class cnt rl notes

7/27/2019 Class Cnt Rl Notes

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Classical Control Notes

M. C. Berg

Version: 31 October 2000


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Residue Theorem

Reference: Sec. 17.3 of Advanced Engineering Mathematicsby C. R. Wylie.

Theorem: If F(s) is an analytic (i.e., differentiable) function of s except at a finite number of poles each of

which lies to the left of a vertical line Real(s) = a, and if sF(s) is bounded as s becomes infinite through the

half plane Real(s) a, then

f(t) = L1 [F(s)] = residues of F(s)est

at each of its poles. (Note that est has only zeros.)

Theorem: If F(s) has a pole of order m at s=p, then the residue of F(s) at s=p is

1

1

1

1( )!lim [( ) ( )]

m

d

dss p F s

s p

m

mm

Key Points:

1. The character of f(t) is completely determined by the poles of F(s).

2. The contribution to f(t) of the portion of f(t) due to each pole of F(s) is a function of the gain and all of

the poles and all of the zeros of F(s).


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Routh-Hurwitz Stability Criterion

The Routh-Hurwitz criterion is used to determine how many roots a polynomial of the form

(s) = ansn + an1sn1 + . . . + a0

has in the right half of the s-plane.

Procedure: Construct the Routh array:

sn: an an2 an4 . . . 0

sn1: an1 an3 an5 . . . 0

sn2: bn2 bn4 bn6 . . . 0

sn3: cn3 cn5 cn7 . . . 0

sn4: dn4 dn6 dn8 . . . 0

..

. ..

. ..

. ..

. ..

. ..

.

s0: x x x . . . 0

where:

bn2 =1

an1det

an an2

an1 an3bn4 =

1an1

det

an an4

an1 an5

bn6 =1

an1det

an an6

an1 an7

. . .

cn3 =1

bn2det

an1 an3

bn2 bn4cn5 =

1bn2

det

an1 an5

bn2 bn6

cn7 =1

bn2det

an1 an7

bn2 bn8 . . .

..

. ..

. ..

.

Then:

1. The number of sign changes in the first column of the array is equal to the number of roots of(s)

in the open right half of the s-plane.

2. If the first element in a row is zero, replace it with a small positive number , then count the sign

changes after completing the array when 0.


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3. If all elements of a row are zero, then (s) has a pair of poles symmetrically located about the

origin (e.g., at j or ). The coefficients in the row immediately preceding the zero row then

define the auxiliary polynomial. The auxiliary polynomial will be a factor of(s) and will have

roots symmetrically located about the origin.


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Root Locus Construction

Consider a polynomial in s of the form

(s) = d(s) + k n(s)

where d(s) and n(s) are polynomials in s with no common roots and k is a parameter. Let

closed-loop poles =

roots of(s)

open-loop zeros =

roots of n(s)

open-loop poles =

roots of d(s)

With

_

(s) =

1 + kn s

d s

( )

( )

=

1 + kb s b s b

s a s am

mm

m

nn

n

+ + +

+ + +

11

0

11

0

K

K

=

1 + kb s z s z s z

s p s p s pm m

n

( )( ) ( )

( )( ) ( )

+ + +

+ + +1 2

1 2

L

L

The zeros of ( )s (i.e., the values of s such that ( )s = 0) coincide one-to-one with the roots of(s) for

0 < |k| < .

Magnitude condition:

1 =

magnitude kn s

d s

( )

( )

Angle condition:

+ =

=( )( )

( ), ,2 1 180 0 1q k

n s

d sqo angle , K

The following rules (for a 180 degree root locus) can be used to sketch the locations of the closed-loop

poles in the s-plane as a function of k, for k 0.

1. For k = 0 the closed-loop poles coincide with the open-loop poles.

2. All sections of the real axis to the left of an odd number of open-loop poles and open-loop zeros are

part of the root loci.


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3. The root loci are symmetric about the real axis.

4. There are as many root locus branches as there are open-loop poles. Each branch starts at an open-

loop pole and goes to an open-loop zero as k. If n > m there are, by definition, nm open-loop

zeros at infinity.

5. The nm branches that go to infinity do so along asymptotes. The angles 1, 2,,nm that the

asymptotes make with the real axis can be determined from

k

k

n m=

2 1180o ; k = 1, 2, . . ., nm

6. All asymptotes intersect the real axis at the point

=(

n

i=1pi) (

m

i=1zi)

n m = (sum of open-loop poles) (sum of open-loop zeros)n m

7. Points of breakaway from or arrival at the real axis may exist. For example, if the real axis between

two real open-loop poles is part of the loci, and if that part of the real axis has no open-loop zeros,

then a breakaway point exists between the two real-axis open-loop poles. At a breakaway point

(arrival point) on the real axis

k = d s

n s

( )

( )

is maximized (minimized). Thus, breakaway and arrival points on the real axis are also real zeros of

dk

ds=

[ ( ) ( ) ( ) ( )]

( )

d s n s d s n s

n s2

8. The angle of departure from any open-loop pole or the angle of approach to any open-loop zero can

be determined by choosing a trial point close to the open-pole or open-zero and applying the angle

condition.


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Nyquist Stability Criterion

Consider a characteristic polynomial

(s) = d(s) + n(s)

where d(s) and n(s) are coprime1 polynomials in s. Let

closed-loop poles =

roots of(s)

open-loop zeros =

roots of n(s)

open-loop poles =

roots of d(s)

Let

( )

( )

( )

( )

s

n s

d s

L s

= +

= +

1

1

The number Z of closed-loop poles in the right half of the s-plane can be determined as follows:

1. Choose an s-plane D-contour that encircles the entire right half of the s-plane and detours around

any open-loop poles on the imaginary axis.

2. Determine the number, P, of open-loop poles within the D contour.

3. Traverse the D-contour in the clockwise direction and map it into the L(s) plane.

4. Count the number, N, of L(s)-plane clockwise encirclements of the 1 point.

5. Determine Z using N = Z P.

1 Two polynominals in s are coprime if no value ofs is a root of both polynomials.


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Determining Gain and Phase Margins

1. Construct a block-diagram representation of the system for which the gain and phase margins are to

be determined.

2. Choose the loop breaking point in the block diagram where the gain and phase margins are to be

determined.

3. Insert a k ej block at the loop breaking point you chose in Step 2.

4. Determine the characteristic polynomial (s) for the resulting system. Write (s) as

( ) ( ) ( )s d s k e n sj= +

Here d(s) the portion of(s) that does not multiply kej and n(s) is the portion that does.

5. Check (e.g., using Matlabs roots command) that all roots of(s) have negative real part (and arewhere you expect them to be) when k = 1 and = 0. If not, then your system is nominally unstable

and it has no gain or phase margins at any loop breaking point.

6. Define

L s

n s

d s( )

( )

( )=

This is the loop transfer function for your system for the loop breaking point you chose in Step 2.

7. Plot either the polar plot or the Bode plot of the frequency response of k e L sj ( ) when k = 1 and = 0

(this will be the same as the polar or the Bode plot of the frequency response ofL(s)).

8. Determine, by inspection of your frequency response plot, the smallest k that is larger than one and

will cause the polar plot of the frequency response of k e L sj ( ) to go through the 1 point when = 0.This k value is the positive gain margin for you system for the loop breaking point you chose in Step 2.

If no k exists that is larger than one and causes the polar plot of the frequency response of k e L sj ( ) togo through the 1 point when = 0, then the positive gain margin for your system for the loop

breaking point you chose in Step 2 is defined to be infinite.

9. Determine, by inspection of your frequency response plot, the largest nonnegative k that is smaller

than one that will cause the polar plot of the frequency response of k e L sj ( ) to go through the 1

point when = 0. This k value is the negative gain margin for your system for the loop breaking pointyou chose in Step 2. If no nonnegative k exists that is smaller than one and causes the polar plot of thefrequency response of k e L sj ( ) to go through the 1 point when = 0, then the negative gain margin

for your system for the loop breaking point you chose in Step 2 is defined to be zero.


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10. Determine, by inspection of your frequency response plot, the smallest positive or negative that will

cause the polar plot of the frequency response of ke L sj ( ) to go through the 1 point when k = 1.The magnitude of this theta value is the phase margin for you system for the loop breaking point you

chose in Step 2.

Notes:

1. Gain margins are most commonly expressed in decibels. This makes the positive gain margin a

positive number of decibels and the negative gain margin a negative number of decibels.

2. The 1 point on the polar plot of the frequency response of a system corresponds to a gain of 1 (zerodecibels) and a phase angle that is ANY odd multiple of 180 degrees. It particularly important to keep

this in mind if you chose to plot only the Bode plot in Step 7.

3. When dealing with a system that has only one feedback loop, the gain and phase margins at all loop

breaking points in that loop will be the same (can you show this?). Thus, when dealing with a single-

loop system, control engineers often refer to the systems gain and phase margins and this can betaken to mean the gain and phase margins of the system for any loop breaking point in the systemsfeedback loop.

4. Matlabsmargin command returns either the positive gain margin or the negative gain margin,

whichever has the smaller magnitude, and the phase margin.


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Root-Locus Lead Compensator Design

1. Translate the design objectives into a specification for the desired dominant closed-loop pole

locations.

2. Assume the compensator form

Gc(s) = K

s + zs + p

3. Choose values for z and p such that the angle criterion is satisfied at the desired dominant closed-loop

pole locations for some value of K.

4. Determine a value for K such that the magnitude criterion is satisfied at the desired dominant closed-

loop pole locations.

5. Test for satisfaction of the design objectives. If the design objectives are not satisfied, repeat the

procedure using different desired dominant closed-loop pole locations and/or different values for z

and p.


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Loop Shaping Lead Compensator Design

1. Translate the design objectives into specifications for2:

(i) The loop transfer function crossover frequency.

(ii) The minimum gain of the loop transfer function at low frequencies.

(iii) The phase margin.


G s Ks

scp

z

z

p

( ) =+

+

3. Construct a Bode plot of the loop transfer function with Gc(s) = K and K set to a value such that

specifications (i) and (ii) are satisified.

4. From your Bode plot, determine the crossover frequency m and the amount of additional phase lead

m that is needed at the crossover frequency to satisfy the phase margin specification.

5. Determine pand z using

m z pp

z

m

m

= =+

and

1

1

sin

sin


procedure selecting different specifications in Step 1.

2 Oftentimes the design objectives will suggest a minimum loop transfer function crossover frequency or a minimum gain of

the loop transfer function at low frequencies, but not both.


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1

0

Bode of with

p

z

z

p

p z

s

s

+

+>

p

z

z

p

Frequency (rad/sec)

Frequency (rad/sec)

Magnitude(LogScale)

Phase(deg)

w

z

w

m

w

p

w

z

w

m

w

p

m

m z p m

p

z

p

z

p

z

m

m

= =

+

=+

sin

sin

sin

1

1

1

1


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Root-Locus Lag Compensator Design

1. Translate the design objectives into specifications for the dominant closed-loop pole locations and the

steady-state tracking error constant.


Gc(s) = K

s + z

s + p

Let Gc(s) = K and construct a root locus of the closed-loop poles versus K. Determine a value for K so

as to achieve the desired dominant closed-loop pole locations. If this is not possible, design a lead

compensator so that it will be possible, then return to Step 1.

3. Using Gc(s) = K and the K value you determined in Step 2, determine the factor by which the

steady-state tracking error constant must be increased in order to satisfy the steady-state tracking

error constant specification.

4. Determine z and p using

=zp

and such that the difference between the angle of the pole and the angle of the zero measured from

the desired dominant closed-loop pole locations is less than 2.


procedure using different specifications for the dominant closed-loop pole locations and the steady-

state tracking error constant.


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Loop Shaping Lag Compensator Design

1. Translate the design objectives into specifications for3:

(i) The loop transfer function crossover frequency.

(ii) The minimum gain of the loop transfer function at low frequencies.

(iii) The phase margin.


G (s) =cs

sz

p

+

+

3. Construct a Bode plot of the loop transfer function with Gc(s) = 1.

4. From your Bode plot, check to see if specifications (i) and (

iii) are satisfied. If not, design a lead

compensator to satisfy them, then return to Step 1.

5. From your Bode plot, determine the factor by which the lag compensator must boost the gain of

the loop transfer function at low frequencies compared to the crossover frequency, in order to satisfy

specification (ii).

6. To make sure that the lag compensator will not add too much phase lag to the loop transfer function

frequency response at the crossover frequency, choose z to be at least a factor of ten less than the

crossover frequency.

7. Determine pusing

= z

p


procedure selecting different specifications in Step 1.

3 Oftentimes the design objectives will suggest a minimum loop transfer function crossover frequency or a minimum gain of

the loop transfer function at low frequencies, but not both.


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1

Magnitude(LogScale)

0

Phase(deg)

Bode of withs

s

z

p

z p

+

+>

z

p

Frequency (rad/sec)

Frequency (rad/sec)

w

p

w

m

w

z

w

p

w

m

w

z

m

m z p m

z

p

z

p

z

p

m

m

= =

+=

+

sin

sin

sin

1

1

1

1

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