Control Systems

Prepared By

Mr.S.Arun

Unit – 1�CONTROL SYSTEM MODELING

SYSTEM

• Define System

A number of components or elements are connected in a sequence to perform a specific function, the group is called as a system

• Define Control system

In a system when the output quantity is controlled by varying the input quantity then the system is called control system.

• Types of System

Two Types of System

• Open Loop System
• Closed Loop System

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Open Loop System

• A system in which the output has no effect on the control action is known as an open loop control system

Closed Loop System

• Control system in which the output has an effect upon the input quantity in order to maintain the desired output value.
• Also called automatic control system

Examples of Control system

Mathematical Model of Control System

• The mathematical model of a control system consists a set of differential equations.
• Linear System, non Linear system, Time variant system, Time invariant system
• The Linear time invariant system can be for analysis using transfer function approach.

Define transfer function

The Transfer function of a system is defined as the ratio of the Laplace transform of output to Laplace transform of input with zero initial conditions.

Types of System in modelling

• Mechanical System
• Mechanical Translational System
• Using Three basic elements and they are ---- Mass(M), Spring(K), Dashpot(B)
• Mechanical Rotational System
• Using Three basic elements and they are ---- Moment of inertia(J), Torsional Spring (K), Dashpot(B)
• Electrical System

Force Balanced equation for Mechanical Translational System

Mass

Mass is the property of a body, which stores kinetic energy. If a force is applied on a body having mass M, then it is opposed by an opposing force due to mass. This opposing force is proportional to the acceleration of the body. Assume elasticity and friction are negligible.

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Where,

F is the applied force

F_m is the opposing force due to mass

M is mass

a is acceleration

x is displacement

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Spring

Spring is an element, which stores potential energy. If a force is applied on spring K, then it is opposed by an opposing force due to elasticity of spring. This opposing force is proportional to the displacement of the spring. Assume mass and friction are negligible.

Where,

F is the applied force, F_k is the opposing force due to elasticity of spring

K is spring constant, x is displacement

With Reference

Without Reference

Dashpot

If a force is applied on dashpot B, then it is opposed by an opposing force due to friction of the dashpot. This opposing force is proportional to the velocity of the body. Assume mass and elasticity are negligible.

Where,

F_b is the opposing force due to friction of dashpot

B is the frictional coefficient

v is velocity

x is displacement

With reference

Without reference

Guide line to determine Transfer function

• The differential equation governing the system are obtained by writing force balance equation of nodes(Mass)
• The linear displacement of masses are assumed by X1, X2, X3….etc
• Draw the free body diagram
• Each mass separately marking
• Opposing force acts in a opposite direction to applied force
• Mass move in applied force, So all elements move in same direction, mark opposing force in opposite direction
• If force not applied in the mass, V,X,a are opposite direction to mass
• Arrange differential equation, equating applied force to opposing force
• Take Laplace transform of D.E

Problem 1

Torque Balanced equation for Mechanical Rotational System

Moment of Inertia

Dashpot

With Reference

Without Reference

Torsional Spring

With Reference

Without Reference

Problem 4

Problem 5

Mathematical Modelling of Electrical Systems

• Most of the electrical systems can be modelled by three basic elements: Resistor, inductor, and capacitor. Circuits consisting of these three elements are analysed by using Kirchhoff’s Voltage law and Current law.

Problem 6

Electrical Analogous of Mechanical Translational Systems

• Two systems are said to be analogous to each other if the following two conditions are satisfied.
• The two systems are physically different
• Differential equation modelling of these two systems are same
• There are two types of electrical analogous of translational mechanical systems. They are
• Force voltage analogy.
• Force current analogy.

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Force Voltage Analogous and Current analogous

Mechanical S/m Force Voltage Force Current

Electrical Analogous of Mechanical Rotational Systems

• Torque-voltage analogy
• The mathematical equations of the rotational mechanical system are compared to the mesh equations of the electrical system.
• Torque current analogy
• The mathematical equations of the rotational mechanical system are compared to the nodal mesh equations of the electrical system.

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Torque Voltage Analogous and Current analogous

Mechanical S/m Torque Voltage Torque Current

Problem 7

Apply changes in equation 3 and 4

Problem 8

Problem 9

BLOCK DIAGRAM REDUCTION

G1

G2

H

G3

G1

G2

H

G3

1/G1

Problem 10

G3/G1

1/(1+G1H)

Problem 11

Problem 12

Problem 13

Problem 14

Signal Flow Graph

• Signal flow graph is a graphical representation of algebraic equations. In this chapter, let us discuss the basic concepts related signal flow graph and also learn how to draw signal flow graphs.
• Basic Elements of Signal Flow Graph
• Nodes and branches are the basic elements of signal flow graph.

Node

• Node is a point which represents either a variable or a signal. There are three types of nodes — input node, output node and mixed node.
• Input Node − It is a node, which has only outgoing branches.
• Output Node − It is a node, which has only incoming branches.
• Mixed Node − It is a node, which has both incoming and outgoing branches.

Branch

• Branch is a line segment which joins two nodes. It has both gain and direction. For example, there are four branches in the above signal flow graph.

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Problem 15

Problem 16

Problem 17

Problem 18

BLOCK DIAGRAM TO SIGNAL FLOW GRAPH METHOD

Problem 19

Problem 20

Transfer function of Armature controlled DC Motor

Transfer function of Field controlled DC Motor

Time Response of Control System

• Time response of the system is defined as the response of the system achieved on providing certain excitation, where the excitation and response must be a function of time.
• Transient response: It is defined as the response of the system as the variation in output of the system before achieving the final value when excited with the input signal.
• Steady-state response: The actual value of the time response, that is achieved after the elimination of the transient response is known as a steady-state response

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Standard Test Signals

Type of a control system

• Type of the system can be defined as the number of poles located exactly at s=0. 

Order of a control system

• The order of the system is given by the order of the differential equation governing the system. It is also given by the maximum power of s in the denominator polynomial of transfer function. The maximum power of s also gives the number of poles of the system and so the order of the system is also given by number of poles of the transfer function.

Find out the type and order for given transfer function

Response of first order system for unit impulse input

Step Response of First Order System

Response of Second Order System

Response of Undamped Second Order System for unit step input

Response of Underdamped Second Order System for unit step input

Response of Critically damped Second Order System for unit step input

Time Domain Specification

• The time domain specifications are
• Delay time (td)
• Rise time (tr)
• Peak time (tp)
• Maximum peakovershoot (Mp)
• Settling time (ts)

• Delay time (t_d)
• The time required to reach at 50% of its final value by a time response signal during its first cycle of oscillation.
• Rise time (t_r)
• The time required to reach at final value by a under damped time response signal during its first cycle of oscillation. If the signal is over damped, then rise time is counted as the time required by the response to rise from 10% to 90% of its final value.
• Peak time (t_p)
• The time required by response to reach its first peak i.e. the peak of first cycle of oscillation, or first overshoot.
• Maximum peak overshoot (M_p)
• difference between the magnitude of the highest peak of time response and magnitude of its steady state. Maximum overshoot is expressed in term of percentage of steady-state value of the response.

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• Settling time (t_s)
• the time required for a response to become steady. It is defined as the time required by the response to reach and steady within specified range of 2 % to 5 % of its final value.
• Steady-state error (e _ss )
• the difference between actual output and desired output at the infinite range of time.

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Expression for time domain specifications

Problem 21

Problem 22

Problem 23

Problem 24

Steady State Error

Problem 25

Problem 26

Problem 27

Problem 28

-1

-2

-3

jω

2jω

3jω

-jω

-2jω

-3jω

Problem 29

-1

-2

-3

jω

2jω

3jω

-jω

-2jω

-3jω

-4

0

l1 =1.35 cm. l2= 1.8 cm. l3=3.5 cm

l1

l2

l3

K = 1.35 * 1.8 * 3.5 / 1 =

Problem 31

-1

-2

-3

jω

2jω

3jω

-jω

-2jω

-3jω

-4

Automatic Controller

• A device which compares the actual value of plant output with the desired value is called as an Automatic Controller. It determines the deviation of the system and produces the control signal that reduces the deviation to 0 and small value. The manner in which the automatic controller produces the control signal is called the control action.

Industrial controller are classified on the basis of control action as

• ON - OFF controller
• Proportional controller (P)
• Integral controller (I)
• Proportional + Integral controller (PI)
• Proportional +Derivative Controller (PD)
• Proportional +Integral + Derivative Controller (PID)

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Proportional Control Action

• In a controller, with proportional control action, there is a continuous relationship between the output of the controller (M) (Manipulated Variable) and Actuating Error Signal E (deviation)

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Integral Control Action

• In a controller, with integral control action, the output of the controller is changed at a rate which is proportional to the actuating error signal. E (t)

Derivative Control action

• In controller with derivative control action the output of the controller depends on the rate of change of the e(t)

P,PI,PID Controller

• P-I Controller:
• P-I controller is mainly used to eliminate the steady state error resulting from P controller. However, in terms of the speed of the response and overall stability of the system, it has a negative impact. This controller is mostly used in areas where speed of the system is not an issue. Since P-I controller has no ability to predict the future errors of the system it cannot decrease the rise time and eliminate the oscillations. If applied, any amount of I guarantees set point overshoot.

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P-D Controller:

• The P-D controller is to increase the stability of the system by improving control since it has an ability to predict the future error of the system response. In order to avoid effects of the sudden change in the value of the error signal, the derivative is taken from the output response of the system variable instead of the error signal. Therefore, D mode is designed to be proportional to the change of the output variable to prevent the sudden changes occurring in the control output resulting from sudden changes in the error signal. In addition D directly amplifies process noise therefore D-only control is not used.

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P-I-D Controller

• P-I-D controller has the optimum control dynamics including zero steady state error, fast response (short rise time), no oscillations and higher stability. The necessity of using a derivative gain component in addition to the PI controller is to eliminate the overshoot and the oscillations occurring in the output response of the system. One of the main advantages of the P-I-D controller is that it can be used with higher order processes including more than single energy storage.

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UNIT-III: FREQUENCY DOMAIN ANALYSIS

• Relationship between time and frequency response, bode plots, polar plots, nyquist plot – Gain and phase margin. Closed-loop frequency response.

Frequency response analysis

Advantages of Frequency Response Analysis

Frequency Domain Specifications

• Resonant Peak(μr)
• The maximum value of the magnitude of closed loop transfer function is called resonant peak.

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Resonant Frequency(ωr)

• The frequency at which the resonant peak occurs is called resonant frequency. The resonant peak is the maximum value of the magnitude of closed loop transfer function

• The slope of the log-magnitude curve near the cut-off frequency is called cut-off rate.

cut-off rate

Gain Margin Kg

Phase Margin (γ)

Frequency Domain Specifications

Frequency response plots

• Open Loop System
• Bode plot
• Polar plot
• Nichols plot
• Closed Loop System
• M and N circles
• Nichols chart

Bode Plot

• The bode plot is a frequency response plot of the transfer function of a system. It consists of two plots-magnitude plot and phase plot. The magnitude plot is a graph between magnitude of a system transfer function in db and the frequency (ω) .The phase plot is a graph between the phase or arguments of a system transfer function in degrees and the frequency (ω). Usually, both the plots are plotted on a common x-axis in which the frequencies are expressed in logarithmic scale.

The advantages are of bode plot.

• The magnitudes are expressed in db and so a simple procedure is to add magnitude of each term one by one.
• The approximate bode plot can be quickly sketched, and the c: can be made at corner frequencies to get the exact plot.
• The frequency domain specifications can be easily determined.
• The bode plot can be used to analyze both open loop and close loop system

Polar plot

• The polar plot of a sinusoidal transfer function G(jω) is a plot of the magnitude of G(jω) versus the phase angle/argument of G(jω) on polar or rectangular coordinates as ω is varied from zero to infinity