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2003 Course Digital Control

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Total No. of Questions : 12] [Total No. of Pages : 5 P1078 [3864]-283 B.E. (Instrumentation and Control) DIGITAL CONTROL (2003 Course) Time : 3 Hours] [Max. Marks : 100 Instructions to the candidates: 1) Answer 3 questions from Section - I and 3 questions from Section - II. 2) Answers to the two sections should be written in separate books. 3) Neat diagrams must be drawn wherever necessary. 4) Use of logarithmic tables slide rule, Mollier charts, electronic pocket calculator and steam tables is allowed. 5) Assume suitable data, if necessary. SECTION - I Q1) a) For the sampled data system shown in Fig. 1 find the Pulse Transfer Function for G(s) = b) 1 1 and H(s) = (T = 1 sec) s s +1 [8] Find the system stability using Jury Stability test whose characteristic equation is given by [8] p(z) = z4 1.3z3 + 0.7z2 + 0.2z 0.006 = 0. OR P.T.O. Q2) a) Obtain the pulse transfer function of the system shown in Fig. 2. [8] b) A plant shown in Fig.3 is described by the Transfer Function k . A unity feedback closed loop system with the plant s ( s + 2) G(s) is stable for all values of k > 0. When the sampling time T = 0.4 sec and T = 3 sec find the stability and Comment on the results. [8] G(s) = Q3) a) Obtain the state model of the following system using Direct programming H(z) = b) z2 + 4z + 3 . ( z 1)( z + 0.9)( z 3) 0 4 1 Diagonalize the following matrix G = 0 3 1 . 0 0 2 [8] [8] OR Q4) a) Obtain the state model of the following system using Parallel programming H(z) = b) ( z + 2) . ( z 1)( z + 0.9)( z 3) [8] 4 1 2 Find the Eigen vectors of the following matrix G = 1 0 2 . [8] 1 1 3 [3864]-283 2 Q5) a) Derive an expression for position form and velocity form of PID controller algorithm. State advantages of velocity form over position form of PID algorithm. [10] b) State and explain the stability of system, asymptotic stability in the large and instability of system in the sense of Liapunov. [8] OR Q6) a) Determine the stability of the equilibrium state of the following system x1 (k + 1) 1 2 x1 (k ) x (k + 1) = 1 4 x (k ) . Also find the Liapunov function. [8] 2 2 b) Obtain the state transition matrix of the following discrete time system 1 0 x(k + 1) = x (k ) + 0.18 0.9 1 1 u(k) and y(k) = [1 0]x(k). [10] SECTION - II Q7) a) Define the terms State Controllability and State Observability. Investigate the system for complete State Controllability and complete State Observability. [8] 1 0 x(k + 1) = x(k) + 4 3 b) 1 0 0 1 u(k) and y(k) = 1 0 0 1 x(k). Consider the system given below. Determine the state feedback gain matrix K such that the system exhibits the deadbeat response. [8] 1 0 x(k) + x(k + 1) = 0.35 1.2 2 If x(0) = find x(2). 3 OR [3864]-283 3 0 1 u(k). Q8) a) b) What is State Observer. Derive an expression for the condition for State Observer. [8] [8] Consider a system 1 1 x(k + 1) = x(k) + 4 3 2 1 u(k) and y(k) = [1 1]x(k). It is desired that the error vector exhibits deadbeat response. Find observer feedback gain matrix Ke. Q9) a) Consider the system x(k + 1) = Gx(k) + Hu(k) and y(k) = Cx(k) where 1 0 G= , H = 0.16 1 0 1 and C = [0 1] Design a full order observer, if the desired eigen values of the observer matrix are z = 0.2 j0.2. [10] b) Explain the Smith predictor and its limitations. [6] OR Q10)Explain the internal model control (IMC) strategies. Design IMC for the system with transfer function G p ( s ) = ~ 2e 3 s . 1 + 10 s [16] Also convert it into conventional controller with approximate dead time Ds 2. as e Ds = Ds 1+ 2 1 Q11)Write a short notes on following system identification methods a) Output Error Method. b) Least Square Method. c) ARX model. OR [3864]-283 4 [18] Q12)Consider the Discrete time control system defined by [18] x(k + 1) = 0.368x(k) + 0.632u(k) and x(0) = 2. Determine the optimal control law to minimize the performance index J= 1 19 [ x(10)]2 + [ x 2 (k ) + u 2 (k )]. 2 2 k =0 Also find Jmin. vvvv [3864]-283 5

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