Chp3 Non Linear Stability_latest PDF

7/26/2019 Chp3 Non Linear Stability_latest PDF

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NON-LINEAR

SYSTEMS

BASED ON THE INPUT AND OUTPUT

CHARECTARISTICS OF THE SYSTEM

LINEAR SYSTEMS

NON LINEAR SYSTEMS


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LINEAR SYSTEMS The systems which satisfy

Homogeneity Principle

Superposition Principle Y1(t)=A1x1(t)

Y2(t)=A2x2(t)

A1x1(t)+ A2x2(t)=y3(t)

Then if y3(t)= Y1(t)+ Y2(t)

Then the system is said be a linear otherwisenon linear system.

Most of the systems are generally non-linear,but in the range of operation of control systemcan be made to linear systems

For example

Spring, Motors, Damper etc.

These non linear systems are representedin linear differential equations ------------------------------------

Transient , Steady State response andstability

Classical methods- Time Domain andFrequency Domain methods

State space analysis


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Non Linear systems and Classification

The systems which does not satisfy principles of linearity

The non linearities are due to many factors, and arebasically classified as Inherently non linearities or Incidental non linearities

Which present in the component inherently

Intentional Non Linearities To make the systems response in desired form we intensionally

introduce some non linearities in the system The following are some of the non linearities

Saturation

Friction

Backlash

DeadZone or DeadBand

Hysterisis Relays


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Non Linear systems to linear systems

Operating a component in restricted region Examples:

Relays, Amplifiers,Thermionic & Semiconductor devices

Electric Drives: 1500 rpm 1500 +/- 50

Automatic Voltage controllers: +/- 5% of the specified Voltage

Some components linear in working range Example: Spring

Advantages of linearisation

Easy to construct mathematical models Easy to design and analyse by use of transfer function basedtechniques(i.e. Time domain and frequency domain)

The non linearties are small than the results obtained from linearanalysis do not differ from actual results

BEHAVIOR OF NON LINEAR SYSTEMS:

Jump resonance hormonic oscillations

Limit cycles ete


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Jump Resonance In the frequency response of non-

linear systems, the amplitude ofthe response (output) may jumpfrom one point to another fromincreasing or decreasing nature offrequency.This phenomenon iscalled jump resonance

Example: A second order systemwith spring damper

M

K = K 1x + K2x3

The frequency response curves of

a) K= 0, b) K>0 Hard spring c) K


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Sub Hormonic oscillations

If a system is excited with a sinusoidal

function, the output is also a sinusoidal withsame frequency but with a different

amplitude and phase angle.

Depends upon system parameters, initial

conditions,amplitude & frequency of forcingfunction

Linear System

A Sin(wt) B Sin (wt+)

Non-Linear System

A Sin(wt) (AnSin (wt)+Bn cos(wt)

Where n= 0 to


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Limits Cycles

The response of the non-linear systems(o/p)

may exhibit oscillations with fixed frequency andamplitude

These oscillations are called Limit Cycles


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Characteristics of non linear

systems The output response to a particular test signal does not

give any information to other inputs

Highly sensitive to input amplitude

Requires more information in addition to the usual

mathematical model i.e. Amplitude of input Initial conditions etc.

Non-Linear Systems posses Limit Cycles

Jump Resonance Harmonics & Sub Harmonics etc.


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Singular points When the all the phase

variables does not

changes with

respective to time i.e

reaching to a constant

point (may be origin,

equilibrium point

Phase trajectory of a second order system for

different values of and for initial condition x1(0)

=1


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Phase Trajectories The state equation of second order system is

dX1/dt= f1(x1,x2) ---(1)dX2/dt= f2(x1,x2) ----(2)

Where X1, X2state variables

dX2/dX1= f2(x1,x2)/ f1(x1,x2)------- (3)

Equations 3 provides slope of the phase trajectory atevery point in phase plane except at singular points

(Which is indeterminate)


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Phase trajectories can be

drawn Analytically

equation (3) is integrated

Numerical method

trajectory for given set of initial conditions

Graphically

Popularly known method are Isoclines method

Delta Method.


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Construction of phase plane trajectory by

isoclines method

Introduction

From equation (3)

Slope of the phase plant at any point is S = dX2/dX1=

f2(X2,X1)/f1(X2,X1) (4)Let any point slope of the phase trajectory is S1, then

F2(X1,X2) = S1f1(X1,X2)

i.e. locus of all such points in phase-plane at which the scope of

the phase trajectory is S1.


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Note: for each set of initial conditions one phase

trajectory can be drawn

A locus passing through the points of same slopein phase plane is called Isocline


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Procedure to construct phase trajectories by

isocline method.

Step1: Phase trajectory starts at a point

corresponding to initial conditions

Note: For each set of initial conditions, one

phase trajectories can be constructed. Step2

Let S1,S2,S3,etc be the slopes of associated

with isoclines 1,2,3 etc Let 1 = Tan

-1(S1)

2 = Tan-1(S2)

3 = Tan

-1(S3

)


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Note: If a straight line is drawn at an angle from a point,

then the slope of the line at that point is tan

step2. Draw two straight lines from point A, one at a slope ofS1

i.e 1=Tan-1(S1)] and the other at a slope of S2 [i.e at angleof 2=tan-1(S2)]

step3.Let these two lines meet the isoclines 2 atp and q.

Now we can say that the trajectory would cross the isoclines2 at midway between p & q. Mark the point B. on theisocline-2 approximately midway between p and q


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step4. The constructional procedure is now

repeated at B to find the crossing point C onthe isocline-3.

By similar procedure the crossing points onthe isoclines are determined

A smooth curve drawn through the crossingpoints on the isoclines are determined

A smoothe curve drawn through the crossingpoints gives the phase trajectory starting at

point A


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Note 2 The accuracy of the trajectory is closely related to the

spacing of the isoclines the phase trajectory will be more

accurate if large number of isoclines are used which are very

close to each other. It should be noted that using a set of

isoclines, any number of isoclines can be constructed.


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(1,-1.5)

(2,-2.5)(.75,-2.5)

(.25,-.8)

(1,-.63)(2,-.86)

(2,1.2)(1,.6)

(1,1.4)

(1.5,2.1)

(.5,2.5)

(.25,1.25)

(1,-.77)

X1

X2

S1=2

S2=1.5

S3=.5S4=0

S5=-2

S6=-1S7=0.5


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Phase Portrait of example


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Constr ction of phase trajectories b


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Construction of phase trajectories by

delta method Consider a second order non-liner or linear system represented

by the equation d2X/dt2+f(X, dX/dt,t) = 0 (1)

The equation (1) can be converted to the form shown below

d2X/dt2+wn2(X+(X, dX /dt,t) = 0

In the above equation is a function of x,

dX/dt ,t but for short intervals, the change in phase variables arenegligible. Hence for a short interval, is considered as aconstant.

d2X/dt2+wn2(X+)= 0 ----(3)

Let us choose as the state variables as

X1=x,x2= dx/dt /wn---(4)

from above equation Dx/dt=wnx2= dx1/dt ----(5)

d2X/dt2 = wn dx2/dt ---(6)

Substituting equation 4 & 5 in 3

dx2/dt = -wn(X1+ )

So the state equations are dx1/dt=wnx2,dx2/dt= -wn(X1+ )


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From the above equations, the slopeequation over short interval can be written as

dx2/dx1= -(x1+)/x2---(8) using above slope equation, a short segment

of the trajectory can the drawn from theknowledge of at any point on the trajectory,

as explained below Procedire:

Let point A be a point on phase trajectorywith co-ordinates(X1,X2) as shown in fig

below (usually the point A will be thestarting point of the trajectory obtained fromthe initial conditions)


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X1

X2

A(1,0)

X2 =4|x2|x2 Center

0(A) 0 C1

-.1875(B) -.140625 C2

-.325(C) -.4225 C3

-.375(D) -.5625 C4

-.4375(E) .765625 C5

..... ...... ......

..... ...... .

.. . .

B(..,-.1875)

C2 C3C1

C(..,-.325)

C4

C5D(..,-.5625)


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