BHARATHIDASAN ENGINEERING COLLEGE

EC6405 CONTROL SYSTEM ENGINEERING

(FOR 4TH SEM ECE)

Lecturer notes

Prepared by

L.Gopinath M.tech

Assistant professor

Unit 1 Control system modeling

Control system

A system is a collection, set or arrangements of elements which performs a desired output

for a given input.

A control system is an interconnection of components forming a system configuration that

will provide a desired system response.

The word control means,

– to regulate

– to direct

– to command or govern

Hence, control system is an arrangement of physical components that connected or related

in such a manner as to command, regulate, direct or govern itself (or) another system.

Open Loop System and Closed Loop System A control system is said to be an open loop system, in which output is dependent on input

but controlling action (or) input is totally independent of the output or changes in output of

system.

A system in which the controlling action or input is dependent on the output or changes in

output is called closed loop system.

OPEN LOOP SYSTEM

In an open loop system, the control action is independent of the desired output. The

actuating signals depend only on the input command and output has no control over it. In

this system the output is neither measured nor fed back for comparison with the input.

Fig. 1.1. Open loop system

The input signal r(t) generates the actuating signal u(t). This signal is also called as error

signal. The error signal is necessary to get the desired controlled output c(t).

Advantages

(i) Such systems are simple and economical.

(ii) It is simple to construct.

(iii) Generally it is a stable system, hence these are not troubled.

(iv) No sensors needed to measure the variables to provide feedback.

Disadvantages

(i) Less accuracy.

(ii) The changes in output due to external disturbances are not corrected automatically.

Hence to maintain the quality and accuracy are much difficult.

1.0.1. CLOSED LOOP SYSTEM

A system in which control action is some how dependent on the output. In this case, the

output is feedback through a feedback element and compared with the reference input.

Thus the actuating signal is the difference of desired output and reference input.

Feedback is that property of a closed loop system which permits the output (or) some other

controlled variable of the system, to be compared with the input to the system. So that the

appropriate control action may be formed as some function of the output and input.

Fig. 1.2. Closed loop system

In closed loop system, the output is compared with the reference input by the feedback

network and if any output changes occur due to disturbance, it can be automatically

corrected by the system itself. Hence, the closed loop control system is also called as

automatic control system.

Feedback Network

It is generally measuring device which measures output and feeds it to comparator.

Examples for Open Loop Control System

All control systems operate on the basis of present timing mechanism are open loop

control systems.

1. Automatic washing machine.

2. The electric switch.

3. The automatic toaster.

Examples for Closed Loop Control System

1. Traffic signal system based on the density of traffic.

2. Liquid level control system.

3. Temperature control system.

Comparison between Open and Closed Loop Control System

S.No. Open Loop Closed Loop

1. Less accurate. More accurate.

2. Generally build easily. Generally complicated and costly.

3. Stability can be ensured. May become unstable at times.

4. Presence of non-linearities cause

malfunctioning.

It usually performs accurately even in

the presence of non-linearities.

5. Any change in the system component

cannot be taken care of automatically.

Change in system component is

automatically taken care of.

6. Input command is the sole factor

responsible for providing the control

action.

The control action is provided by the

difference between the input

command and the corresponding

output.

7. The control adjustment depends upon

human judgement and estimate.

The control adjustment depends on

output and feedback element.

TRANSFER FUNCTION

Transfer function is defined as for a linear system, the ratio of Laplace transform of output to

the Laplace transform of input with zero initial condition.

It is also defined as the Laplace transform of the impulse response of the system with zero

initial conditions.

Transfer function (T, F)= C(s)R(s)

Mechanical system

Mechanical systems are classified into two categories based on the motion.

(i) Mechanical translation system

The translation systems are related to force and translational motion.

(ii) Mechanical rotational system

The rotational systems are related to torque and angular motion.

(i)Mechanical Translation System

Basic elements of translational system

(i) Inertia force (fM)

It is a force which occurs due to mass M. The weight of the mechanical system is

represented by the element mass and it is assumed to be concentrated at the centre of the

body.

Force balance equation of mass

Consider an ideal mass element shown in Fig. Let us apply a force on it. The mass will

offer an opposing force which is proportional to acceleration of the body.

Let f = Applied force on mass

f M = Opposing force due to mass

Fig. 1.3. Ideal mass element

By Newton’s second law of motion,

f M = Mass Acceleration

= M d2x

dt2

f M = M d2x

dt2

(ii) Damping force ( f b)

Whenever two physical bodies are in motion, there exists a friction. The friction existing

in rotating mechanical system can be represented by the damper. The clamper is a piston

moving inside a cylinder filled with viscous fluid. The force exists due to the damper are

said to be damping force.

There are three types of friction.

(a) Static friction: It is observed only when the body is not in motion and its tendency is to

prevent the body from moving. The direction of static friction is always opposite to the

direction in which the body tends to move.

(b) Viscous friction: It is the frictional force experienced when the body is in motion and

this force is proportional to the velocity of the body.

(c) Coulomb friction: It is experienced when the body is in motion. This friction opposes

the motion but has no dependence upon the magnitude of velocity (or) acceleration of the

body.

Force balance equation

Consider an ideal frictional element damper. Let us apply a force on it. The damper will

offer an opposing force which is proportional to velocity of the body.

Let f = Applied force

fb = Opposing force due to friction.

Fig. 1.4. Ideal damper

By Newton’s second law,

fb Velocity fb dx

dt

fb = B dx

dt

When the damper has exists between two displacements. Let us having two displacement

in the following Fig.1.5.

Fig. 1.5. Ideal damper in between displacement

Let, displacement be x1 and x2.

fb d

dt (x1 – x2)

fb = B d

dt (x1 – x2)

(iii) Spring force ( fk)

The elastic deformation of the body can be represented by a spring. The force due to this is

said to be spring force (fk).

Force balance equation

Consider an ideal spring element shown in Fig.1.6, with one end fixed displacement. Let a

force be applied on it. The spring will offer an opposing force which is proportional to the

displacement of the body.

Fig. 1.6. Ideal spring

Let f = Applied force

fk = Opposing force due to spring

By Newton’s second law,

fk x

fk = k x

When the spring has displacement at both ends as shown in Fig.1.7.

Fig. 1.7. Ideal spring with displacement at both ends

Let us be the displacement as x1 and x2 .

fk (x1 – x2)

fk = k (x1 – x2)

D’Alembert’s Principle for Translational System

It states that “for any body, the algebraic sum of the externally applied force and the

forces restraining motion in any given direction and at any instant is zero.”

i.e., Sum of applied force = Sum of opposing forces

Procedure to find the transfer function of mechanical translational

system

Step 1: Draw the free body diagram of the system. The free body diagram should be

obtained by drawing each mass and then marking all the forces acting on that mass. The

direction of applied force is always opposite to the direction of opposing force.

Step 2: Let us assign the displacement for each mass as x1, x2, x3, etc.

Step 3: For each free body diagram of mass, apply the D’Alembert’s principle.

Step 4: For each force in the free body diagram, write the differential equation governing

the system which are obtained by writing force balance equations.

Step 5: Take Laplace transform of those differential equation and rearrange the equation to

eliminate the unwanted variable and obtain the ratio of output to the input in Laplace

transform, i.e., Transfer function.

Example Write the differential equations governing the mechanical system shown

in Fig.1.8 and determine the transfer function.

Fig. 1.8.

Solution: In the given system, the applied force f (t) is the input and the displacement x

is the output.

Transfer function = Laplace transform of output

Laplace transform of input

Input = f (t), Output = X

L [input] = F(s), L [output] = X(s)

Transfer function = X(s)

F(s)

Step 1: Draw the free body diagram and mark all the force acting on it and the direction of

applied force is always opposite to the opposing force.

Fig. 1.9. Free body diagram

Step 2: Displacement is assigned as X.

Step 3: Apply D’Alembert’s principle

Sum of applied forces = Sum of opposing forces

f (t) = f M + fK + fb

Step 4: Write the force balance equation for the above equation to get the differential

equation.

f (t) = M d2X

dt2 + KX + B

dX

dt

Step 5: Taking Laplace transform to the above differential equation,

F(s) = M s2 X(s) + K X(s) + B s X(s)

Rearrange the equation as the ratio of output to the input.

F(s) = M s2 X(s) + K X(s) + B s X(s)

F(s) = X(s) [ M s2 + K + B s ]

X(s)

F(s) =

1

M s2 + K + B s

Thus the differential governing the system is

f (t) = M d2X

dt2 + KX + B

dX

dt

The transfer function of the system is

X(s)

F(s) =

1

M s2 + K + B s

Electrical Analogous of Mechanical Translational System

The electrical system has two types of input either voltage or current sources. According

to this input, there are two types of analogies. They are:

(i) Force-voltage analogy by Loop analysis.

(ii) Force-current analogy by Nodal analysis.

Loop Analysis Nodal Analysis

Voltage (V) Current (I)

Current (I) Voltage (V)

Charge (q) Flux ()

Resistance (R) Conductance (G)

Capacitance (C) Inductance (L)

Inductance (L) Capacitance (C)

Force-Voltage Analogy

Consider the following translational mechanical system.

According to D’Alembert’s principle,

Sum of opposing forces = Sum of applied forces

f M1 + fK + fB = f (t)

By writing the force balance equation with respect to velocity,

M dv

dt + K v dt + B v = f (t) (1)

Consider the following RLC electrical system.

By applying KVL, we get

R i + 1

C i dt + L

di

dt = v(t) (2)

Comparing equation (1) and (2),

f (t) = M dv

dt + K v dt + B v

v(t) = L di

dt +

1

C i dt + R i

Thus the equation (1) and (2) made analogous the force of mechanical system to the

voltage of electrical system. This analogy is known as Force-Voltage (or) F-V analogy.

Mechanical

Translational System Electrical System

Force f Voltage V

Velocity v Current I

Displacement x Charge q

Mass M Inductance L

Damping coefficient B Resistance R

Spring K Inverse of capacitance

1

C

Table 1.1. F-V analogous

Force-Current Analogy

For current analogy, consider the parallel RLC electrical circuit shown below.

By applying KCL, I(t) = V(t)

R + C

d V(t)

dt +

1

L V(t) dt

I(t) = G V(t) + C d V(t)

dt +

1

L V(t) dt (3)

where Conductance G = 1

R Comparing equations (1) and (3),

f (t) = M dv

dt + B v + K v dt

where v = velocity

I(t) = G V(t) + C d V(t)

dt +

1

L V(t) dt

where V = voltage

Thus the equation (1) and (3) made analogous the force of mechanical system to the

current of electrical system. This analogy is known as Force-Current (or) F-I analogy.

Mechanical

Translational System Electrical System

Force f Current I

Velocity v Voltage V

Displacement x Flux

Mass M Capacitance C

Damping coefficient B Conductance G =

1

R

Spring K Inverse of inductance

1

L

Table 1.2. F-I analogous

The analogous of mechanical translation system to electrical system for both input

voltage and current are listed below.

Mechanical

Translational System F-V F-I

Force f Voltage V Current I

Velocity v Current I Voltage V

Displacement x Charge q Flux

Mass M Inductance L Capacitance C

Damping coefficient B Resistance R Conductance G =

1

R

Spring K Inverse of Capacitance

1

C

Inverse of inductance

1

L

Table 1.3. Analogous of mechanical to electrical system

Procedure to Draw the Electrical Analogy of Mechanical Translation System

Step 1: Draw the free body diagram of the system. The free body diagram should be

obtained by drawing each mass and then marking all the forces acting on that mass. The

direction of applied force is always opposite to the direction of opposing force.

Step 2: Let us assign the displacement for each mass as x1 x2 x3, etc.

Step 3: For each free body diagram of mass, apply the D’Alembert’s principle.

Step 4: For each force in the free body diagram, write the differential equation

governing the system are obtained by writing force balance equations.

Step 5: For each differential equation, write the electrical analogous with respect to

voltage and current are known as force-voltage and force-current respectively by referring

the Table 1.3.

Step 6: Draw the electrical circuit with the analogous equation of force-voltage and

force-current as shown below.

(ii) Mechanical Rotational System

In rotational system, there are three basic elements as in the translational system. They are:

(i) Inertia torque

(ii) Damping torque

(iii) Spring torque

(i) Inertia Torque (Tj)

Inertia torque Tj is proportional to angular acceleration

i.e., Tj d 2

dt2

Torque balance equation for Inertia Torque

Consider an ideal inertia element shown in Fig.1.10, with angular displacement and

applied torque.

Fig. 1.10. Ideal inertia element

Let T(t) = Applied torque

Tj = Opposing torque due to moment of inertia

Here Tj d 2

dt2

Tj = J d 2

dt2

(ii) Damping Torque (TB)

Damping torque is proportional to angular velocity.

i.e., TB dq

dt [ Angular velocity =

dq

dt ]

Torque balance equation for Damping Torque

Consider an ideal frictional element shown in Fig.1.11, with applied torque and angular

acceleration at one side.

Fig. 1.11. Ideal friction element with one angular acceleration

Let T = Applied torque

TB = Opposing torque due to friction (or) Damping torque

TB dq

dt

TB = B dq

dt

Consider an ideal friction element shown in Fig.1.12, with applied torque and angular

acceleration at both the sides.

Fig. 1.12. Ideal friction element with two angular accelerations

Let T = Applied torque

TB = Opposing torque due to friction

TB d

dt (1 – 2)

TB = B d

dt (1 – 2)

(iii) Spring Torque (TK)

Spring torque is proportional to angular displacement.

i.e., TK [ Displacement = ]

Torque balance equation for Spring Torque

Consider an ideal spring element shown in Fig.1.13, with applied torque and angular

displacement at one side.

Fig. 1.13. Ideal spring element with one displacement

Let T = Applied torque

TK = Opposing torque due to spring

TK

TK = K

Consider an ideal spring element shown in Fig.1.14, with applied torque and angular

displacement of both sides.

Fig. 1.14. Ideal spring element with displacement at both sides

Let T = Applied torque

TK = Opposing torque due to spring

TK (1 – 2)

TK = K (1 – 2)

Block Diagram A control system consists of number of components. The function of each component in a system

is represented by a block. All blocks are interconnected by lines with arrows indicating the flow of

signals from the output of block to the input of another block is said to be a block diagram.

A block diagram of a system is a diagrammatic representation of the function of each component

and the flow of signals.

The major elements of block diagram representation are,

(i) Block

(ii) Summing point

(iii) Branch point

Block

A block is used to represent the function of each component in a system. The functional block (or)

block is a symbol for the mathematical operation on the input signal to the block that produces the

output.

A simple block is shown below.

The arrowhead point towards the block indicates the input, and the arrowhead leading away from

the block represents the output.

Summing Point

Summing points are used to add two (or) more signal in the system. Example summing point is

shown below.

The plus (or) minus sign at each arrowhead indicates whether the signal is to be added or

subtracted. It is important that the quantities being added or subtracted have the same dimensions

and the same units.

Branch Point (or) Take off Point

A branch point is a point from which the signal from a block goes concurrently to other blocks or

summing points.

RULES TO REDUCE THE BLOCK DIAGRAM

The transfer function of the system can be obtained by reducing the block diagram. To

reduce the given block diagram, the following rules should be used. The following rule

made some modification and finally made it to single block to find the transfer function of

the block diagram.

Rule 1: Combine the two cascade blocks

Rule 2: Combine the parallel (or) Feed forward blocks

Rule 3: Reduction of feedback loop

[ For positive feedback loop

C(s)

R(s) =

G(s)

1 – G(s) H(s)

For negative feedback loop

C(s)

R(s) =

G(s)

1 + G(s) H(s) ]

Rule 4: Moving a branch point ahead of the block

Rule 5: Moving a branch point before the block

Rule 6: Moving the summing point ahead of the block

Rule 7: Moving the summing point before the block

Rule 8: Interchanging the summing point

Rule 9: Splitting the summing point

Rule 10: Combining the summing point

SIGNAL FLOW GRAPH

The signal flow graph is used to represent the control system graphically and it was

developed by S.J. Mason. Mason proposed the Mason’s gain formula to find the transfer

function of the system.

A signal flow graph is a diagram which represents a set of simultaneous equations. It

consists of line network in which various nodes are connected by means of directed lines

called branches.

A signal flow graph is a graphical representation of the relationship between variables of a

set of linear algebraic equations.

Signal flow graphs are particularly useful for feedback control system because feedback

theory is primarily concerned with the flow and processing of signals in systems.

Terms used in Signal Flow Graphs

Terms Explanation

Node A node is a point representing a variable or signal

Branch A branch is directed line segment joining two nodes. The

arrow on the branch indicates the direction of signal flow and

the gain of a branch is the transmittance.

Transmittance The gain acquired by the signal when it travels from one node

to another is called transmittance. The transmittance can be

real or complex.

Input node

(or) Source

node

It is a node that has only outgoing branches.

Output node

(or) sink node

It is a node that has only incoming branches

Mixed node It is a node that has both incoming and outgoing branches

Path A path is a traversal of connected branches in the direction of

Terms Explanation

the branch arrows. The path should not cross a node more than

once.

Open path A open path starts at a node and ends at another node.

Closed path Closed path starts and ends at same node.

Forward path It is a path from an input node to an output node that does not

cross any node more than once.

Forward path

gain

It is the product of the branch transmittances of a forward path.

Individual

loop

It is a closed path starting from a node and after passing

through a certain part of a graph arrives at same node without

crossing any node more than once.

Loop gain It is the product of the branch transmittances of a loop

Non-touching

loops

If the loops do not have a common node then they are said to

be non-touching loops.

1.0.1. PROPERTIES OF SIGNAL FLOW GRAPH

(i) Signal flow graph is applicable to linear systems only.

(ii) A node in the signal flow graph represents the variable or signal.

(iii) The algebraic equations which are used to construct signal flow graph must be in

the form of cause and effect relationship.

(iv) A branch indicates the functional dependence of one signal upon another. A

signal passes only in the direction specified by the arrow of branch.

(v) A node adds all signals of all incoming branches and transmits this sum to all

outgoing branches. This is known as additive rule.

(vi) A mixed node which has both incoming and outgoing branches may be treated as

an output node by adding an outgoing branch of unity gain.

(vii) The signal travel along branches only in the marked direction.

(viii) For system a signal flow graph is not unique.

1.0.2. RULES TO REDUCE SIGNAL FLOW GRAPH

The following rules are used to reduce the signal flow graph and to find the transfer

function of the system.

Rule 1: Incoming signal to a node through a branch is given by the product of a signal

at previous node and the gain of the branch.

i.e.,

Here the source node is X1. Sink node is X2. Gain is ‘a’

X2 = Product of a signal of previous node and gain

X2 = a X1

Similarly

X4 = a1 X1 + a2 X2 + a3 X3

Rule 2: Cascaded branches can be combined by the product of the gain of the branches

to give a single node.

i.e.,

Rule 3: Parallel branches can be combined by adding the gain of the branches to give a

single node.

i.e.,

Rule 4: A mixed node can be eliminated by multiplying the transmittance of outgoing

branch to the transmittance of all incoming branches to the mixed node.

Rule 5: A loop may be eliminated by the following condition.

The signal flow graph of a system can be reduced either by using the rules of a signal

flow graph or by using Mason’s gain formula.

For signal flow graph reduction using the rules of signal flow graph, write equations at

every node and then rearrange these equations to get the transfer function.

Hence Mason’s gain formula is the easy method to find the transfer function compared

to the reduction method using the signal flow graph rules.

Mason’s Gain Formula

The Mason’s gain formula is used to determine the transfer function of the system from

signal flow graph.

Mason’s gain formula, T = 1

1

K PK K

where, T = Transfer function of the system

K = Number of forward path

PK = Forward path gain of Kth forward path

= 1 –

Sum of

individual

loop gain +

Sum of gain product

of all possible two

non-touching loops –

Sum of gain product

of all possible

three non-touching

loops

+

K = for that path of the graph which is not touching Kth forward path.

Unit 2 Time Response Analysis

Definition: The time response of a system is the output of the system as a function of time

to the given input.

In time response analysis basically system is tested in time instead of the frequency

method. When a system is subjected to an input, its stability (or) response is named as

1. Transient response

2. Steady state response

Transient Response

One of the most important characteristic of control system is transient response. The

transient response is the response of the system as a function of time when the input

changes from one state to another state. Because the purpose of control system is to

provide a desired response. The transient response of control system often must be

adjusted until it is satisfactory.

Whenever there is a input change, the system cannot response immediately. It requires

some time gap to response. This time gap is referred as transient response. So in transient

response, the system is checked for its speed of response.

Stead State Response

The steady state response is the response of the system for a given input after a very long

time. The steady state response of any system gives an idea of the accuracy of the system.

So in steady state we check for the system stability for the input.

Standard Test Signals

For testing of time response analysis in the laboratory, four types of standard test input

signals are used. They are:

1. Step input

2. Impulse input

3. Ramp input

4. Parabolic input

STEP INPUT

A step is a signal whose value changes from one level (usually

zero) to another level in zero time i.e., a sudden change.

Mathematically a step input is given by

r(t) = A for t > 0

r(t) = 0 for t < 0

The Laplace transform of step signal is Fig. 2.15. Step input

L[r(t)] = R(s) = A

s

Unit step input: If the magnitude of A is unity (i.e., A = 1) then

the step signal is said to be unit step signal. i.e.,

r(t) = A for t > 0

where A = 1 for unit step

r(t) = 1 for t > 0

The Laplace transform of unit step signal is Fig. 2.16. Unit step input

L[r(t)] = R(s) = 1

s

IMPULSE INPUT

A signal which has zero amplitude everywhere except at the

origin (i.e., at zero time) then the signal is said to be impulse

input. Also it is said to be a signal having a large amplitude and

it exists for a short time such that the area under the curve.

Fig. 2.17. Impulse input

Mathematically an impulse signal is given by

r(t) = ∞ at t = 0

r(t) = 0 at t 0

i.e., – E

E

A f (t) dt = A

The Laplace transform of impulse signal is given by

L [A f (t)] = R(s) = A

Unit impulse input: A unit impulse function is a signal having a large amplitude and it

exists for a short time such that the area under the curve is unity. It is also known as ‘Delta

Function’ f (t).

Fig. 2.18. Unit impulse input

For unit impulse input A = 1

i.e., – E

E

A f (t) dt = A where A = 1

– E

E

f (t) dt = 1

The Laplace transform of unit impulse signal is

L [ f (t)] = R(s) = 1

RAMP INPUT

The ramp is a signal which starts at a value of zero and

increases linearly with time i.e., constant velocity.

Mathematically a ramp input is given by

r(t) = A t for t > 0

r(t) = 0 for t < 0 Fig. 2.19. Ramp input

Laplace transform of ramp input is given by

L [r(t)] = R(s) = A

s2

Unit ramp signal: The slope of A is unity (i.e., A = 1) of a ramp

signal, then the signal is said to be unit ramp signal.

Mathematically a unit ramp input is given by

r(t) = A t for t > 0

where A = 1 for unit ramp

r(t) = t for t > 0

The Laplace transform of unit ramp is given by Fig. 2.20. Unit ramp input

L [r(t)] = R(s) = 1

s2

PARABOLIC INPUT

The instantaneous value of a parabolic signal varies as square

of the time from an initial value of zero at t = 0.

i.e., It is a integral of ramp.

Mathematically a parabolic signal is given by

r(t) = A t2

2 for t > 0 Fig. 2.21. Parabolic input

r(t) = 0 for t < 0

The Laplace transform of parabolic signal is given by

L [r(t)] = R(s) = A

s3

Unit parabolic input: For a unit parabolic input the

instantaneous value of A = 1.

Mathematically the unit parabola is given by

r(t) = t2

2 for t > 0

The Laplace transform of unit parabola is

L [r(t)] = R(s) = 1

s3 Fig. 2.22. Unit parabolic input

Order of the system

Since we studied about the modelling of system in different ways. In that transfer function

is one of the methods. The transfer function is obtained by the differential equation

governing the system.

Thus the input and output relationship of a control system can be expressed by nth order

differential equation is given by

a0 d n

dtn c(t) + a1

d n–1

dt n–1 c(t) + a2

d n–2

dt n–2 c(t) + + an – 1

d

dt c(t) + an (t) =

b0 d m

dtm r(t) + b1

d m–1

dt m–1 r(t) + + bm – 1

d

dt r(t) + bm r(t) (1)

where r(t) = input and c(t) = output (or) response

Taking Laplace transform for equation (1), we get

a0 sn C(s) + a1 sn–1 C(s) + a2 sn–2 C(s) + + an – 1 s C(s) + an C(s) =

b0 sm R(s) + b1 sm–1 R(s) + + bm – 1 s R(s) + bm R(s) (2)

By rearranging the above equation, we get

Transfer function = C(s)

R(s) =

b0 sm + b1 sm–1 + b2 sm–2 + + bm – 1 s + bm

a0 sn + a1 sn–1 + a2 sn–2 + + an – 1 s + an

(3)

The order of the system is given by the maximum power of s in the denominator

polynomial R(s).

Here R(s) = a0 sn + a1 sn–1 + a2 sn–2 + + an – 1 s + an

Now, n is the order of the system.

When n = 0, the system is zero order system.

When n = 1, the system is first order system.

When n = 2, the system is second order system.

This order can be specified for both open loop and closed loop system.

The numerator and denominator polynomial of equation (3) can be expressed in the

factorized form as

T(s) = C(s)

R(s) =

(s + z1) (s + z2) (s + zm)

(s + p1) (s + p2) (s + pn) (4)

where z1 , z2 , zm are the zeros of the system

p1 p2 pn are the poles of the system

Now, the value of n gives the number of poles in the transfer function. Hence the order is

also given by the number of poles of the transfer function.

Steady state error

Steady state error is an important measure of the accuracy of a control system. Basically

these errors occurs from the nature of inputs, non-linearities present in the system, etc.

The steady state error ess is the difference between the input and the output of a closed

loop system for a known input as t ∞.

Mathematically, ess = lim

t ∞ e(t) =

limt ∞

[ r(t) – C(t) ]

where e(t) = Error signal in time response

According to final value theorem

ess = lim

t ∞ e(t) =

lims ∞

s E(s)

where E(s) = Error signal in s-domain

Consider a closed loop system with error E(s).

Closed loop transfer function = C(s)

R(s) =

G(s)

1 + G(s) H(s)

Response of the system C(s) = G(s)

1 + G(s) H(s) R(s)

According to the given system, the response is also given by

C(s) = E(s) G(s)

E(s) = C(s)

G(s)

E(s) = G(s) R(s)

1 + G(s) H(s)

1

G(s) [ C(s) =

G(s) R(s)

1 + G(s) H(s)]

E(s) = R(s)

1 + G(s) H(s)

Hence the steady state error ess is given by

ess = lim

s 0 s E(s)

ess = lim

s 0

s R(s)

1 + G(s) H(s)

In order to make steady state error analysis, several independents of system under

consideration, the output of system will be referred as ‘position’. Rate of change of output

will be referred as ‘velocity’ and double the rate of change of output will be referred as

‘acceleration’.

Classification of Steady State Error

According to the error and information about the error, the steady state error are classified

into two types.

(i) Static error constants

(ii) Dynamic error coefficients (or) Generalized error coefficient

Static Error Constants

Static errors are the error occur due to the present input function. According to the input

function, the error is classified into three types.

(i) Position error constant (Kp ): It is a measure of steady state error between the

input and output of the system, when the system is subjected to a unit step function.

(ii) Velocity error constant (Kv ): It is a measure of steady state error between the

input and output of the system, when the system is subjected to a ramp function.

(iii) Acceleration error constant (Ka ): It is a measure of steady state error between the

input and output of the system, when the system is subjected to a parabolic

function.

Generalized Error Coefficient

The static error constants Kp , Kv, Ka are used only for particular input. But using

generalised error coefficient, we can apply system to any arbitary input. Moreover Kp ,

Kv, Ka gives definite values for errors, either zero, a finite value (or) ∞. They do not give

information regarding the variation of error with time. The dynamic error coefficient gives

complete picture of error, as a function of time.

Error e(t) = C0 r(t) + C1

1 r ′(t) +

C2

2 r ′ ′(t) (1)

The above equation is known as the “Error Series”.

C0 , C1 , C2 are generalised (or) dynamic error constants.

We know that, E(s) = R(s)

1 + G(s) H(s) (2)

where 1 + G(s) H(s) are characteristic polynomial.

The reciprocal of characteristic polynomial = 1

1 + G(s) H(s)

e(s) = 1

1 + G(s) H(s) (3)

Substitute equation (3) in equation (2), we get

E(s) = R(s) e(s)

Taking Laplace inverse transform, we get

e(t) = L–1 [ R(s) e(s) ]

By convolution integral theorem,

e(t) = 0

t

r(t – ) e() d

where = Dummy variable

Expand r(t – ) using Taylor’s series

r(t – ) = r(t) –

1 r ′(t) +

2

2 r ′ ′(t) ∞

e(t) = 0

t

r(t – ) e() d

e(t) = 0

t

r(t) –

1 r ′(t) +

2

2 r ′ ′(t)

e() d

We know that

ess(t) = Ltt ∞ e(t)

ess(t) = Ltt ∞

0

t

r(t) –

1 r ′(t) +

2

2 r ′ ′(t)

e() d

= 0

∞

r(t) –

1 r ′(t) +

2

2 r ′ ′(t)

e() d

ess(t) = r(t) 0

∞

e() d + r ′(t)

1 0

∞

– e() d + r ′ ′(t)

2 0

∞

2 e() d

ess(t) = C0 r(t) + C1

1 r ′(t) +

C2

2 r ′ ′(t)

where C0 = (–1)0 0

∞

e() d

C1 = (–1)1 0

∞

e() d

C2 = (–1)2 0

∞

2 e() d

Cn = (–1)n 0

∞

n e() d

By basic definition of Laplace transform

F(s) = 0

∞

e – st f (t) dt

e(s) = 0

∞

e – s e() d

Taking the limit s 0 on both sides

Lt

s 0 e(s) =

Lts 0

0

∞

e – s e() d = 0

∞

e– 0 e() d

= 0

∞

e() d = C0

Lt

s 0 e(s) = C0

Differentiate e(s) with respect to s ,

e′(s) = d

ds (e(s)) =

d

ds

0

∞

e – s e() d

e′(s) = 0

∞

– e – s e() d

Taking limit s 0 on both sides, we get

Lt

s 0 e′(s) =

Lts 0

0

∞

– e – s e() d

= 0

∞

– e– 0 e() d

= 0

∞

– e() d = C1

Lt

s 0 e′(s) = C1

Similarly, C2 = Lt

s 0 e′′(s)

Controllers

The controllers are used in the system to produce a control signal necessary to reduce the

error signal to zero (or) to a small value. The control action may operate through

mechanical, pneumatic, hydraulic (or) electrical means.

Most of the controllers implements through some control actions. Some of the control

actions are shown below.

Consider a closed loop control system shown in Fig.

where R(s) = Input reference signal

e = Error signal

m = Manipulated variable

B(s) = Feedback signal

G1(s), G2(s) = Feed forward transfer function

H(s) = Feedback transfer function

Classification of controllers

(i) ON-OFF controller (or) Two position controller

(ii) Proportional controller

(iii) Integral controller

(iv) Proportional plus integral controller

(v) Proportional plus derivative controller

(vi) PID controller

ON-OFF CONTROLLER

In ON-OFF controller, the actuating element is capable of assuming only tow positions,

with either zero (or) maximum input to the process.

Example: An electric switch is open (or) closed depending on whether the controlled

variable is above (or) below the set point.

PROPORTIONAL CONTROLLER (P)

Proportional action is a mode of controller action in which there are continuous linear

relation between values of deviation (error signal) and manipulated variable.

It is defined as the action of a controller in which the output signal M(t) is proportional to

the measured actuating error signal e(t).

M(t) e(t)

M(t) = Kp e(t)

where Kp is termed proportional sensitivity or the gain

Taking Laplace transform

M(s) = Kp E(s)

Kp = M(s)

E(s)

Kp = Change in the controller output

Change in deviation

So the proportional controller is essentially an amplifier with an adjustable gain A.

The proportional sensitivity Kp is the change of M caused by unit change of deviation.

Proportional band = 100

Kp

The range of error to cover 0% – 100% controller output is called proportional band (PB).

The important points to be remembered are:

If error = 0, the output is a constant equal to M.

If there is error for every 1% of error, a correction of Kp % is added to or subtracted

from M depending upon the reverse on direct action of the controller.

There is a band of error about zero of magnitude PB within which the output is not

saturated at 0 or 100%.

A proportional controller is simply an amplifier with adjustable gain.

The function of an automatic control is to maintain the variable at a constant value. Large

shift due to proportional control give the instability of the system.

INTEGRAL CONTROLLER (I)

In a controller with integral control action, the value of output M(t) is proportional to the

measured actuating error signal e(t).

d M(t)

ds = Ki e(t)

M(t) = Ki 0

t

e(t)

where Ki is an adjustable constant, the transfer function of integral controller is

M(s)

E(s) =

Ki

s

For zero actuating error, the value of M(t) remains stationary. The integral control action

is called reset control (or) reset action.

PROPORTIONAL PLUS INTEGRAL CONTROLLER (PI)

In a PI controller, the controller output M(t) is proportional to a linear combination of

actuating signal e(t) and its time derivative.

M = Kp

Ti e + Kp

de

dt

where de

dt is the rate of change of e with respect to time.

Kp is the proportional sensitivity and Ti is the integral time.

In integral form, M = Kp

T1 e dt + Kp e + M

In operational form, M = Kp

1

Ti s + 1 e

The transfer function of PI controller is

M(s)

e(s) = Kp

1 + 1

Ti s

PROPORTIONAL PLUS DERIVATIVE CONTROLLER (PD)

The PD controller refers to the control action where a derivative control action is added to

the proportional control action.

M(t) = Kp e(t) + Kp (Td) de(t)

dt

Taking Laplace transform

M(s) = Kp (1 + Td s) e(s)

M(s)

e(s) = Kp (1 + Td s)

where Kp is proportional sensitivity and Td derivative time.

PID CONTROLLER

A PID controller is the combination of proportional control action, derivative control

action and integral control action with an adjustable gain for each action.

The equation of PID controller with this combined action is given by

M(t) = Kp e(t) + Kp Td de(t)

dt +

Kp

Ti 0

t

e(t) dt

Taking Laplace transform

M(s)

e(s) = Kp

1 + Td s + 1

Ti s

Comparison of

Proportional, Integral and Derivative Action

(i) Proportional-derivative control provides the smallest maximum error because the

derivative part of the response increases the proportional sensitivity to a high

value.

Stabilization point is also smallest in PD control.

(ii) PID control has the large maximum deviation (error), then controller with

derivative action, offset is also large.

(iii) Proportional control has the next smallest maximum deviation and offset is

eliminated due to presence of integral control.

(iv) Proportional integral control has no offset. It has the large maximum derivation

and persisting deviation.

(v) Integral control is best suited for control of process having little of no energy

storage. It also has large maximum derivation and a long stabilization time.

Unit 3 Frequency Response Analysis

Bode Plot

A sinusoidal function may be represented by two separate plots. One giving the magnitude

versus frequency and the other the phase angle versus frequency.

Bode plot is a frequency response plots of G(S) H(S) . It gives

magnitude and phase angle details separately.

ADVANTAGES OF BODE PLOT

1. It is the simplest method.

2. The multiplication of magnitudes can be converted into addition.

3. Transfer function can be determined easily.

4. The low frequency and high frequency characteristics of the transfer function can be

determined.

MAGNITUDE PLOT

function is named as “Bode Magnitude Plot”.

PHASE ANGLE PLOT

ransfer function is

named as “Bode Phase Angle Plot”.

SEMI LOG SHEET

The magnitude plots and phase plots of bode plot are drawn on the semi log sheet. The

magnitude of G(j

dB = 20 log10 M

The range of 1 and 2 where 2 1 is

called as “one decade”.

Generally there are 5 decades available on semilog sheet.

Slope Triangle

STANDARD FORM OF OPEN LOOP TRANSFER FUNCTION FOR BODE PLOT

G(S) H(S) =

K (1 + s Tz1 ) (1 + s Tz2

) ……

sn (1 + s Tp1 ) (1 + s Tp2

) ……

where K = Bode gain,

Tz1 , Tz2

, Tp1 , Tp2

= Time constants of different poles and zeros.

IMPORTANT DEFINITIONS

Gain cross over frequency : It is the frequency at which magnitude of G(j

decibels.

Phase cross over frequency : It is the frequency at which the phase angle of G(j

– 180 line.

Gain Margin : It is the magnitude in decibels at phase cross over frequency.

Phase Margin : It is the phase angle at gain cross over frequency.

SIGN CONVERSION FOR PHASE MARGIN AND GAIN MARGIN

TO FIND STABILITY OF THE SYSTEM BY USING GM AND PM

1. If both GM and PM are positive, system is stable.

2. If any one of the two (or) both are negative, the system is unstable.

Polar Plot

It is a plot of magnitude versus phase angle for different values of frequency. It is a plot of

the magnitude of G(j versus phase angle of G(j

from zero to infinity. So the polar plot is the locus of vectors | G(j | G(j

from zero to infinity. Both magnitude | G(j | and phase G(j

directly for eac

as Nyquist plot.

Step 1 : Only open loop transfer function of a system must be considered.

Step 2 : If the system has unity feedback consider only G(S).

Step 3 : If block diagram is given, multiply the forward block gain with feedback path

gain to get open loop transfer function.

Step 4 : Substitute s = j j = –1

Step 5 : Write magnitude and phase equation for the given function using the following

standards.

(a) G(S) = s + a

Substitute s = j

Magnitude M = | G(j | = 2 + a2

= tan–1

a

(b) G(S) = 1

s + a

Substitute s = j

G(j = 1

j a

Magnitude M = | G(j | = 1

2 + a2

= – tan–1

a

(c) G(S) = 1

a – s

Substitute s = j

G(j = 1

a – j

Magnitude M = | G(j | = 1

a2 + 2

= – tan–1

–

a = tan–1

a

(d) G(S) = 1

s – a

Substitute s = j

G(j = 1

j – a

Magnitude M = | G(j | = 1

2 + a2 – tan–1

a

(e) G(S) = s

Substitute s = j

G(j = j

Magnitude M =

= + 90

( ) G(S) = s2

Substitute s = j

G(j = j2 2 = – 2

Magnitude M = 2

= + 180

In general, G(S) = sn

then, M = n

= 90 * n

(g) G(S) = 1

s

Substitute s = j

G(j = 1

j =

– j

Magnitude M = 1

= – 90

In general, G(S) = 1

sm , M =

1

m – 90 * m

(h) G(S) = K

Magnitude M = k

= 0 if k > 0

= 180 if k < 0

Summary of Magnitude and Phase Angle Equations

Function Magnitude Equations Phase Angle Equation

s + a 2 + a2 tan–1

a

1

s + a

1

2 + a2 – tan

a

1

s – a

1

2 + a2 –

180 – tan–1

a

1

a – s

1

a2 + 2 tan–1

a

s 90

sn n n * 90

1

s

1 – 90

1

sm

1

m – m * 90

k k 0 if k > 0

– 180 if k < 0

Step 6 :

.

Step 7 : Draw the polar plot. It is usually drawn on a polar graph sheet. It has concentric

circles and the radial line indicates the phase angle of plot. Positive angles are measured in

anticlockwise direction and negative angles are measured in clockwise direction.

Step 8 : Gain Cross Over : The point where the plot touches unit circle is called Gain

gco.

Phase cross over : The point where the plot touches – 180 line is called phase cross over.

The magnitude measured at this point is called as phase cross over magnitude

| G(j pco) |.

Gain margin (Kg ) : It is defined as the reciprocal of the magnitude of the open loop

pco.

Kg = 1

| G(j pco) |

Phase margin ( ) : It is defined as the additional phase angle lag at gain cross over

required to bring the system to the limit of instability.

= 180 gco

Step 9 : To find stability of the system :

Case (i) : When both phase margin and gain margin are ‘+’ve, then the system is stable.

Case (ii) : Either phase margin (or) gain margin are ‘–’ve, then the system is unstable.

Corner Frequency : It is the frequency at which the polar plot changes its direction and

magnitude.

Example : G(S) = 1

1 + sT

Corner frequency = 1

T rad / sec

G(S) = 1

1 + 10 s

Corner frequency = 1

10 = 0.1 rad / sec

Compensation

The Control system is designed to meet a desired response. The response of the system are

usually specified as performance specifications. These specifications are generally related

to accuracy, relative stability and speed of response.

To meet the desired response, we need to adjust the gain of system specification. In

practical, the adjustment of the gain alone will not be sufficient to meet the given

specification. In many cases, only increasing of gain may also result in poor stability or

instability. In those cases, it is necessary to introduce additional devices or components in

the system to alter the behaviour to meet the desired response.

Such a redesigning or addition of a suitable device is called compensation.

A device added into the system for the purpose of compensation is called compensator.

The compensation scheme used for feedback control system is either series compensation

or parallel compensation (or) series-parallel compensation.

(i) Series Compensation (Cascade)

In series compensation method, the compensating network is introduced in forward path of

a control system.

Let the compensation be GC(s).

(ii) Parallel Compensation (Feedback)

In feedback compensation method, the compensating network is introduced in the

feedback path of a control system.

(iii) Series-Parallel Compensation

In this method, two compensating network are introduced to get the desired output. First

one is introduced in forward path and second one is introduced in feedback path.

The compensator may be electrical, mechanical, hydraulic, pneumatic or other type of

device or network. Usually, an electric network or electronic device serves as compensator

in many control systems.

The different types of electrical (or) electronic compensator used are:

(i) Lag-compensator

(ii) Lead-compensator

(iii) Lag-Lead compensator

Need of Compensation

It is necessary to compensate an unstable system to make a stable system.

It may be necessary to improve an existing system to satisfy (or) to meet

specification on the steady state and transient performance.

It also used with a fixed configuration to control a particular component of a system.

It will give absolute desired stability as well as satisfactory specification.

The design method is carried out in frequency domain and the choice has been either to

use lag (or) lead (or) lag-lead network.

Unit 4 Stability Analysis

Concept of stability

The concept of stability can be explained by considering a prism placed on a plane

horizontal surface. If the prism is resting on its base, the position is said to be stable. If the

prism is resting on its tip and released, in this case, the prism will fall down without any

external application of forces. This position is said to be unstable.

The stability of the system is given by the following definitions. The stability is analysis

with the output of the system with respect to input.

Stable System

A system is said to be a stable system, if the system has bounded output for any bounded

input.

A system is asymptotically stable, if in the absence of the input, the output tends toward

zero irrespective of initial conditions.

Limitedly Stable

For a bounded input signal, if the output has constant amplitude oscillations, then the

system may be stable or unstable under some limited constraints. Such a system is called

limitedly stable.

Absolutely Stable

If a system output is stable for all variations of its parameters, then the system is called

absolutely stable system.

Conditionally Stable System

If a system output is stable for a limited range of variations of its parameters, then the

system is called conditionally stable system.

Relative Stability

It is a quantitative measure of how fast the transients die out in a system.

Unstable System

A system is said to be unstable if its response to a bounded input ultimately becomes an

infinite amplitude.

Stability Condition

Control loop transfer function = C(s)

R(s) =

G(s)

1 + G(s) H(s)

G(s) = 0

∞

g(t) e – st dt

where g(t) is the impulse response of the system.

Taking the magnitude of the sides of equation

| G(s) | = 0

∞

| g(t) | e – st dt

1 + G(s) H(s) is known as characteristic polynomial and when equated to zero, it is

characteristic equation.

C.E = 1 + G(s) H(s) = 0

C.E = 1 + open loop T.F = 0

Remarks on Stability

(i) If all the roots of characteristic equation have negative real parts, then the system

is said to be stable.

(ii) If any root of C.E has ‘+ve” real part, then the system is unstable.

(iii) If few non-repeating roots on imaginary axis and the remaining roots on L.H.S of

s-plane, the system is ‘Limitedly (or) Marginally Stable System’.

Remarks on Coefficients of Characteristic Polynomials

(i) If all the coefficients are positive and if no coefficients is zero, then all the roots

are in the left half of s-plane.

(ii) If any coefficient ai is equal to zero, then some of the roots may be on the

imaginary axis or on the right left of s-plane.

(iii) If any coefficient ai is negative, then atleast one root is in the right half of

s-plane.

It is concluded that the negativeness of any of the coefficient of a characteristic

polynomial indicates that the system is either unstable or at most marginally stable. Hence

for a stable system, all the coefficients of its characteristic polynomial be positive.

Routh Hurwitz criterion

The Routh-Hurwitz stability criterion is an analytical procedure for determining whether all the

roots of a polynomial have negative real part or not.

Routh-Hurwitz (R-H) criterion uses s-plane concept for getting result about the stability of a

given system.

The stability of the system is examined for the characteristic polynomial of the system. And it

is a method for determining system stability that can be applied to on nth order characteristic

equation of the form.

Characteristic equation = 1 + G(s) H(s) = 0

1 + G(s) H(s) = 0 = a0 sn + a1 sn – 1

+ + an – 1 s1 + an

R-H criterion is based on arranging the coefficients of characteristic equation into an array

called ‘Routh Array’.

C.E = 1 + G(s) H(s) = 0

C.E = a0 s6 + a1 s5 + a2 s4 + a3 s3 + a4 s2 + a5 s1 + a6 s0

Routh Array

sn a0 a2 a4 a6

sn – 1

a1 a3 a5

sn – 2

A B C

sn – 3

D E 0

s0

where A = a1 a2 – a0 a3

a1 ; B =

a1 a4 – a0 a5

a1

C = a1 a6 – a0 0

a1 a6

D = A a3 – B a1

A ; E = A a5 – C a1

A

The table is continued till the coefficients for s0 is obtained.

For a system to be stable, the necessary condition is “There should be no changes in sign of 1st

column of the array”.

If the first column of the array has a sign change, then the system is unstable and the number

of sign changes of the terms of first column correspond to the number of roots lying in the right

half of s-plane.

Nyquist plot

The concept of Nyquist plot is based on the polar plot which can be conveniently applied

to the stability analysis of any kind of system. The following section will briefly explain

the Nyquist plot.

(i) Pole-zero configuration

(ii) Concept of encirclement, enclosed and number of encirclement

(iii) Analytic function and its singularities.

(iv) Mapping theorem (or) Principle of argument

(v) Nyquist stability criterion

5.0.1. POLE-ZERO CONFIGURATION

Any function which can be expressed as a ratio of two polynomials has its own poles and

zeros.

Consider function G(s) H(s) called open loop transfer function of a system.

Poles of G(s) H(s) are called open loop poles.

Zeros of G(s) H(s) are called open loop zeros.

Now consider the closed loop transfer function C(s)

R(s) as

C(s)

R(s) =

G(s)

1 + G(s) H(s)

The poles of this transfer function are the roots of the characteristic equation 1 + G(s)

H(s) = 0 and are called closed loop poles of a system.

Let us take G(s) H(s) = 6

s (s + 5) then open loop poles are s = 0, – 5.

Open loop zeros are absent.

While 1 + G(s) H(s) = 0 is characteristic equation.

i.e., 1 + 6

s (s + 5) = 0

i.e., s2 + 5 s + 6 = 0

Closed loop poles are the roots of s2 + 5 s + 6 = 0.

Now consider the mathematical function F(s) = 1 + G(s) H(s).

This expression is expressed as ratio of two separate polynomials in as,

F(s) = 1 + G(s) H(s) = P(s)

Q(s)

(i) Now P(s) = 0 are the zeros of 1 + G(s) H(s) and Q(s) = 0 are the poles of

1 + G(s) H(s). The LCM of 1 + G(s) H(s) is always the denominator of

G(s) H(s). So Q(s) polynomial is always the denominator of the function G(s)

H(s). So Q(s) = 0 gives the roots which we have called open loop poles of a

system. Hence, “Poles of 1 + G(s) H(s) = open loop poles of a system”.

(ii) Now consider P(s) = 0 gives the zeros of 1 + G(s) H(s). But P(s) = 0 gives us

an equation which is nothing but the characteristic equation of the system. This

gives closed loop poles of the system. Hence “Zeros of 1 + G(s) H(s) = Closed

loop poles of a system”.

(iii) Now F(s) = 1 + G(s) H(s)

G(s) H(s) = 6

s (s + 5)

F(s) = 1 + 6

s (s + 5)

s2 + 5 s + 6

s (s + 5)

P(s)

Q(s)

So Q(s) = 0 gives poles of 1 + G(s) H(s) i.e., s = 0, – 5 which are the open loop poles,

while P(s) = 0 gives zeros of 1 + G(s) H(s). i.e., roots of s2 + 5 s + 6 which are the

closed loop poles. Hence,

“The system is absolutely stable if all the zeros of 1 + G(s) H(s) i.e., closed loop poles

of the system are located in the left half of s-plane”.

5.0.2. CONCEPT OF ENCIRCLEMENT AND ENCLOSED

It is important to distinguish between the concept of encircled and enclosed which are

frequently used to apply Nyquist criterion.

Encircled: A point is said to be encircled by a closed path if it is found inside the path.

With reference to Fig.1, the point is encircled in the clockwise direction and the point B is

not encircled.

Enclosed: Any point or region is said to be enclosed by a closed path, if it is found to lie

to the right of the path when the path is traversed in the prescribed direction. The shaded

regions in Fig.2 and 3 are the regions enclosed by the closed path. With reference to Fig.2,

the point A is enclosed by closed path and the point B is not enclosed. With reference to

Fig.3, the point A is not enclosed by closed path but point B is enclosed.

COUNTING NUMBER OF ENCIRCLEMENT

Consider a closed path in s-Plane

In the above Figure, it is easy to conclude that the number of

encirclements of point A and 0 is 1 in anticlockwise direction.

But for complicated cases use the following methods to find

number of encirclements.

Steps to get number of encirclements

(i) Draw a vector from a point whose encirclements are to be determined, in such a

way to join any point outside that closed path in any direction.

(ii) Identify the number of intersections of this vectors with a closed path.

(iii) Make these intersections with small arrow on the same vector indicating direction

of closed path at the time of intersection.

(iv) Cancel the oppositely directed encirclements. The remaining arrows gives us the

number of encirclements of that point.

For counting of encirclement anticlockwise are treated as positive and clockwise are

treated as negative.

Root Locus It is the locus (or) path of the roots traced out on the s-plane as its parameter is changed.

Root Locus Technique

It is a method of plotting the locus of the root of the characteristic equation in s-plane

when the gain of the system is varied over the entire range.

5.0.1. CONCEPT OF ROOT LOCUS TECHNIQUE

Consider a unity feedback system as shown below.

Fig. 5.23. Second order system

The open loop transfer function of this system is

G(s) H(s) = K

s (s + a)

where K and a are constants.

The open-loop transfer function has two poles – one at origins s = 0 and the other at

s = – a .

The closed loop transfer function of this system is,

C(s)

R(s) =

K

s2 + as + K

The characteristic equation of the system is

s2 + as + K = 0

System is always stable for positive values of ‘a’ and K, but its dynamic behaviour is

controlled by the roots of characteristic equation, the roots are given by

s1 s2 = – a

2 ±

a

2

2 – K

Here any of the system parameters ‘a’ or K varies, the roots of the characteristic equation

change.

Case (i):

Here the encirclement of point A is one and it is in anticlockwise direction. Hence the

number of encirclement, N = + 1.

Case (ii):

Here the encirclement of point A is two and it is in clockwise direction. Hence the number

of encirclement, N = – 2.

Case (iii):

Here the path is closed around the point A. Then encirclement of point A is two, but these

two encircles are in opposite direction. One circle is in clockwise direction.

i.e., Its value is +1 and another encircle is in anticlockwise direction. i.e., its value is – 1.

So these two encirclements cancel each other at the time of counting. Hence the number of

encirclement N = 0.

PRINCIPLE OF ARGUMENT

Principle of argument also called as mapping theorem, it states that the mapped locus ′(s)

encircles the new origin of F-plane as many times as the difference between the number of

zeros and poles of F(s) which are encircles by (s) path in s-plane mathematically,

N = z – p

where N = Encirclements of origin of F-plane by ′(s) path

p = Number of poles of F(s) encircled by (s) path in s-plane

z = Number of zeros of F(s) encircled by (s) path in s-plane

Let p and z be the number of poles and zeros of F(s) which are encircled by (s) path. We

are not interested in all the poles and zeros of F(s) but only those which are encircled by

(s) path in s-plane. So hereafter

p = Number of poles of F(s) encircled by (s)

z = Number of zeros of F(s) encircled by (s)

According to mapping procedure, the closed path (s) in s-plane can be mapped into other

plane say F-plane to get a closed path say ′(s).

PROCEDURE TO CONSTRUCT ROOT LOCUS

Step 1: Calculation of poles and zeros

From the transfer function, calculate the poles and zeros.

To find poles, Equate denominator of G(s) H(s) = 0

To find zeros, Equate numerator of G(s) H(s) = 0

Step 2: Location of poles and zeros

Plot the poles and zeros on an ordinary graph.

Number of poles = p

Number of zeros = z

Step 3: Number of separate root loci

Case (i): Number of separate root loci = Number of poles when p > z

Case (ii): Number of separate root loci = Number of zeros when z > p

Step 4: Starting point and ending point of the root locus

The root locus starts from poles and end at zeros. This point is denoted by K, usually K

starts at K = 0 and end at K = ∞.

Step 5: Asymptotes

To locate zeros at infinite, we draw guiding lines which direct the root locus branches

to meet at ∞. These lines are called asymptotes.

Number of asymptotes (Na )

Na = p – z

where p = Number of poles

z = Number of zeros

Point of intersection (a ) (centroid)

a = poles – zeros

p – z

Angle of asymptotes (k )

k = 180 (2 K + 1)

p – z

where K = 0, 1, 2, 3, Na – 1

If Na = 3, then K = 0, 1, 2,

Step 6: Break-away point and Break-in point

The Break-away or Break-in points either lie on real axis or exist as complex conjugate

pairs.

If there is a root locus on real axis between 2 poles, then there exist a Break-away

point.

If there is a root locus on real axis between 2 zeros, then there exists a Break-in point.

The Break-away point can be obtained as roots of characteristic equation after making.

d K

d s = 0

Selection of Breakaway point

(i) It is to be found when two poles (or) two zeros lie adjacent to each other.

(ii) It is to be selected such that the number of poles and zeros to right of break-away

point should be odd.

Step 7: Intersection of root locus on j axis

By using R-H criterion, we can get the value of ‘K’ and ‘’ so as the plot touches j

axis.

Step 8: Angle of departure at complex poles

If there are complex poles determine the angle of departure.

Angle of departure, dp = 180 –

Sum of angles of

vector to complex

pole from poles +

Sum of angles

of vector

to complex pole

from zeros

= 180 – p + e

Example: Consider the following graph.

Here p1 and p2 are the complex poles, then the angle of departure for pole p1 is

dp1 = 180 – (p2 + p3 + p0) + (z1 + z2)

Step 9: Angle of arrival at complex zeros

If there is a complex zeros, then determine the angle of arrival at the complex zero.

Angle of arrival, ap = 180 –

Sum of angles of

vector to complex

zero from

other zeros

+

Sum of angles

of vector

to complex zero

from poles

ap = 180 – z + p

Step 10: Operating point

When damping ratio is given, then take = cos . Draw a line at angle of = cos–1

on second quadrant in clockwise direction from origin. The point where this line touches

the plot, is called operating point.

The value of Ksd at operating point is given by magnitude condition

Ksd = | Mp1 Mp2 Mp3 Mpn |

| Mz1 Mz2 Mz3 Mzn |

where Mp1 = Magnitude of pole

It is calculated by measuring at the distance of pole p1 from point sd .

where Mz1 = Magnitude of zero

It is calculated by measuring at the distance of zero z1 from point sd .

Similarly other poles and zeros can be calculated.

If the absence of zeros (or) no zeros, then the denominator is given

i.e., | Mz1 Mz2 Mzn | = 1

Step 11: Plot the values in an ordinary graph

EFFECT OF ADDING POLES AND ZEROS

(i) Effect of adding poles

Relative stability of a closed loop system is reduced due to addition of poles to the

function.

Consider G(s) = K

s (s + p1)

Poles s = – p1

Corresponding root locus is

If we add a real pole at s = – p2 , root locus will change as

It will be observed that due to addition of pole,

(i) Root loci have bent towards right.

(ii) Breakaway point has shifted to the right.

Now, if one more real pole at s = – p3 is added, then the root loci will be

Root locus are further pushed to right, the system become less table.

Now consider, G(s) = K

s2 + 2 s + 2

Complex poles at s = – 1 ± j1.

This system is always stable. It is not crossing imaginary axis.

If a real pole at s = – p1 is added, new root locus will be,

Root loci have bend to the right.

For K > critical value system becomes unstable.

If one more real pole at s = – p2 is added, the new root locus will be

The system becomes less stable.

(ii) Effect of adding zeros

It results in bending the root loci towards left. The system tends to become more stable.

For G(s) = K

s (s + p1) , the root locus will be

If we add zero s = – z1 , then the root locus will be

G(s) = K (s + z1)

s (s + p1)

If we add complex zeros, then the root locus will be

G(s) = K (s + z1 + j1) (s + z1 – j1)

s (s + p1)

Adding a complex zero also result in stabilizing system more.

Unit 5 State Variable Analysis

The state variable approach completely replace the classical approaches. In fact, the

classical approaches provide the control engineer with a deep physical insight into the

system and greatly aid the preliminary system design where a complex system is

approximated by a more manageable model.

State: The state of a system, which is a set of variables along with current time

summarizes the current configuration of a system.

State variable: The state variables of a dynamic control system are the variables which

constitute the smallest set of variables that determines the state of the system. The state

variables describe the output response of a control system for specified inputs and existing

state.

State vector: The state vector is defined as a column vector x(t ) that describes the state of system for t t 0 .

State space: If x 1(t ), x2(t ), xn(t ) are the minimum number of state variables

which are necessary to describe the dynamics of a control system, then the n -dimensional

space whose coordinates axis consists of x 1-axis, x2-axis xn-axis is called a state

space.

State Model of Linear system

State space representation of a given system consists of two equation (i) state equation,

(ii) output equation. Both the equations must be represented as function of state variables

and input.

The state and output equations constitute the state model of the system.

Fig. 6.1. Structure of a general control system

Derivative of each state variable of a linear system is given by a linear combination of

system states and inputs i.e.,

•

= a11 x1 + a12 x2 + + a1n xn + b11 u1 + b12 u2 + b1m um x1

•

= a21 x1 + a22 x2 + + a2n xn + b21 u1 + b22 u2 + b2m um x2

• = an1 x1 + an2 x2 + + ann xn + bn1 u1 + bn2 u2 + bnm um

where are coefficients cij and dij are constants.

Then, y (t ) = C X(t ) + D u(t )

where y(t ) is n 1 output vector.

C is n n output matrix defined by

c

11 c

12

c

1n

C = c

21 c

22

c

2n

cn1 cn2 cnn

and D is n m transmission matrix defined by

d

11 d

12

d

1m

D =

d

21 d

22

d

2m

dn1 dn2 dnm

The output vector matrix of a state model is given by

y

1(t

)

c11

c12

c

1n x

1(t

)

d11

y

2(t

)

c

21 c

22

c

2n x

2(t

)

d21

= +

yn(t ) cn1 cn2 cnn xn(t ) dn1

d12

d

1m

d22

d

2m

dn2

dnm

u

1(t

)

u

2(t

)

um(t )

Thus the state model of linear time-invariant systems is given by

•

X(t ) = A X(t ) + B u(t ) State equation y (t ) = C

X(t ) + D u(t ) Output equation

State Space Representation

ELECTRICAL SYSTEM

Procedure to obtain the State-Equation

Step 1: Write KVL for the given electrical network.

Step 2: Choose current through inductor and voltage across the capacitor as a state

variable.

Step 3: Choose the exciting sources as input u .

Step 4: Choose voltage drop across the resistor as output y .

Step 5: Substitute the state variable in KVL equation and rearrange the equation to

obtain state space model.

MECHANICAL SYSTEM

Procedure to obtain the State Equation

Step 1: Apply D’Alembert’s principle for mechanical system.

Step 2: Choose displacement, velocity and acceleration as state variables and applied force

as input u .

Step 3: Substitute the state variables in differential equation.

Rearrange the equation to obtain the state equation.

Step 4: Choose the displacement as output y .

BLOCK DIAGRAM

Procedure to obtain State Equation

Step 1: Choose the output of each integrator as state variable as X1(s), X2(s ),

Step 2: Take inverse Laplace transform to each state variable.

Step 3: Rearrange the equation to obtain state equation

SIGNAL FLOW GRAPH

Procedure to obtain State Equation

Step 1: Choose the output of each integrator as state variables as X1(s ), X2(s),

Step 2: Take inverse Laplace transform to each state variable.

Step 3: Rearrange the equation to obtain state equation.

State Space Representation using phase variable

An alternate state-space representation of control system is using phase variables as state

variables. The phase variables state model is easily determined if the system model is already

known in the differential equation or transfer function form.

The phase variables are defined as the state variables which are obtained from one of the

system variables and its derivatives.

There are four methods of modelling a system by using phase variables, they are,

1. Controllable canonical form

2. Observable canonical form

3. Diagonical canonical form

4. Jordan canonical form

6.6.3. PROPERTIES OF STATE TRANSITION MATRIX

Property 1: (0) = e A

0 = I

Property 2: (t ) = e A t = (e– A t )–1 = [ (– t ) ]– 1

(or)

–1

(t ) = (– t )

Property 3: (t 1

+ t 2

) = e A (t

1 +

t 2

) = e

A t e

A t 2

= (t1)(t2) =(t2)(t1)

Concept of Controllability and Observability

Kalman was first to introduce the concept of controllability in 1960. These

concepts are used in optimal control theory to arrive at an optimal control solution.

The concept of controllability is linked with the equation

•

X(t ) = A X(t ) + B u(t )

and the concept of observability is linked with the

equation y (t ) = C X(t ) + D u(t

)

The concept basically depends on the input u(t ) and its affect on state vector X(t )

and output y (t ).

6.7.1. CONTROLLABILITY

Ability of input u (t ) to exercise control on the state-variables forming state vector

X(t ) describes the controllability of control system. Controllability means total or

complete control on the system.

A system is said to be controllable, if all the states are completely controllable. A

system can also be considered as controllable, if every state of the system can be

exercised control in such a manner that the states are transferred from an initial state

to a desired final state in some finite time.

Definition:

“The state X(t ) at t = t 0 is said to be controllable, if the state can be driven to a desired state X(tf ) in some finite time t = t f by application of continuous control input u (t )”.

Controllability Test

•

A linear time invariant system described by dynamic equation X(t ) = A X(t ) + B u (t ) is controllable, if and only the controllability matrix QC (n np) is of rank n , the order of the system.

Controllability Matrix Q C = [ B AB A2B An – 1 B ]

If | QC | 0, then the system is completely controllable and QC = rank

Condition for complete state controllability in s-plane

It can be proved that a necessary and sufficient condition for complete state

controllability is that no cancellation occurs in the transfer function (or) transfer

matrix. If cancellation occurs, the system cannot be controlled in the direction of the

concelled mode.

For example, consider

Y(s )

=

(s + 2)

u (s ) (s + 2) (s – 1)

The factor (s + 2) occurs in the numerator and denominator, so cancel each other.

Because of this cancellation, the system is not completely state controllable.

Necessary and sufficient conditions for controllability

(i) The matrix B must have no rows with all zeros.

(ii) If any row of the matrix is zero, then the corresponding state variable is

uncontrollable.

(iii) If QC is singular matrix, then the system is not completely state controllable. If

QC is non-singular, then the system is completely state controllable.

6.7.2. OBSERVABILITY

Observability is the concept coupled with the states and the output, and is

specified by the matrices C and D.

A system is considered to be observable if the system states are observable. This

implies that every state variable of the system affects some of the outputs.

Definition:

“The state X(t 0) at t = t 0 for a system subjected to control input u (t ) is said to be observable, if for a desired finite time t = tf t 0. Knowledge of u (t ) and output y (t ) over the interval t 0 t tf determines the state (Xt0)”.

Observability Test

Observability of a control system can be tested from observability matrix and from

matrices A and C.

A linear time-invariant system described by dynamic equations

•

A X(t ) + B u(t ) X(t ) =

y (t ) = C X(t ) + D u(t )

is observable, if and only if observability matrix Q0

Q = [ CT AT CT (AT)2 CT (AT)n – 1 CT ]

0

If | Q0 | 0, then the system is completely observable and Q0 = Rank of n , the

order of the system

Conditions for Complete Observability in s-Plane

The necessary and sufficient conditions for complete observability is that non

cancellation occurs in the transfer function (or) transfer matrix. If cancellation occurs,

the cancelled mode cannot be observed in the output.

For example, consider

Y(s

)

=

(s + 2) (s +

1)

u (s

)

(s + 2) (s +

3)

Clearly, the two factors (s + 2) cancel each other. This means that there are non-

zero initial states X(0), which cannot be determined from the measurement of y(t ).

Necessary and Sufficient Conditions for Observability

(i) For completely observability is that none of the column of the matrix C be

zero.

(ii) If any column of C has all zeros, then the corresponding state variable is not

observable.

Sampled Data control system

Sampled data technique is most appropriate for control systems requiring long distance

data transmission. Pulse amplitude modulated (PAM) data is easily transmitted by means of a

carrier over a transmission channel and the data reconstructed at the receiving end.

Signal sampling reduces the power demand made on the signal and is therefore helpful for

signals of weak power origin.

A simple control scheme employing a digital controller is shown below.

Fig. 6.2. Sampled-data control system (digital controller)

A digital controller in which either a special purpose computer or a general purpose computer forms

the heart, is therefore an ideal choice for complex control systems.

A digital controller also has the versatility that its control function can be easily modified by changing

a few program instructions or even the entire program and a change in instruction can be done manually

or automatically under control of a function.

Digital controllers used in digital control systems have the inherent characteristic that they accept the

data as short duration pulses i.e., sampled or discrete data and produce a similar kind of output as control

signal.

A sampler and analog to digital converter (ADC) is needed at the computer input. The sampler

converts the continuous time signal into a discrete time signal which are then expressed in numerical

code.

Numerically coded output data of digital computers are decoded into continuous time signal by

digital-to-analog converter (DAC) and hold circuit. This continuous-time signal then controls the

continuous-time system.

The overall system is hybrid in which the signal is in sampled form in the digital controller and in

continuous form in the rest of the system. It is referred as a sampled-data control system.

Sampling Process

Sampling is the conversion of a continuous-time signal into a discrete-time signal obtained by taking

samples of the continuous time signal at discrete time instants.

If f (t ) is the input to the sampler, then the output is f (KT), where T is called the sampling

1

interval or sampling period. The reciprocal of T i.e., T = FS is called the sampling rate. This

type of sampling is called periodic sampling

SAMPLING THEOREM

A band limited continuous time signal with highest frequency (bandwidth) f m hertz, can be uniquely

recovered from its samples provided that the sampling rate FS is greater than or equal to 2 f m samples per

second.Sampled-data signal which has been modified by a digital controller must be converted into

analog form for use in the continuous part of the system. This is accomplished by means of various types

of hold circuits.The simplest hold circuit is the zero-order hold (ZOH) in which the reconstructed signal

acquires the same value as the last received sample for the entire sampling period.