Root Locus Method

Root Locus Method

Root Locus Method

Root Locus Method

,1::numerator of Roots:Zeros

,1: r,denominato theof Roots:Poles

i

j

Zeros

Poles

Roots of the characteristic equationDepends on Kc (tuning) of the loop.

1- This control loop will never go unstable.2- When Kc=0, the root loci originates from The OLTF poles:-1/3, and -13- The number of root loci/branches=number Of OLTF poles=24- As Kc increases, the root loci approaches infinity

1- This control loop can go unstable.2- When Kc=0, the root loci originates from The OLTF poles:-1/3, -1, -23- The number of root loci/branches=number Of OLTF poles=34- As Kc increases, the root loci approaches infinity

1- This control loop can never go unstable. As Kc increases the root loci move away from I-axis, and D- mode adds a lead to the loop makes it more stable. Addition of lag reduces stability2- When Kc=0, the root loci originates from the OLTF poles:-1, -1/33- The number of root loci/branches=number Of OLTF poles=24- As Kc increases, one rout locus approaches – infinity and the other -5, the zero of the OLTF

The rout locus must satisfy the MAGNITUDE and the ANGLE conditions

The rout locus must satisfy the MAGNITUDE and the ANGLE conditions

)i(θeq'y

xand)i(θqexyier'ydic

iθrexbia

MAGNITUDE CONDITION

ANGLE CONDITION

)]2sin()2[cos(1)( kkeyx

yx i

MAGNITUDE CONDITION

ANGLE CONDITION

Example:

Matlab comands:rlocus Evans root locus Syntax rlocus(sys)rlocus(sys,k)rlocus(sys1,sys2,...)

[r,k] = rlocus(sys)r = rlocus(sys,k)

)h(rlocus

;3])2[11],5tf([2h;32ss15s2sh(s)

;)s(d)s(n)s(h

2

2

Matlab comands:Find and plot the root-locus of the following system. h = tf([2 5 1],[1 2 3]);Rlocus(h, k)

Frequency Response Technique

Process Identification: A- Step Test Open-Loop Response B- Frequency Response.

Frequency Response Technique

B- Frequency Response.

Frequency Response Technique

Recording from sinusoidal testing

Frequency Response Technique

Mathematical Interpretation:

Frequency Response Technique

Mathematical Interpretation (Continued):

Amplitude of the response

radian degrees

Frequency Response Technique

Mathematical Interpretation (Continued):

Amplitude of the response

Amplitude Ratio Magnitude Ratio

Frequency Response Technique Mathematical Interpretation (Continued): All these terms (AR, MR, and Phase angle)

are functions of Frequency response is the study of how

AR(MR) and phase angle of different components change as frequency changes.

Methods of Generating Frequency Response: A- Experimental Method B- Transforming the OLTF after a sinusoidal

input

Frequency Response Technique Methods of Generating Frequency

Response: B- Transforming the OLTF after a

sinusoidal input

Frequency Response Technique Methods of Generating Frequency

Response: B- Transforming the OLTF after a

sinusoidal input

Frequency Response Technique Methods of Generating Frequency

Response: B- Transforming the OLTF after a

sinusoidal input

Frequency Response Technique Methods of Generating Frequency

Response: B- Transforming the OLTF after a

sinusoidal input

Long time

Frequency Response Technique Methods of Generating Frequency

Response: B- Transforming the OLTF after a

sinusoidal input

Frequency Response Technique Methods of Generating Frequency

Response: B- Transforming the OLTF after a

sinusoidal input

Frequency Response Technique Example:

Frequency Response Technique Example:

Frequency Response Technique Example:

Frequency Response Technique

Generalization

Frequency Response Technique

Generalization

Frequency Response Technique

1- Bode Plots, 2-Nyquist Plots, and 3- Nichols Plots

1- Bode Plots

Frequency Response Technique

1- Bode Plots

Frequency Response Technique

1- Bode Plots

Frequency Response Technique

1- Bode Plots

Frequency Response Technique

1- Bode Plots

Frequency Response Technique

1- Bode Plots

Bode Plots

Bode Plots

Frequency Response Technique

1- Bode Plots

Frequency Response Technique

1- Bode Plots

EXAMPLE:

Frequency Response Technique

1- Bode Plots

EXAMPLE:

Frequency Response Technique

1- Bode Plots

EXAMPLE:

Frequency Response Technique

1- Bode PlotsFrequency Response Stability Criterion

Frequency Response Technique

1- Bode PlotsFrequency Response Stability Criterion

Frequency Response Technique

1- Bode PlotsFrequency Response Stability Criterion

Frequency Response Technique

1- Bode PlotsFrequency Response Stability Criterion

... ,(1.05) Time Third ,(1.05)A time Second1.05by increased is Amplitude means This

1.05)Kc0.0524(0.8AR25Kc If. sustainedis noscillatio and unchanged, is E(s)controller the to connected is C(t) and 0,Tset(t) 0,t at

-πθ 23.8,Kc 1,AR

equal are Amplitudes

)sin(0.219tπ)sin(0.219tC(t))sin(0.219t(t)T

32

set

Frequency Response Technique

1- Bode PlotsFrequency Response Stability Criterion

unstable is system the,-180at 1AR if stable; is

system the,-180at 1AR If rads). (180- is

angle phase theunity when than less bemust ratio amplitude thestable, be tosystem control aFor

o

oo

Frequency Response Technique

1- Bode PlotsFrequency Response Stability CriterionEXAMPLE:

Frequency Response Technique

1- Bode PlotsFrequency Response Stability CriterionEXAMPLE:

Without dead time

With dead time

ωu=0.160 rad/s

It is easier for the process with dead time to go unstable

Frequency Response Technique

1- Bode PlotsFrequency Response Stability CriterionEXAMPLE:

Without dead time

With dead time

ωu=0.160 rad/s

It is easier for the process with dead time to go unstable

MATLAB CONTROL TOOL BOX

MATLAB CONTROL TOOL BOX

MATLAB CONTROL TOOL BOX

Bode(num, den)