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Topic includes
- Introduction to the automatic control
- Basic elements of control system
- Open and closed loop with advantage and disadvantage with examples
- Concept of feedback
- Dead time
- Laplace transform
- It's properties and inverse Laplace
- Transfer function with procedure
- Poles , zeroes and characteristic equation
- Transfer function of mechanical system
- Time response with unit ramp and step
- Stability of Hurwitz criteria and Routh criteria
- Root locus with procedure
- Bode plot, gain and phase margin
- Important questions
Introduction to the automatic control
Control system is very important in the field of engineering as it finds wide application not only in engineering but also in other fields .
The first automatic control system is a fly ball governer to control speed of a steam engine.
If engine speed drops below desired value, the centrifugal force of the fly ball decreases.
Basic elements of control system
A control system consists of several elements where each element performs a particular task. These elements are connected in a proper sequence and facilitates signal to flow through them.
Elements of control system depends on the type of control system. Main types are open loop control system and closed loop control system.
Open loop control system
It has no feedback element. Hence, control action is independent of output.
The basic elements of an open loop control system are the controller and the controlled process. The controller may be amplifier, filter circuits etc.
Input is applied to the controller where some control action are performed and a signal is obtained from the controller. The controller is connected to a controlled process which is a regulating system. The direction of flow of signal is unidirectional. Input signal flows in forward direction from the controller to the controlled process and finally output signal is obtained.
Examples are
- A washing machine
Here time of washing is predefined and the machine stop after set time.
- Field control and armature control of DC machine .
Advantages
- Construction is simple in nature
- Cost of the system is less.
- It is generally stable.
Disadvantages
- It is slow and less reliable..
- It has less accuracy
Closed Loop System
A closed loop system is one in which the output has an effect on controller and controlled process through a feedback element. Thus, this system is called feedback control system.
its elements are :
Feedback Element
It is a device which convert output signal to another suitable variable feedback signal.
This feedback signal is compared with the input signal in the error detector of controller.
Controller
It consists of two elements - error detector and control element .
The error detector compares the feedback signal and input signal. Thus an error signal is obtained from the error detector. Error signal is the difference of input signal and feedback signal.
Error signal is fed to the control elements to produce control. The control system consists of amplifier and a power stage error signal is usually at low lower power stage.
Controlled process
The plant or process that produces the desired output is called controlled process. Control signal obtained from the controller is fed to the controlled process to produce desired output.Example A room AC which regulates both temperature and humidity of a room at a particular level.
Advantages
- Performance of closed loop control system is less affected by non linear elements.- It is more reliable.
- It is faster.
Disadvantages
- Closed loop system is complex in contruction.- Maintenance is difficult.
- It is unstable under certain Conditions.
Concept of feedback
http://rahulsinghmar.blogspot.com/?m=1
For example: Lets eliminate a route of temperature control of a down-to-earth cistern having irrigate in it.
Dead time
Dead time is the delay from when a controller output (CO) signal is issued until when the measured process variable (PV) first begins to respond. The presence of dead time,Өp, is never a good thing in a control loop.
Laplace transform
It is a mathematical transformation which transform a time variable function into a complex variable function .
The left side of equation read as Laplace transform and the right side read as function of s.
Properties of Laplace transform
1. Linearity
2. Time delay
3. Initial value theorem
4. Final value theorem
5. Laplace transform of differentiation
6. Laplace transform of integration
Inverse Laplace transform
The time domain function can be obtained from it's Laplace transform by a process called inverse Laplace transform.
Importance of Laplace transform
- It transforms time domain integral and differential equations into algebraic equation in s domain .
- Characteristics equation of a system is an algebraic equation.
- The solution in time domain can be obtained by applying inverse laplace transform.
- Initial and final value theorem can be easily determined.
Application of Laplace transform
A control system may be Electri, mechanical , hydraulic , thermal system etc. These systems are represented by corresponding differential equations according to the laws of the system.
Laplace transform of differential equation gives an algebraic equation of s.
Time domain solution is obtained by applying inverse Laplace transform.
Laplace transform of differential equation gives an algebraic equation of s.
Time domain solution is obtained by applying inverse Laplace transform.
Transfer function
The transfer function of a control system is defined as the ratio of Laplace transform of output to the Laplace transform of input taking all initial conditions as zero.
Cases
If m=n then the transfer function is proper
If m >n then the transfer function is improper
If m< n then the transfer function is strictly proper.
Properties of transfer function
- It is valid only for linear time invariant system.
- All initial conditions are taken as zero while obtaining transfer function.
- Transfer function doesn't depend on input of the system.
- It cannot be defined non - linear system.
Procedure of transfer function
- Write the differential equations for the given system.
- Find the Laplace transform of the differential equation assuming zero initial conditions.
- Find the ratio of Laplace transform of output to Laplace transform of input.
- This ratio is transfer function of the system.
Poles of transfer function
It is defined as the value of s which when substituted in the denominator of transfer function, it makes the value of transfer function infinity.
Number of poles =P=m
Zeroes of transfer function
It is defined as the value of s which when substituted in the numerator of transfer function zero.
If z<p then number of zero at infinity P-z
If z> p then number of zero at infinity z-P
Characteristics equation
It is defined as the equation obtained by equating denominator of transfer function to zero value.
Transfer function of mechanical system
There are two types of linear mechanical system namely, translational and rotational . To find transfer function of mechanical system, differential equations governing the system are to be deducted by using Newton's second law of motion and D' Alembert's principle.
Translation system
The motion of a linear translational system takes place in a straight line. There are three types of forces which resits the motion .
These are inertia force , damping force, and spring force .
Rotational system
The rotational motion of a body is due to the rotation of the body about a fixed axis. There are three torques which resist rotational motion.
These are inertia torque, damping torque and spring torque.
D' Alembert's principle
This Principle is used to write the equations of motion of a mechanical system.
It states that for any body the algebraic sum of externally applied forces and resisting forces in any given direction is zero.
A reference direction is chosen. The forces in this direction are taken as positive and in opposite direction are taken as negative.
Laplace transform of unit step and unit ramp function.
Time response
It is defined a the variation of output with respect of time when an input signal is applied to the system. It is a plot of output versus time.
Time response is of two types transient response and steady state.
Transient response
It is for very short time duration, identified in terms of time constant of the system.
Steady state response
It is a state of output when time approaches infinity .
Order of system
It is the highest power of s in the denominator of system transfer system like first order , second order etc.
Derive input functions like parabolic ...
Important exam point of view
Stability of control system
Stable system
If all the roots of characteristic equation lie on the left side of s plane or second and third quadrant then the bounded input, the response is bounded and the system is stable.
The roots of characteristic equations are the pole of closed loop transfer function.
Unstable system
If the root of characteristic equation lies on the right half of s plane, then the system response is in bounded for a bounded input and the system is unstable.
So a system is unstable if any poles of closed loop transfer function lies on right half of s plane .
Marginally stable system
If roots of characteristic equation lie on left half of s plane except one or more repeated roots on imaginary axis and then the system is marginally stable.
Necessary conditions for stability
- There should be no missing coefficient otherwise system is unstable.
- All the coefficient are positive otherwise the system is unstable .
A Hurwitz and E.J. Rough have suggested different method for investigating sufficient condition for stability of a system. These are called Hurwitz criteria and Routh criteria respectively.
The root locus technique provides the information about the location of roots of characteristic equation graphically.
It was introduced in 1948.
Root locus of a control system is a plot of the roots of the characteristic equation when gain k is varied from zero to infinity.
Concept important for exam
Rules and procedure
Bode plot
Stability analysis of control system can be made by time domain and frequency domain approach. Routh Hurwitz and root locus technique are time domain approach.
Bode plot is a graphical plot to determine stability of a system by frequency domain approach .
It is powerful method for stability analysis.
Concept of bode plot
It is graphical representation of sinusoidal open loop transfer function to determine the stability of a control system. The sinusoidal open loop transfer function is obtained by replacing s by jw.
Procedure for drawing bode plot
- Bode plot is drawn in semi log paper with frequency w on log.
- Identify the type of system from which initial slope of the bode plot is obtained.
- Identify the corner frequencies due to open loop or zero.
- Draw a line with initial slope upto first corner frequency.
- Draw a line from first to second corner frequency by adding the slope due to next pole or zero.
- Process is continued. This gives magnitude plot of bode plot.
- Calculate phase angle for different frequency and draw phase plot of bode plot.
Nyquist stability criteria
It was developed by H. Nyquist in 1932 to determine the stability of closed loop system.
Important question for Board
- open loop and closed loop with examples
- concept of feedback
- Laplace transform with unit step and ramp fun.
- properties of Laplace transform, basic numerical
- inverse Laplace with applications
- poles, zeroes concept with numerical
- transfer function of mechanical system
- time response and order of system and damping
- stability analysis by Routh Hurwitz criteria
- root locus concept a, rules and procedure
- bode plot concept , gain and phase margin with procedure and stability system
- Nyquist stability concept and procedure of polar plot
Very good notes for exam preparation 👍
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