unit2.doc

Control SystemsUnit 2: Transfer Function Analysis and Reduction TechniquesUnit 2: Transfer Function and Block Diagram AnalysisContent:S.No.ConceptPage No.

1Learning Objectives

2Impulse Response and Transfer Function

3Impulse Response and Transfer Function

4Assumptions and Limitations of Transfer Function

5Assumptions and Limitations

6Examples

7Block Diagrams

8Block Diagrams

9Block Diagram of a Closed Loop System

10Block Diagram of a Closed Loop System

11Block Diagram Reduction Rules

12Block Diagram Reduction Rules

13Examples

14Multi Input Multi Output Systems

15Multi Input Multi Output Systems

16Examples

17Keywords to Remember

18Keywords to Remember

19Summary

20Summary

21Solved Problems

22Solved Problems

23Review Questions

24Review Questions

25Match the following

26Match the following - I

27Match the following - II

28True or False

29True or False

30Drag and Drop

31Drag and Drop

32Model Questions

33Model Questions

34Exercises

35Exercises

36Assignments

37Assignments

Concept

TitleLearning Objectives

ContentInstruction to aniamtorGraphics

By the completion of this unit the learner will able to Understand the impulse response of a system Understand the transfer function for linear time invariant systems Know the procedure for deriving transfer function Determine the block diagram of a given system Understand Multi Input Multi Output Systems

Display the points one by one

Concept Assumptions and Limitations of Transfer Function

Title:Assumptions and Limitations

ContentInstruction to animatorGraphics

Assumptions for Deriving Transfer Function

The following assumptions are made in deriving transfer functions of physical systems.

1. It is assumed that there is no loading, i.e., no power is drawn at the output of the system.

2. If the system has more than one non loading element in tandem, then the transfer function of each element can be determined independently and the overall transfer function of the physical system is determined by multiplying the individual transfer functions.

3. The system should be approximated by a linear lumped constant parameters model by making suitable assumptions.

Limitations:

1. Transfer function is considered only for the system which does not possess any initial conditions i.e., when initial conditions are zero.

2. It is applicable to only linear time invariant systems.

3. It does not give any information about the structure of the system.

4. External conditions lose its importance, as initial conditions are taken zero.

Concept:Assumptions and Limitations of Transfer Function

Title:Examples

ContentInstruction to animatorGraphics

Examples:

1. Armature Control of DC motor

Consider the armature - controlled d.c. motor as shown in the figure

The transfer function of the armature controlled d.c. motor is

In this system,

= resistance of armature ()

= inductance of armature winding (H)

= armature current (A)

= field current (A)

= applied armature voltage (V)

= back e.m.f (volts)

= torque developed by motor (Nm)

= angular displacement of motor - shaft (rad)

J = equivalent moment of inertia of motor and load referred to motor shaft (kg-m2).

= equivalent viscous friction coefficient of motor and load referred to motor shaft .

We know that, in linear range of the magnetization curve

(1)

where is a constant.

The torque developed by the motor is proportional to the product of the armature current and air gap flux, i.e.,

. (2)

where is a constant.

In the armature controlled DC motor, the field current is kept constant.

(3)

where is known as motor torque constant.

The back e.m.f of the DC motor is given by

(4)

where is the back emf constant.

By applying KVL in the armature loop

(5)

The torque equation is

(6)

Taking the Laplace transforms of eqns. (3) to (6), assuming zero initial conditions, we get

.. (7)

. (8)

(9)

From the above equations, the transfer function of the system is obtained as

2. Field Control of DC motor

Consider a field - controlled d.c. motor as shown in the figure

The transfer function of the field controlled d.c. motor is

Where = motor gain constant;

= time constant of field circuit;

= mechanical time constant.

In the field controlled motor, the armature current is fed from a constant current source.

EMBED Equation.3 Where is a constant.

Apply KVL in filed loop, we get

The torque equation is

Taking the Laplace transform of above eqns. Assuming zero initial conditions, we get

From the above equations the transfer function of the motor is obtained as

Where = motor gain constant; = time constant of field circuit; = mechanical time constant.

3. Find the transfer function for the system shown in figure.

where, M is the mass of the system

K is the spring deflection

B is the coefficient of viscous damping

Solution:

Consider the free body diagram of the given system

Let be the force acting due to spring constant K.

be the force due to damping

be the force due to mass (M) element.

Force due to mass, M is

Force due to friction,

Force due to elastic spring,

By Newtons second law,

Force f(t) = Sum of individual forces acting on the system.

Apply Laplace transform to the above equation

Insert fig. 1 below 1st line 1st exampleInsert fig. 2 below 1st line 2nd example

Insert fig. q3.1 below q3Insert fig. q3.2 below 1st line of solution

Fig. 1

Fig. 2

Fig. q3.1

Fig. q3.2

Concept:Block Diagrams

Title:Block Diagrams

ContentInstruction to AnimatorGraphics

Block Diagrams A block diagram of a system is a pictorial representation of the functions performed by each component and of the flow of signals.

In general, a block diagram consists of a specific configuration of four types of elements: Blocks

Summing points

Takeoff points

Arrows representing unidirectional signal flow.Block: The functional block or simply block is a symbol for the mathematical operation on the input signal to the block to produces the output. Summing point: Summing points are used to add two or more signals in the system. The plus or minus sign at each arrowhead indicates whether the signal is to be added or subtracted.

Takeoff point / Branch Point: A branch point is a point from which the signal from a block concurrently to other blocks or summing points.Insert fig. 1 below 2nd paragraph

Design fig such that by clicking on the heading block, summing point, takeoff point the corresponding matter will come in the figure.

Fig. 1

Concept:Block Diagrams

TitleExample


1. Represent the armature controlled DC motor in Block diagram

Answer:

The armature controlled DC motor transfer function is given by

Block diagram representation of armature controlled DC motor is,

2. Represent the field controlled DC motor in Block diagram form

Answer:

The field controlled DC motor transfer function is given by

Block diagram representation of field controlled DC motor is,

Insert fig. 1 below Block diagram representation of armature controlled DC motor is,

Insert fig. 2 below Block diagram representation of field controlled DC motor is,

1q

Fig. 1

Q2

Concept:Block Diagram of a Closed Loop System

TitleBlock Diagram of a Closed Loop System


The figure shows the block diagram of a negative feedback system. The following terminology is defined with reference to diagram.r(t), R(s) = Reference input (command)

y(t), Y(s) = C(s) = Output (controlled variable)

b(t), B(s) = Feedback signal

u(t), U(s) = Actuating signal = error signal e(t), E(s), when H(s) = 1H(s) = Feedback transfer function

G(s)H(s) = L(s) = B(s)/E(s) = Loop transfer function

G(s) = C(s)/E(s) = Forward - Path transfer function

T(s) = C(s)/R(s) = Closed - loop transfer function.

The closed - loop transfer function T(s) can be expressed as a function of G(s) and H(s)From the figure shown, we write

Y(s) = G(s) U(s) . (1)and

B(s) = H(s) Y(s) . (2)The actuating signal is written, U(s) = R(s) - B(s) (3)Substituting eq. (3) into eq. (1) yields

Y(s) = G(s)R(s) - G(s)B(s) (4)Substituting eq. (2) into eq. (4) and then solving for Y(s)/R(s) gives the closed loop transfer function:

The characteristic equation of the above system is

Insert fig. 1 above 1st paragraph.

Insert fig. 1 here also

Fig. 1

Concept:Block Diagram Reduction Rules

TitleBlock Diagram Reduction Rules


A complex block diagram configuration can be simplified by certain rearrangements of block diagram using the rules of block diagram algebra.

Some of the important rules are given in the table shown

Concept:Block Diagram Reduction Rules

TitleExamples


1. Reduce the block diagram of the figure shown Solution:

Step1: Move the branch point after the block

Step 2: Eliminate the feedback path and combine blocks in cascade.

Step 3: Combing parallel blocks

Step 4: Combining blocks in cascade

Fig. 1.0

Fig. 1.1

Fig. 1.2

Fig. 1.3

Fig. 1.4

Fig. 1.5

Concept:Multi Input Multi Output Systems

TitleMulti Input Multi Output Systems


When multiple inputs are present in a linear system, each input can be treated independently of the others. Complete output of the system can then be obtained by superposition, i.e., outputs corresponding to each input along are added together.Consider a two-input linear system shown in the figure.

The response to the input R1(s) can be obtained by assuming R2(s) = 0. The corresponding block diagram is

The response to the input R2(s) can be obtained by assuming R1(s) = 0. The corresponding block diagram is

The response to the simultaneous application of R(s) and U(s) can be obtained by adding the two individual responses.

In case of MIMO system with r inputs and m outputs shown in the figure, the ith output Ci(s) is given by

where is the jth input and is the transfer function between the ith output and jth input with all the other inputs reduced to zero. From the figure shown,

This can be expressed in compressed matrix notation asC(s) = G(s)R(s)

Where R(s) = input vector (r x1)

G(s) = transfer function matrix ( m x r)

C(s) = output vector (m x1)

The block diagram of the open loop MIMO system is

The block diagram of the closed loop MIMO system is

The transfer function of the closed loop MIMO system is given by,T(s) = C(s)/R(s) = G(s)[I + G(s)H(s)]-1

Insert fig.1 below the statement Consider a two-input linear system shown in below figure

Insert fig. 2 above eqn. CR1(s).

Insert fig. 3 above eqn. CR2(s).

Insert fig. 4 above the eqn. Ci(s).

Insert fig. 5 below the statement open loop MIMO system is shown below

Insert fig. 6 below the statement closed loop MIMO system is shown below

Fig.4

Concept:Multi Input Multi Output Systems

TitleExamples


Example:

1. For the system represented by block diagram shown, determine the output C1/R1 and C2/R1.

Solution: Case(i): To find C1/R1

In this case set R2 = 0 and consider only one output C1Step1: Eliminating the feedback pathStep2: Combining the blocks in cascade and splitting the summing point.

Step3: Eliminating the feedback path

Step4: Combining the blocks in cascade



Therefore,

Case (ii): To find C2/R1In this case set R2 = 0 and consider only one output C2.

Step1: Eliminating the feedback pathStep2: Combining the blocks in cascade and splitting the summing point.





Therefore,

Fig. 1.0

Fig. 1.1.1

Fig. 1.1.2

Fig. 1.1.3

Fig. 1.1.4

Fig. 1.1.5

Fig. 1.1.6

Fig. 1.1.7

Fig. 1.2.1

Fig. 1.2.2

Fig. 1.2.3

Fig. 1.2.4

Fig. 1.2.5

Fig. 1.2.6

Fig. 1.2.7

Concept:Keywords to Remember

TitleKeywords to Remember


Impulse response

Transfer function

Block diagrams

Blocks

Summing point

Takeoff point/ Branch point

Characteristic equation

Concept:Summary

TitleSummary


The transfer function of a linear time - invariant system is defined as the ratio of the Laplace transform of the output variable to the Laplace transform of the input variable under the assumption that all initial conditions are zero.

The transfer function of a linear time - invariant system is defined as the Laplace transform of the impulse response, with all the initial conditions set to zero. Transfer function is considered only for the system which does not possess any initial conditions i.e., when initial conditions are zero.

Transfer function is applicable to only linear time invariant systems.

Transfer function does not give any information about the structure of the system.

A block diagram of a system is a pictorial representation of the functions performed by each component and of the flow of signals.

In general, a block diagram consists of a specific configuration of four types of elements:

Blocks

Summing points

Takeoff points

Arrows representing unidirectional signal flow.

Block is a symbol for the mathematical operation on the input signal to the block to produces the output.

Summing points are used to add or subtract two or more signals in the system.

A branch point is a point from which the signal from a block concurrently to other blocks or summing points.

The characteristic equation of the closed loop system is

Concept:Solved Problems

TitleSolved Problems


1. Reduce the block diagram shown in the figure and find its transfer function.Solution:Step1: Moving takeoff point before the blockStep2: Combining blocks in cascade



Step 5: Eliminating the feedback path


Step 7: Eliminating the feedback path

The transfer function is

2. Find the transfer function for the given block diagram shownSolution:Step1: From the given system the gain at the takeoff point 1 is equal to the gain at the takeoff point 2, hence we can interchange the takeoff point.Step 2: Eliminate the feedback pathsStep3: Combining blocks in parallel

The transfer function is

Fig. 1.0

Fig. 1.1

Fig. 1.2

Fig. 1.3

Fig. 1.4

Fig. 1.5

Fig. 1.6

Fig. 1.7

Fig. 1.8

Fig. 2.0

Fig. 2.1

Fig. 2.2

Fig. 2.3

Concept:Review Questions

TitleReview Questions


1) Consider a system represented by the block diagram

The closed - loop transfer function is

a) c)

b) d) None of these

2) Consider the block diagram shown in the figure

The closed loop transfer function is

a) c)

b) d)

(Answer (b)) 3) Consider the differential equation

Where and u(t) is a unit step. The poles of this system are:

a) c)

b) d) None of these

(Answer (b))4) A car of mass m = 1000kg is attached to a truck using a spring of stiffness K = 20000 N/m and a damper of constant b = 200 Ns/m. The truck moves at a constant acceleration of a 0.7 m/s2.

The transfer function between the speed of the truck and the speed of the car is:

a) c)

b) d) None of the above

5) Consider the closed - loop system shown in the figure

Compute the closed - loop transfer function and the closed - loop zeros and poles.

a)

b)

c)

d)

(Answer (c))6) For the block diagram shown in figure, C(s)/R(s) is given bya) c)

b) d)

(Answer (a))

7) The block diagram of a control system is shown in figure. The transfer function G(s)=Y(s)/ U(s) of the system is

a) c)

b) d)

(Answer (b))8) For the block diagram shown in figure, the transfer function is equal to

a)

b)

c) d)

(Answer (c))

9. If the initial conditions for a system are inherently zero, what does it physically mean? a) The system is at rest but stores energy.

b) The system is working but does not store energy.

c) The system is at rest or no energy is stored in any of its parts.

d) The system is working with zero reference input.

10. A system can be completely described by a transfer function if it is

a) Nonlinear and continuous

b) Linear and time- varying

c) Nonlinear and time- invariant

d) Linear and time- invariant

11. For a system, zero initial condition means

a) Zero stored energy

b) No initial movement of moving parts

c) System is at rest and no energy is stored in any of its components

d) Input reference signal is zero.

12. The transfer function of a system can be used to evaluate

a) Time constant

b) Feedback

c) Type of system

d) Output for a given input

Fig. q6

Fig. q7

Fig. q8

Concept:Match the following

TitleMatch the following


1st answer

A

3

B

4

C

1

D

2

2nd answer

A

3

B

1C

4D

2

Concept:True or False

TitleTrue or False


1. Very few physical systems are linear within some range of the variables.

False

2. The roots of the characteristic equation are the zeros of the closed-loop system. False

3. A linear system satisfies the properties of superposition and homogeneity True

4. The transfer function is the ratio of the Laplace transform of the output variable to the Laplace transform of the input variable, with all initial conditions equal to zero.

True

5. Linear system not obeys the superposition principle. False

6. Transfer function of a control system depends on initial conditions of input and output.False

7. Transfer function of control system depends on the system parameters alone.

True

Concept:Drag and Drop

TitleDrag and Drop


1. A unit impulse function on differentiation results in

Answer: Unit doublet2. The transient function of the system and response in terms of Laplace transform are same for which of the following input function?

Answer: Unit impulse function3. Laplace transform of unit doublet is Answer: s4. The overall gain of the block diagram Answer:

5. The overall gain of the block diagram Answer:

Concept:Model Questions

TitleModel Questions


1. The dynamic behavior of the system is described by the equation,, where e is the input and C is the output. Determine the transfer function of the system.

Solution: Given that, the equation describing dynamic behavior of the system is

Apply Laplace transform to the above equation by assuming all initial conditions are zero

The desired transfer function is

2. Define the transfer function and determine the transfer function of RLC series circuit if voltage across the capacitor is output variable and input is voltage source V(s).

Solution:

Transfer Function of Series RLC Circuit:

By applying KVL in the above loop

. (1)

The output voltage is given by, (2)By applying Laplace transform to eqn. (1), we get

(3)By applying Laplace transform to eqn. (2), we get

(4)From Eqns. (3) & (4)

The above equation is our desired transfer function.3. Find the transfer function of the following system, shown in the figure.

Solution:

Let assume the given network is,The voltage at node 1 is VC1 and node 2 is Vo.Apply KCL at node 1,

. (1)Apply KCL at node 2

(2)

By applying Laplace transform to eqn. (1)

. (3)By applying Laplace transform to eqn. (2)

.. (4)

Substitute eqn. (3) in eqn. (4)

In the given problem and

4. Derive the transfer function of the following network shown.

Solution: By applying KVL in the 1st loop, we get

. (1)

By applying KVL in the 2nd loop, we get

.. (2)

and the output is given by,

(3)

By applying Laplace transform to the above equations, we get

.. (4)

. (5)

.. (6)

From eqn. (4)

Substitute I1(s) in eqn. (5)

By substituting eqn. (6) in the above equation and by rearranging we get,

5. Obtain the transfer function of the following system.

Solution:From the given system two displacements are there. There are two nodesat node (2)Free body diagram of mass M is,

By Newtons second law, we have, f(t) = fk2 + fM

By taking Laplace transform to the above equation.

(1)At node (1), assume a dummy mass element with M = 0( Free body diagram at node (1) is,

By Newtons second law, we have fm + fk1 + fk2 = 00 + K1x1 + K2(x1 x) = 0By applying Laplace transform to the above equation,K1X1(s) + K2[X1(s) X(s)] = 0 (K1 + K2) X1(s) K2 X(s)] = 0

(2)

Substitute equation (2) in equation (1)

( The required transfer function is,

6. Find the transfer function of the following system shown

Solution:Here input is applied torque T and output is the displacement .

( The desired transfer function is

1. Consider the moment of inertia J1

According to Newtons second law,

Taking Laplace of the above equation, we get

(1)

2. Consider the moment of inertia J2


Taking Laplace to the above equation we get

(2)

Substitute equation (2) in equation (1)

( Desired transfer function

7. Find the transfer function of the following system in figure.

Solution:The given system consists of two nodes.At node (1)The free body diagram of mass M1 is


Taking Laplace transform, we get

(1)

At note (2)The free body diagram of mass M2 is,


Taking Laplace transform, we get

(2)Substitute equation (1) in equation (2)

( Desired transfer function

8. Obtain the transfer function of the following system shown in the figure.

Solution:In the given n/w, there is one node but there is no mass element to represent it. Hence, assuming a fictions mass element with M = 0. The given figure is modified as

Considering mass element and the opposing forces acting on it.

Taking Laplace transform of the above equation, we get

9. Explain the properties of block diagrams.

Aug/ Sep - 2008, 2007

or

Explain the rules of block diagram reduction.

Answer: Covered in screen

10. Find the transfer function for the figure shown, using block diagram reduction technique.

Solution:

The given block diagram is redrawn as

Step1: Combining parallel blocks Step2: Moving takeoff point ahead the block and Combining blocks in cascade


Step 4: Moving takeoff point ahead to the block Step 5: Combining blocks in cascade and eliminating the feedback path

Step 6: Eliminating the feedback path Step 7: Eliminating the feedback path

The required transfer function is

11. Determine the transfer function for the following block diagram.

Answer: Step1: Moving summing point ahead to the block Step2: Interchange the summing points and Combining blocks in cascade

Step3: Eliminating the feedback paths



12. Determine the transfer function for the following block diagram.Answer:

Step1: Moving takeoff point behind the block

Step2: Combining parallel blocks Step3: Moving takeoff point ahead to the block

Step 4: Eliminating the feedback path Step 5: Combining blocks in cascade Step 6: Eliminating the feedback path


Fig. q3Fig. q5.1

Fig. q5.2

Fig. q5.3

Fig. q6.1

Fig. q6.2

Fig. q6.3

Fig. q7.1

Fig. q7.2

Fig. q7.3

Fig. q8.1

Fig. q8.2

Fig. q8.3

Fig. q10

Fig. q10.1

Fig. q10.2

Fig. q10.3

Fig. q10.4

Fig. q10.5

Fig. q10.6

Fig. q10.7

Fig. q10.8

Fig. q11.1

Fig. q11.2

Fig. q11.3

Fig. q11.4

Fig. q11.5

Fig. q12.1

Fig. q12.2

Fig. q12.3

Fig. q12.4

Fig. q12.5

Fig.q12.6

Fig. q12.7

Concept:Exercises

TitleExercises


1. Find the transfer functions, , for the network as shown the figureAnswer:

2. Find the transfer function for the block diagram as shown the figure Answer:

3. Write the differential equations for the mechanical system as shown in the figure. Answer:

4. Obtain the transfer function of the mechanical system as shown in figureAnswer:

Insert fig. 1,2,3, 4 below the questions 1,2, 3,4 respectively.

Fig. 2

Fig. 3

Fig. 4

Concept:Assignments

TitleAssignments


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unit2.doc

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Transcript of unit2.doc