Basic Control System ConceptsTo understand how PID controllers are used, it is important to first understandsome basic control system principles. The image below illustrates the 2 basic types of control system.

The open loop systemcan clearly be seen to have no feedback, therefore if the load changes on the motor, the motor speed will change. The control unit cannot command the driver to increase or decrease the power to the motor, as it has no knowledge of the speed change induced in the motor, by the change in load.The closed loop systemhowever has feedback from the motor. So if the motors speed were to decrease due to an increase in load, the control unit could command the driver to increase the power to the motor, keeping a constant speed. Common direct current motor control is achieved via PWM, and simply increasing or decreasing the duty cycle will increase or decrease the motor speed.Before I introduce you about various controllers in detail, it is very essential to know the uses of controllers in the theory of control systems. The important uses of the controllers are written below:

Controllers improve steady state accuracy by decreasing the steady state errors.1. As the steady state accuracy improves, the stability also improves.2. They also help in reducing the offsets produced in the system.3. Maximum overshoot of the system can be controlled using these controllers.4. They also help in reducing the noise signals produced in the system.5. Slow response of the over damped system can be made faster with the help of these controllers.Now what are controllers? A controller is one which compares controlled values with the desired values and has a function to correct the deviation produced.Types of ControllersLet us classify the controllers. There are mainly twotypes of controllersand they are written below:Continuous Controllers:The main feature of continuous controllers is that the controlled variable (also known as the manipulated variable) can have any value within the range of controllers output. Now in the continuous controllers theory, there are three basic modes on which the whole control action takes place and these modes are written below. We will use the combination of these modes in order to have a desired and accurate output.1. Proportional controllers.2. Integral controllers.3. Derivative controllers.Combinations of these three controllers are written below:4. Proportional and integral controllers.5. Proportional and derivative controllers.Now we will discuss each of these modes in detail.Proportional ControllersWe cannot usetypes of controllersat anywhere, with each type controller, there are certain conditions that must be fulfilled. Withproportional controllersthere are two conditions and these are written below:1. Deviation should not be large, it means there should be less deviation between the input and output.2. Deviation should not be sudden.Now we are in a condition to discuss proportional controllers, as the name suggests in a proportional controller the output (also called the actuating signal) is directly proportional to the error signal. Now let us analyze proportional controller mathematically. As we know in proportional controller output is directly proportional to error signal, writing this mathematically we have,

Removing the sign of proportionality we have,

Where Kpis proportional constant also known as controller gain.It is recommended that Kpshould be kept greater than unity. If the value of Kpis greater than unity, then it will amplify the error signal and thus the amplified error signal can be detected easily.Advantages of Proportional ControllerNow let us discuss some advantages of proportional controller.1. Proportional controller helps in reducing the steady state error, thus makes the system more stable.2. Slow response of the over damped system can be made faster with the help of these controllers.Disadvantages of Proportional ControllerNow there are some serious disadvantages of these controllers and these are written as follows:1. Due to presence of these controllers we some offsets in the system.2. Proportional controllers also increase the maximum overshoot of the system.

Integral ControllersAs the name suggests inintegral controllersthe output (also called the actuating signal) is directly proportional to the integral of the error signal. Now let us analyze integral controller mathematically. As we know in an integral controller output is directly proportional to the integration of the error signal, writing this mathematically we have,

Removing the sign of proportionality we have,

Where Kiis integral constant also known as controller gain. Integral controller is also known as reset controller.

Advantages of Integral ControllerDue to their unique ability they can return the controlled variable back to the exact set point following a disturbance thats why these are known as reset controllers.Disadvantages of Integral ControllerIt tends to make the system unstable because it responds slowly towards the produced error.Derivative ControllersWe never usederivative controllersalone. It should be used in combinations with other modes of controllers because of its few disadvantages which are written below:1. It never improves the steady state error.2. It produces saturation effects and also amplifies the noise signals produced in the system.Now, as the name suggests in a derivative controller the output (also called the actuating signal) is directly proportional to the derivative of the error signal. Now let us analyze derivative controller mathematically. As we know in a derivative controller output is directly proportional to the derivative of the error signal, writing this mathematically we have,

Removing the sign of proportionality we have,

Where Kdis proportional constant also known as controller gain. Derivative controller is also known as rate controller.

Advantages of Derivative ControllerThe major advantage of derivative controller is that it improves the transient response of the system.Proportional and Integral ControllerAs the name suggests it is a combination of proportional and an integral controller the output (also called the actuating signal) is equal to the summation of proportional and integral of the error signal. Now let us analyze proportional and integral controller mathematically. As we know in a proportional and integral controller output is directly proportional to the summation of proportional of error and integration of the error signal, writing this mathematically we have,

Removing the sign of proportionality we have,

Where Kiand kpproportional constant and integral constant respectively.

Advantages and disadvantages are the combinations of the advantages and disadvantages of proportional and integral controllers.Proportional and Derivative ControllerAs the name suggests it is a combination of proportional and a derivative controller the output (also called the actuating signal) is equals to the summation of proportional and derivative of the error signal. Now let us analyze proportional and derivative controller mathematically. As we know in a proportional and derivative controller output is directly proportional to summation of proportional of error and differentiation of the error signal, writing this mathematically we have,

Removing the sign of proportionality we have,

Where Kdand kpproportional constant and derivative constant respectively.

Advantages and disadvantages are the combinations of advantages and disadvantages of proportional and derivative controllers

Sometimes, the control element has only two position either it is fully closed or fully open. This control element does not operate at any intermediate position, i.e. partly open or partly closed position. The control system made for controlling such elements, is known ason off control theory. In this control system, when process variable changes and crosses certain preset level, the output valve of the system is suddenly fully opened and gives 100% output.

Generally in on off control system, the output causes change in process variable. Hence due to effect of output, the process variable again starts changing but in reverse direction. During this change, when process variable crosses certain predetermined level, the output valve of the system is immediately closed and output is suddenly reduced to 0%.As there is no output, the process variable again starts changing in its normal direction. When it crosses the preset level, the output valve of the system is again fully open to give 100% output. This cycle of closing and opening of output valve continues till the said on-off control system is in operation.A very common example ofon-off control theoryis fan controlling scheme of transformer cooling system.When transformer runs with such a load, the temperature of theelectrical power transformerrises beyond the preset value at which the cooling fans start rotating with their full capacity.As the cooling fans run, the forced air (output of the cooling system) decreases the temperature of the transformer.When the temperature (process variable) comes down below a preset value, the control switch of fans trip and fans stop supplying forced air to the transformer. After that, as there is no cooling effect of fans, the temperature of the transformer again starts rising due to load.Again when during rising, the temperature crosses the preset value, the fans again start rotating to cool down the transformer.Theoretically, we assume that there is no lag in the control equipment. That means, there is no time day for on and off operation of control equipment. With this assumption if we draw series of operations of an ideal on off control system, we will get the graph given below.

But in practical on off control, there is always a non zero time delay for closing and opening action ofcontrollerelements.This time delay is known as dead time. Because of this time delay the actual response curve differs from the above shown ideal response curve.Let us try to draw actual response curve of an on off control system.

Say at time TOthe temperature of the transformer starts rising. The measuring instrument of the temperature does not response instantly, as it requires some time delay for heating up and expansion of mercury in temperature sensor bulb say from instant T1the pointer of the temperature indicator starts rising. This rising is exponential in nature. Let us at point A, thecontrollersystem starts actuating for switching on cooling fans and finally after period of T2the fans starts delivering force air with its full capacity. Then the temperature of the transformer starts decreasing in exponential manner.

Definition of Control SystemAs the human civilization is being modernized day by day the demand of automation is increasing accordingly. Automation highly requires control of devices.

Acontrol systemis a system of devices or set of devices, that manages, commands, directs or regulates the behavior of other device(s) or system(s) to achieve desire results. In other words thedefinition of control systemcan be rewritten asA control system is a system, which controls other system.In recent years,control systemsplays main role in the development and advancement of modern technology and civilization. Practically every aspects of our day-to-day life is affected less or more by some control system. A bathroom toilet tank, a refrigerator, an air conditioner, a geezer, an automatic iron, an automobile all are control system. These systems are also used in industrial process for more output. We find control system in quality control of products, weapons system, transportation systems, power system, space technology, robotics and many more. Theprinciples of control theoryis applicable to engineering and non engineering field both.Requirement Of Good Control SystemAccuracy:Accuracy is the measurement tolerance of the instrument and defines the limits of the errors made when the instrument is used in normal operating conditions. Accuracy can be improved by using feedback elements. To increase accuracy of any control system error detector should be present in control system.Sensitivity:The parameters of control system are always changing with change in surrounding conditions, internal disturbance or any other parameters. This change can be expressed in terms of sensitivity. Any control system should be insensitive to such parameters but sensitive to input signals only.Noise:An undesired input signal is known as noise. A good control system should be able to reduce the noise effect for better performance.Stability:It is an important characteristic of control system. For the bounded input signal, the output must be bounded and if input is zero then output must be zero then such a control system is said to be stable system.

Bandwidth:An operating frequency range decides the bandwidth of control system. Bandwidth should be large as possible for frequency response of good control system.Speed:It is the time taken by control system to achieve its stable output. A good control system possesses high speed. The transient period for such system is very small.Oscillation:A small numbers of oscillation or constant oscillation of output tend to system to be stable.Types Of Control SystemsThere are two maintypes of control system. They are as follow1. Open loop control system2. Closed loop control systemOpen Loop Control SystemA control system in which the control action is totally independent of output of the system then it is calledopen loop control system. Open loop system is also called as Manual control system. Fig 1 shows the block diagram of open loop control system in which process output is totally independent of controller action.

Practical Examples Of Open Loop Control System1. Electric Hand Drier Hot air (output) comes out as long as you keep your hand under the machine, irrespective of how much your hand is dried.2. Automatic Washing Machine This machine runs according to the pre-set time irrespective of washing is completed or not.3. Bread Toaster This machine runs as per adjusted time irrespective of toasting is completed or not.4. Automatic Tea/Coffee Maker These machines also function for pre adjusted time only.5. Timer Based Clothes Drier This machine dries wet clothes for pre adjusted time, it does not matter how much the clothes are dried.6. Light Switch lamps glow whenever light switch is on irrespective of light is required or not.7. Volume on Stereo System Volume is adjusted manually irrespective of output volume level.Advantages Of Open Loop Control System1. Simple in construction and design.2. Economical.3. Easy to maintain.4. Generally stable.5. Convenient to use as output is difficult to measure.Disadvantages Of Open Loop Control System1. They are inaccurate.2. They are unreliable.3. Any change in output cannot be corrected automatically.Closed Loop Control SystemControl system in which the output has an effect on the input quantity in such a manner that the input quantity will adjust itself based on the output generated is calledclosed loop control system. Open loop control system can be converted in to closed loop control system by providing a feedback. This feedback automatically makes the suitable changes in the output due to external disturbance. In this way closed loop control system is called automatic control system. Figure below shows the block diagram of closed loop control system in which feedback is taken from output and fed in to input.

Practical Examples Of Closed Loop Control System1. Automatic Electric Iron Heating elements are controlled by output temperature of the iron.2. Servo Voltage Stabilizer Voltage controller operates depending upon outputvoltageof the system.3. Water Level Controller Input water is controlled by water level of the reservoir.4. Missile Launched & Auto Tracked by Radar The direction of missile is controlled by comparing the target and position of the missile.5. An Air Conditioner An air conditioner functions depending upon the temperature of the room.6. Cooling System in Car It operates depending upon the temperature which it controls.Advantages OF Closed Loop Control System1. Closed loop control systems are more accurate even in the presence of non-linearity.2. Highly accurate as any error arising is corrected due to presence of feedback signal.3. Bandwidth range is large.4. Facilitates automation.5. The sensitivity of system may be made small to make system more stable.6. This system is less affected by noise.Disadvantages Of Closed Loop Control System1. They are costlier.2. They are complicated to design.3. Required more maintenance.4. Feedback leads to oscillatory response.5. Overall gain is reduced due to presence of feedback.6. Stability is the major problem and more care is needed to design a stable closed loop system.Comparison of Closed Loop And Open Loop Control SystemSR. NO.OPEN LOOP CONTROL SYSTEMCLOSED LOOP CONTROL SYSTEM

1The feedback element is absent.The feedback element is always present.

2An error detector is not present.An error detector is always present.

3It is stable one.It may become unstable.

4Easy to construct.Complicated construction.

5It is an economical.It is costly.

6Having small bandwidth.Having large bandwidth.

7It is inaccurate.It is accurate.

8Less maintenance.More maintenance.

9It is unreliable.It is reliable.

10Examples: Hand drier, tea makerExamples: Servovoltagestabilizer, perspiration

Feedback Loop Of Control SystemA feedback is a common and powerful tool when designing a control system. Feedback loop is the tool which take the system output into consideration and enables the system to adjust its performance to meet a desired result of system.In any control system, output is affected due to change in environmental condition or any kind of disturbance. So one signal is taken from output and is fed back to the input. This signal is compared with reference input and then error signal is generated. This error signal is applied to controller and output is corrected. Such a system is called feedback system. Figure below shows the block diagram of feedback system.

When feedback signal is positive then system called positive feedback system. For positive feedback system, the error signal is the addition of reference input signal and feedback signal. When feedback signal is negative then system is called negative feedback system. For negative feedback system, the error signal is given by difference of reference input signal and feedback signal.Effect Of FeedbackRefer figure beside, which represents feedback system whereR = Input signalE = Error signalG = forward path gainH = FeedbackC = Output signalB = Feedback signal

1. Error between system input and system output is reduced.2. System gain is reduced by a factor 1/(1GH).3. Improvement in sensitivity.4. Stability may be affected.5. Improve the speed of response

At point B, the controller system starts actuating for switching off the cooling fans and finally after a period of T3the fans stop delivering force air. Then the temperature of the transformer again starts rising in same exponential manner.N.B.: Here during this operation we have assumed that, loading condition of theelectrical powertransformer, ambient temperature and all other conditions of surrounding are fixed and constant.Linear Control SystemsIn order to understand thelinear control system, we should know the principle of superposition. The principle ofsuperposition theoremincludes two the important properties and they are explained below:Homogeneity:A system is said to be homogeneous, if we multiply input with some constant A then output will also be multiplied by the same value of constant (i.e. A).Additivity:Suppose we have a system S and we are giving the input to this system as a1 for the first time and we are getting output as b1 corresponding to input a1. On second time we are giving input a2 and correspond to this we are getting output as b2. Now suppose this time we giving input as summation of the previous inputs ( i.e. a1+ a2) and corresponding to this input suppose we are getting output as (b1+ b2) then we can say that system S is following the property of additivity. Now we are able to define thelinear control systemsas thosetypes of control systemswhich follow the principle of homogeneity and additivity.Examples of Linear Control SystemConsider a purely resistive network with a constant dc source. This circuit follows the principle of homogeneity and additivity. All the undesired effects are neglected and assuming ideal behavior of each element in the network, we say that we will get linearvoltageand current characteristic. This is the example oflinear control system.Non-linear SystemsWe can simply definenon linear control systemas all those system which do not follow the principle of homogeneity. In practical life all the systems are non-linear system.Examples of Non-linear SystemA well known example of non-linear system is magnetization curve orno load curve of a dc machine. We will discuss briefly no load curve of dc machines here: No load curve gives us the relationship between the air gap flux and the field winding mmf. It is very clear from the curve given below that in the beginning there is a linear relationship between winding mmf and the air gap flux but after this, saturation has come which shows the non linear behavior of the curve or characteristics of thenon linear control system.

Analog or Continuous SystemIn thesetypes of control systemwe have continuous signal as the input to the system. These signals are the continuous function of time. We may have various sources of continuous input signal like sinusoidal type signal input source, square type of signal input source, signal may be in the form of continuous triangle etc.Digital or Discrete SystemIn these types of control system we have discrete signal (or signal may be in the form of pulse) as the input to the system. These signals have the discrete interval of time. We can convert various sources of continuous input signal like sinusoidal type signal input source, square type of signal input source etc into discrete form using the switch.Now there are various advantages of discrete or digital system over the analog system and these advantages are written below:1. Digital systems can handle non linear control systems more effectively than the analog type of systems.2. Power requirement in case of discrete or digital system is less as compared to analog systems.3. Digital system has higher rate of accuracy and can perform various complex computations easily as compared to analog systems.4. Reliability of digital system is more as compared to analog system. They also have small and compact size.5. Digital system works on the logical operations which increases their accuracy many times.6. Losses in case of discrete systems are less as compared to analog systems in general.Single Input Single Output SystemsThese are also known as SISO type of system. In this the system has single input for single output. Various example of this kind of system may include temperature control, position control system etc.Multiple Input Multiple Output SystemsThese are also known as MIMO type of system. In this the system has multiple outputs for multiple inputs. Various example of this kind of system may include PLC type system etc.Lumped Parameter SystemIn these types of control systems the various active (resistor) and passive parameters (likeinductorand capacitor) are assumed to be concentrated at a point and thats why these are called lumped parameter type of system. Analysis of such type of system is very easy which includes differential equations.Distributed Parameter SystemIn thesetypes of control systemsthe various active (resistor) and passive parameters (likeinductorand capacitor) are assumed to be distributed uniformly along the length and thats why these are called distributed parameter type of system. Analysis of such type of system is slightly difficult which includes partial differential equations.Mathematical Modelling of Control SystemThere are various types of physical systems namely we have

1. Mechanical system.2. Electrical system.3. Electronic system.4. Thermal system.5. Hydraulic system.6. Chemical system etc.Before I describe these systems in detail let us know, what is the meaning of modeling of the system?Mathematical modelling of control systemis the process of drawing the block diagram for these types of systems in order to determine the performance and the transfer functions. Now let us describe mechanical and electrical type of systems in detail. We will derive analogies between mechanical and electrical system only which are most important in understanding the theory of control system.Mathematical Modelling of Mechanical SystemsWe have two types of mechanical systems. Mechanical system may be a linear mechanical system or it may be a rotational mechanical type of system.In linear mechanical type of systemswe have three variables 1. Force which is represented by F.2. Velocity which is represented by V.3. Linear displacement represented by XAnd also we have three parameters-1. Mass which is represented by M.2. Coefficient of viscous friction which is represented by B.3. Spring constant which is represented by K.In rotational mechanical type of systemswe have three variables-1. Torque which is represented by T.2. Angular velocity which is represented by 3. Angular displacement represented by And also we have two parameters 1. Moment of inertia which is represented by J.2. Coefficient of viscous friction which is represented by B.Now let us consider the linear displacement mechanical system which is shown below-

We have already marked various variables in the diagram itself. We have x is the displacement as shown in the diagram. From the Newtons second law of motion, we can write force as-

From the diagram we can see that the,

On substituting the values of F1, F2and F3in the above equation and taking the Laplace transform we have the transfer function as,

This equation ismathematical modelling of mechanical control system.

Mathematical Modelling of Electrical SystemIn electrical type of systemswe have three variables

1. Voltage which is represented by V.2. Current which is represented by I.3. Charge which is represented by Q.And also we have three parameters which are active and passive elements 1. Resistancewhich is represented by R.2. Capacitance which is represented by C.3. Inductance which is represented by L.Now we are in condition to derive analogy between electrical and mechanical types of systems. There are two types of analogies and they are written below:Force Voltage Analogy :In order to understand this type of analogy, let us consider a circuit which consists of series combination ofresistor,inductorand capacitor.

AvoltageV is connected in series with these elements as shown in the circuit diagram. Now from the circuit diagram and with the help of KVL equation we write the expression forvoltagein terms of charge,resistance,capacitorandinductoras,

Now comparing the above with that we have derived for the mechanical system we find that-

1. Mass (M) is analogous toinductance(L).2. Force is analogous tovoltageV.3. Displacement (x) is analogous to charge (Q).4. Coefficient of friction (B) is analogous toresistanceR and5. Spring constant is analogous to inverse of thecapacitor(C).This analogy is known as forcevoltageanalogy.Force Current Analogy:In order to understand this type of analogy, let us consider a circuit which consists of parallel combination ofresistor,inductorand capacitor.

AvoltageE is connected in parallel with these elements as shown in the circuit diagram. Now from the circuit diagram and with the help of KCL equation we write the expression for current in terms of flux,resistance,capacitorandinductoras,

Now comparing the above with that we have derived for the mechanical system we find that,

1. Mass (M) is analogous to Capacitor (C).2. Force is analogous to current I.3. Displacement (x) is analogous to flux ().4. Coefficient of friction (B) is analogous toresistance1/ R and5. Spring constant K is analogous to inverse of theinductor(L).This analogy is known as force current analogy.Now let us consider the rotational mechanical type of system which is shown below we have already marked various variables in the diagram itself. We have is the angular displacement as shown in the diagram. From the mechanical system, we can write equation for torque (which is analogous to force) as torque as,

From the diagram we can see that the,

On substituting the values of T1, T2and T3in the above equation and taking the Laplace transform we have the transfer function as,

This equation is mathematical modelling of electrical control system.

When we study the analysis of thetransient state and steady state response of control systemit is very essential to know a few basic terms and these are described below.

Standard Input Signals :These are also known as test input signals. The input signal is very complex in nature, it is complex because it may be a combination of various other signals. Thus it is very difficult to analyze characteristic performance of any system by applying these signals. So we use test signals or standard input signals which are very easy to deal with. We can easily analyze the characteristic performance of any system more easily as compared to non standard input signals. Now there are various types of standard input signals and they are written below:Unit Impulse Signal :In the time domain it is represented by (t). TheLaplace transformationof unit impulse function is 1 and the corresponding waveform associated with the unit impulse function is shown below.

Unit Step Signal :In the time domain it is represented by u (t). TheLaplace transformationof unit step function is 1/s and the corresponding waveform associated with the unit step function is shown below.

Unit Ramp signal :In the time domain it is represented by r (t). TheLaplace transformationof unit ramp function is 1/s2and the corresponding waveform associated with the unit ramp function is shown below.Unit Ramp SignalParabolic Type Signal :In the time domain it is represented by t2/ 2. TheLaplace transformationof parabolic type of the function is 1 / s3and the corresponding waveform associated with the parabolic type of the function is shown below.

Sinusoidal Type Signal :In the time domain it is represented by sin (t).TheLaplace transformationof sinusoidal type of the function is / (s2+ 2) and the corresponding waveform associated with the sinusoidal type of the function is shown below.

Cosine Type of Signal :In the time domain it is represented by cos (t). TheLaplace transformationof the cosine type of the function is / (s2+ 2) and the corresponding waveform associated with the cosine type of the function is shown below,

Now are in a position to describe the two types of responses which are a function of time.Transient Response of Control SystemAs the name suggeststransient response of control systemmeans changing so, this occurs mainly after two conditions and these two conditions are written as follows- Condition one :Just after switching on the system that means at the time of application of an input signal to the system. Condition second :Just after any abnormal conditions. Abnormal conditions may include sudden change in the load, short circuiting etc.Steady State Response of Control SystemSteady state occurs after the system becomes settled and at the steady system starts working normally.Steady state response of control systemis a function of input signal and it is also called as forced response.Now the transient state response ofcontrol systemgives a clear description of how the system functions duringtransient state and steady state response of control systemgives a clear description of how the system functions during steady state. Therefore the time analysis of both states is very essential. We will separately analyze both the types of responses. Let us first analyze the transient response. In order to analyze the transient response, we have some time specifications and they are written as follows:Delay Time :This time is represented by td. The time required by the response to reach fifty percent of the final value for the first time, this time is known as delay time. Delay time is clearly shown in the time response specification curve.Rise Time :This time is represented by tr. We define rise time in two cases:1. In case of under damped systems where the value of is less than one, in this case rise time is defined as the time required by the response to reach from zero value to hundred percent value of final value.2. In case of over damped systems where the value of is greater than one, in this case rise time is defined as the time required by the response to reach from ten percent value to ninety percent value of final value.Peak Time :This time is represented by tp. The time required by the response to reach the peak value for the first time, this time is known as peak time. Peak time is clearly shown in the time response specification curve.Settling Time :This time is represented by ts. The time required by the response to reach and within the specified range of about (two percent to five percent) of its final value for the first time, this time is known as settling time. Settling time is clearly shown in the time response specification curve.Maximum Overshoot :It is expressed (in general) in percentage of the steady state value and it is defined as the maximum positive deviation of the response from its desired value. Here desired value is steady state value.Steady State Error :It can be defined as the difference between the actual output and the desired output as time tends to infinity.Now we are in position we to do a time response analysis of a first order system.Transient State and Steady State Response of First Order Control SystemLet us consider the block diagram of the first order system.

From this block diagram we can find overall transfer function which is linear in nature. The transfer function of the first order system is 1/((sT+1)). We are going to analyze the steady state and transient response ofcontrol systemfor the following standard signal.1. Unit impulse.2. Unit step.3. Unit ramp.Unit impulse response :We have Laplace transform of the unit impulse is 1. Now let us give this standard input to a first order system, we have

Now taking the inverse Laplace transform of the above equation, we have

It is clear that thesteady state response of control systemdepends only on the time constant T and it is decaying in nature.

Unit step response :We have Laplace transform of the unit impulse is 1/s. Now let us give this standard input to first order system, we have

With the help of partial fraction, taking the inverse Laplace transform of the above equation, we have

It is clear that the time response depends only on the time constant T. In this case the steady state error is zero by putting the limit t is tending to zero.

Unit ramp response :We have Laplace transform of the unit impulse is 1/s2. Now let us give this standard input to first order system, we have

With the help of partial fraction, taking the inverse Laplace transform of the above equation we have

On plotting the exponential function of time we have T by putting the limit t is tending to zero.

Transient State and Steady State Response of Second Order Control SystemLet us consider the block diagram of the second order system.

From this block diagram we can find overall transfer function which is nonlinear in nature. The transfer function of the second order system is (2) / ( s ( s + 2 )). We are going to analyze thetransient state response of control systemfor the following standard signal.Unit impulse response :We have Laplace transform of the unit impulse is 1. Now let us give this standard input to second order system, we have

Where is natural frequency in rad/sec and is damping ratio.

Unit step response :We have Laplace transform of the unit impulse is 1/s. Now let us give this standard input to first order system, we have

With the help of partial fraction, taking the inverse Laplace transform of the above equation we have

Now we will see the effect of different values of on the response. We have three types of systems on the basis of different values of .

1. Under damped system :A system is said to be under damped system when the value of is less than one. In this case roots are complex in nature and the real parts are always negative. System is asymptotically stable. Rise time is lesser than the other system with the presence of finite overshoot.2. Critically damped system :A system is said to be critically damped system when the value of is one. In this case roots are real in nature and the real parts are always repetitive in nature. System is asymptotically stable. Rise time is less in this system and there is no presence of finite overshoot.3. Over damped system :A system is said to be over damped system when the value of is greater than one. In this case roots are real and distinct in nature and the real parts are always negative. System is asymptotically stable. Rise time is greater than the other system and there is no presence of finite overshoot.4. Sustained Oscillations :A system is said to be sustain damped system when the value of zeta is zero. No damping occurs in this case.Now let us derive the expressions for rise time, peak time, maximum overshoot, settling time and steady state error with a unit step input for second order system.Rise time :In order to derive the expression for the rise time we have to equate the expression for c(t) = 1. From the above we have

On solving above equation we have expression for rise time equal to

Peak Time :On differentiating the expression of c(t) we can obtain the expression for peak time. dc(t)/ dt = 0 we have expression for peak time,

Maximum overshoot :Now it is clear from the figure that the maximum overshoot will occur at peak time tp hence on putting the valye of peak time we will get maximum overshoot as

Settling Time :Settling time is given by the expression

Steady state error :The steady state error is diffrerence between the actual output and the desired output hence at time tending to infinity the steady state error is zero.

For any control system there exists a reference input termed as excitation or cause which operates through a transfer operation termed astransfer functionand produces an effect resulting in controlled output or response. Thus the cause and effect relationship between the output and input is related to each other through atransfer function.

It is not necessary that the output will be of same category as that of the input. For example in case of anelectrical motor, the input is an electrical quantity and output is a mechanical one. In control system all mathematical functions are converted to their corresponding Laplace transforms. So the transfer function is expressed as a ratio of Laplace transform of input function to Laplace transform of output function.

The transfer function can be expressed as,

While doing Laplace transform, while determining transfer function we assume all initial conditions to be zero.

The transfer function of a control system is defined as the ration of the Laplace transform of the output variable to Laplace transform of the input variable assuming all initial conditions to be zero.Procedure for determining the transfer function of a control system are as follows :1. We form the equations for the system2. Now we take Laplace transform of the system equations, assuming initial conditions as zero.3. Specify system output and input4. Lastly we take the ratio of the Laplace transform of the output and the Laplace transform of the input which is the required transfer function

Methods of obtaining a Transfer function: There are major two ways of obtaining a transfer function for the control system .The ways are Block diagram method : It is not convenient to derive a complete transfer function for a complex control system. Therefore the transfer function of each element of a control system is represented by a block diagram. Block diagram reduction techniques are applied to obtain the desired transfer function. Signal Flow graphs : The modified form of a block diagram is a signal flow graph. Block diagram gives a pictorial representation of a control system . Signal flow graph further shortens the representation of a control system.The transfer function of a system is completely specified in terms of its poles and zeroes and the gain factor. Let us know about the poles and zeroes of a transfer function in brief.

Where, K = system gain,z1, z2, zm= zeros of the transfer functionp1, p2, pn= poles of the transfer function

Putting the denominator of equation (i) equal to zero we get the poles value of the transfer function. For this the T.F is infinity.

Putting the numerator of equation (ii) equal to zero we get the value of zero of the transfer function. For this T.F is equal to zero.

There are two types of transfer functions :-i) Open loop transfer function( O.L.T.F) : Transfer function of the system without feedback path or loop.ii) Closed loop transfer function (C.L.T.F) : Transfer function of the system with feedback path or loop.

Theroot locus technique in control systemwas first introduced in the year 1948 by Evans. Any physical system is represented by a transfer function in the form of

We can find poles and zeros from G(s). The location of poles and zeros are crucial keeping view stability, relative stability, transient response and error analysis. When the system put to service strayinductanceandcapacitanceget into the system, thus changes the location of poles and zeros. Inroot locus technique in control systemwe will evaluate the position of the roots, their locus of movement and associated information. These information will be used to comment upon the system performance.

Now before I introduce what is a root locus technique, it is very essential here to discuss a few of the advantages of this technique over other stability criteria. Some of the advantages of root locus technique are written below.Advantages of Root Locus Technique1. Root locus technique in control system is easy to implement as compared to other methods.2. With the help of root locus we can easily predict the performance of the whole system.3. Root locus provides the better way to indicate the parameters.Now there are various terms related to root locus technique that we will use frequently in this article.1. Characteristic Equation Related to Root Locus Technique :1 + G(s)H(s) = 0 is known as characteristic equation. Now on differentiating the characteristic equation and on equating dk/ds equals to zero, we can get break away points.2. Break away Points :Suppose two root loci which start from pole and moves in opposite direction collide with each other such that after collision they start moving in different directions in the symmetrical way. Or the break away points at which multiple roots of the characteristic equation 1 + G(s)H(s)= 0 occur. The value of K is maximum at the points where the branches of root loci break away. Break away points may be real, imaginary or complex.3. Break in Point :Condition of break in to be there on the plot is written below :Root locus must be present between two adjacent zeros on the real axis.4. Centre of Gravity :It is also known centroid and is defined as the point on the plot from where all the asymptotes start. Mathematically, it is calculated by the difference of summation of poles and zeros in the transfer function when divided by the difference of total number of poles and total number of zeros. Centre of gravity is always real & it is denoted by A.

Where N is number of poles & M is number of zeros.5. Asymptotes of Root Loci :Asymptote originates from the centre of gravity or centroid and goes to infinity at definite some angle. Asymptotes provide direction to the root locus when they depart break away points.6. Angle of Asymptotes :Asymptotes makes some angle with the real axis and this angle can be calculated from the given formula,

Where p = 0, 1, 2 . (N-M-1)N is the total number of polesM is the total number of zeros.7. Angle of Arrival or Departure :We calculate angle of departure when there exists complex poles in the system. Angle of departure can be calculated as 180-{(sum of angles to a complex pole from the other poles)-(sum of angle to a complex pole from the zeros)}.8. Intersection of Root Locus with the Imaginary Axis :In order to find out the point of intersection root locus with imaginary axis, we have to use Routh Hurwitz criterion. First, we find the auxiliary equation then the corresponding value of K will give the value of the point of intersection.9. Gain Margin :We define gain margin as a by which the design value of the gain factor can be multiplied before the system becomes unstable. Mathematically it is given by the formula

10. Phase Margin :Phase margin can be calculated from the given formula:

11. Symmetry of Root Locus :Root locus is symmetric about the x axis or the real axis.How to determine the value of K at any point on the root loci ? Now there are two ways of determining the value of K, each way is described below.

1. Magnitude Criteria :At any points on the root locus we can apply magnitude criteria as,

Using this formula we can calculate the value of K at any desired point.2. Using Root Locus Plot :The value of K at any s on the root locus is given by

Root Locus PlotThis is also known as root locus technique in control system and is used for determining the stability of the given system. Now in order to determine the stability of the system using the root locus technique we find the range of values of K for which the complete performance of the system will be satisfactory and the operation is stable.Now there are some results that one should remember in order to plot the root locus. These results are written below:1. Region where root locus exists :After plotting all the poles and zeros on the plane, we can easily find out the region of existence of the root locus by using one simple rule which is written below,

Only that segment will be considered in making root locus if the total number of poles and zeros at the right hand side of the segment is odd.2. How to calculate the number of separate root loci ? :A number of separate root loci are equal to the total number of roots if number of roots are greater than the number of poles otherwise number of separate root loci is equal to the total number of poles if number of roots are greater than the number of zeros.Procedure to Plot Root LocusKeeping all these points in mind we are able to draw theroot locus plotfor any kind of system. Now let us discuss the procedure of making a root locus.1. Find out all the roots and poles from the open loop transfer function and then plot them on the complex plane.2. All the root loci starts from the poles where k = 0 and terminates at the zeros where K tends to infinity. The number of branches terminating at infinity equals to the difference between the number of poles & number of zeros of G(s)H(s).3. Find the region of existence of the root loci from the method described above after finding the values of M and N.4. Calculate break away points and break in points if any.5. Plot the asymptotes and centroid point on the complex plane for the root loci by calculating the slope of the asymptotes.6. Now calculate angle of departure and the intersection of root loci with imaginary axis.7. Now determine the value of K by using any one method that I have described above.By following above procedure you can easily draw theroot locus plotfor any open loop transfer function.8. Calculate the gain margin.9. Calculate the phase margin.10. You can easily comment on the stability of the system by using Routh array.

Bode plotswere first introduced by H.W. Bode, when he was working at Bell labs in the United States. Now before I describe what are this plots it is very essential here to discuss a few advantages over other stability criteria. Some of the advantages of this plot are written below:

Advantages of Bode Plot1. It is based on the asymptotic approximation, which provides a simple method to plot the logarithmic magnitude curve.2. The multiplication of various magnitude appears in the transfer function can be treated as an addition, while division can be treated as subtraction as we are using a logarithmic scale.3. With the help of this plot only we can directly comment on the stability of the system without doing any calculations.4. Bode plotsprovides relative stability in terms ofgain marginandphase margin.5. It also covers from low frequency to high frequency range.Now there are various terms related to this plot that we will use frequently in this article.1. Gain Margin:Greater will thegain margingreater will be the stability of the system. It refers to the amount of gain, which can be increased or decreased without making the system unstable. It is usually expressed in dB.2. Phase Margin:Greater will thephase margingreater will be the stability of the system. It refers to the phase which can be increased or decreased without making the system unstable. It is usually expressed in phase.3. Gain Cross Over Frequency:It refers to the frequency at which magnitude curve cuts the zero dB axis in the bode plot.4. Phase Cross Over Frequency:It refers to the frequency at which phase curve cuts the negative times the 180 degree axis in this plot.5. Corner Frequency:The frequency at which the two asymptotes cuts or meet each other is known as break frequency or corner frequency.6. Resonant Frequency:The value of frequency at which the modulus of G (j) has a peak value is known as resonant frequency.7. Factors:Every loop transfer function (i.e. G(s) H(s)) product of various factors like constant term K, Integral factors (j), first order factors ( 1 + jT)( n)where n is an integer, second order or quadratic factors.8. Slope:There is a slope corresponding to each factor and slope for each factor is expressed in the dB per decade.9. Angle:There is an angle corresponding to each factor and angle for each factor is expressed in the degrees.Bode PlotThese are also known as logarithmic plot (because we draw these plots on semi-log papers) and are used for determining the relative stabilities of the given system. Now in order to determine the stability of the system using bode plot we draw two curves, one is for magnitude called magnitude curve another for phase calledBode phase plot.Now there are some results that one should remember in order to plot the Bode curve. These results are written below: Constant term K:This factor has a slope of zero dB per decade. There is no corner frequency corresponding to this constant term. The phase angle associated with this constant term is also zero. Integral factor 1/(j)n:This factor has a slope of -20 n (where n is any integer)dB per decade. There is no corner frequency corresponding to this integral factor. The phase angle associated with this integral factor is -90 n here n is also an integer. First order factor 1/ (1+jT):This factor has a slope of -20 dB per decade. The corner frequency corresponding to this factor is 1/T radian per second. The phase angle associated with this first factor is -tan- 1(T). First order factor (1+jT):This factor has a slope of 20 dB per decade. The corner frequency corresponding to this factor is 1/T radian per second. The phase angle associated with this first factor is tan- 1(T) . Second order or quadratic factor : [{1/(1+(2/)} (j) + {(1/2)} (j)2)]:This factor has a slope of -40 dB per decade. The corner frequency corresponding to this factor is nradian per second. The phase angle associated with this first factor is tan-1{ (2 / n) / (1-( / n)2)} .Keeping all these points in mind we are able to draw the plot for any kind of system. Now let us discuss the procedure of making a bode plot:

1. Substitute the s = j in the open loop transfer function G(s) H(s).2. Find the corresponding corner frequencies and tabulate them.3. Now we are required one semi-log graph chooses a frequency range such that the plot should start with the frequency which is lower than the lowest corner frequency. Mark angular frequencies on the x-axis, mark slopes on the left hand side of the y-axis by marking a zero slope in the middle and on the right hand side mark phase angle by taking -180 degrees in the middle.4. Calculate the gain factor and the type or order of the system.5. Now calculate slope corresponding to each factor.For drawing theMagnitude curve:(a) Mark the corner frequency on the semi log graph paper.(b)Tabulate these factors moving from top to bottom in the given sequence.1. Constant term K.2. Integral factor 1/(j)n.3. First order factor 1/ (1+jT).4. First order factor (1+jT).5. Second order or quadratic factor : [{1/(1+(2/)} (j) + {(1/2)} (j)2)](c) Now sketch the line with the help of corresponding slope of the given factor. Change the slope at every corner frequency by adding the slope of the next factor. You will get magnitude plot.(d) Calculate the gain margin.For drawing theBode phase plot:1. Calculate the phase function adding all the phases of factors.2. Substitute various values to above function in order to find out the phase at different points and plot a curve. You will get a phase curve.3. Calculate the phase margin.Stability Conditions of Bode PlotsStability conditions are given below :1. For Stable System :Both the margins should be positive. Or phase margin should be greater than the gain margin.2. For Marginal Stable System :Both the margins should be zero. Or phase margin should be equal to the gain margin.3. For Unstable System :If any of them is negative. Or phase margin should be less than the gain margin.