ADVANCE CONTROLSYSTEMSS

7/30/2019 ADVANCE CONTROLSYSTEMSS

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Code No: R5410205 1IV B.Tech I Semester(R05) Supplementary Examinations, May/June 2009

ADVANCED CONTROL SYSTEMS(Electrical & Electronics Engineering)

Time: 3 hours Max Marks: 80Answer any FIVE Questions

All Questions carry equal marks

1. Define Minimum energy control. State the theorem and prove the same. [4+4+8]

2. For the system

x =

0 12 3

x

find a suitable Lyapunov function V(x). Find an upper bound on time that it takes the system to get

from the initial condition x(0) =

11

to within the area defined by x21 + x

22 = 0.1 . [16]

3. (a) Consider the system with

i. Consider the system with

A =

1 0 00 2 00 0 3

B =

0 11 01 1

andC =

1 2 00 0 1

Obtain equivalent system in controllable companion form

ii. Obtain equivalent observable companion form for the system given in (a).[8+8]

Obtain equivalent system in controllable companion form

(b) Obtain equivalent observable companion form for the system given in (a)[8+8]

4. (a) Explain output Regulator problem?

(b) Consider the linear plant of a system characterized by the transfer function G(s)0=100/s2

. Makethe output C(t) follow a unit step input r(t) minimizing.

J=0

{(x(t) c(t)2 + 0.25u2(t)}dt where u(t) is the actuating signal of the plant. [8+8]

5. (a) Derive the Euler Lagrange equation and the boundary condition to be satisfied by the extermalof the functional

J(x) =t1t0

g(x,

x, t)dt

(b) Find the extermal of the functional

J(x) =

/4

0

(x2x2

)dt

x(0) = 0,x(4 ) is free. [4+4+8]

6. (a) State and explain the principle of optimality.

(b) Obtain the Hamilton Jacobi equation for the system described by

x = u(t) subject to the initial condition x(0) = x0 Find the control law that minimizes

j = 1

2x2 (t1) +t10

x2 + u2

dt : t1 Specified [2+6+8]

7. (a) Explain the effect of inherent nonlinearities on static accuracy.

(b) Derive the Describe function for an on-off nonlinearity with hysteresis. [6+10]

8. Draw a phase-plane portrait of the following system:..

x +.

x + |x| = 0. [16]


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1. Convert the system

x(t) =

1 00 2

x(t)+

11

u(t)

y(t) = [ 1 1 ] x(t)

(a) Find, if possible, a control law, which will derive the system

X(0) = 0 to x1

=

11

in2sec. from

(b) Find, if possible, the state x(0) wheny(t) =1

2e2t + 3

2for u(t) = 1, t > 0 [8+8]

2. (a) Consider the characteristic equation6+25+84+123+202+16+16 = 0Conduct Rouths test and comment on the Lyapunovs stability.

(b) What are the conditions for stability of a system in terms of coefficients of characteristic equation. [6+4+6]

3. (a) Given the system X = Ax + Bu where

A =

1 0 00 2 00 0 3

B =

1 00 11 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. [10+6]

4. (a) Explain Tracking Problem?

(b) Explain Minimum fuel probelm? [8+8]5. Illustrate with an example the problem with terminal time t1 fixed and x (t1) free.

[16]

6. (a) State and explain the principle of optimality.

(b) Obtain the Hamilton Jacobi equation for the system described by

x = u(t) subject to the initial condition x(0) = x0 Find the control law that minimizes

j = 1

2x2 (t1) +t10

x2 + u2

dt : t1 Specified [2+6+8]

7. Obtain the Describing function analysis for the system shown in Figure 1

Figure 1:

8. Draw a phase-plane portrait of the system defined by.

x 1 = x1 + x2.

x 2 = 2x1 + x2. [16]


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1. Define Minimum energy control. State the theorem and prove the same. [4+4+8]

2. State, prove and explain Lyapunovs stability theorm. Also explain what are the sufficient conditions of stability.[2+2+4+8]

3. (a) Explain the linear system with full-order state observer with neat block diagram.

(b) Design a full-order state observer for the given state model.

Given A =

1 1

2 1

; C = [1, 0]

and given values are 1= 5,2= 5. [8+8]

4. A plant is described by the equations

x =

0 10 0

x +

01

ux1(0) = 1x2(0) =Choose the feedback law u=K[x1+x1]

Find the value of k so that J=1/20

(x21 + x22 + u

2)dt is minimized when

(a) = 0

(b) = 1Also determine the values of minimum J in two cases. [4+4+4+4]

5. (a) Derive the transversality condition in extermination of functions.

(b) Prove that for the functional

J(x) =t1t0

A(x, t)

1 +

2

x dt

the transversality condition reduces to orthogonlity is

x

y = 1 where y(t) is the curve on which the movableright points lies. [8+8]

6. For the system

x = u, with |u| 1, find the control which drives the system from an arbitrary initial state to the origin and

minimizes J =t10

|u(t)| dt, t1 is free. [16]

7. (a) Explain the popular nonlinearities.

(b) For the system shown in Figure 1, determine the amplitude and frequency of the limit cycle. [6+10]

Figure 1:

8. A linear second-order servo is described by the equation..

e + 2 n

.

e + 2ne = 0

where = 0.15, n = 1 and e(0) = 1.5,

e(0) = 0Determine the singular points. Construct the phase trajectory, using the method of isoclines. [16]


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1. Consider a system described by the state equation0x = A(t)x(t) + bu(t)

Where A(t) =

1 et

0 1

; b =

01

Is this system controllable at t=0? If yes, find the minimum-energy control to drive it from x(0)=0 to

x =

11

at t=1

[8+8]

2. For the system

x =

0 12 3

x

find a suitable Lyapunov function V(x). Find an upper bound on time that it takes the system to get

from the initial condition x(0) =

11

to within the area defined by x2

1+ x2

2= 0.1 . [16]

3. (a) For a Mixing tank system design a deadbeat observer.

(b) Give the differences between deadbeat state feedback controller and deadbeat observer.

4. A first - order system is described by the differential equation x(t)=2x(t) + u(t)Find the control lawthat minimizes the performance index

J=1/2tf0

3x2 + 1

4u2

dx where tf=1 sec. [16]

5. Illustrate with an example the problem with terminal time t1 fixed and x (t1) free.[16]

6. (a) State and prove optimal control problem based on dynamic programming in discrete time system

(b) Explain the principles of causality and invariant imbedding. [2+6+8]

7. (a) Explain the subharmonic oscillations and self-excited oscillations.

(b) Derive the Describe function for Dead-Zone nonlinearity. [8+8]

8. Obtain the trajectory representing the response of the system shown in Figure 1, when it is subjectedto the inputr(t) =R1 1(t) + R2 1(t-2) + R3 1(t-3)where 1(t - ti) is a unit-step function occurring at t = ti . Assume that the system is at rest initially.[16]

Figure 1:

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ADVANCE CONTROLSYSTEMSS