state space form : [4] 1 0 x1 ( k ) 0 1 0 u1 ( k ) x ( k + 1) = 0 0 1 x2 ( k ) + 1 0 u2 ( k ) 0 0.5 1.5 x3 ( k ) 2 1 b) c) Also comment on the system stability. Define the concepts Controllability and observability of discrete-data Control System . [4] Investigate the controllability and observability of the following system : 1 1 x1 ( k ) 0 x ( k + 1) = + x(k ) 0 1 x 2 ( k ) 1 y ( k ) = [1 1]. x ( k ) [3864] - 232 [8] -4-

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