Reporte 7 Lugar Raices

8/13/2019 Reporte 7 Lugar Raices

1/6

1

Abstract In this paper is presented the geometric place of

poles and zeros of three transfer functions using the Root Locus

Analysis Method. Also is presented a brief analysis using the

Ruth-Hurwitz theorem to determine critical stability points.

I. NOMENCLATURE

G(S) = Transfer Function.

= Zero.=Poles.= Angle of the Asymptote.

= Friction Coefficient.

K= Gain

II. INTRODUCTION

f we consider the Closed Loop (CL) System like the shown

in figure 1, we can de observe that many of its

characteristics are determined by the CL poles. The CL poles

are the roots of the characteristic polynomial. To find the

roots, we must factorize the equation, but sometimes, this can

be very laborious. The Root Locus Analysis its a graphic

method that we can use to approach the poles place of the

system at any K gain parameter in the system.

Fig. 1.

III. ROOT LOCUS METHOD

To graphically determine the roots place, we need to follow

the next steps:

1. The graph will have N branches, where N isthe number of poles of the CL system.

2. Draw the poles and zeros in a Cartesian coordinatesystem, where the X axis represent the Real

numbers and Y axis the imaginary numbers. If

there is less zeros than poles, we consider that the

missing zeros have infinite magnitude. The

branches always go from poles to zeros.

3. Determine if there exist root place in the real axis.To the left of an impair number of zero/pole, it

exist the root place. Fig. 2.

Fig. 2.

4. The branches that do not tend to the zeros of CL,tend asymptotically to the zeros with infinite

magnitude. So we calculate the angle of the

asymptote () by using:

5. The point in the real axis in which asymptote begin(Fig.3) is defined by:

Fig. 3.

6. We must calculate all the input/output angle for allthe pole/zero of interest, and then solve from the

phase condition:

Where:

Root Locus AnalysisProfessor: Dr. Luis Amzquita Brooks, Control Engineering, CIIIA-UANL,

Student: Lic. Fernando Guerrero Vlez, CIIIA - UANL

I


2/6

2

If and input/output angles are in the wrongdirection (This is, if the output angle of a pole arent pointing to the zero direction), we mustadd 180 to

and

.

7. Calculate the input/output points from the real axisusing the Closed Loop Denominator (CLD), where

the values of K remain real, this is, considering the

CL system:

Then:

Where, the real solutions of the following

equation:

(

)

Are the input/output points that were looking for.

8. Finally, we must calculate the point where theasymptote of CLD cross over the imaginary axis

(Critical Stability), by using the Routh-Hurwitz

Stability Criterion.

IV. EXAMPLE PROBLEMS

Solve the following problem using the Root Locus Method.

a)

Step 1:If we rewrite G(s) as:

Then we can observe that exist 3 poles

and 2 zeros. Thus, exist 3 branches.

Step 2-3:Drawing the poles/zeros and finding the

root place.

Fig. 4.Where:

Step 4:

Step 5:

Then, the graphic is:

Fig. 5.

Step 6:Input angles to Z1:

() ( ) () ()


3/6

3

Using the phase condition:

Then:

Now, obtaining the output angles of :

( ) ( )

() ()

( )

Using again the phase condition, then we

find:

Step 7:To find the input/output point from real

axis, then:

(

)

Then, finding the numerator roots:

So, the output point from the real axis is

located in (-0.53,0). Finally the graphic is:

Fig. 6.

Step 8:Using Routh-Hurwitz to calculate the

imaginary axis cross point:

Where:

That lead to the following values of k:

b)





root place.

Fig. 7.

Where:


4/6

4

Step 4:

Step 5:


Fig. 8.

Step 6:Input angles to Z1:

() () ( ) ( )

Using the phase condition:

Then:

Now, obtaining the output angles of :

( )

()

Using again the phase condition, then we

find:


axis, then:

(

)



located in (-8.24,0). Finally the graphic is:

Fig. 9.



Where:


5/6

5

Finally, the values of kmust be:

c)





root place.

Fig. 10.

Where:

Step 4:

Step 5:


Fig. 11.

Step 6:Because a pole always must end in a zero,

and because we have the poles/zeros

arranged in pair, then we can note that the

pair of poles P3,P4will show a trajectory

starting among themselves, and ending

between the pair of zeros. The pair of

poles P1,P2 will approach to the asymptote

in (-4,0), to end in the zeros at theinfinite. This is, the input/output angle for

de poles P3,P4 and the zeros Z1,Z2 are

multiples of 180. The input/output point

of these trajectories will be calculated in

step 7.


axis, then:

(

)



located in (1.45,0) and (-6.95,0), the input

point is located in (-1.92,0). Finally thegraphic is:


6/6

6

Fig. 12.



Where:

Finally, the values of kmust be:

V. RESULTS AND CONCLUSIONWe are now presenting the plots obtained for the following

parameters:

g = 9.8

l = 2

m = 1

M = 3

b = 0.3

As wee see in the past plots, the pendulum exhibit a

sinoidal behavior, and it tends to be in its acceleration,

velocity and position equilibrium state.

VI. REFERENCIAS

[1] Dinmica de Sistemas, Katsuhiko Ogata, !era edicin en Espaol

Documents

Reporte 7 Lugar Raices