R05321303-advanced-control-systems

Set No. 1

Code No: R05321303

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. A feedback system is characterized by the closed loop transfer function s2 +3s+3 T (s) = s3 +2s 2 +3s+1 Construct a state model for this system and also give the block diagram representation for the same. [16] 2. (a) Consider  the system    −1 0 1 0 ˙    1 −2 0 X= X + 1 U 0 0 −3 1 and output  y= 1 1 1 X Transform the system into observable canonical form. (b) The state equations of a system are given below. Determine whether the system  andobservable.   is completely controllable 2 −6 −18 −6 3 1  X +  −3  u X˙ =  2 7  −4 −8  −3 y= 1 3 1 X [8+8] 3. What are the different types of non-linearities. Explain each of them in detail. [16] 4. Determine the type of singularity for each of the following differential equations. Also locate the singular points on the phase plane x¨ + 3x˙ + 2x = 0. Draw the phase trajectory. [16] 5. Determine the stability of the origin of the following system. x˙ 1 = x2 x˙ 1 = −g(x1 ) + x2

[16]

6. A single-input sytem  is described  the following state equation.  by  10 −1 0 0 X˙ =  1 −2 0  X +  1  u 0 0 1 −3 Design a state feedback controller which will give closed loop poles at -1±j2, -6. Draw a block diagram of the resulting closed loop system. [16] 7. (a) Find the curve with minimum arc length between the point x(0) = 0 and the curve θ(t) = t2 − 10t + 24 (b) Discuss state variable and control inequality constraints. 1 of 2

[8+8]

Set No. 1

Code No: R05321303

8. (a) Explain formulation of the optimal control problem. (b) Explain formulation of the optimal control problem for the minimum-time problem. [8+8] ⋆⋆⋆⋆⋆

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Set No. 2

Code No: R05321303

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. Obtain the state space representation of the system shown in Figure 1 in which x1 , x2 , x3 constitute the state vector. Also find transfer function matrix. [16]

Figure 1 2. (a) Consider    the system  0 −1 0 1 ˙    1 −2 0 X + 1 U X= 1 0 0 −3 and output  y= 1 1 1 X Transform the system into observable canonical form. (b) The state equations of a system are given below. Determine whether the system  andobservable.   is completely controllable 2 −6 −18 −6 ˙ u    −3 2 3 1 X + X= 7  −4 −8  −3 y= 1 3 1 X [8+8]

3. Derive the expression of describing function for deadzone non-lineartiy.

[16]

4. Consider a system with an ideal relay as shown in figure 4b. Determine the singualar point. Construct phase trajectories, corresponding to initial conditions, (a) c(0)=2, c(0) ˙ = 1 and (b) c(0) = 2, c(0) ˙ = 15 . Take r=2 Volts and M=1.2 Volts.

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[16]

Set No. 2

Code No: R05321303

Figure 4b 5.  Find  a Lyapunov function   for   the following system . x˙ 1 x1 −1 1 = x˙ 2 2 −3 x2

[16]

6. What is a Reduced-orfder observer? Derive the equations of Reduced order observer? [16] 7. (a) Given x˙ = −x + u x(0) = x0 , x(2) = x1 Find u* that minimizes R2 J = (x2 + u2 ).dt 0

(b) Discuss minimization of function.

[8+8]

8. Suppose that the system x˙ 1 (t) = x2 (t); x˙ 2 (t) = µ(t) is to be controlled to minimize the performance measure the R2 J(x1 , u) = 12 u2 dt. Find a set of necessary condtions for optimal control. [16] 0

⋆⋆⋆⋆⋆

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Set No. 3

Code No: R05321303

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. For the system shown in Figure 1, choose υ1 (t) and υ2 (t) as state variables and obtain the state variable representation. The parameters of the system are given as R=1MΩ; C=1µF. Find the state transition matrix. [16]

Figure 1 2. Consider    the system  0 −1 0 1 ˙    1 −2 0 X − 0 U X= 1 0 0 −3 and output  y= 1 1 0 X Transform the system into (a) controllable canonical form and (b) observable canonical form.

[8+8]

3. Derive the expression of describing function for deadzone non-lineartiy.

[16]

4. Linear second order servo is described by the equation e¨ + 2τ wn e˙ + wn2 e = 0, where τ =0.15, wn = 1 rad/sec e(0)=1.5 and e˙ (0) = 0. Determine the singular point. Construct the phase trajectory, using the method of isoclines. [16] 5. Determine whether the folowing quardratie form in negative difenate or positive definite. Q = −x2 − 3x22 − 11x23 + 2 x1 x2 − 4x2 x3 − 2x1 x3 [16] 6. What is a Reduced-orfder observer? Derive the equations of Reduced order observer? [16] 7. (a) The functional given by Rt1 J(x) = (2x + 12 x˙ 2 ) dt 1

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Set No. 3

Code No: R05321303 x(1) = 2, x(t1 ) = 2, Final the extremals.

t1 > 1 is free.

(b) Discuss the application of Euler-Lagrange equation and derive the equation. [8+8] 8. Formulate the two point boundary -value problem which when solved, yields the optimal control u∗ (t) for the system x˙ 1 = x2 x˙ 2 = x1 + (1 − x21 ) x2 + u x(0) = [ 1 0 ]T R2 J = 12 (2x21 + x22 + u2 ) dt 0

when (a) u(t) is not bounded and (b) |u(t)| ≤ 1.0

[8+8] ⋆⋆⋆⋆⋆

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Set No. 4

Code No: R05321303

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. A feedback system is characterized by the closed loop transfer function s2 +3s+3 T (s) = s3 +2s 2 +3s+1 Construct a state model for this system and also give the block diagram representation for the same. [16] 2. Block diagram representation of a linear time-invariant system is given in figure 2. Verify its complete controllable and observable conditions. [16]

Figure 2 3. The block diagram of a system with hysteresis is shown in Figure 3. Using describing function method, determine whether limit cycle exists in the system. If limit cycles exists, determine their amplitude and frequency. [16]

Figure 3 4. Linear second order servo is described by the equation e¨ + 2τ wn e˙ + wn2 e = 0, where τ =0.15, wn = 1 rad/sec e(0)=1.5 and e˙ (0) = 0. Determine the singular point. Construct the phase trajectory, using the method of isoclines. [16] 5. Determine the stability of the following system using Lyapunov method. x˙ 1 = x2 x˙ 2 = −x1 − x21 .x2 1 of 2

[16]

Set No. 4

Code No: R05321303

6. (a) Show that the zero‘s of a scaler system are invariant under linear state feedback to the input. (b) For a single input system explain pole placement by state feedback.

[8+8]

7. (a) Given x˙ = −x + u x(0) = x0 , x(2) = x1 Find u* that minimizes R2 J = (x2 + u2 ).dt 0

(b) Discuss minimization of function.

[8+8]

8. (a) Explain formulation of the optimal control problem. (b) Explain formulation of the optimal control problem for the minimum-time problem. [8+8] ⋆⋆⋆⋆⋆

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