Control systems

Dr.Y.NARASIMHA MURTHY Ph.D
yayavaram@yahoo.com

CONTROL SYSTEMS
INTRODUCTION :

A control system is one which can control any quantity of interest in a machine, mechanism or
other equipment in order to achieve the desired performance or output.(or) A control system is an
interconnection of components connected or related in such a manner as to command, direct, or
regulate itself or another system. For example consider, the driving system of an automobile.
Speed of the automobile is a function of the position of its accelerator. The desired speed can be
maintained (or a desired change in speed can be achieved) by controlling pressure on the
accelerator pedal. This automobile driving system (accelerator, carburetor and engine-vehicle)
constitutes a control system.

Control systems find numerous and widespread applications from everyday to extraordinary in
science, industry, and home. Here are a few examples:

(a) Home heating and air-conditioning systems controlled by a thermostat
(b) The cruise (speed) control of an automobile
(c) Manual control
(i) Opening or closing of a window for regulating air temperature or air quality
(ii) Activation of a light switch to regulate the illumination in a room
(iii) Human controlling the speed of an automobile by regulating the gas supply to the engine
(d) Automatic traffic control (signal) system at roadway intersections
(e) Control system which automatically turns on a room lamp at night, and turns it off in
Day light

The general block diagram of a control system is shown below.
.

Fig.1 .Block diagram of a control system.

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The above method of representation of a control system is known as block diagram
representation where in each block represents an element, a plant, mechanism, device etc.,
whose inner details are not indicated. Each block has an input and output signal which are linked
by a relationship characterizing the block. It may be noted that the signal flow through the block
is unidirectional
Basic Control System components : The basic control system components are objectives i.e
inputs or actuating signals to the system and the Output signals or controlled variables etc.The
control system will control the outputs in accordance with the input signals.The relation beteen
these components is shown in the block diagram.

The components of the control system changes as we move from openloop control system to
closed loop control systems. In a closed loop control system ,the feedback control network play
an important role in getting the correct output.

The general block diagram of a control system with feed back is shown below. The error detector
compares a signal obtained through feedback elements, which is a function of the output
response, with the reference input. Any difference between these two signals gives an error or
actuating signal, which actuates the control elements. The control elements in turn alter the
conditions in the plant in such a manner as to reduce the original error.

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Fig.2 .General block diagram of an automatic control system.

Types of control systems :

There are basically two types of control systems (i) the open loop system and the (ii) closed loop
system. They can both be represented by block diagrams. A block diagram uses blocks to
represent processes, while arrows are used to connect different input, process and output parts.
Open loop Control System : Asystem which do not possess any feed back network ,and
contains only the input and output relationship is known as a open loop control system

.

Examples of the open loop control systems are washing machines, light switches, gas ovens,
burglar alarm system etc.
The drawback of an open loop control system is that it is incapable of making automatic
adjustments. Even when the magnitude of the output is too big or too small, the system can‟t
make the necessary adjustments. For this reason, an open loop control system is not suitable for
use as a complex control system.

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Closed loop control system : A closed loop system is one which uses a feed back control
between input and output. A closed loop control system compares the output with the expected
result or command status, then it takes appropriate control actions to adjust the input signal.
Therefore, a closed loop system is always equipped with a sensor, which is used to monitor the
output and compare it with the expected result.

The output signal is fed back to the input to produce a new output. A well-designed feedback
system can often increase the accuracy of the output.

Examples for closed loop systems are air conditioners, refrigerators, automatic rice cookers,
automatic ticketing machines, etc. For example An air conditioner, uses a thermostat to detect
the temperature and control the operation of its electrical parts to keep the room temperature at a
preset constant.
One advantage of using the closed loop control system is that it is able to adjust its output
automatically by feeding the output signal back to the input. When the load changes, the error
signals generated by the system will adjust the output suitably. The limitation of a closed loop
control systems is they are generally more complicated and thus also more more expensive to
design.
Linear versus Nonlinear Control Systems : Linear feedback control systems are idealized
models fabricated by the analyst purely for the simplicity of analysis and design

When the magnitudes of signals in a control system are limited to ranges in which system
components exhibit linear characteristics (i.e., the principle of superposition applies), the system
is essentially linear.
But when the magnitudes of signals are extended beyond the range of the linear operation,
depending on the severity of the nonlinearity, the system should no longer be considered linear.
For instance, amplifiers used in control systems often exhibit a saturation effect when their input

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signals become large; the magnetic field of a motor usually has saturation properties. Other
common nonlinear effects found in control systems are the backlash or dead play between
coupled gear members, nonlinear spring characteristics, nonlinear friction force or torque
between moving members, and so on. Quite often, nonlinear characteristics are intentionally
introduced in a control system to improve its performance or provide more effective control.
Also the analysis of linear systems is easy and lot of mathematical solutions are available for
their simplification.
Nonlinear systems, on the other hand, are usually difficult to treat mathematically, and there are
no general methods available for solving a wide class of nonlinear systems. In practice , first a
linear-system is modeled by neglecting the nonlinearities of the system and the designed
controller is then applied to the nonlinear system model for evaluation or redesign by computer
simulation.
Distinguish between Openloop and Closed loop control systems

Time-Invariant control Systems :

When the parameters of a control system do not change with respect to time during the
operation of the system, the system is called a time-invariant system. In practice, most physical
systems contain elements that drift or vary with time. For example, the winding resistance of an
electric motor will vary when the motor is first being excited and its temperature is rising.
Another example of a time-varying system is a guided-missile control system in which the mass

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of the missile decreases as the fuel on board is being consumed during flight. Although a time-
varying system without nonlinearity is still a linear system, the analysis and design of this class
of systems are usually much more complex than that of the linear time-invariant systems.

Continuous and Discrete Data Control Systems : A continuous-data system is one in which
the signals at various parts of the system are all functions of the continuous time variable t. The
signals in continuous-data systems may be further classified as ac or dc. In control systems the
ac control system, means that the signals in the system are modulated by some form of
modulation scheme. A dc control system, on the other hand, simply implies that the signals are
unmodulated, but they are still ac signals according to the conventional definition. The schematic
diagram of a closed loop dc control system is shown below. Typical waveforms of the signals in
responseto a step-function input are shown in the figure. Typical components of a dc control
system are potentiometers, dc amplifiers, dc motors, dc tachometers, and so on.

In ac control systems , the signals are modulated i.e the information is transmitted by an ac
carrier signal. Here the output controlled variable behaves similarly to that of the dc system. In
this case, the modulated signals are demodulated by the low-pass characteristics of the ac motor.
Ac control systems are used extensively in aircraft and missile control systems in which noise

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and disturbance often create problems. By using modulated ac control systems with carrier
frequencies of 400 Hz or higher, the system will be less susceptible to low-frequency noise.
Typical components of an ac control system are synchros, ac amplifiers, ac motors, gyroscopes,
accelerometers etc.
Discrete Data Control System
If the signal is not continuously varying with time but it is in the form of pulses then the control
system is called Discrete Data Control System.If the signal is in the form of pulse data, then the
system is called Sampled Data Control System. Here the information supplied intermittently at
specific instants of time. This has the advantage of Time sharing system. On the other hand, if
the signal is in the form of digital code, the system is called Digital Coded System. Here use of
Digital computers, micro processors or microcontrollers are made use of such systems and are
analyzed by the Z- transform theory.
Block Diagrammatic Representation :
It is a representation of the control system giving the inter-relation between the transfer function
of various components. The block diagram is obtained after obtaining the differential equation &
Transfer function of all components of a control system. The arrow head pointing towards the
block indicates the i/p & pointing away from the block indicates the o/p.
Suppose G(S) is the Transfer function then G(S) = C(S) / R(S)

After obtaining the block diagram for each & every component, all blocks are combined to get a
complete representation. It is then reduced to a simple form with the help of block diagram
algebra.
Basic elements of a block diagram

 Blocks
 Transfer functions of elements inside the blocks
 Summing points
 Take off points
 Arrow

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A control system may consist of a number of components. A block diagram of a system is a
pictorial representation of the functions performed by each component and of the flow of signals.
The elements of a block diagram are block, branch point and summing point.
Block :
In a block diagram all system variables are linked to each other through functional blocks. The
functional block or simply block is a symbol for the mathematical operation on the input signal
to the block that produces the output.

Summing point :
The blocks are used to identify many types of mathematical operations, like addition and
subtraction and represented by a circle, called a summing point. As shown belowdiagram a
summing point may have one or several inputs. Each input has its own appropriate plus or minus
sign. A summing point has only one output and is equal to the algebraic sum of the inputs

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A takeoff point is used to allow a signal to be used by more than one block or summing point

Arrow – associated with each branch to indicate the direction of flow of signal

Advantages of Block Diagram Representation :
It is always easy to construct the block diagram even for a complicated system
Function of individual element can be visualized
Individual & Overall performance can be studied
Over all transfer function can be calculated easily

Limitations of a Block Diagram Representation

No information can be obtained about the physical construction
Source of energy is not shown

Block diagram reduction technique: Because of the simplicity and versatility, the block
diagrams are often used by control engineers to describe all types of systems. A block diagram
can be used simply to represent the composition and interconnection of a system. Also, it can be
used, together with transfer functions, to represent the cause-and-effect relationships throughout
the system. Transfer Function is defined as the relationship between an input signal and an
output signal to a device.
Procedure to solve Block Diagram Reduction s :
Step 1: Reduce the blocks connected in series
Step 2: Reduce the blocks connected in parallel
Step 3: Reduce the minor feedback loops
Step 4: Try to shift take off points towards right and Summing point towards left
Step 5: Repeat steps 1 to 4 till simple form is obtained
Step 6: Obtain the Transfer Function of Overall System
Block diagram rules
(1) Blocks in Cascade [Series] : When two blocks are connected in series ,their resultant transfer
function is the product of two individual transfer functions.

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(2) Combining blocks in Parallel: When two blocks are connected parallel as shown below ,the
resultant transfer function is equal to the algebraic sum (or difference) of the two transfer
functions.This is shown in the diagram below.

(3) Eliminating a feed back loop:The following diagram shows how to eliminate the feed back
loop in the resultant control system

(4) Moving a take-off point beyond a block: The effect of moving the takeoff point beyond a
block is shown below.

(5) Moving a Take-off point ahead of a block: The effect of moving the takeoff point ahead of a
block is shown below.

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Applications of the control systems : There are various applications of control systems which
include biological propulsion; locomotion; robotics; material handling; biomedical, surgical,
and endoscopic ; aeronautics; marine and the defense and space industries. There are also many
household and industrial application examples of the control systems, such as washing machine,
air conditioner, security alarm system and automatic ticket selling machine, etc.
(i) Washing machine :
The most commonly used house hold application is the washing machine.It comes under
automatic control system ,where the machine automatically starts to pour water, add washing
powder, spin and wash clothes, discharge wastewater, etc. After the completion of all the
procedures, the washing machine will stop the operation.
However, this kind of machine only operates according to the preset time to complete the
whole washing process. It ignores the cleanness of the clothes and does not generate feedback.
Therefore, this kind of washing machine is of open loop control system.
(ii). Air conditioner
The air conditioner is used to automatically control the temperature of the room.In the air
conditioner the coolant circulated in the machine will absorb heat indoor, then it will be
transported from the vaporization device to cooling device. The hot air is then blown to outdoor
by a fan. There is an adjustable temperature device equipped in the air conditioner for the users
to adjust the extent of cooling. When the temperature of the cool air is lower than the preset one,
the controller of the air conditioner will stop the operation of the compressor to cease the
circulation of the coolant. The temperature sensor installed near the vaporization device will
continuously measure the indoor temperature, and send the results to the controller for further
processing.This operation will come under closed loop control system.Thesimple block diagram
of air conditioner system is shown below.

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Feed back Control System : The feed back control system is represented by the following
block diagram .In the diagram feed back signal is denoted by B(S) and the output is C(S).The
input function is denoted by R(S).

The open loop gain of the system is G(S) and the feed back loop gain H(S). Then the feed
back signal b(s) is given by

B(S) = H(S). C(S)

Transfer function : The input- output relationship in a linear time invariant system is given by
the transfer function.For a time invariant system it is defined as the ratio of Laplace transform
of the out to the Lapalce transform of the input

The important features of the transfer functions are,

The transfer function of a system is the mathematical model expressing the differential
equation that relates the output to input of the system.
The transfer function is the property of a system independent of magnitude and the nature
of the input .
The transfer function includes the transfer functions of the individual elements. But at the
same time, it does not provide any information regarding physical structure of the system
If the transfer function of the system is known, the output response can be studied for
various types of inputs to understand the nature of the system
It is applicable to Linear Time Invariant system.
It is assumed that initial conditions are zero.
It is independent of i/p excitation.
It is used to obtain systems o/p response.
If the transfer function is unknown, it may be found out experimentally by applying
known inputs to the device and studying the output of the system

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From the above block diagram G(S) = C(S) / E(S) & E(S) = R(S) – B(S)

So, C(S) = G(S) .E(S)
= G(S)[ R(S)- B(S)
= G(S) [ R(S) – H(S).C(S)
Therefore

This is the transfer function of the closed loop control system

Properties of Systems : For any control system to understand its performance the following
properties are very important.

(i).Linearity : A system is said to be linear if it follows both the law of addtivity and law of
homogeneity. The system which do not follow the law of homogeneity and additivity is called a
non-linear system.

If input x1(t) produces response y1(t) and input x2(t) produces response y2(t) then the scaled
and summed input a1x1(t) +b1x2(t) produces the scaled and summed response a1y1(t) +b1y2(t)
where a1 and a2 are real scalars. It follows that this can be extended to an arbitrary number of
terms, and so for real numbers .

(ii) Time Invariance : A system with input x(t) and output y(t) is time-invariant if x(t- t0) is
creates output y ( t – t0) for all inputs x and shifts t0.

(iii). Causality : A system is causal, if the output y(t) at time t is not a function of future inputs
and it depends only on the present and past inputs . All analog systems are causal and all
memeoryless systems are causal .

If the system is causal, then this implies h(t) = 0, t < 0. Alternatively, h[n]=0, n < 0.

(iv).Stability : A system is said to be a stable if for every bounded-input there exists a bounded
output .

Transfer Function: For a open loop control system shown below the transfer function is the
ratio of Laplace transform of the out-put to the Laplace transform of the input.

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The Laplace transform of the Input is R(S) and the Laplace transform of the output
is C(S) .

So,the Transfer function of the system is G(S) = C(S) / R(S)

Example : Find the transfer function of the following RC circuit

By definition ,the transfer function is G(S) =

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The Laplace transformed Network is shown above.From the circuit we can write that

Poles and Zeros : In the transfer function of a control system both numerator and
denominator will be polynomials. If these numerator and denominator are solved , the roots of
the numerator are called Zeros and the roots of the denominator are called the Poles. These
poles of the transfer function decides the stability of the control system.

The transfer function provides a basis for determining important system response
characteristics without solving the complete differential equation. As defined, the transfer
function is given by the following expression with variable s = σ + jω,

It is always easy to factor the polynomials in the numerator and denominator, and to write

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the transfer function in terms of those factors .then the above transfer function can be written
as

where the numerator and denominator polynomials, N(s) and D(s), have real coefficients
defined by the system’s differential equation and K = bm/an.

As written in the above equation the Zi s are the roots of the equation N(s) = 0, and are
defined to be the system Zeros, and the pi s are the roots of the equation D(s) = 0, and are
defined to be the system poles.

All of the coefficients of polynomials N(s) and D(s) are real, therefore the poles and zeros must

be either purely real, or appear in complex conjugate pairs.

The stability of a linear system may be determined directly from its transfer function. An nth
order linear system is asymptotically stable only if all of the components in the homogeneous
response from a finite set of initial conditions decay to zero as time increases, or

where the pi are the system poles. In a stable system all components of the homogeneous
response must decay to zero as time increases. If any pole has a positive real part there is a
component in the output that increases without bound, causing the system to be unstable.

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Example : A linear system is described by the differential equation

Find the system poles and zeros.

Solution: From the differential equation the transfer function is

So,the system has a single real zero at s = −1/2, and a pair of real poles at s = −3 and s = −2.

SIMULATION DIAGRAMS: The simulation diagram is similar to the diagram used to represent
the system on an analog computer.The basic elements used are ideal integrators, ideal
amplifiers, and ideal summers, shown in below diagram. Additional elements such as
multipliers and dividers may be used for nonlinear systems.

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To obtain the simulation diagram ,the following steps are to be followed.

1. Start with differential equation.

2. On the left side of the equation put the highest-order derivative of the dependent variable. A
first-order or higher-order derivative of the input may appear in the equation. In this case the
highest-order derivative of the input is also placed on the left side of the equation. All other
terms are put on the right side.

3. Start the diagram by assuming that the signal, represented by the terms on the left side of
the equation, is available. Then integrate it as many times as needed to obtain all the lower-
order derivatives. It may be necessary to add a summer in the simulation diagram to obtain the
dependent variable explicitly.

4. Complete the diagram by feeding back the approximate outputs of the integrators to a
summer to generate the original signal of step 2.Include the input function if it is required.

Example: Draw the simulation diagram for the series RLCcircuit of below figure in which the
output is the voltage across the capacitor.

SIGNAL FLOW GRAPHS(SFG) : The block diagram method is a useful tool for simplifying the
representation of a control system. But when there are more than two feed back loops and if
there exists inter-coupling between feedback loops, and when a system has more than one
input and one output, the block diagram approach is very complex. Hence an alternate method
is proposed by S.J. Mason. This method is called signal flow graphs. In these graphs each node

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represents a system variable & each branch connected between two nodes acts as Signal
Multiplier. The direction of signal flow is indicated by an arrow.

A signal flow graph is a diagram that represents a set of simultaneous equations. It consists of a
graph in which nodes are connected by directed branches. The nodes represent each of the
system variables. A branch connected between two nodes acts as a one-way signal multiplier: the
direction of signal flow is indicated by an arrow placed on the branch, and the multiplication
factor (transmittance or transfer function) is indicated by a letter placed near the arrow.

So,in the figure above , the branch transmits the signal x1 from left to right and multiplies it by
the quantity a in the process. The quantity a is the transmittance, or transfer function.

Flow-Graph Definitions : A node performs two functions:

Addition of the signals on all incoming branches and
Transmission of the total node signal (the sum of all incoming signals) to all outgoing
branches

There are three types of nodes .They are Source nodes , Sink nodes and Mixed nodes

Source nodes (independent nodes) : These represent independent variables and have only
outgoing branches. In Fig. 5.21, nodes u and v are source nodes.

Sink nodes (dependent nodes): These represent dependent variables and have only incoming
branches. In Fig (a), nodes x and y are sink nodes.

Mixed nodes (general nodes): These have both incoming and outgoing branches. In Fig. (a),
node w is a mixed node. A mixed node may be treated as a sink node by adding an out going
branch of unity transmittance, as shown in Fig (b), for the equation x = au +bv

and w = cx = cau + cbv

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Fig(a) Fig (b)

A path is any connected sequence of branches whose arrows are in the same direction and A
forward path between two nodes is one that follows the arrows of successive branches and in
which a node appears only once. In Fig.(a) the path uwx is a forward path between the nodes
u and x.

Flow-Graph Algebra : The following rules are useful for simplifying a signal flow graph:

Series paths (cascade nodes). Series paths can be combined into a single path by multiplying
the transmittances as shown in Fig ( A ).

Path gain. The product of the transmittances in a series path.

Parallel paths. Parallel paths can be combined by adding the transmittances as shown in Fig(B).

Node absorption. A node representing a variable other than a source or sink can be eliminated
as shown in Fig (C).

Feedback loop. Aclosed path that starts at a node and ends at the same node.

Loop gain. The product of the transmittances of a feedback loop.

These results are shown diagrammatically in the following figures (A) ,(B) and C) where the
original diagram and equivalent diagrams are shown.

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Masons gain formula: The relationship between an input variable and an output variable of a
signal flow graphis given by the net gain between input and output nodes and is known as overall
gain ofthe system. Masons gain formula is used to obtain the over all gain (transfer function) of
signal flow graphs. According to Mason‟s gain formula Gain is given by

Where, Pk is gain of k th forward path and Δ is determinant of graph. Here the Δ is given by
Δ = 1-(sum of all individual loop gains)+(sum of gain products of all possible combinations of
two non touching loops –sum of gain products of all possible combination of three non touching
loops)
Δk is cofactor of kth forward path determinant of graph with loops touching k th forward path.
It is obtained from Δ by removing the loops touching the path Pk.
Finding transfer function from the system flow graphs is explained below by example.

Example1 : Obtain the transfer function of the system whose signal flow graph is shown below.

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There are two forward paths: One is Gain of path 1 : P1=G1 and the other is Gain of path 2:
P2=G2

There are four loops with loop gains :

L1=-G1G3, L2=G1G4, L3= -G2G3, L4= G2G4

There are no non-touching loops.

Δ = 1+G1G3-G1G4+G2G3-G2G4

Forward paths 1 and 2 touch all the loops. Therefore, Δ1= 1, Δ2= 1

So,the transfer function T is given by

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Example 2 : Obtain the transfer function of C(s) /R(s) of the system whose signal flow graph
shown below.

From the system flow graph it is clear that

There is one forward path, whose gain is: P1=G1G2G3

There are three loops with loop gains:

L1=-G1G2H1, L2=G2G3H2, L3= -G1G2G3

There are no non-touching loops: Δ = 1-G1G2H1+G2G3H2+G1G2G3

Forward path 1 touches all the loops. Therefore, Δ1= 1.

The transfer function T is given by

System Stability:

The study of stability of a control system is very important to understand the performance .
This means that the system must be stable at all times during operation. Stability may be used to
define the usefulness of the system. Stability studies include absolute & relative stability.
Absolute stability is the quality of stable or unstable performance. Relative Stability is the
quantitative study of stability.

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The stability study is based on the properties of the Transfer Function. In the analysis, the
characteristic equation is very important ,which describe the transient response of the system.
From the roots of the characteristic equation, following conclusions about the stability can be
drawn.

(1) When all the roots of the characteristic equation lie in the left half of the S-plane, the system

response due to initial condition will decrease to zero at time Thus the system will be termed as
a stable system.

(2) When one or more roots lie on the imaginary axis & there are no roots on the RHS of S-
plane, the response will be oscillatory without damping. Such a system will be termed as
critically stable.

(3) When one or more roots lie on the RHS of S-plane, the response will exponentially increase

in magnitude and there by the system will be Unstable.

Stability – Definitions :

 A system is stable, if its o/p is bounded for any bounded i/p . or A system is stable, if
it‟s response to a bounded disturbing signal vanishes ultimately as time ‘ t ‘ approaches
infinity.

 A system is un stable, if it’s response to a bounded disturbing signal results in an o/p of
infinite amplitude or an Oscillatory signal.

 If the o/p response to a bounded i/p signal results in constant amplitude or constant
amplitude oscillations, then the system may be stable or unstable under some limited
constraints. Such a system is called Limitedly Stable system.
 If a system response is stable for a limited range of variation of its parameters, it is
called
Conditionally Stable System.
 If a system response is stable for all variation of its parameters, it is called Absolutely
Stable system.
Routh-Hurwitz Stability Criterion : This criterion is derived from the theory of equations
and is an algebraic method to determine the number of roots of a given equation with positive
real part.

The Routh-Hurwitz criterion is a method of finding whether a linear system is stable or not by
examining the locations of the roots of the characteristic equation of the system. In fact, the

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method determines only if there are roots that lie outside of the left half plane; it does not
actually compute the roots.

To determine whether this system is stable or not, check the following conditions

Two necessary but not sufficient conditions that all the roots have negative real parts are

a) All the polynomial coefficients must have the same sign.

b) All the polynomial coefficients must be nonzero.

A sufficient condition for a system to be stable is that each & every term of the column of

the Routh array must be positive or should have the same sign. Routh array can be obtained as
follows

Consider the Characteristic equation of the form,

Similarly rest of the elements, can be evaluated.

The limitations of the Routh-Hurwitz stability criteria are

(1) It is valid only if the Characteristic equation is algebraic.

(2) If any co-efficient of the Characteristic equation is complex or contains power of „e‟ this
criterion can not be applied.

(3) It gives information about how many roots are lying in the RHS of S-plane; values of the
roots are not available. Also it cannot distinguish between real & complex roots.

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Special cases in Routh-Hurwitz criteria :

(1) When the 1st term in a row is zero, but all other terms is non-zeroes then substitute a small
positive number ε for zero & proceed to evaluate the rest of the elements. When the 1 st column
term is zero, it means that there is an imaginary root.

(2) All zero row: In the case, write auxiliary equation from preceding row, differentiate this
equation & substitute all zero row by the co-efficient obtained by differentiating the auxiliary
equation. This case occurs when the roots are in pairs. The system is said to be limitedly stable.

Application of Routh’s Criteria : Routh‟s criterion can be applied to determine range of certain
parameters of a system to ensure stability. For example, it is usually of interest to find the range
of the open loop gain K for closed loop stability.

Example 1 : Find the stability of a system whose characteristic equation is given below.

From the above table it is clear that the no. of sign changes in the 1st column = zero. No roots are
lying in the RHS of S-plane. So, the given System is Absolutely Stable.

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Example 2: Find the stability of a system whose characteristic equation is given below .

From the above table it is clear that the no. of sign changes in the 1st column = 2 and two roots
are lying in the RHS of S-plane. So , the given System is unstable

ROOT LOCUS :

Root locus is the plot of the loci of the root of the complementary equation when one or more
parameters of the open-loop Transfer function are varied, mostly the only one variable available
is the gain „K‟ The negative gain has no physical significance hence varying „K‟ from „0‟ to „∞‟ ,
the plot is obtained called the “Root Locus Point”.

Root locus gives the complete dynamic response of the system. It provides a measure of
sensitivity of roots to the variation in the parameter being considered. It is applied for single as
well as multiple loop system

Rules for the Construction of Root Locus :

(1) The root locus is symmetrical about the real axis.

(2) The no. of branches terminating on „∞‟ equals the no. of open-loop pole-zeroes.

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(3) Each branch of the root locus originates from an open-loop pole at „K = 0‟ & terminates at
open-loop zero corresponding to „K = ∞‟.

(4) A point on the real axis lies on the locus, if the no. of open-loop poles & zeroes on the real
axis to the right of this point is odd.

(5) The root locus branches that tend to „∞‟, do so along the straight line.

Asymptotes making angle with the real axis is given by

Where, n=1,3,5,…………………

P is the number of poles and z is the number of zeros

(6) The asymptotes cross the real axis at a point known as Centroid

(7) The break away or the break in points [Saddle points] of the root locus are determined from
the roots of the equation dk /ds = 0.

(8) The intersection of the root locus branches with the imaginary axis can be determined by the
use of Routh-Hurwitz criteria or by putting s= jω in the characteristic equation & equating the
real part and imaginary to zero. To solve for ω and K i.e., the value of „ω‟ is intersection point
on the imaginary axis & „K‟ is the value of gain at the intersection point.

9) The angle of departure from a complex open-loop pole θd is given by θd = 1800 + ∟GH

Nyquist Criteria:

A stability test for time invariant linear systems can also be derived in the frequency domain. It is
known as Nyquist stability criterion. It is based on the complex analysis result known as
Cauchy‟s principle of argument.

The Nyquist stability criterion relates the location of the roots of the characteristic equation to
the open-loop frequency response of the system. In this, the computation of closed-loop poles is
not necessary to determine the stability of the system and the stability study can be carried out

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graphically from the open-loop frequency response. Therefore experimentally determined open-
loop frequency response can be used directly for the study of stability. When the feedback path is
closed. The Nyquist criterion provides the information on absolute and relative stability of the
system.

Let us suppose that the system transfer function is a complex function. By applying Cauchy‟s
principle of argument to the open-loop system transfer function, we will get information about
stability of the closed-loop system transfer function and arrive at the Nyquist stability criterion

The importance of Nyquist stability lies in the fact that it can also be used to determine the
relative degree of system stability by producing the so-called phase and gain stability margins.
These stability margins are needed for frequency domain controller design techniques

The Nyquist plot is a polar plot of the function D(s) = 1 = G(s).H(s)
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The Nyquist criterion states that the number of unstable closed-loop poles is equal to the
number of unstable open-loop poles plus the number of encirclements of the origin of the
Nyquist plot of the complex function D(s).

Continuous Time Feedback Control Systems : If the signals in all parts of a control system are
continuous functions of time, the system is classified as continuous time feedback control
system. Typically all control signals are of low frequency and if these signals are un modulated,
the system is known as a d.c. control system. These systems use potentiometers as error
detectors, d.c amplifiers to amplify the error signal, d.c. servo motor as actuating device and d.c
tachometers or potentiometers as feedback elements. If the control signal is modulated by an
a.c carrier wave, the resulting system is usually referred to as an a.c control system. These
systems frequently use synchros as error detectors and modulators of error signal, a.c
amplifiers to amplify the error signal and a.c servo motors as actuators. These motors also serve
as demodulators and produce an un modulated output signal.

Discrete Data Feedback Control Systems

Discrete data control systems are those systems in which at one or more pans of the feedback
control system, the signal is in the form of pulses. Usually, the error in such system is sampled
at uniform rate and the resulting pulses are fed to the control system. In most sampled data
control systems, the signal is reconstructed as a continuous signal, using a device called 'hold
device'. Holds of different orders are employed, but the most common hold device is a zero
order hold. It holds the signal value constant, at a value equal to the amplitude of the input
time function at that sampling instant, until the next sampling instant .These systems are also
known as sampled data control systems.

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Fig : Discrete Data Feedback Control Systems

Discreet data control systems, in which a digital computer is used as one of the elements, are
known as digital control systems. The input and output to the digital computer must be binary
numbers and hence these systems require the use of digital to analog and analog to digital
converters

Time response analysis : It is an equation or a plot that describes the behavior of a system and
gives information about it with respect to time response specification as overshooting, settling
time, peak time, rise time and steady state error. Time response is formed by the transient
response and the steady state response.
Time response = Transient response + Steady state response.

Transient time response or Natural response describes the behavior of the system in its first short
time until arrives the steady state value. If the input is step function then the output or the
response is called step time response and if the input is ramp, the response is called ramp time
response .. etc.
Transient Response: The transient response is defined as the part of the time response that
goes to zero as time becomes very large. Thus yt(t) has the property

Lim y(t) = 0
t -->∞
The time required to achieve the final value is called transient period. The transient response may
be exponential or oscillatory in nature. Output response consists of the sum of forced response
(form the input) and natural response (from the nature of the system).The transient response is
the change in output response from the beginning of the response to the final state of the
response and the steady state response is the output response as time is approaching infinity (or
no more changes at the output). The behavior of a system in transient state is shown below.

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Steady State Response: The steady state response is the part of the total response that remains
after the transient has died out. For a position control system, the steady state response when
compared to with the desired reference position gives an indication of the final accuracy of the
system. If the steady state response of the output does not agree with the desired reference
exactly, the system is said to have steady state error.
Response to a Unit Step Input –First Order

Consider a feedback system with G)s) = 1/τs as shown below

The closed loop transfer function of the system is given by

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For a unit step input R (s) = 1 / s and the output is given by

Inverse Laplace transformation yields

The plot of c(t) Vs t is shown below

The response is an exponentially increasing function and it approaches a value of unity as
t --- > ∞

At t = τ the response reaches a value,

which is 63.2 percent of the steady value. This time, τ is known as the time constant of the
system. One of the important characteristics about the system is its speed of response or how
fast the response is approaching the final value. The time constant τ is indicative of this measure
and the speed of response is inversely proportional to the time constant of the system.Another
important characteristic of the system is the error between the desired value and the actual value
under steady state conditions. This quantity is known as the steady state error of the - system and
is denoted by ess.

The error E(s) for a unity feedback system is given by

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For the system under consideration G(s) = 1 / τs and R(s) = 1 /s Therefore

As t ~ ∞ e (t) ~ 0 . Thus the output of the first order system approaches the reference input,
which is the desired output, without any error. In other words, we say a first order system tracks
the step input without any steady state error.
Response to a Unit Ramp Input :

To study the response of a unit ramp let us consider a feedback system with G)s) = 1/τs as shown below

The response of the system for which , R(s) = 1 / s2 is given by

The time response is obtained by taking inverse Laplace transform of above equation

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Differentiating the above equation we get

This is the response of the system to a step input. Thus no additional information about the
speed of response is obtained by considering a ramp input.So, no additional information about
the speed of response is obtained by considering a ramp input.

But the effect on the steady state error is given by

Thus the steady state error is equal to the time constant of the system. The first order system,
therefore, can not track the ramp input without a finite steady state error. If the time constant is
reduced not only the speed of response increases but also the steady state error for ramp input
decreases. Hence the ramp input is important to the extent that it produces a finite steady state
error.
The response of a first order system for unit ramp input is shown below.

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Response to a Unit Step Input - Second Order System

Let us consider a type 1, second order system as shown in Fig.below. Since G(s) has one pole at
the origin, it is a type one system.

The closed loop transfer function is give by,

The transient response of any system depends on the poles of the transfer function T(s). The
roots of the denominator polynomial in s of T(s) are the poles of the transfer function. Thus the
denominator polynomial of T(s), given by
D(s) = τs2 + S + K

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is known as the characteristic polynomial of the system and D(s) = 0 is known as the
characteristic equation of the system. The above Eqn. is normally put in standard from, given
by,

The poles of T( s), or, the roots of the characteristic equation

S2 + 2δ ωn s + ωn 2 = 0

are given by,

Where is known as the damped natural frequency of the system. If δ > 1,
the two roots s1, s2 are real and we have an over damped system. If δ = 1, the system is known as
a critically damped system. The more common case of δ < 1 is known as the under damped
system.

Steady State Errors : One of the important design specifications for a control system is the
steady state error. The steady state output of any system should be as close to desired output as
possible. If it deviates from this desired output, the performance of the system is not satisfactory

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under steady state conditions. The steady state error reflects the accuracy of the system. Among
many reasons for these errors, the most important ones are the type of input, the type of the
system and the nonlinearities present in the system. Since the actual input in a physical system is
often a random signal, the steady state errors are obtained for the standard test signals, namely,
step, ramp and parabolic signals.
Error Constants : Let us consider a feedback control system as shown below.

The error signal E (s) is given by E (s) = R (s) - H (s) C (s)
But C (s) = G (s) E (s)

From the above equations we have

Applying final value theorem, we can get the steady state error ess as,

The above equation shows that the steady state error is a function of the input R(s) and the open
loop transfer function G(s). Let us consider various standard test signals and obtain the steady
state error for these inputs.
Proportional, Integral and Derivative Controller (PID Control) :

A Proportional–Integral–Derivative (PID) controller is a three-term controller which is
considered as a standard controller in industrial settings. . It can be found in virtually all kinds of

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control equipments, either as a stand-alone (single-station)controller or as a functional block in
Programmable Logic Controllers (PLCs)and Distributed Control Systems (DCSs).
An integral control eliminates steady state error due to a velocity input, but its effect on dynamic
response is 'difficult to predict as the system order increases to three. It is known that a
derivative term in the forward path improves the damping in the system. One of the best-known
controllers used in practice is the PID controller, where the letters stand for proportional,
integral,and derivative. The integral and derivative components of the PID controller have
individual performance implications, and their applications require an understanding ofthe basics
of these elements. Hence a suitable combination of integral and derivative controls results in a
proportional, integral and derivate control, usually called PID control. The transfer function of
the PID controller is given by,

The diagram below gives the ideal PID controller.

The overall forward path transfer function is given by,

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and the overall transfer function is given by,

Proper choice of Kp, KD and Kr results in satisfactory transient and steadystate responses. The
process of choosing proper Kp, KD, at Kr for a given system is known as tuning of a P ID
controller.
Bode plots : Bode plots are graphs of the steady state response of stable continuous time Linear
time invariant systems for sinusoidal inputs,plotted as change in magnitude and phase versus
frequency on logarithmic scale.Bode plots are a visual description of the system.So,these Bode
plots are used for the representation of sinusoidal transfer function . In this representation the
magnitude of G(jω) in db, i.e, 20 log G(jω) is plotted against 'log ω'.Similarly phase angle of
G(jω) is plotted against 'log ω' .Hence the abscissa is logarithm of the frequency and hence the
plots are known as logarithmic plots. The plots are named after the famous mathematician H. W.
Bode.
The transfer function G(jω) can be written as G(jω) = G(jω)∟Ф( ω) Where Ф( ω) is the angle
G(jω) .

Since G(jω) consists of many multiplicative factors in the numerator and denominator it is
convenient to take logarithm of G(jω) to convert these factors into additions and subtractions,
which can be carried out easily.

To plot the magnitude plot , magnitude is plotted against input frequency on a logarithmic
scale.It can be approximated by two lines and it forms the asymptotic (approximate) magnitude
Bode plot of the transfer function:

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(i) for angular frequencies below ωc it is a horizontal line at 0 dB since at low frequencies the
term ω / ωc is small and can be neglected, making the decibel gain equation above equal to
zero,

(ii) for angular frequencies above ωc it is a line with a slope of −20 dB per decade since at
high frequencies the term ω / ωc dominates and the decibel gain expression above simplifies to
-20 log (ω / ωc ) which is a straight line with a slope of −20 dB per decade.

These two lines meet at the corner frequency. From the Bode plot, it can be seen that for
frequencies well below the corner frequency, the circuit has an attenuation of 0 dB,
corresponding to a unity pass band gain, i.e. the amplitude of the filter output equals the
amplitude of the input. Frequencies above the corner frequency are attenuated – the higher the
frequency, the higher the attenuation.

Phase plot

The phase Bode plot is obtained by plotting the phase angle of the transfer function given by
–
Φ = - tan (ω/ωc ) Versus ω , where ω and ωc are the input and cutoff angular frequencies
respectively.
For input frequencies much lower than corner, the ratio ω/ωc is small and therefore the phase
angle is close to zero. As the ratio increases the absolute value of the phase increases and
becomes –45 degrees when ω= ωc As the ratio increases for input frequencies much greater
than the corner frequency, the phase angle asymptotically approaches −90 degrees. The
frequency scale for the phase plot is logarithmic.

The main advantage of using the Bode plots is that multiplication of magnitudes can be
converted into addition. Also a simple method of plotting an approximate log-magnitude curve is

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obtained. It is based on asymptotic approximations. Such approximation by straight line
asymptotes is sufficient if only rough information on the frequency response characteristics is is
needed.The phase angle curves can be drawn easily if a template for the phase angle curve of
1+jω is available. Expanding the low frequency range ,by use of a logarithmic scale for the
frequency is very advantageous ,since characteristics at low frequencies are most important in
practical systems. The only limitation is ,it is not possible to draw the curve ,right down to zero
frequency because of the logarithmic frequency (log 0 = ∞) .But this do not give any serious
problem.

Polar plot : This plot is drawn to study the frequency response in polar co-ordinates. The polar
plot of a sinusoidal transfer function G(jω) is a plot of the magnitude of G(jω) Vs the phase of
G(jω) on polar co-ordinates as ω is varied from 0 to ∞. Polar graph sheet has concentric circles
and radial lines. Concentric circles represent the magnitude. Radial lines represents the phase
angles. In polar sheet +ve phase angle is measured in ACW from 00 and -ve phase angle is
measured in CW from 00 .
To sketch the polar plot of G(jω) for the entire range of frequency ω, i.e., from 0 to infinity, there
are four key points that usually need to be known:

The start of plot where ω = 0,
The end of plot where ω = ∞,
where the plot crosses the real axis, i.e., Im(G(jω)) = 0, and
where the plot crosses the imaginary axis, i.e., Re(G(jω)) = 0.

Procedure: The following steps are involved in the plotting of Polar graphs

Express the given expression of OLTF in (1+sT) form.
Get the expressions for | G(jω)H(jω)| & G(jω)H(jω).
Tabulate various values of magnitude and phase angles for different values of ω ranging
from 0 to ∞.
Usually the choice of frequencies will be the corner frequency and around corner
frequencies. Choose proper scale for the magnitude circles.
Fix all the points in the polar graph sheet and join the points by a smooth curve.
Write the frequency corresponding to each of the point of the plot.

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Gain Margin: Gain Margin is defined as “the factor by which the system gain can be increased
to drive the system to the verge of instability”.

For stable systems, ωgc < ωpc
│G(j)H(j)│ at ω= ωpc < 1
GM = in positive dB .
More positive the Gain Margin , more stable is the system

For marginally stable systems, ωgc = ωpc
│G(j)H(j)│ at ω=ωpc = 1
GM = 0 dB

For Unstable systems, ωgc > ωpc
│G(j)H(j)│ at ω=ωpc > 1
GM = in negative dB

Gain is to be reduced to make the system stable

The important conclusions are :

 If the gain is high, the GM is low and the system‟s step response shows high overshoots
and long settling time.
 On the contrary, very low gains give high GM and PM, but also causes higher ess(Error),
higher values of rise time and settling time and in general give sluggish response.

 Thus we should keep the gain as high as possible to reduce ess and obtain acceptable
response speed and yet maintain adequate GM and PM.

 As a thumb rule an adequate GM of 2 ( ie 6 dB ) and a PM of about 30 is generally
considered sufficient enough.

 If the gain of the system is increased by a factor 1/β, then the G(j)H(j) at ω=ωpc becomes
β(1/β) = 1 and hence the │G(j)H(j)│ locus pass through -1+j0 point driving the system to
the verge of instability

PHASE MARGIN: Phase Margin is defined as “ the additional phase lag that can be introduced
before the system becomes unstable”.

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Let „A‟ be the point of intersection of │G(j)H(j)│ plot and a unit circle centered at the
origin.
Draw a line connecting the points „O‟ & „A‟ and measure the phase angle between the
line OA and +ve real axis. This angle is the phase angle of the system at the gain cross
over frequency. G(jωgc)H(jgc) = Фgc
If an additional phase lag of PM is introduced at this frequency, then the phase angle
G(jωgc)H(jωgc) will become 180 and the point „A„ coincides with (-1+j0) driving the
system to the verge of instability.
This additional phase lag is known as the Phase Margin.

For a stable system, the phase margin is positive.

A Phase margin close to zero corresponds to highly oscillatory system

A polar plot may be constructed from experimental data or from a system transfer
function
If values of ω are marked along the contour, a polar plot gives the same information as a
Bode plot
Usually, the shape of a polar plot is very important as it gives much information about the
system.

Relative and absolute stability: A stable system is a dynamic system with a bounded response
to a bounded input. A necessary and sufficient condition for a feedback system to be stable is
that all the poles of the system transfer function have negative real parts

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A system is considered marginally stable if only certain bounded inputs will result in a bounded
output.

In practical systems, it is not sufficient to know that the system is stable but a stable system must
meet the specifications of relative stability which is a quantitative measure of how fast the
transients die out in the system.

Relative stability of a system is usually defined in terms of two design parameters-phase margin
and gain margin.

The relative stability of a system can be defined as the property that is measured by the relative
real part of each root or pair of roots. The relative stability of a system can also be defined in
terms of the relative damping coefficients of each complex root pair and, therefore, in terms of
the speed of response and overshoot instead of settling time.

Time Domain Analysis Vs Frequency Domain Analysis

Variable frequency, sinusoidal signal generators are readily available and precision
measuring instruments are available for measurement of magnitude and phase angle. The
time response for a step input is more difficult to measure with accuracy.
It is easier to obtain the transfer function of a system by a simple frequency domain
test.Obtaining transfer function from the step response is more tedious.

If the system has large time constants, it makes more time to reach steady state at each
frequency of the sinusoidal input. Hence time domain method is preferred over frequency
domain method in such systems.

In order to do a frequency response test on a system, the system has to be isolated and the
sinusoidal signal has to be applied to the system. This may not be possible in systems
which can not be interrupted. In such cases, a step signal or an impulse signal may be
given to the system to find its transfer function. Hence for systems which cannot be
interrupted, time domain method is more suitable.
The design of a controller is easily done in the frequency domain method than in time
domain method. For a given set of performance measures in frequency domain , the
parameters of the open loop transfer function can be adjusted easily.
The effect of noise signals can be assessed easily in frequency domain rather than time
domain.
The most important advantage of frequency domain analysis is the ability to obtain the
relative stability of feedback control systems. The Routh Hurwitz criterion is essentially
a time do main method which determines the absolute stability of a system.

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Since the time response and frequency response of a system are related through Fourier
transform , the time response can be easily obtained from the frequency response. The
correlation between time and frequency response can be easily established so that the
time domain performance measures can be obtained from the frequency domain
specifications and vice versa.

Frequency Response of a Control System : To study the frequency response of a control
system let us consider second order system with the transfer function,

The steady state sinusoidal response is obtained by substituting s = j ω in the above equation

Normalising the frequency ω, with respect to the natural frequency ωn by defining a variable

We have,

From the above equation the magnitude and angle of the frequency response is obtained as,

and

The time response for a unit sinusoidal input with frequency OJ is given by,

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The magnitude and phase of steadystate sinusoidal response for variable frequency can be
plotted and are shown in the Fig. (a) and (b).

An important performance measure, in frequency domain, is the bandwidth of the system.
From Fig. , we observe that for δ < 0.707 and U > Ur the magnitude decreases monotonically.
The frequency Ub where the magnitude becomes 0.707 is known as the cut off frequency. At
this frequency, the magnitude will be

Control system Compensators:
A control system is usually required to meet three time response specifications, namely, steady
state accuracy, damping factor and settling time. To get the desired design with minimum errors
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and to adjust the parameters of the overall system to satisfy the design criterion an additional
subsystem called 'Compensator „ must be used.
This compensator may be used in series with the plant in the forward path or in the feedback
path shown in Fig. below. The compensation in the forward path is known as series or cascade
compensation and the later is known as feedback compensation. The compensator may be a
passive network or an active network.

Series compensation Feedback compensation
In general the series compensation is much simpler than the feed back compensation.But the
series compensation frequently require additional amplifiers to increase the gain and to provide
isolation.i.e to avoid power dissipation the series compensator is inserted in the lowest energy
point in the feed forward path.In general the number of components required in feed back
compensation will be less than the number of components in series compensation.
There are three important types of compensators.(i) Lead Compensator (ii) Lag compensator and
(iii) Lag-Lead compensator.
In a network ,when a sinusoidal input signal ei is applied at its input and if the steady state
output eo has a phase lead then the network is called a lead network. Similarly In the network for
a sinusoidal input ei ,if the steady state output has a phase lag ,then the network is called a lag
network.
Similarly in a network for a sinusoidal input .if the steady state out put has both phase lag and
lead ,but different frequency regions ,then the network is called lag-lead network.Generally the
phase lag occurs at low frequency regions and the phase lead occurs at higher frequency
region.A compensator having the characteristics of a lead network ,lag network or lead-lag
network is called a lead compensator ,lag compensator or lag-lead compensator.
The lead compensators, lag compensators and lag-lead compensators are be designed either
using electronic components like operational amplifiers or using Root Locus methods. The first

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type are called electronic lag compensators .The second type are Root locus compensators. The
Root locus compensators are have many advantages over electronic type.
Lead compensation basically speeds up the response of the control system and increase the
stability of the system. Lag compensation improves the steady –state accuracy of the system but
reduces the speed of the response. If improvements in both transient response and steady state
response is required ,both lead compensator and lag compensator are used simultaneously .In
general using a single lag-lead compensator is always economical.
Lag-lead compensation combines the advantage of lag and lead compensation. Since, the lag –
lead compensator possesses two poles and two zeros ,such a compensation increases the order of
the system by 2 ,unless the cancellation of the poles and zeros occurs in the compensated system.

Lead Compensator : It has a zero and a pole with zero closer to the origin. The general form of
the transfer function of the load compensator is

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Subsisting

Transfer function s

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Lag Compensator :

It has a zero and a pole with the zero situated on the left of the pole on the negative real axis. The
general form of the transfer function of the lag compensator is

Therefore, the frequency response of the above transfer function will be

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Now comparing with

Therefore,the transfer function is

Lag-Lead Compensator
The lag-lead compensator is the combination of a lag compensator and a lead compensator. The
lag-section is provided with one real pole and one real zero, the pole being to the right of zero,

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whereas the lead section has one real pole and one real zero with the zero being to the right of
the pole.
The transfer function of the lag-lead compensator will be

The figure below shows the lag lead compensator

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The above transfer functions are comparing with

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Then

Therefore, the transfer function is given by

state variable analysis :
Analysis and the design of feedback control systems by the classical design methods (root locus
and frequency domain methods) based on the transfer function approach are inadequate and not
convenient. So, the development of the state-variable approach, took place .This methos has
the following advantages over the classical approach.

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1. It is a direct time-domain approach. Thus, this approach is suitable for digital computer
computations
2. It is a very powerful technique for the design and analysis of linear or nonlinear, time-variant
or time-invariant, and SISO or MIMO systems.
3. In this technique, the nth order differential equations can be expressed as „n‟ equations of first
order. Thus, making the solutions easier.
4. Using this approach, the system can be designed for optimal conditions with respect to given
performance indices.
Definitions:
State : The state of a dynamical system is a minimal set of variables x1(t), x2(t)x3(t) …xn(t)
such that the knowledge of these variables at t = t0 (initial condition), together with the
Knowledge of inputs u1(t), u2(t), u3(t)… um (t) for t ≥0, completely determines the behavior of
the system for t < t0.

State-Variables :
The variables x1(t),x2(t),x3(t)… xm(t) such that the knowledge of these variables at t = t0 (initial
condition), together with the knowledge of inputs u1(t), u2(t), u3(t)… um(t) for t ≥ t0, completely
determines the behavior of the system for t < t0 ; are called state-variables. In other words, the
variables that determine the state of a dynamical system, are called state-variables.
State-Vector :

If n state variables x1(t), x2(t), x3(t)… xn(t) are necessary to determine the behavior of a
dynamical system, then these n state-variables can be considered as n components of a vector
x(t), called state-vector.

State-Space : The n dimensional space, whose elements are the n state-variables, is called state-
space. Any state can be represented by a point in the state-space.

State equation of a linear time-invariant system :
For a general system of the Figure shown below the state representation can be arranged in the
form of n first-order differential equations as

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……(1)
Integrating equation 1 , we get

Where I = 1,2,3……..n
Thus, the n state-variables and, the state of the system can uniquely be determined at any t < tn,
provided each state-variable is known at t = tn and all the m control forces are known throughout
the interval t 0 to t.
The n differential equations of 1 may be written in vector form as

………… (2)

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Equation 2 is the state equation for time-invariant systems. However, for time-varying systems,
the function vector f(.) is dependent on time as well, and the vector equation may be given as

……………….(3)
Equation (3) is the state equation for time-varying systems.
The output y(t) can, in general, be expressed in terms of the state vector x(t) and input vector u(t)
as
For time-invariant systems :

------------------------ -----(4)

For time-varying systems: ----------------(5)

STATE MODEL OF A LINEAR SINGLE-INPUT-SINGLE-OUTPUT SYSTEM:
The state model of a linear single-input-single-output system can be written as

(1)

(2)
Where

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d = Transmission Constant , u(t) = Input or Control Variable (scalar)
and, y(t) = Output Variable (scalar).
The block-diagram representation of the state model of linear single-input-single-output system
is shown below.

A very important conclusion is that the derivatives of all the state-variables are zero at the
equilibrium point. Thus, the system continues to lie at the equilibrium point unless otherwise
disturbed.

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SOLUTION OF STATE EQUATIONS FOR LINEAR TIME-INVARIANT SYSTEMS :

The state equation of a linear time-invariant system is given by

For a homogeneous (unforced) system

So , we have ………..(1)
Take the Laplace Transform on both sides

…………… (2)

The above may also be written as

Taking the inverse Laplace transform --------------(3)
This equation (3) gives the solution of the LTI homogeneous state Equation.It can be observed
that the initial state x(0) at t = 0, is driven to a state x(t) at time t. This transition in state is carried
out by the matrix exponential eAt. Because of this property, eAt is termed as the State Transition
Matrix and is denoted by ø(t).

Determination of transfer function using SVA:
Let us consider the following state space equation ,

Now, take the Laplace Transform (with zero initial conditions)

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We want to solve for the ratio of Y(s) to U(s), which gives the transfer function.So, we need to
remove Q(s) from the output equation. We solve the state equation for Q(s)

The matrix Φ(s) is called the state transition matrix. Now , put this into the output equation

Now solving for transfer function ,we get

This is the method of determining the transfer function from state variable analysis (SVA).

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Acknowledgment: Dear Reader, I don’t claim any ownership for this material, as it is a
collection from various books and websites and other articles. I have simply added my
experience and try to provide this material for those people who are preparing for NET/UGC
and other Engineering exams.Please download this material for your preparation .But never
make it commercial. It is purely meant for students who are in need of this material.

References: 1:Automatic control systems - B.C. Kuo, Prentice-Hall of India.

2. Modern Control Engineering- K. Ogata, Prenticd-Hall of India.

3. Control Systems Engineering - I.G. Nagrath, M. Gopal; Wiley Eastern Ltd

4. Control Systems -Second Edition - Dr. N.C. Jagan
And many more people‟s study material on the net.

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