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)
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