Rr322204-digital-and-optimal-control-systems

Set No. 1

Code No: RR322204

III B.Tech Supplimentary Examinations, Aug/Sep 2008 DIGITAL AND OPTIMAL CONTROL SYSTEMS (Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. Given the transfer function 4(z−1)(z 2 +1.2z+1) G(z) = (z+0.1)(z 2 −0.3z+0.8) , Obtain (a) a series realization diagram and

[8]

(b) a parallel realization diagram, using pure delay elements z −1 .

[8]

2. Consider the discrete control system represented by the following transfer function −1 G(z) = 1−z1+0.8z −1 +0.5z −2 Obtain the state representation of the system in the observable canonical form. Also find its state transition matrix. [16] 3. Investigate the controllability and observability of the following system         1 −1 x1 (k) 1 −2 x1 (k + 1) u(k) + = 0 0 x2 (k) 1 −1 x2 (k + 1)       x1 (k) 1 0 y1 (k) = x2 (k) 0 1 y2 (k) [16] 4. Show that the transfer function U(s) / E(s) of the PID controller shown below.    1 U (s) T1 + T2 T1 T2 · s 1+ = Ko + E(s) T1 (T1 + T2 ) T1 + T2 Assume the gain k is very large compared with unity, or k >>1as shown in the figure 4 [16]

Figure 4 5. Control a system, defined by .

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Set No. 1

Code No: RR322204 X = AX + Bu

Y = CX where,   0 1 A = −2 −3

B =



0 2



C =



1 0



It is desired to have eigenvalues at -3.0 and -5.0 by using a state feedback control u = - KX. Determine the necessary feedback gain matrix k and the control signal u. [16] 6. (a) State and explain the minimum - time problems. Describe its performance index. [6] (b) Let f (X) = -x1 x2 and let g(X) =x21 + x22 − 1. What are the potential candidates forminima   of f subject  to the constraint g=0?. Show that the points 1 1 1 1 √ , √ and − √2 , - √2 actually provide the minima. [10] 2 2 7. (a) Explain the concepts of variational calculus. (b) Explain the formulation of Variational Calculus using Himiltonian method. [8+8] 8. (a) Explain the minimal D.O. realization algorithm (b) The state space triple for a multivariable control system is given by     0 0 −1 0 0 0 1 0  1 0 2  0 1  0 0 0       0 1 0   0 1  0 3 0    A=  0 0 −2 0 −4 0  ; B =  1 1  ;      0 0 0  0 1  1 −1 0  0 0 1 0 −1 −1 0 1 C=



0 0 1 0 0 0 0 0 2 0 1 0



Find the corresponding transfer matrix without calculating inverse of any matrix of order more than 2. [8+8] ⋆⋆⋆⋆⋆

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Set No. 2

Code No: RR322204

III B.Tech Supplimentary Examinations, Aug/Sep 2008 DIGITAL AND OPTIMAL CONTROL SYSTEMS (Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Obtain the Z-transform of i. f1 (t) = a1 (1 − e−at ) where ‘a’ is a constant
ii. f2 (t) = 9k 2k−1 − 2k + 3 k≥0

(b) Show that   z k (k − 1) ..... (k − h + 1) ak−h =

(h!)z . (z−a)h+1

2. (a) Obtain a state space presentation of the following system: Y (z) z −1 +2z −2 = 1+0.7z −1 +0.12z −2 U (z) Choose state variation state matrix as a diagonal matrix.

[4] [4] [8]

[9]

(b) State and prove the properties of the state transition matrix of discrete time system [7] 3. Investigate the stability of the following system and calculate the range of K, over which the system is stable. Select the sampling period T = 0.4 sec(figure3) [16]

Figure 3 4. What is a dead beat response? Explain the characteristic of the poles of the transfer function of a system that has a dead beat response. [16] 5. A discrete time  the plant equation  regulator system has 4 2 −1 u(k) X(k) + X(k + 1) = 3 −1   1 y(k) = 1 1 X(k) + 7u(k) Design a state feedback control algorithm with u(k) = -KX(k) which places the closed loop characteristic roots at ±j0.5. [16] 6. (a) What are the major theoretical approaches for design of optimal control. Explain one of the approaches in detail. (b) Find the q extremalfor the functional • Rt1 1+x2 j(x) = dt x(0) + 0; x(t1 ) = t1 − 5 x 0

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[8+8]

Set No. 2

Code No: RR322204

7. (a) State and explain the Pontryagin’s minimum principle. (b) Find the points in the three-dimensional euclidean space that minimizes the function f (x1 , x2 , x3 ) = x21 + x22 + x23 and lie on the intersection of the surfaces x3 = x1 x2 + 5 x1 + x2 + x3 = 1 [8+8] 8. (a) Derive the transfer matrix relation from state space representation. [6] (b) The state space triple  0  A= 2 1

(A, B, C) of a system is given by      1 0 0 0 1 1 0 3 0 ; B =  1 0 ; C = 0 0 1 0 1 1 1

Calculate the input and output decoupling zeros, if any. Is the matrix A cyclic? Find out the transfer matrix T(s). [10] ⋆⋆⋆⋆⋆

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Set No. 3

Code No: RR322204

III B.Tech Supplimentary Examinations, Aug/Sep 2008 DIGITAL AND OPTIMAL CONTROL SYSTEMS (Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. For the digital control system shown below determine C(z) and C(s). (figure 1) [16]

Figure 1 2. Obtain the Jordan canonical form realization for the following transfer function G(z) =

3z 2 − 4z + 6
3 z − 31

[16]

3. Explain Liapunov stability criterion for the linear time variant systems. [16] 4. Write notes on the following: (a) Digital controller design using Bilinear transformation.

[6]

(b) Difference equation solution using Z-transform method.

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(c) Dead beat response.

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5. Write notes on the following (a) Pole placement design of multi input- multi output system by state feedback (b) Pole placement design by incomplete feedback 6. Consider the system x˙ 1 = x2 + u1 x˙ 2 = u2 Find the optimal control u*(t) for the functional R4 J = 21 (u21 + u22 )dt .

[8+8]

0

Given x1 (0)=x2 (0) = 1; and x1 (4)=0.

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[16]

Set No. 3

Code No: RR322204

7. (a) With suitable diagrams illustrate the one point is fixed end, terminal time t1 free and x(t1 ) is specified problem and derive the necessary conditions of variational calculus. (b) For the system d2 y =u dt2 with |u| ≤ 1, find the control which drives the system from an arbitrary initial state to the origin in a condition satisfying |y| ≤ 0.5 in the minimum time. [8+8] 8. (a) With step-by-step procedure explain the observable realization algorithm. (b) The transfer matrix of a system is T (s) = R(s)P

−1

(s) =



s+1 0 1 1



s2 0 −1 s − 1

−1

Obtain controllable realization in the state space form. Is the realization expected to be minimal? [8+8] ⋆⋆⋆⋆⋆

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Set No. 4

Code No: RR322204

III B.Tech Supplimentary Examinations, Aug/Sep 2008 DIGITAL AND OPTIMAL CONTROL SYSTEMS (Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. A slowly changing continuous-time signal f(t) is sampled every T seconds. (a) Assuming that the changes in f(t) are very slow compared to the sampling frequency, show that in the z-domain, 1−z −1 T

corresponds to differentiation. Obtain a block diagram of this differentiator using a pure delay element z −1 . [8] (b) What is an equivalent integrator in the z-domain? Draw a graph of the output in each case when the input is a unit step sequence. [8] 2. For the following discrete control system represented by z −1 (1+z −1 ) G(z) = (1+0.5z−1 )(1−0.5z−1 ) Obtain the state representation of the system in the controllable canonical form. Also find its state transition matrix. [16] 3. Define controllability and observability of discrete time systems. For the following system, Y (z) z −1 (1 + 0.8z −1 ) = U (z) 1 + 1.3z −1 + 0.4z −2 Determine whether the system is observable and controllable. [16] 4. What is a dead beat response? Explain the characteristic of the poles of the transfer function of a system that has a dead beat response. [16] 5. (a) Draw the schematic diagram of a state feedback control system and explain how stability can be improved by state feedback. (b) State and explain the necessary and sufficient conditions for arbitrary pole placement. [8+8] 6. (a) State and explain the minimum - time problems. Describe its performance index. [6] (b) Let f (X) = -x1 x2 and let g(X) =x21 + x22 − 1. What are the potential candidates forminima   of f subject  to the constraint g=0?. Show that the points 1 1 1 1 √ , √ and − √2 , - √2 actually provide the minima. [10] 2 2 7. (a) State and explain the constrained optimization problem. 1 of 2

Set No. 4

Code No: RR322204

(b) Consider the system x˙ 1 (t) = x2 (t) x˙ 2 (t) = −x2 (t) + u(t) Find the optimal control u*(t) which has to minimize Rt1 J = 12 (x21 + u2 )dt t0

and the control inequality constrains are given by |u(t)| ≤ 1 for t ∈ [t0 , t1 ]

[8+8] 8. (a) Derive the transfer matrix relation from state space representation. [6] (b) The state space triple  0  A= 2 1

(A, B, C) of a system is given by      0 0 1 0 1 1 0    3 0 ; B= 1 0 ; C= 0 0 1 0 1 1 1

Calculate the input and output decoupling zeros, if any. Is the matrix A cyclic? Find out the transfer matrix T(s). [10] ⋆⋆⋆⋆⋆

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