Rr321303-advanced-control-systems

Set No. 1

Code No: RR321303

III B.Tech II Semester Supplimentary Examinations, Aug/Sep 2008 ADVANCED CONTROL SYSTEMS (Electronics & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. Suppose you are given a n-dimensional linear time-invariant system. How do you transform into a observability canonical form. State and prove the theorem used. [16]   0 1 x find a suitable Lyapunov function V(x). Find 2. For the system x˙ = −2 − 3 an upper on time that it takes the system to get from the initial condition   bound 1 to within the area defined by x21 + x22 = 0.1. [16] x(0) = 1 3. (a) Explain the different methods of determination of observer gain matrix. (b) Consider the system described by the state model:   • −1 +1 ; C = [1, 0] where A = X = Ax 1 −2 Y = Cx Obtain the state observer gain matrix using all 3 methods. The desired given values for the observer matrix are µ1 = −5, µ2 = −5. [6+10] 4. For the discrete - time system given Gp (s) = 1/S. −ST G0 (s) = 1−eS , r(t)=unit step T=1 sec. Find optimal transfer function T*(z) so that output C(t) follows input r(t) minimizing: [16] α P Je = [r(kT ) − c(kT )]2 with k=0

Je =

α P

u2 (kT ) = 0.5.

k=0

5. Illustrate with an example the problem with terminal time t1 and x(t1 ) free. [16] 6. Express the following T(s) given below as P0−1 (s)Q0 (s), with P0 (s) and Q0 (s) relatively left prime and P0 (s) row proper. Find the minimal realization. Determine the values of the controllability index and observability index for the minimal realization. [6+5+5] 7. Write MATLAB commands for drawing Nyquist plot and obtaining the stability of the system whose transfer function is given by: [8+8] 1 of 2

Set No. 1

Code No: RR321303 G(s)(H(s) =

K(s+1) (s+1)(s+4)

8. (a) What is MATLAB ?Explain its merits and demerits and give some its features? (b) Explain the following in connection with MATLAB: i. ii. iii. iv.

Command window Command line editing Format command Starting MATLAB. ⋆⋆⋆⋆⋆

2 of 2

[8+8]

Set No. 2

Code No: RR321303

III B.Tech II Semester Supplimentary Examinations, Aug/Sep 2008 ADVANCED CONTROL SYSTEMS (Electronics & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. Convert the system     1 −1 0 u(t) x(t) + x(t) = 1 0 −2 y(t) =



 1 1 x(t)

(a) Find, if possible, acontrol  law, which will derive the system from 1 X(0) = 0 to x1 = in 2 sec. 1 (b) Find, if possible, the state x(0) when y(t) = 21 e−2t +

3 2

f or u(t) = 1, t > 0. [8+8]

2. (a) Define Lyapunovs stability and Instability Theorem. (b) Suppose you are given a linear continuous time autonomous system, how do you decide whether a system is globally asymptotically stable? [8+8] 3. (a) Draw the block diagram and deduce the expression of transfer function for the controller-observer. (b) Consider the system defined by:     • 1 −1 1 u x + X= 0 0 2 Show that this system cannot be stabilized by the state feedback control µ = −kx whatever matrix k is chosen. [8+8] •

4. (a) A first order system x = (−x + u) is to be controlled to minimize: R1 J = 1/2 (x2 + u2 )dt 0

Find the optimal control law.

(b) Explain the limitations of transfer function approach for optimal control problem. [8+8] 5. Illustrate with an example the problem with terminal time t1 fixed and x(t1 ) free. [16] 6. (a) Derive the relations required for obtaining an observable realization algorithm of a given transfer matrix T(s). 1 of 2

Set No. 2

Code No: RR321303

(b) Obtain state space controllable realization of a system with transfer matrix. [6+10] −1   s + 4 2(s + 1) 2(s − 1) s + 1 T (s) = 0 s2 − s + 4 4 −s 7. Write the MATLAB Programme for finding the error constants for: (a) step (b) ramp. (c) parabolic inputs and steady state error of the system for all the inputs whose transfer function is given by: G(s)H(s) =

10(s+4) . (s+1) (s+3) (s+5)

[4+4+4+4]

8. (a) How do you perform the following operations using MATLAB? i. To find eigen values ii. Matrix multiplication Illustrate with examples. (b) Write short notes on: i. Relational and logic operations ii. Matrices operations and functions using MATLAB techniques. ⋆⋆⋆⋆⋆

2 of 2

[8+8]

Set No. 3

Code No: RR321303

III B.Tech II Semester Supplimentary Examinations, Aug/Sep 2008 ADVANCED CONTROL SYSTEMS (Electronics & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. State and prove canonical decomposition theorem. [16]   0 1 2. (a) For the system x˙ = x, find a suitable Lyapunov function v(x). −1 − 1 Obtain an upper bound on the response time such that it takes the system to go from a point on the boundary of the closed curve V(x) = 100 to a point within the closed curve V(x) = 0.05. (b) What are the different types of stability in the sense of Lyapunov and briefly explain each of them. [8+8] 3. (a) Draw the block diagram and deduce the expression of transfer function for the controller-observer. (b) Consider the system defined by:     • 1 −1 1 u x + X= 0 0 2 Show that this system cannot be stabilized by the state feedback control µ = −kx whatever matrix k is chosen. [8+8] 4. (a) Explain output Regulator problem? (b) Consider the linear plant of a system characterized by the transfer function G(s) = 100/s2 . Make the output C(t) follow a unit step input r(t) minimizing. Rα J = {(x(t) − c(t)2 + 0.25u2 (t) }dt where u(t) is the actuating signal of the 0

plant.

[6+10]

5. Illustrate with an example the problem with terminal time t1 free and x(t1 ) fixed. [16] 6. Break up the following transfer matrices into R(s) and P(s): (a) T (s) = R(s)P −1 (s) (b) R(s) and P(s) are relatively right prime, (c) P(s) is column proper:  s+1 s+2  s2 s2 +1 i. T (s) = 2 2s+3 s

s2 +1

1 of 2

[6+5+5]

Set No. 3

Code No: RR321303 

 ii. T (s) = 

(s−2)(s+1) s(s−1)2 − 1s 2 s(s−1)

1 (s−1)2

0 1 s−1



 .

7. Write the MATLAB commands for drawing root locus for the following system with, [16] G(s)H(s) =

K (s+1) (s+3) (s+4)

8. (a) How do you perform the following operations using MATLAB? i. To find eigen values ii. Matrix multiplication Illustrate with examples. (b) Write short notes on: i. Relational and logic operations ii. Matrices operations and functions using MATLAB techniques. ⋆⋆⋆⋆⋆

2 of 2

[8+8]

Set No. 4

Code No: RR321303

III B.Tech II Semester Supplimentary Examinations, Aug/Sep 2008 ADVANCED CONTROL SYSTEMS (Electronics & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. State the basic theorem for determining the concept of controllability of time varying system utilizing state transition matrix. Explain the same with proof. [16]   0 1 x find a suitable Lyapunov function V(x). Find 2. For the system x˙ = −2 − 3 an upper on time that it takes the system to get from the initial condition   bound 1 to within the area defined by x21 + x22 = 0.1. [16] x(0) = 1 3. (a) Given the system X = Ax + Bu where     1 0 0 1 0 2 0  B = 0 1  A= 0 0 0 3 1 1

Design a linear state variable feedback such that the closed-loop poles are located at -1,-2 and -3.

(b) Explain the concept of Stabilizability. 4. (a) The second order linear System: X˙ 1 (t) = −0.2x1 (t) + x2 (t) + u(t) X˙ 2 (t) = −0.5x2 (t) + 0.5u(t) Minimize the performance index. R5 J = 1/2 x21 (t) + x22 (t) + u2 (t) dt.

[10+6]

0

(b) Draw block diagram of optimal control system for (a).

[8+8]

5. Illustrate with an example the problem with terminal time t1 free and x(t1 ) fixed. [16] 6. (a) Derive the relations required for obtaining a controllable realization algorithm of a given transfer matrix T(s).         0 0 0 1 0 −2 1 2 0 0     (b) Given A = 2 1 0 ; B = 1 0 ; C = ;E = . −2 2 0 1 1 0 1 0 0 1 Find the corresponding transfer matrix. [8+8] 7. Obtain Bode plot for the following system and write a programme in MATLAB. [16] G(s)H(s) =

100 s(1+0.1s) (1+0.05s)

1 of 2

Set No. 4

Code No: RR321303

8. (a) What is MATLAB ?Explain its merits and demerits and give some its features? (b) Explain the following in connection with MATLAB: i. ii. iii. iv.

Command window Command line editing Format command Starting MATLAB. ⋆⋆⋆⋆⋆

2 of 2

[8+8]