signals and systems fall 2003 lecture #21 25 november 2003 1.feedback a)root locus b)tracking...
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Signals and SystemsFall 2003
Lecture #2125 November 2003
1. Feedbacka) Root Locus b) Trackingc) Disturbance Rejection d) The Inverted Pendulum
2. Introduction to the Z-Transform
The Concept of a Root Locus
• C(s),G(s) — Designed with one or more free parameters• Question: How do the closed-loop poles move as we vary these
parameters? — Root locus of 1+ C(s)G(s)H(s)
)()()(1
)()()(
sHsGsC
sHsCsQ
The “Classical” Root Locus ProblemC(s) = K— a simple linear amplifier
A Simple Example
In either case, pole is at so = -2 - K
Sketch where pole movesas |K| increases...
Becomes more stable Becomes less stable
1)( ,2
1)(
sG
ssH
22
1
2)( (a)
Ks
K
s
Ks
K
sQ2
1
21
2
1
)( (b)
Ks
s
KssQ
What Happens More Generally ?
• For simplicity, suppose there is no pole-zero cancellation in G(s)H(s)
Closed-loop poles are the solutions of
That is
— Difficult to solve explicitly for solutions given any specific value of K, unless G(s)H(s) is second-order or lower.
— Much easier to plot the root locus, the values of s that are solutions for some value of K, because:
1) It is easier to find the roots in the limiting cases for K = 0, ±∞.
2) There are rules on how to connect between these limiting points.
)()(1
)()(
sHsKG
sHsQ
0)()(1 sHsKG
KsHsG
1)()(
Rules for Plotting Root Locus
• End points— At K = 0, G(so)H(so) = ∞⇒ so are poles of the open-loop system function G(s)H(s).
— At |K| = ∞, G(so)H(so) = 0⇒so are zeros of the open-loop system function G(s)H(s). Thus:
Rule #1:A root locus starts (at K = 0) from a pole of G(s)H(s) and ends (at
|K| = ∞) at a zero of G(s)H(s).Question: What if the number of poles ≠ the number of zeros? Answer: Start or end at ±∞.
KsHsG
1)()(
Rule #2: Angle criterion of the root locus
• Thus, s0 is a pole for some positive value of K if:
In this case, s0 is a pole if K = 1/|G(s0) H(s0)|.
• Similarly s0 is a pole for some negative value of K if:
In this case, s0 is a pole if K = -1/|G(s0) H(s0)|.
:)12()()(0 00 nsHsGK
nsHsGK 2)()(0 00
Example of Root Locus.
One zero at -2, two poles at 0, -1.
)1(
2)()(
ss
ssHsG
))()(( sHsGrlocus
Tracking
In addition to stability, we may want good tracking behavior, i.e.
for at least some set of input signals.
We want C(jω)P(jω) to be large in frequency bands in which we want good tracking
ttytxte as 0)()()(
)()()(1
1)(
)()()(1
1)(
jXjPjC
jE
sXsPsC
sE
Tracking (continued)
Using the final-value theorem
Basic example: Tracking error for a step input
)()()(1
lim)(limlim00
sXsPsC
sssEte
sst
ssXtutx
1)()()( Suppose
)()()(1
lim)(lim Then 0
sXsPsC
ste
st
Disturbance Rejection
There may be other objectives in feedback controls due to unavoidable disturbances.
Clearly, sensitivities to the disturbances D1(s) and D2(s) are much reduced when the amplitude of the loop gain
)()()(1
1)(
)()(1
)()(
)()(1
1)( 21 sD
sHsCsD
sHsC
sHsX
sHsCsE
1)()( sHsC
Internal Instabilities Due to Pole-Zero Cancellation
However
)(
Unstable
)33(
2)(
)()(1
)(
)(
Stable33
1)(
)()(1
)()( )(
2)( ,
)1(
1 )(
2
2
sXsss
ssX
sHsC
sCW(s)
sXss
sXsHsC
sHsCsY
s
ssH
sssC
Inverted Pendulum
m=Mass, I=moment of inertia
balance)(Moment )(
)(cos)(
)(sin2
2
2
2
dt
tdIt
dt
txdmltmgl
1 cos,sinsmall is assuming Linearize :
2
2
2
2 )()(
)(
dt
txdmltmgl
dt
tdI
)()(
)(
2
2
sXmglIs
mlss
sH
──Unstable!
Feedback System to Stabilize the Pendulum
• PI feedback stabilizes θ
• Subtle problem: internal instability in x(t)!–Additional PD feedback around motor / amplifier
centers the pendulum
Root Locus & the Inverted Pendulum
• Attempt #1: Negative feedback driving the motor
• Root locus of M(s)G(s)
– Remains unstable!
after K. Lundberg
)1)(1(
/
)(
)()(
2
ss
gs
sX
ssG
LL
)1()(
)()(
ss
k
sV
sXsM
M
m
Root Locus & the Inverted Pendulum
• Attempt #2: Proportional/Integral Compensator
• Root locus of K(s)M(s)G(s)
– Stable for large enough K
after K. Lundberg
ssK
k1
1)(
Root Locus & the Inverted Pendulum
• BUT – x(t) unstable:
System subject to drift...
• Solution: add PD feedback around motor and compensator:
after K. Lundberg
)()()(1
)()(
)(
)(
sGsMsK
sMsK
x
sX
C
)1)(/()1)(1(
)1)(1(122
22
2 sgkss
ssk
s KMLMK
LKM
The z-Transform
Motivation: Analogous to Laplace Transform in CT
We now do not restrict ourselves
just to z = ejω
The (Bilateral) z-Transform
][nx ][nh ][ny
nn zzHnyznx )(][ ][
LTI DTfor ionEigenfunct
convergesit assuming ][)(
n
nznhzH
]}[{][)(][ nxZznxzXnxn
n
The ROC and the Relation Between zT and DTFT
—depends only on r = |z|, just like the ROC in s-plane only depends on Re(s)
• Unit circle (r = 1) in the ROC ⇒ DTFT X(ejω) exists
jrez ||, zr
n
njn
n
njj erenxrenxreX )][()]([)(
}][{ nrnx F
n
nj rnxrez |][|at which ROC
•
Example #1
This form for PFE and inverse z- transform
That is, ROC |z| > |a|, outside a circle
This form to findpole and zero locations
sided-right- nuanx n
-n
)( nn znuazX
0
1)(n
naz
||||,i.e.,1 If 1 azaz
az
z
az
11
1
Example #2:
Same X(z) as in Ex #1, but different ROC.
sided-left-]1[ nuanx n
1
1
1 )(
n
nn
n
nn
za
znuazX
0
11 n
n
n
nn zaza
,
11
11
1
1
1
az
zza
za
za
||||.,.,1 If 1 azeiza
Rational z-Transforms
x[n] = linear combination of exponentials for n > 0 and for n < 0
— characterized (except for a gain) by its poles and zeros
)(
)(
rational is )(
zD
zNX(z)
zX
Polynomials in z
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