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2003 Course Digital Control

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Total No. of Questions : 12] P1357 [Total No. of Pages :4 [3764] - 283 B.E. (Instrumentation and Control) DIGITAL CONTROL (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) Assume suitable data, if necessary. SECTION - I Q1) a) Derive a mathematical model of ZOH. Derive the pulse transfer function of the closed loop control system shown by Fig. 1 [8] b) Examine the stability of following system by using Jury stability test. [8] P(z) = 2z4 + 7z3 + 10z2 + 4z + 1 OR Q2) a) Obtain the pulse transfer function of the system shown in Fig. 2 [6] P.T.O. b) Consider the system described by following equation y(k) 0.6y(k 1) 0.81 y(k 2) + 0.67 y(k 3) 0.12y (k 4) = x(k) where x(k) is input and y(k) is output of the system. Determine the stability of the system using Jury s Stability criteria. [10] Q3) a) Explain the properties of Dead Beat Controller. Design the deadbeat controller and find its unit step response. [10] b) Diagonalize the following matrix [6] 1 0 0 G= 0 0 1 6 11 6 OR Q4) a) Obtain the state transition matrix of the following discrete time system[8] 1 x1 (k + 1) 0 = x (k + 1) 0.24 1 2 x1 (k ) 1 x (k ) 1 + u (k ) and y (k ) = [ 0] 1 x (k ) 1 2 x2 (k ) Derive an expression for the pulse transfer function of discrete time system. b) Determine the stability of the equilibrium state of the following system[8] 1 x1 (k ) x1 (k + 1) 0 = x (k + 1) 0.5 1 x (k ) . Also find the Liapunov function. 2 2 Q5) a) b) Derive an expression for velocity form of PID controller algorithm. State advantages of velocity form over position form of PID algorithm. [10] 1 . Design a s(s + 1) deadbeat controller GD(z) such that the closed loop system will exhibit a deadbeat response to a unit step input. [8] Consider the digital control system with G p (s ) = [3764]-283 2 OR Q6) a) Derive an expression for the stability of an LTI discrete time system in the sense of Liapunov. [8] b) What is mean by Hermitian Matrix. Explain the sylveter s Criterion for positive / negative definiteness of matrix. [10] SECTION - II Q7) a) Derive an expression for Ackermann s formula for pole placement. [8] b) Consider the system x(k + 1) = Gx(k) + Hu(k) and y(k) = Cx(k) where[8] 2 0 0 0 1 0 2 0 , H = 1 0 and C = 1 0 0 . G= 0 1 0 0 3 1 0 1 Check the system for complete state controllability. OR Q8) a) Consider the system x(k + 1) = Gx(k) + Hu(k) and y(k) = Cx(k) where[8] 0 0.16 0 G= , H = and C = [0 1] 1 1 1 Design a full order observer, if the desired eigen values of the observer matrix are z = 0.2+ j0.2. b) Determine the Liapunov function V(x) for the following system x(k + 1) = G.x(k) where [8] 1 1.2 G= 0 0.5 Q9) a) b) Explain the effect of dead time on the performance of system. Explain the Smith predictor for compensation of dead time. Also state the important features of deadtime compensator. [10] Discuss the design steps for state regulator. OR [3764]-283 3 [6] Q10)Explain the internal model control (IMC) strategies. Design IMC for the system with transfer function [16] 2e 5 s ~ G p (s ) = . Also convert it into conventional controller with approximate 1 + 20s Ds dead time as e Ds 2. = Ds 1+ 2 1 Q11)State the Quadratic Optimal Control problem. Consider the discrete time control system defined by [18] x1 (k + 1) 1 1 x1 (k ) x (0 ) 1 = and 1 = x2 (k + 1) a 1 x2 (k ) x2 (0 ) 0 Where 0.25 < a < 0. Determine the optimum value of a that will minimize the following performance index 1 * J = x (k )Qx(k ). 2 k =0 OR Q12)Write a short notes on following system identification methods (Any three) : [18] a) ARX model. b) ARMA model. c) Least Square Estimation. d) Output error method. kbkb [3764]-283 4

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