control systems--the last basic course, pt iii

8/3/2019 Control Systems--The Last Basic Course, Pt III

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The Stability of Linear Feedback Systems

The Concept of Stability

Absolute Stability is when it can be said with certainty that the closed loop system in

consideration is either stable or unstable.Relative stability refers to addressing a system withdegree of stability. For example, a fighter plane is lesser stable than a passenger plane which

makes it easily manoeuverable.

A stable system is that dynamic system which always gives a bounded response to a

bounded input.

Poles in left half plane give decreasing response for the disturbance inputs; poles on

imaginary axis gives neutral response and when in the right half plane, give increasing

response.

Loud speaker systems for public addressing are an example of positive feedback system.Speaker's distance from microphone affects the sensed amplitude of useful signal. When the

speaker is too close to it, the amplitue sensed it high and following amplification produces a

distortion of signal and an oscillatory squeal due to excessive amplification.

'A'(not 'the',meaning there can be more) necessary and sufficient condition for a

feedback control system to be stable is all poles of transfer function having a negative

real part.

The system is not stable when all the roots are in left-hand plane.

When the simplepoles are on imaginary axis, the steady state response for bounded

input is a sustained oscillation; a sinusoidal input(which is consideredbounded) of the

frequency equal to the frequency of these simple poles(the value on imaginary axis is

some j which implicitly correspond to a frequency) makes the response unbounded.Hence, such a system is marginally stablebecause only a few inputs make it unstable.

An unstable system will have one of more poles(i.e. Roots of the characteristic

equation) in the right half plane or repeated j roots. Such a system will give unbounded

response to just any bounded input.

Is our system stable? Finding the roots to characteristic equation gives information more

than required. There are three established methods to answer this just about enough:1) s-

plane approach; 2) frequency(j) plane approach; 3) time domain approach.

The Routh-Hurwitz Stability Criteria(late 1800s; was published independently by both)

The basis is characteristic equation, which is used to only indicate whether there are rootsin right half s-plane.

The criteria is based upon the necessary condition of all coeffients of characteristic equation


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q(s) being of same sign. When q(s) is written in factored form,

the equality of signs of coefficients is very rough approximation for the algebraic

manipulations might be generating same sign, yet keeping some root on right half s-plane.

Although, why all coefficients must have same sign must become clear the moment one

makes all ri -ve.

Now the tabulation of coefficients is performed in the following way


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The criteria is that the number of roots of q(s) with their real parts lying in right half s-

plane is equal to the number of times the sign changes in first column of table.

Four distinct cases or configurations of 1st column which need to be taken care of:

case 1No element in 1st column is zero: this is independent of the order of q(s). In

this case, no special manipulations are required.

Case 2There is a zero in the first column, but some other elements of the row

containing the zero in the first column are nonzero: the zero element is replaced witha small number which is allowed to tend to zero then.

Case 3There is a zero in the first column, and the other elements of the row con-

taining the zero are also zero: this happens when the system has singularities

symmetrically located around the origin; factors like (s j)(s + j) would be there. In

this case auxillary fucntion U(s) is used which is the polynomial comprising the table

entries just the zero having row. The order of U(s) is always even and it indicates

symmetrical root pairs.

Case 4Repeated roots on imaginary(j) axis: system is neither stable nor unstable.

Routh-Hurwitz criteria does not display this unstability.

The Relative Stability of Feedback Control Systems

It is necessary to measure relative damping of each rootof the characteristic equation.

Relative stability of a system can be defined as a property which is measured by the relative

real parts of roots or a pair of roots.

Routh-Hurwitz can be used in this analysis too by changing the variable which involves

shifting the vertical axis.

The Stability of State Variable Systems

This, again, is all about analyzing the characteristic equation (s).

The solution to an unforced system is exponential function.

Eigen values or characteristic roots are simply roots of the characteristic equation.

The nth order equation in resulting from the determinant manipulations is the

charactersitic equation.


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The Root Locus Method(introduced in 1948)

Relative stability and transient response are directly related with position of the

characteristic roots in s-plane.

Oftentimes, location of the roots are needed to be set when some parameter changes. So, it is

worthwhile to know the locus of the roots of characteristic equation in s-plane whensome parameters are varied.

Root locus is path of roots of characteristic equation traced out in s-plane when a

system parameter is varied from zero to infinity.

Closed loop roots and Open loop poles: first of all, these are just ways to address twoimportant things which come out of same characteristic equation. When roots of complete

characteristic equation are considered(i.e with constant offset), it is closed loop root; whenpoles of only the KG(s) part are considered, it is open loop poles because we are acting as if

we ignored the feedback loop.

Analysis of a second order system:

values possible for s are s1 ans s2.

Values of are for the angle the vector from open loop pole at real axis to the position of

closed loop root(s1, s2 or complements) makes.

Equation 7.9 is simply mathematical summation of angles formed as shown below in thediagram.

Locus is always ofclosed loop roots.


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Actually, the form of the locus cannot be ascertained just like that; a testpoint in s-

plane(YES, THERE ARE INFINITE POINTS IN S-PLANE WHICH REQUIRES US TO

VERIFY WHETHER THE POINT ISA ROOT AT OUT LOCUS) by the equations of the

type of eq 7.9. This angle criteria only gives the locus.

The Root Locus Procedure(psychology of motion of roots as we alter/vary some parameter)

Consider equation 7.25 below. Following results are generated

When K = 0, all the roots of characteristic equation are poles

when K tends to infinity, all the roots are zeros. If we divide the complete eq by K and

then tend it to infinity, (s pi) part is nullified and only zeros are to be found.

It can be concluded, hence, that locus of roots of characteristic eq 1 + KP(s) = 0

starts from the poles when K = 0 and ends at zeroes when K tends to an infinitely

large positive number.

In many cases, the zeroes will be at infinity because our P(s) have more poles than

zeroes. With n poles, M zeroes and n > M, the locus of roots will have n M branchesapproaching n M zeroes at infinity. This simply means that since the locus starts fromeach pole, it has got to end somewhere which is infinity in this case because n M is in

fact the number of poles in excess of zeroes we have. These are separate loci(SL). The locus of roots on the real axis always lie on section of real axis to the left of an odd

number of poles and zeros. This result is exceedingly valuable because it assures the


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locus is going to lie on real axis only while there are an infinite number of paths

possible for a locus to pass through two points. This result can be thought of as Nyquist

Theorem of locus of roots(set of all values of roots). This result follows from the angle

criteria in equation 7.17(also as seen in eq 7.9 before) below.

Root loci must be symmetrical with respect to the horizontal real axis.

Loci reach the zeros at infinity along asymptotes centered at A and with angle A.

These always originate from a point at real axis

Explaining Eq 7.30:

the net phase at any root locus segment is always 180 degree. So, consider a remote

point at an SL away from bothfinitepoles and zeros

as the point is far away from both poles and zeros, the angles from both poles and zeros,, are essentially equal. Hence the net angle is simply (n M) , which must be 180.


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accounting for all such remote points at remote root locus segments gives us eq 7.30. we

have = 180/(n M)

Asymptote Centroid(ref. Modeling and control of dynamic systems by Narciso F. Macia,

George Julius Thaler, Google Books):

aysmptotes, when extended, cross the real axis at same location in s-plane.

Although each pair of asymptotes cross the real axis at same point, when the order of

system is increased, all asymptotes 'essentially' meet at same point. There are several ways of deriving this centroid point which is given by

In characteristic equation, only high powers of s need to be considered and this reduces

it to something like

It is tough to decide what the nature of path of roots is going to be. It is only a series of

spontaneous steps which begins with finding characteristic equation and then its roots.


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In the formula for centroid,

minus sign has been put before

poles and zeros to give a

location in left hand s-plane.

Consider the plot for a fourth

order system below.

Look at figure (b) above. The locus start from -4 and from -2, and they breakaway from real

axis along respective asymptotes to end at zeros somewhere at infinity(which is only a

concept).

The point on imaginary axis which SL happen to cross is given by Routh-Hurwitzcriteria.

Finding Breakaway Point: This happens when the net change in angle due to small


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displacement is zero. Loci break away where ever there is multiplicity of roots.

The angle criteria gives that the tangents to loci at breakaway pointare equally spaced

over 360 degrees, which is to keep the netphase difference to 180 degree.

Consider this method in which K is isolated and characteristic equation is rewritten and

example(exciting indeed!). Equation 7.33 is differntiated wrt s to get an equation of

degree n + M 1 and is solved to find point of maximum, which would be ourbreakaway point.

The angle of departure(obviously from a pole) and arrival(more obviously to a zero):

for a particular pole or zero, it is given by the subtracting all the terms of all poles and

zeros except this one from 180, using the angle criteria(again!).

Knowing the angle helps completing the locus, particularly in case of complex poles or

zeros.

Departure from a conjugate is going to be negative of the departure of the other pole or

zero.

Example 7.4 of Book from Dorf & Bishop summerizes it all up. Look at page number 461.


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Dominant Roots: The root of system closest to origin of s-plane for they dominate the

transient response.

Parameter Design: After all the analysis it looks like root locus is a single parameter

method. What if more than one parameters vary and their effect over the nature of roots is to

be seen? This is done by parameter design which is nothing but isolating each parameter in

the characterisitc equation and doing root locus analysis separately for it. Example for this is

on page number 469 of the book from Dorf & Bishop.

Sensitivity and the root locus:

When system gain T(s) is independent of K, logarithmic sensitivity and root sensitivity are

directly proportional. For analysis using root locus, root sensitivity is given as

Needless to say, a number of examples of dynamic systems of different order should be read

to understand the patterns involved.


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PID Controllers

these are controllers(Gc) which are widely in use.

Another way to look at this is that a PID controller is a cascade of a PI(when Kd = 0) and

PD(when Ki = 0) controllers.


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Negative Gain Root Locus

The characteristic equation is 1 + KG(s) = 0. This implies, G(s) = - (1/K) and that |KG(s)| =

1. Also that KG(s) = 0 + k360, because K is -ve.

All the seven steps of sketching locus of roots keep same. Just the positive values of K give

locus segments in right hand s-plane making system unstable.

Only those values of K are to be used which give segments in left hand s-plane.

Page numbers 519, 520 and 521 present loci of roots of some commen functions.

control systems--the last basic course, pt iii

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