Advanced Control Systems Rr

Set No. 1

Code No: RR321303

III B.Tech II Semester Supplementary Examinations, Aug/Sep 2007 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 in 2 sec. X(0) = 0 to x1 = 1 (b) Find, if possible, the state x(0) when y(t) = 21 e−2t + [8+8]

3 2

f or u(t) = 1, t > 0

2. (a) Explain the procedure to formulate a Lyapunov function and to investigate the stability of a linear system. (b) Check the stability of the system described by x˙ 1 = x2 x˙ 2 = −x1 − x21 x2

[6+10]

3. (a) Consider the system defined by     0 0 1 0    0  0 0 1 B = A = 1 −1 − 5 − 6

By using the state feedback control µ = −kx it is desired to have the closedloop poles at S = −2 ± j4 and S = −10. Determine the state feedback gain matrix.

(b) Explain design steps for pole placement.

[10+6]

4. A first - order system is described by the differential equation x(t)=2x(t) + u(t). Find the control law that minimizes the performance index
Rtf 3x2 + 41 u2 dx where tf = 1 sec . J = 1/2 0

[16]

5. Illustrate with an example the problem with terminal time t1 fixed and x(t1 ) free. [16] 6. (a) Derive the transfer matrix relation from state space representation 1 of 2

Set No. 1

Code No: RR321303

(b) The state space triple (A, B, C,) of a system is given by [6+10]       0 0 0 1 0 1 1 0     A = 2 3 0 ;B = 1 0 ;C = 0 0 1 0 1 1 1 1 Calculate the input and output decoupling zeros, if any. Is the matrix A cyclic? Find out the transfer matrix T(s). 8 7. Design a phase lead compensator such that a system with G(s) = s(s+1)(s+4) has 0 a phase margin of 45 and a steady state error of 0.25 due to ramp input Write a MATLAB Programme for the above problem. [16]

8. (a) Write short notes on the following in MATLAB i. String evaluation ii. Switch giving suitable examples. (b) Describe about error and warning message in MATLAB ⋆⋆⋆⋆⋆

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[8+8]

Set No. 2

Code No: RR321303

III B.Tech II Semester Supplementary Examinations, Aug/Sep 2007 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 + [8+8]

3 2

f or u(t) = 1, t > 0

2. (a) Determine the stability of the origin of the following system: x˙ 1 = −x1 + x2 − x1 (x21 + x22 ) x˙ 2 = −x1 − x2 + x2 (x21 + x22 ) (b) Consider the second-order system     x1 −1 − 2 ˙ X = AX, where X = , A = x2 1 −4

[8+8]

Find the real symmetric matrix P, which satisfies stability condition of Lyapunov’s method. 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 Minimum - Time problem? (b) Explain State Regulator problem in brief?

[8+8]

5. Illustrate with an example the problem with terminal time t1 and x(t1 ) free. [16] 6. (a) Derive the condition to be satisfied for the system nonminimal. Also define input-decoupling zeros and output decoupling zeros. 1 of 2

Set No. 2

Code No: RR321303

s (b) For the given transfer matrix T(s) of dimension p × m be T (s) = s2 +s+1 with p=m=1 Verify the following state space models realise the this transfer matrix?     0 0 1 0 1    0  0 1 1   0 ; i. A =  ; B =  1   −1 −2 −2 1  0 0 0 0 −3       0 0 1 ;B = ;C = 0 1 ii. A = [6+10] −1 −1

7. Obtain polar plot for the following system by writing MATLAB Programme. [16] G(s)H(s) =

40 s(1+0.2s) (1+0.01s)

8. (a) Explain some MATLAB commands and explain how they will be used with examples. (b) Write short notes about the following commands with examples. i. ii. iii. iv.

Path Clear Who load ⋆⋆⋆⋆⋆

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[8+8]

Set No. 3

Code No: RR321303

III B.Tech II Semester Supplementary Examinations, Aug/Sep 2007 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] 2. (a) Define Lyapunov?s 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) Consider the system with    0 1 0 0    1 0 2 0 B = A= 1 0 0 3

  1 1  0 and C = 0 1

2 0

0 1



Obtain equivalent system in controllable companion form.

(b) Obtain equivalent observable companion form for the system given in (a) [8+8] 4. Consider described    by the equation  a system • 0 0 1 u x1 (0) = x2 (0) = 1.Choose the f eedback law u = x + X = 1 0 0 −x1 − kx2 . Rα (a) Find the value of K so that J = 1/2 (x21 + x22 ) dt 0

(b) Find Sensitivity of J with respect to K.

[8+8]

5. (a) Find the points in the three-dimensional eudidean space that extermize the function f (x1 , x2 , x3 ) = x21 + x22 + x23 and lies on the intersection of the surfaces x3 = x1 x2 + 5 x1 + x2 + x3 = 1 (b) Explain about neighborhood of a function. 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, 1 of 2

[10+6] [6+5+5]

Set No. 3

Code No: RR321303 (c) P(s) is column proper  s+1 s+2  s2 s2 +1 i. T (s) = 2 2s+3 2  s s +1  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. Obtain polar plot for the following system by writing MATLAB Programme. [16] G(s)H(s) =

40 s(1+0.2s) (1+0.01s)

8. Explain the following terms in MATLAB (a) M - files (b) MATLAB - work space (c) Expansion of MATLAB.

[6+5+5] ⋆⋆⋆⋆⋆

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Set No. 4

Code No: RR321303

III B.Tech II Semester Supplementary Examinations, Aug/Sep 2007 ADVANCED CONTROL SYSTEMS (Electronics & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. Explain observability test for continuous varying system. Define the theorem and prove it. [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) 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. Consider described    by the equation  a system • 0 0 1 u x1 (0) = x2 (0) = 1.Choose the f eedback law u = x + X = 1 0 0 −x1 − kx2 . Rα (a) Find the value of K so that J = 1/2 (x21 + x22 ) dt 0

(b) Find Sensitivity of J with respect to K.

[8+8]

5. (a) Find the Euler Lagrange equations and the boundary conditions for the extremal of the functional π

J(x) =

R2

(x˙ 21 + 2x1 x2 + x˙ 22 ) dt

0

x1 (0) = 0, x1 ( π2 ) is tree x2 (0) = 0 x1 ( π2 ) = −1 (b) What is a Hamiltonian. Formulate the optimal control problem in terms of Hamiltonian. [8+8] 6. (a) Derive the transfer matrix relation from state space representation 1 of 2

Set No. 4

Code No: RR321303

(b) The state space triple (A, B, C,) of a system is given by [6+10]       0 0 0 1 0 1 1 0     A = 2 3 0 ;B = 1 0 ;C = 0 0 1 0 1 1 1 1 Calculate the input and output decoupling zeros, if any. Is the matrix A cyclic? Find out the transfer matrix T(s). 7. Explain how the P I D controller is designed by writing programme in MATLAB. [16] 8. (a) Explain about input data formats in MATLAB with suitable examples. (b) Describe about file operators in MATLAB with examples. ⋆⋆⋆⋆⋆

2 of 2

[8+8]