World€¦ · model.obtain the canonical form of state model = + u and y= Evaluate 2 4 convert the...

OBJECTIVE:

This subject deals with state space, describing function, phase plane and stability analysis including

controllability and observability .it also deals with modern control and optimal control systems.

GROUP-I (SHORT ANSWER TYPE QUESTIONS)

S.No QUESTION

BLOOMS

TAXONOM

Y LEVEL

COURSE

OUTCOME

UNIT-I

STATE SPACE ANALYSIS

SHORT ANSWER TYPE QUESTIONS

1 What are the advantages of state space analysis? Evaluate 1

2 What are draw backs of transfer function model analysis Analyze 1

3 What is state and state variable? Remember 1

4 What is state vector?

.

Understand 1

5 What state model of nth

order system? Understand 1

6 What is state space? Understand 1

7 What is i/p and o/p space? Understand 1

8 What are the advantages of state space model using physical variable?. Remember 2

9 What are the properties of state transition matrix? Understand 1

10 Write resolving matrix? Understand 1

LONG ANSWER TYPE QUESTIONS

1 Explain the state variable and state transition matrix? Understand 1

2 What is state? Write about concept of state variables? Understand 1

3 What is state? Write about concept of state variables? Understand 1

4 Write shot notes on formulation of state equations? Analyse 1

5 Discuss about state transition matrix? Understand 1

6 Derive the expression for the calculation of the transfer function from the

state variables for the analysis of system?

Understand 1

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7 Write short notes on canonical form of representation .list its advantages

and disadvantages?

Understand 2

8 derive the controllable canonical form for the following transfer function

= Evaluate

2

9 derive the observable canonical form for the following transfer function

= Understand

2

10 obtain the Jordan canonical form of state space representation for the

following transfer function

=

Understand 2ANALYTICAL QUESTIONS

1 linear time invariant system is described by the following

statemodel.obtain the canonical form of the state model.

= + u and y=

Understand 1

2 convert the following system matrix to canonical form

A=

Evaluate 2

3 a linear time invariant system is described by the following state

model.obtain the canonical form of state model

= + u and y=

Evaluate

2

4 convert the following system matrix to canonical form and hence

calculate the STM A=

Analyse 2

5 consider the transfer function system = .obtain the state space

representation of the system in

(a) controllable canonical form

(b) observable canonical form

Evaluate 2

6 for the state equation =Ax

Where A= .find the intial condition vector x(0) which

will excite only the mode corresponding to eigen value with the most

negative real part.

Evaluate 1

7 a system variables for the state variable representation of the system are,

A= ,B=

Determine the complete state response and the output response of the

system for the intial state X(0)=

Evaluate

1

8 obtain the time response of the following systems,

[x]+ u and Y=

Where u(t) is unit step input and the intial condition

(0)=0, (0)=0,

Evaluate 1

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9 find the state response of the system shown in figure

=

+

u,

=

Understand

1

10 consider the differential equation system given by

y(0)=0.1,y(0)=0.05.

Obtain the response y(t),subjected to the given intial condition

Understand 1

UNIT-II

CONTROLANBILITY AND OBSERVABILITY


1 Define controllability?

Remember

3

2 State condition for controllability by Kalman’s method? Analyze

3

3 State condition for controllability by Kalman’s method? Analyze

3

4 Define observability?

Analyze

4

5 State condition for by observabilityKaman’s method? Evaluate

4

6 What is the advantage and disadvantage in Kaman’s test for

observability?

Understand 4

7 State the duality between controllability and observability? Remember 4

8 How the modal matrix can be determined?

Remember 3,4

9 What are Eigen values? Evaluate

3,4

10 What is need for observability test? Remember

4


1 state and explain controllability and observability? Understand 3,4

2

Write the necessary and sufficient conditions for complete state

controllability and observability?

Understand 3,4

3 Derive the condition for complete state controllability? Understand 3

4 Derive the condition for complete state controllable output? Analyse 3

5 Derive the condition for complete state observability? Understand 4

6 State the basic theorem for determining the concept of controllability of

time varying system utilizing state transition matrix. Explain the same

with proof?

Understand

3

7 State and explain the principle of duality? Understand 3,4

8 Derive the controllable canonical form for the following transfer function

=

Evaluate

3

9 Derive the observable canonical form for the following transfer function

=

Understand 4

10 Derive the jordan canonical form for the following transfer function

=

Understand 3,4

ANALYTICAL QUESTIONS

1 .Evaluate the controllabillity of the sytem with the matrix

A=

,B=

Evaluate

3

2 investigate the controllability and observability of the system

A=

,B=

C=

Understand 3,4

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3 consider the system described by the state equation

X(t)=

x(t)+

Verify controllability.

Remember 3

4 determine the state controllability and observability of the following

system

=

+

u

C=[0 1]

Apply 3.4

5 examine the observability of the system given below using canonical

form

=

+ u

Y=[3 4 1]

Remember 2,3

6 A feedback system has a closed loop transfer function

=

.construct three different state models for this system and give block

diagram representation for each state model.

Create 1

7 determine the state controllability and observability for the systems

represented by following

State equation :

=

+

u

y=[0 1]

Apply 3,4

8 State whether the system is controllable at t=0 or not. if yes ,find the

minimum energy control to drive it from x(0)= to

Evaluate 3

UNIT-III

DESCRIBING FUNCTION ANALYSIS


1 What are linear systems and nonlinear systems? Give examples? Understand 5

2 How the nonlinearities are classified? Give examples?

Understand 5

3 What is the purpose of introducing nonlinearities into the system?

.

Understand 5

4 Write any two properties of non linear systems?

Analyze

5

5 Write short notes on sub harmonicoscillations?

Understand 5

6 What are limit cycles?

.

Understand 5

7 Difference between sub harmonic &self excited oscillations?

.

Understand 5

8 What is frequency entertainment?

.

Understand 5

9 What is asynchronous quenching?

Analyze 5

10 What is Saturation? Give examples? Remember

5


1 Discuss the characteristics of non-linear system. Analyze 5

2 List out the types of non-linearities are to be found in practical control

system. Explain in detail

Understand

5

3 Discuss about describing function. Give its limitations Understand 5

4 Derive the describing function of saturation non-linearity. Analyse 5

5 Derive the describing function of dead zone of non-linearity Understand 5

6 Derive the describing function of relay with dead zone. Understand 5

7 Derive the describing function of on-off non-linearity. Understand 5

8 Derive the describing function of an on-off non-linearity with hysteresis. Understand 5

9 Derive the describing function of dead zone and saturation of non-

linearity.

Understand 5

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10 Explain about the stability analysis with describing function. Understand 5


1 find the curve with minimum arc length between the point x(0)=1 and the

line t1=4.

Understand

5

2 find the curve with minimum arc length between the point x(0)=0 and the

curve -10t+24

Evaluate 5

UNIT-1V

PHASE-PLANE ANAYLSIS


1 What is phase plane?

Understand 6

2 What is phase trajectory?

.

Understand 6

3 What is singular point?

Analyze 6

4 How the singular points are classified?

Understand 6

5 Draw the phase portrait of a stable node?

Analyze 6

6 What are the differences in stability analysis of linear and nonlinear

systems?

Evaluate

6

7 Define stability of nonlinear system at origin?

Remember 6

8 What is stable in the large?

Analyze

6

9 How limit cycles are determined from phase portrait?

Evaluate 6

10 What are the methods available for constructing phase trajectories’?

Remember

6


1 Describe the isoclines method of drawing phase plane trajectory. Analyze 6

2 Discuss about Phase Plane Understand 6

3 Describe the isoclines method of drawing Phase plane trajectory. Understand 6

4 What are Singular points? Explain the classification of singular points

based on the location of Eigen values of the system.

Analyse

6 5 Explain about the control system with linear gain and show the input

output characteristics

Evaluate

6 6 Describe limit cycles in phase portrait Understand 6

7 Describe the delta method of drawing Phase plane trajectory. Understand 6

8 Describe analytic method of drawing Phase plane trajectory and also

write procedure for phase plane trajectory.

Understand 6

9 Discuss Phase Trajectory? Understand 6

10 Discuss phase portrait? Understand 6


1 draw the phaseplane trajectory for the following equation using isocline

method

+2 + x=0.given that , find the point (6,0)

Evaluate 6

2 determine the kind of singularity for the differential equation

+3 +2y=0

Evaluate 6

3 find out the singular point for the following syste m

+3 -10=0

Evaluate 6

4 a linear second order servo is described by the equation +2 +

x=0.where , rad/sec,x(0)=1.5, =0.determine the

singular point.

Analyse

6

5 find the trajectories in the (t,x) plane which wil extremize

J(X)=

+ )dt

In each of the following cases

(a)t1=1,x(o)=1,x(1)=5

(b)t1=1,x(o)=1,x(1) is free

Evaluate

6

6 a simple servo is described by the following equations reaction

torque= +0.5

Drive torque=2 sign(e+0.5 ) e= -

e(0)=2 and (0)=0

construt the phase trajectory using delta method.

Evaluate

6

7 a second order servo containing a relay with dead-zone and hysteresis is

shown in figure 2.construct the phase trajectory of the system with intial

conditions e(0)=0.65 and (0)=0

Evaluate

6

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UNIT-V

STABILITY ANALYSIS


1 Define free system?

Evaluate 7

2 Define forced system?. Understand 7

3 Explain Liapunov’s stability theorem?

Remember 7

4 Explain sufficient conditions for stability? Apply 7

5 Define asymptotic stability?

Remember 7

6 Define Liapunov’s function? Create 7

7 Define Liapunov’s instability theorem? Apply 7

8 Write state equation of linear autonomous system? Evaluate 7

9 List different methods for constructing liapunov’s functions for non-

linear systems?

Remember 7

10 What is krasovskii’s method?

remember

7


1 Explain the stability in the sense of liapunov. Analyze 7

2 Explain Liapunov stability theorem? And explain sufficient conditions

for stability

Evaluate

7 3 Define term .Asymptotic stability Evaluate 7

4 Explain Liapunov instability theorem Analyse 7

5 Discuss the advantages and limitations of Liapunov stability Understand 7

6 Explain the direct method of Liapunov for the linear continuous time

autonomous system

Understand

7 7 Explain briefly the construction of Liapunov function using variable

gradient method

Understand

7 8 Explain briefly the construction ofLiapunov’s function using

Krasovskii’s method.

Understand

7 9 Discuss about stability? Understand 7

10 Discuss about Asymptotic stability in the large. Understand 7


1 Consider a non linear system described by equations

Investigate the stability of the equilibrium state.

Evaluate

7

2 Check the stability of the system described by

Evaluate 7

3

.Consider the second order system described by

=

Clearly the equilibrium state is the origin. Determine the stability of this

state.

Evaluate 7

4 Consider a nonlinear system described by the equation

, . Find the lyapunov function.

Evaluate 7

5 For the system

. Find a suitable lyapunov function V(x).

Evaluate 7

6 Determine the stability of the system described by where A =

Evaluate 7

7 For the system

. Find a suitable lyapunov function V(x).

Evaluate 7

8 Find the lyapunov function for the following system

=

Evaluate 7

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9 Find the lyapunov function for the following system

=

Evaluate 7

10 Check the stability of the system described by

Use variable gradient method

Evaluate 7

UNIT-VI

MODAL CONTROL


1 What is pole placement by state feedback?

Understand 8

2 What are necessary conditions to be satisfied for design using state

feedback?

Understand 8

3 What are the advantages of control system in space?

Creating

8

4 Draw the block diagram o f a systemwith state feedback?

Apply 8

5 What is control law?

Analyse

8

6 What is state observer?

Apply 9

7

What is full-order state observer?

Creating 9

8 What is reduced-order state observer?

Analyse

9

9 What is minimum -order state observer?

.

Apply 9

10 What is necessary condition for design of state observer?

Apply 8


1 State and prove the effect of state feedback on controllability of closed

loop systems.

Analyze 8

2 Explain the effect of state feedback on the observability Understand 8

3 State and prove the necessary and sufficient conditions for the design of

state feedback control through the pole placement.

Understand

8,9 4 Explain the design of state feedback control through pole placement in

detail.

Analyse 8,9

5 Explain with the help of block diagram full order observer Evaluate 9

6 Consider the system defined by

=

U .show that

this system cannot be stabilized by state feedback control U = -Kx.

Whatever matrix is chosen.

Evaluate

8,9

7 Consider a single input/ single output system =

U,

Y =

Determine the observability of the system under state feedback.

Evaluate

8,9

8 Consider the system where A

=

B =

. It is desired to place the poles at S = -2+j4,

S = -2-j4, S = -10. Determine the state feedback matrix

Evaluate

8,9

9 Explain reduced order observer. Derive the equation. Understand 9

10 Derive the error dynamics of the full order observer Understand 9


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1 Consider a single input/ single output system =

U,

Y =

Determine the observability of the system under state feedback.

analyse

8

2 Consider the system defined by

=

U. show that

this system cannot be stabilized by state feedback control U = -Kx.

Whatever matrix is choosen

Evaluate 8,9


=

B =

. It is desired to place the poles at S = -2+j4,

S = -2-j4, S = -10. Determine the state feedback matrix.

Evaluate 8

4 Consider the system defined by =

=

U. show

that this system cannot be stabilized by state feedback control U = -Kx.

Evaluate 8,9

5 Consider the system where A =

B =

. It is desired to place the poles at S = -2+j4, S

= -2-j4, S = -10. Determine the state feedback gain.

Evaluate 8,9


=

B =

C = . The closed loop poles at

S = -2+j2 ,S = -2-j2 , S = -6.

Evaluate 8,9

7 Consider the system , Y = CX , A =

; C = .

Design a full order state observer matrix are =-5, = -5.

Evaluate 9

8 Consider the system X =AX, Y = CX, A =

; C = .

Design a full order state observer matrix are μ_1 =-5, μ_2 = -5.

Evaluate 9

UNIT-VII

CALCULUS OF VADRIATIONS


1 State the fundamental theorem of calculus of variations? Apply

10

2 Explain control variable inequality constraints?

Evaluate

10

3 Discuss constrained minimization?

Creating

10

4 Discuss constrained minimum principle?

Apply

10

5 Write the traversality condition of the calculus of variation?

Analyse 10

6 What is proper field of external of the function?

Apply

10

7 What is central field of externals of the function?

Analyse

10

8 Explain state variable inequality constraints? Analyse 10

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9 Explain about equality constraints Evaluate 10

10 Explain about in equality constraints Apply 10


1 State and prove the fundamental theorem of calculus of variations. Analyze 10

2 Explain the Euler-Lagrange equation Understand 10

3 Derive the necessary conditions for an extremal of the function

f(x)= x(t),ẋ(t),t)dt. Where both terminal time t1 and x(t1) free.Understand 10

4 Derive the traversality condition of the calculus of the variation.. Analyse 10

5 Discuss the constrained minimization principle Understand 10

6 Explain the control variable inequality constraints Understand 10

7 Explain state variable inequality constraints. Understand 10

8 Explain the control variable equality constraints Understand 10

9 Discuss the constrained minimization principle Understand 10

10 Discuss importance constraints Understand 10



J(x) =

where both terminal time &X( ) are

free.

.

Analyse 10


J(x) =

where both terminal time specifed

&X( ) are free.

Evaluate 10

3 Find the extremal for the function V = with

boundary conditions X(0) = 0 & X(1) =1

Evaluate 10

4 Find the extremal for the function J(x) = with

boundary conditions (0) = 0 & (π/4) =1, (0) = 0, (π/4) =-1

Evaluate 10

5 Find the extremal for the function V = , with the

boundary conditions X(0) = 0 & X(π/2) =1.

Evaluate 10

6 Find the curve with minimum arc length between the point X(0) =1 and

the line = 4.

Evaluate 10

7 Given , X(0) = , X(2) = . Find the U* that minimize

J =

Evaluate 10

UNIT-VIII

OPTIMAL CONTROL


1 Define continuous time systems? Analyze 10

2 Define discontinuous time systems? Understand 10

3 Discuss the concept of formulation of the optimal control problem? Understand 10

4 Discuss minimum time problem? Analyse 10

5 Discuss minimum energy control? Understand 10

6 Discuss minimum fuel problem? Understand 10

7 Discuss output regulator problem? Understand 10

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8 Discuss tracking problem? Evaluate 10

9 Discuss continuous time state regulator problem? Understand 10

10 Discuss parameter optimization for regulators? Understand 10


1 Explain the concept of formulation of the optimal control problem. Understand 10

2 Explain about minimum time problem. Evaluate 10

3 Explain about minimum energy control. Evaluate 10

4 Explain about minimum fuel problem? Analyse 10

5 Explain about state regulator problem? Understand 10

6 Explain about tracking problem? Understand 10

7 Explain about continuous time regulator problem? Understand 10

8 Explain about discrete time regulator problem? Understand 10

9 Explain about continuous time regulator problem? Understand 10

10 Explain about continuous time state regulator problem? Understand 10

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World€¦ · model.obtain the canonical form of state model = + u and y= Evaluate 2 4 convert the...

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