presentation for Controller

7/27/2019 presentation for Controller

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Response of a Control System, c(t):

Consider block diagram of a simple closed-loop control systemwhich is given as:

Here, C(s) = response of the control system in space domain

c(t) = response of the control system in time domain.

c(t) = ct(t) + css(t)

Now,


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ct(t)= transient response. It remains for a short

time.

css(t) = steady state response. It remains as time,t

approaches infinity.

Where,


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Test Signals

(1) Unit Step Signal: It is defined as:

r(t)=1 for t> 0

R(s)=

s

1

Taking the Laplace

transformation w

e

have


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(2) Unit Ramp Signal: It is defined as:

r(t)=t for t> 0

Taking the Laplacetransformation we

have

R(s)= 2s

1


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(3) Unit Parabolic Signal: It is defined as:

r(t)= for t> 0

Taking the Laplacetransformation we

have

R(s)= 3s

1

2

t2


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FIRST- ORDER SYSTEM:

Physically a first order system may represent an RC circuit,

BK arrangement, thermal system and so on.

RCs1

R)s(G

Lets take an example of any transfer function which

is given by:

RCs1

R

)s(R

)s(C,OR


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UNIT STEP RESPONSE OF FIRST- ORDER

SYSTEM:

APPLYING UNIT STEP INPUT TO THE ABOVE FIRST ORDER SYSTEM WE HAVE

Putting in the above transfer function we haveR(s)=

s

1

C(s) = R(s)G(s)

Or, C(s)=

RCs1R

s

1)s(G)s(R

The above value of C(s) can be written as:

RCs1

CsR

s

R)s(C

2


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STEADY STATE ERROR(ess):

It is a measure of the accuracy of the control

system.

Any physical control system inherently suffers

steady state error in response to certain types ofinputs.

A system may have no steady state error to a

step input but the same system may exhibit non-

zero steady state error to a ramp input.


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GENERALLY STEADY STATE ERROR SHOULD BE AS LOW

AS POSSIBLE.

Steady state error depends upon:

(1)Type of the input: That is input is Step, Ramp, Parabolic etc.

(2)Type of the system: Type zero, Type one, Type two etc.

(3)Non-linearities of system components: Static friction, Backlash,

etc.

Steady State Error (ess) is mathematically given by:


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State space representation is a mathematical model of a physical system as

a set of input, output and state variables related by first-order differential

equations.

To abstract from the number of inputs, outputs and states, the variables are

expressed as vectors.

"State space" refers to the space whose axes are the state variables.If the dynamic system is linear and time invariant, the differential and

algebraic equations may be written in matrix form. The state space

representation (also known as the "time-domain approach") provides a

convenient and compact way to model and analyze systems with multiple

inputs and outputs.The concept of the state of a dynamic system refers to a minimum set of

variables, known as state variables, that fully describe the system and its

response to any given set of inputs

State variables and State space

representation


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The set of n equations define the derivatives

of the state variables to be a weighted sum of

the state variables and the system inputs.


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Transient Response Specifications

The desired performance characteristics of control systems arespecified in terms of time-domain quantities.

Systems with energy storage cannot respond instantaneously and

will exhibit transient responses whenever they are subjected toinputs or disturbances.

Frequently, the performance characteristics of a control system are

specified in terms of the transient response to a unit -step input

since it is easy to generate and is sufficiently drastic.


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Following specifications are commonly used to specify the transient

response characteristics of a control system to a unit step input:

(i) Delay time, td : It is the time required for the response to reach

half the final value(desired value) the very first

time.

(ii) Rise time, tr: It is the time required for the response to reach

from 0% to 100% of the final value(desired value)

for underdamped system.

(iii) Peak time, tp : It is the time required for the response to reach

the first peak of the overshoot.


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(v) Settling time, ts : It is the time required for the response curve toreach and stay within a range usually 2% or 5%

of the final or desired value.


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Automatic Controllers

An automatic controller compares the actual value of the plant

output with the reference input (desired value), determines the

deviation, and produces a control signal that will reduce the

deviation to zero or to a small value.


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The controller detects the actuating error signal, which is usually ata very low power level, and amplifies it to a sufficiently high level.

The output of an automatic controller is fed to an actuator.

The actuator is a power device such as an electric motor, a

hydraulic motor, or a pneumatic motor or valve that produces the

input to the plant according to the control signal.


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Based on the control action(1) Two-position or on-off controllers

(2) Proportional controllers

(3) Integral controllers(4) Proportional-plus-integral controllers

(5) Proportional-plus-derivative controllers

(6) Proportional-plus-integral-plus-derivative

controllers(PID controllers)

Types of Controllers:


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Based on the type of Power employed

Types of Controllers:

(1)Pneumatic controllersemploys pressurised

gas or air

(2) Hydraulic controllersemploys pressurised liquid

(3) Electronic controllers-employs electricity,

and so on


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(1) Two-position or on-off controllers:

eg. Electrical devices

u(t) = U1 for e(t) >0

= U2 for e(t)


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(2) Proportional Controllers: essentially an

amplifier with adjustable gainEg. Fan regulator, brake, flush tank

u(t) = Kp e(t)

Or, in Laplace transformed quantities,

Where Kp is termed the proportional gain.

Let, e(t) be the actuating error signal and u(t) be the output signal from the

controller also called control signal.

pK)s(E

)s(U


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(3) Integral Controllers:


Where Ki is an adjustable constant

Let, e(t) be the actuating error signal and u(t) be the output signal from the

controller also called control signal.

t0i dt)t(eK)t(u

s

K

)s(E

)s(Ui


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(4) Proportional plus Integral Controllers: Its control action is

defined by:


Where Ti is called the integral time

t0i

p

pdt)t(e

T

K)t(eK)t(u

sT

11K

)s(E

)s(U

i

p


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(5) Proportional plus Derivative Controllers: Its control action is

defined by:


Where Td is called the derivative time

dt

)t(deTK)t(eK)t(udpp

sT1K)s(E

)s(Udp


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(6) Proportional plus Integral plus Derivative Controllers: Its

control action is defined by:


Where Kp is the proportional gain, Ti is called the integral time

and Td is called the derivative time.

dt

)t(deTKdt)t(e

T

K)t(eK)t(u

dp

t

0i

p

p

sTsT

11K

)s(E

)s(Ud

i

p


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The output of a PID controller, equal to the control input to the plant, in the time-

domain is as follows:


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Suppose we have a simple mass, spring, and damper problem.

The modeling equation of this system is:

FkxxbxM


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kbsMssF

sX

2

1

)(

)(

Its transfer function is given by:

kbsMs

2

1 )(sX)(sF

Its block diagram representation is given by:


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Let,M = 1 kg, b = 10 N s/m, k = 20 N/mPlugging these values into the above transfer function,

We have:


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If we use the unit step input,F(t) = 1 N; Or, F(s) = 1/s

then,

)2010(

11)(

2

ssssX

% MATLAB COMMAND TO OBTAIN RESPONSE

s = tf('s');

P = 1/(s^2 + 10*s + 20);

step(P)

If we use the following MATLAB command we get the response as shown in fig. on

next slide:


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0 0.5 1 1.5 2 2.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05Step Response

Time (sec)

Amplitude

Open-Loop Step Response


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Block Diagram of an Automatic closed loopcontrol system

CONTROLLER

(thermostat)PLANT

(air conditioner)

INPUT

(desired

temperature)

OUTPUT

(temperature)

ERROR

DETECTOR(processor)

FEEDBACK ELEMENT

(thermometer)

Signal to

thermostat

Temperature Control System of a Room


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EFFECT OF PROPORTIONAL CONTROLLER:

The closed-loop transfer function of the above system with a proportional controller

is:


s = tf('s');

P = 300/(s^2 + 10*s +320);

step(P)


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The plot shows that

the proportional

controller:

reduced both the

rise time and thesteady-state error,

increased the

overshoot, and

decreased thesettling time by small

amount.0 0.2 0.4 0.6 0.8 1 1.2 1.4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (sec)

Amplitude

Closed-Loop Step Response with

Proportional Controller(Kp):


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EFFECT OF PROPORTIONAL DERIVATIVE (PD) CONTROLLER:


%Assume Kp = 300 and Kd = 10;

s = tf('s');

P = (300+10*s)/(s^2 + 20*s +320);

step(P)


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Closed-Loop Step Response with

Proportional Derivative(PD)Controller(Kp+sKd):

The plot shows that the

proportional derivative

controller:

reduced both theovershoot and the

settling time,

small effect on rise

time and steady state

error.

0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4Step Response

Time (sec)

Amplitude


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EFFECT OF PROPORTIONAL CUM INTEGRAL (PI) CONTROLLER:


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CLOSED

LOOP

RESPONSE

RISE TIME OVERSHOOTSETTLING

TIME

STEADY STATE

ERROR

Kp Decrease Increase SmallChangeDecrease

Ki Decrease Increase Increase Eliminate

Kd Small Change Decrease Decrease No Change

The effects of each of controller parameters, Kp , Kd and Ki on a closed-loop system

are summarized in the table below.

EFFECTS OF EACH OF CONTROLLER PARAMETERS:


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The goal of this problem is to show you how each ofKp , Kd and Ki

contributes to obtain

Fast rise time

Minimum overshoot

No steady-state error


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Plot of PV vs time, for three values of Kp (Ki and Kd held constant)


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Plot of PV vs time, for three values of Ki

(Kp

and Kd

held constant)


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Plot of PV vs time, for three values of K (K and K held constant)

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