ABSOLUTE AND RELATIVESTABILITY OF CONTROL

SYSTEMSCONTROL SYSTEMS TERM PAPER

SUBMITTED BY:

B. KUMARASWAMY

2011JE0642

STABILITY OF CONTROL SYSTEM

Concept of stability:

Stability is probably the most important consideration when designing control systems. One of the most important characteristics of a control system is that the output must follow the desired signal as exactly aspossible. The stability of a system is determined by the form of the response to any input or disturbance. Absolute stability refers to whether a system is stableor unstable. In a stable system the response to an input will arrive at and maintain some useful value. Inan unstable system the output will not settle at the desired value and may oscillate or increase towards some high value or a physical limitation. For example, a system is stable if the response to an impulse input approaches zero as time approaches infinity. “If any oscillations setup in a system in a consequence to application of an input are damped out with respect to time, the system is said to be stable” The term absolute stability is used in relationto qualitative analysis of stability and the term relative stability is used in relation to comparative analysis of stability or in other terms it can be referred as degree of stability of system

The stability of a system depends much on the poles ofits transfer function. Let’s consider the following transfer function G(s) of a system

G(s) =N (s )D (s )

=

…………… (1)

And the system response is determined, except for theresidues, by the poles of G(s), that is by the solutions of the following characteristic equation of the system Sn +a1 Sn-1 +a2 Sn-2 +…………+an-1 S+ an =0 ……………… (2) Equation (2) can be written as (S-P1) (S-P2)……………………. (S-Pn) =0

Where p1, p2,…, pn. are roots of characteristic equation (2), or the finite poles of transfer function (1). The system is asymptotically stable if and only if the roots of the characteristic equation are negative (for the real poles) or have negative real parts (for the complex poles).

The case of purely imaginary roots, like complex roots, all purely imaginary roots occur in conjugate pairs. Let’s call the multiplicity of the roots of (3) is r. If the multiplicity r is equal to 1 (r = 1), the

corresponding response is oscillatory with a constant amplitude. On the other hand, if the multiplicity of the imaginary roots r > 1 the corresponding response terms are oscillatory but the oscillations are of increasing amplitude, if the multiplicity of the imaginary root r <1 the corresponding response terms are oscillatory but the oscillations are of decreasing amplitude . Hence, for r <1the system is stable and forr = 1 the system is marginally stable, but for r > 1 the system is unstable. Thus the absolute stability can be determined by examining the sign of real part of the roots of a system characteristic equation, the location of the roots being in the s-plane. Table: Relation between poles(roots) stability

Type of

root

S-plane graph Response graph Remarks

real and

negativeAsymptotically

stable

real and

positive Unstable

ZeroMarginally

stable

Conjugate

and

complex

with

negative

real part

Asymptotically

stable

Conjugate

imaginary

(multiplic

ity

=1)

Marginally

stable

Conjugate

with

positive

real part

Unstable

Conjugate

imaginary

multiplic

ity 2

Unstable

Roots of

multiplic

ity 2 at

the

origin

Unstable

Table: Relation between poles (roots) stability

There are several methods to determine the stability of a control system

Routh – Hurwitz Criterion:

This is an algebraic method that provides information on the absolute stability of a linear time-invariant system that has a characteristic equation with constant coefficients. The criterion tests whether any ofthe roots of C.E. lie in the right-half s-plane.The number of roots that lie on the jω-axis and in the right-half plane is also indicated. In order to determine the existence ofa root having positive real part for a polynomial equation given by An Sn +an-1 Sn-1 +an-2 Sn-2 +…………+a1 S+ a0

=0

The following array is formed

From the above array so formed the number of changes of sign in the first column elements of Routh array gives the number of positive real part roots of the polynomial. Therefore, for a stable system there should be no change of sign in the first column of Routh array formed from the coefficients of the characteristic equation expressed in polynomial form.

Four distinct cases or configurations of the first column array much be considered, and each must be treated separately and requires suitablemodifications of the array calculation procedure:

1. No element in the first column is zero;

2. There is zero in the first column, but some other element of the row containing the zero in the first column are nonzero; >> Solution: Take entry as small value ε>0 and proceed, taking ε>0 in subsequent calculations. After completing the process let approach to zero. Example:

C1 = -12/ ε d1=6

The system is unstable and the two roots of the equation lie in the right half of the s-plane

3. There is a zero in the first column, and other elements of the row containing the zero are also zero; >> Solution: When all the elements in the row vanish and the application of routh criterion fails this difficulty faced is overcome by forming an

auxiliary equation using elements of the last but one vanishing row. The derivative of this auxiliary equation is taken with respect to s and the coefficients of the differentiated equation are taken as the elements of the following row.Example:Consider the equation A(s) = 0 where A(s) is S5+2S4+24S3+48S2-50The Routh array starts off as S5 1 24 -25S4 2 48 -50 ……………….> auxiliary polynomial P(s)S3 0 0

The auxiliary polynomial P(s) is

Which indicates that A(s) = 0 must have two pairs of roots of equal magnitude and opposite sign, which are also roots of the auxiliary polynomial equation P(s) = 0. Taking the derivative of P(s) with respect to s we obtain dP(S)÷ds= S3+ 96 S

So the S3 row is as shown below and the Routh array is

……> coefficients of dP(S)÷ds

As there is a sign change in the first column the system is unstable.

Another method for determining the stability of control system is Nyquist criterion

NYQUIST CRITERION:

A stability test for time invariant linear systems can also be derived in the frequency domain. It is known as Nyquist stability criterion. It is based on the complex analysis result known as Cauchy’s principle of argument. Note that the system transfer function is a complex function. By applying Cauchy’s principle of argument to the open loop system transfer function, we will get information about stability of the closed-loop system transfer

function and arrive at the Nyquist stability criterion. Nyquist criterion is used to identify the presence of roots of a characteristic equation of a control system in aspecified region of s-plane .from the stability view of point the specified region being the entire right hand side beyond the imaginary axis of complex s-plane. Nyquist criterion is similar to Routh-Hurwitz criterion but the approach differsin following respect: (1) The open loop transfer function G(S)H(S)is considered instead of closed loop characteristic equation (2) Inspection of graphical plot of G(S) H(S) enables to get more than Yes or No answer of Routh-Hurwitz method pertaining to stability of control systems.

Cauchy’s Principle of Argument:

Let F(s) be an analytic function in a closed region of the complex plane given in Figure (1) except at a finite number of points (namely, thepoles of F(s)). It is also assumed that F(s) is analytic at every point on the contour. Then, astravels around the contour in the - plane in theclockwise direction, the function F(s) encircles

the origin in the {Re(F(s)),Im(F(s))}- plane inthe same direction N times (see Figure (1)), with N given by N= Z-P Where Z and P stand for the number of zeros and poles (including their multiplicities) of the function F(s) inside the contour.The above result can be also written as Arg{F(s)}=(Z-P) 2π = 2Πn

Which justifies the terminology used, “the principle of argument”.

By using Nyquist criterion.

1. The stability of open loop system can be found by studying the behavior of the Nyquist

plot for G(s)H(s) in relative to the origin of G(s)H(s)-plane although the poles of G(s)H(s) are not known.

2. The stability of closed loop system can be found by studying the behavior of Nyquist plot for G(s)H(s) in relative to the (-1,j0) point.

The Nyquist stability criterion methods can besummarized as follows:

1. The Nyquist path is determined in s-plane.

2. Nyquist plot for G(s)H(s) is sketched in the G(s)H(s)-plane with s value equals to thepoints value along the Nyquist-path.

3. The open-loop and closed-loop stability for a system can be determined by observing the behavior of the Nyquist plot for G(s)H(s)relative to the origin (0,j0) and point (-1,j0).

Steps in determining the stability using Nyquist stability criterion:

1. From the characteristic equation, F(s)=1+G(s)H(s), the Nyquist path on the S-plane is

constructed from the behavior of zero pole ofG(s)H(s) at first.

2. Sketch the Nyquist plot for G(s)H(s) on the G(s)H(s)-plane.

3. Determine the value of N0 and N-1 from thebehavior of Nyquist plot for G(s)H(s) with respect to origin point (0,j0) and point(-1,j0).

4. Obtain the value of P0 from N0 = Z0 -P0 (Z0 is known)If P0 =0, then the open loop system is stable.

5. Then, after P0 is known, obtain the valueof P-1 by P0 = P-1

6. Obtain Z-1 from N-1 = Z-1 – P-1.

If Z-1 = 0 then, the closed loop system is stable.

The above two methods are used to determine the absolute stability of control system. From the above results Nyquist plot is appropriate to determine the stability of control system.

RELATIVE STABILITY OF CONTROL SYSTEM:

Now we look into the relative stability of control systems there are several methods for determining the relative stability. i.e gain margin, phase margin, and can be determined from the bode plot also.

Gain margin and phase margin: Gain Margin: (GM) The gain margin is the change in open-loop gain, expressed in decibels (dB), required at 1800 of phase shift tomake the closed-loop system unstable or additional gain that makes the system on the vergeof instability. Phase Margin: The phase margin is the change in open-loop phase shift, required at unity gain to make the closed-loop system un- stable or additional phase lag that makes the system on theverge of instability. Gain, and phase margins give a measure of“the degree of stability” for closed loop systems. Phase margin is the most widely used measure of relative stability when working in the frequency domain.  On a Nyquist plot we examine the unit circle (which is just all thosepoints that have a magnitude of 1) and we can see that the system we intuitively think of as less stable is closer to the -1 point when we measure distance along the unit circle (which

goes through -1). Similarly Gain margin is another widely used measure of relative stability when working in the frequency domain. On a Nyquist plot we examine the -180o crossing and we can see that the system we intuitively think of as less stable is closer to the -1 point when we measure distance along the negative real axis. Definitions of the some of the terms which are used in the determination of phase and gain margin areGain Crossover:Occurs at the point where 20Log10G(jw)H(jw) = 0dB.The frequency here is called the phase margin frequency (wf). Phase Crossover:Occurs when frequency = -180°, at the gain-margin frequency (wc).

Figure: Nyquist diagram showing gain and phase margins

Figure: Nyquist plot for two systems

From the above figure it shows that both the systems are stable as they are inside the unit circle in consideration with phase margin the system with dotted line has less phase margin w.r.t the other system so the the system with dotted line is less stable.Similarly if the gain margin of system is greater than the other system the system with more gain margin is relatively more stable than the system with lesser gain margin. In decibel gain margin is given by G.M= 20Log10(1÷∨G (jw)H (jw )∨¿) db

Where wis the phase cross over frequency For stable system ¿G (jw )H (jw)∨¿ <1, the gain margin in decibel is positive. For marginally stable system |G (jw)H (jw)∨¿ =1, the gain margin in decibel is zero. For stable system ¿G (jw )H (jw)∨¿ >1, the gain margin in decibel is negative. And the gain is to be reduced to make the system stable.

The phase margin is given by P.M =1800+ <G (jw)H (jw)

Where w is the gain cross over frequency and the above angle is measured in clockwise direction.

Bode plot: The Bode plot method gives a graphical procedure for determining the stability of a control system based on sinusoidal frequency response. The transfer function of a control system for a sinusoidal input response can be determined by substituting jw in place of Laplace operator “s”. therefore , if the open loop transfer function of a system is G(s)H(s), the corresponding sinusoidal open loop transfer function is G(s)H(s) which can be expressed in the form of magnitude and phase angle. Relative stability of a closedloop control system can be conveniently assessedby plotting open loop transfer function by bode plot method. The gain margin and phase margin isdetermined directly from bode plot.

Bode Plot deals with the frequency response of a system simultaneously interms of magnitude and phase. More precisely, the log-magnitude and phase frequency response

curves are known as Bode Plots. Such plots are useful due to the following reasons:

– For designing lead compensators– For finding stability, gain and phase margin– For system identification from the frequency response

From Nyquist Criteria you know that instability occurs if there is encirclement of -1 corresponding to a phase of 1800.• This implies that for stability the magnitude of the transfer function must be less than unityat a frequency which corresponds to 1800 phase. In fact the actual Gain in dB corresponding to this frequency provides the gain margin.• In a similar manner the phase value corresponding to the frequency where gain of thesystem is 0 dB provides the phase margin.

The stability of a control system is determined from bode plot i.eWe find the phase margin and gain margin as shown in the following figure.

1). For stable systems: the gain cross over frequency < phase cross over frequency2). For unstable systems: the gain cross over frequency > phase cross over frequency

3). For marginally stable systems : the gain cross over frequency = phase cross over frequency

Above Figure shows the gain margin and phase margin from bode –plot.

Comparison between stable and unstable system with the help of phase and gain margin.

Relative analysis by using Nyquist plot:

On investigating stability, one should be more have an accurate around |G(jw)| = 1 and <G(jw) =-1800 to obtain more accurate results for gain margin and phase margin.

The above figure explains about the relative stability of control system by determining the phase margin and gain margin from Nyquist-plot.

Root Locus Analysis: The relative stability and the transient response of a closed-loop system is completely determined by the location in the s-plane of the closed-loop system poles and zeros.This shows if the system is stable and also whether there is any oscillatory behavior in thetime response. Therefore, it is worthwhile to determine how the roots of the characteristic equation as a system parameter is varied. It is a graphical method for system analysis and design. Root-locus plots are used toplot the system roots over the range of a

variable to determine if the system will become unstable, or oscillate. It is useful to determine the locus of roots in s-plane as a parameter varied since the roots is a function of the system’s parameter. The root locus technique may be used to great advantage in conjunction with the Routh-Hurwitz criterion.

Conjugate

and

complex

with

negative

real part

Asymptotically

stable

Conjugate

imaginary

(multiplic

ity

=1)

Marginally

stable

Conjugate

with

positive

real part

Unstable

For a system to be stable the roots of its characteristic equation should lie in the left

hand side of the s-plane. From the location of the roots in s-plane the nature of time responseand system stability can be ascertained.

References:

1.http://www.facstaff.bucknell.edu/mastascu/econtrolhtml/freq/freq6.html

2.http://www.ece.rutgers.edu/~gajic/psfiles/nyquist.pdf3.http://en.wikipedia.org/wiki/Routh

%E2%80%93Hurwitz_stability_criterion4.http://www.egr.msu.edu/classes/me451/jchoi/2012/

notes/ME451_L10_RouthHurwitz.pdf5.http://www.facstaff.bucknell.edu/mastascu/

econtrolhtml/freq/freq5.html6.http://www.mathworks.com/videos/using-bode-plots-

phase-and-gain-margins-3-of-5-77059.html7. http://www.me.ust.hk/~mech261/index/Lecture/

Chapter_7.pdf8. Linear Control Systems by B.S. Manke