Control Theory (035188)lecture no. 2

Leonid Mirkin

Faculty of Mechanical EngineeringTechnion — IIT

Outline

Goals and means of loop shaping

M- and N-circles

Nichols chart

Bode’s gain-phase relation

Philosophical remark: Bode’s sensitivity integral

Typical requirements on loop L.j!/

low frequencies

high frequencies

0Mag

nitude

(dB)

crosover region

�180

Phase(deg

)

!c

Frequency (rad/sec)

Typical operations with L.j!/

low frequencies

high frequencies

0

Mag

nitude

(dB)

crosover region

�180

Phase(deg

)

!c

Frequency (rad/sec)

We might need to:

I set required crossover freq. !c

(means: proportional controller)

I increase phase around !c

(means: lead controller)

I increase low-frequency gain(means: lag controller)

I decrease high-frequency gain(means: low-pass filter)

Lead controller

General form: Clead.s/ Dp˛ s C !m

s Cp˛ !m, where ˛ > 1 is a parameter.

Mag

nitude

(dB)

Phase(deg

)

Frequency (rad/sec)

10 lg ˛

0

0

�10 lg ˛

�m ´ arcsin ˛�1˛C1

0:01!m 0:1!m !m 10!m 100!m

Bode Diagram of Clead.s/

Lead controller for different ˛

Frequency (rad/sec)

−15

−10

−5

0

5

10

15

10−2

10−1

100

101

102

0

30

60

90

Mag

nitude

(dB)

Phase(deg

)

Bode diagrams of Clead.s/ for !m D 1 and ˛ D f2; 4; 10; 20g

Phase lead of lead controllers

Maximal phase lead, �m D arcsin ˛�1˛C1 , is achieved at ! D !m:

1 2 3 4 5 7 10 14 20 30 ˛0

19:5

30

36:941:8

48:6

54:960:164:869:3

�m

.˛/

Practically, the valuesI ˛ > 20 unsuitable,

because of implementation problems and the low slope of the curve abovefor large ˛. If �m > 60ı required, use 2 or more lead controllers.

Clead: properties and usage

Pros:I increases �ph,

thus helping to increase the distance of L.j!/ from the critical point.

Cons:I decreases the low-frequency gainI increases the high-frequency gainI decreases �g

(because, typically, !c < !� and jClead. j!/j > 1 for ! > !m)

Usage (some rules of thumb):I choose !m � the intended crossover frequency !c

I choose ˛ < 20(if phase lead > 60ı needed, several lead controllers may be used)

Lag controller

General form: Clag.s/ D10s C !m10s C !m=ˇ

, where ˇ > 1 is a parameter.

Mag

nitude

(dB)

Phase(deg

)

Frequency (rad/sec)

20 lg ˇ

10 lg ˇ

0

� �5:7ı

��m

0:001!m 0:01!m 0:1!m=p

ˇ 0:1!m !m 10!m

Bode Diagram of Clag.s/

Lag controller for different ˇ

0

10

20

30

40

50

10−3

10−2

10−1

100

−90

−45

0

Mag

nitude

(dB)

Phase(deg

)

Frequency (rad/sec)

Bode diagrams of Clag.s/ for !m D 1 and ˇ D f2; 4; 10; 30; 1g

Clag: properties and usage

Pros:I increases the low-frequency gain of L.j!/

Cons:I adds phase lag

thus might decrease the distance of L.j!/ from the critical point.

Usage (rule of thumb):I choose !m � the intended crossover frequency !c

(this will add a phase lag of at most 5:7ı at !c)

I add � 5:7ı to lead controller (if lag to be added after lead)

Low-pass filters

Might be of the form Clp.s/ D!b

s C !bor Clp.s/ D

!2bs2 C 2�!bs C !2b

, with

Bode Diagrams of Clp1

and Clp2

(with ζ=0.5)

−40

−20

0

−180

−135

−90

−45

0

Mag

nitude

(dB)

Phase(deg

)

Frequency (rad/sec)

0:1!b !b 10!b

Clp: properties and usage

Pros:I decreases the high-frequency gain of L.j!/

Cons:I adds phase lag

thus might decrease the distance of L.j!/ from the critical point.

Usage (rule of thumb):I choose !b � decade above crossover frequency !c

(this will still add some phase lag at !c)

I add additional phase lead to lead controller

Loop shaping design: possible steps

The design steps:

1. Add proportional gain to get required crossover frequency !c

2. Add (if necessary) lead controller to improve �ph

3. Add (if necessary) lag controller to provide desired low-frequency gain

4. Add (if necessary) low-pass filter to decrease high-frequency gain

5. Run simulations to verify that everything is as expected

Remark:Sometimes (e.g., if an integral action is required) steps 2. and (3.,4.) may beexchanged

Outline

Goals and means of loop shaping

M- and N-circles

Nichols chart

Bode’s gain-phase relation

Philosophical remark: Bode’s sensitivity integral

M-circles

M-circles are contours of constant closed-loop magnitude on Nyquist plane.

Let L.j!/ D x C jy. Then T .j!/ D xC jy1CxC jy . Hence,

jT .j!/j2 DM 2 ” M 2.1C x/2 CM 2y2 D x2 C y2” .1 �M 2/x2 � 2M 2x C .1 �M 2/y2 DM 2:

Then two cases are possible:

M D 1 then x D �12

(vertical line)

M ¤ 1 then .1 �M 2/�x2 � 2 M2

1�M2x ˙ M4

.1�M2/2C y2� DM 2, so we get:

�x � M2

1�M2

�2 C y2 D � M

1�M2

�2

(circle centered at M2

1�M2 with radius Mj1�M2j )

M-circles (contd)

−2 −1.5 −1 −0.5 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 dB

−20 dB

−6 dB

−3 dB

−12 dB

20 dB

6 dB

3 dB

12 dB

Real Axis

Ima

gin

ary

Axis

M-circles on Nyquist diagram

M-circles: how to read

−1 0

0

−12dB

−6dB

−3dB

3dB6dB

12dB

Re

Im

L.j!/

!1!2

!3

−20

−12

−6

−3

0

3

6

12

Frequency (rad/sec)

Ma

gn

itu

de

(d

B)

jT .j!/j

!1 !2 !3

N-circles

N-circles are contours of constant closed-loop phase on Nyquist plane:

−2 −1.5 −1 −0.5 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−350o

−10o

−315o

−45o

±90o

Real Axis

Ima

gin

ary

Axis

N-circles on Nyquist diagram

Outline

Goals and means of loop shaping

M- and N-circles

Nichols chart

Bode’s gain-phase relation

Philosophical remark: Bode’s sensitivity integral

Motivation

For L.s/ D 12960000.sC5/.s2C0:8sC16/.s2C1:52sC1444/.sC10/.sC100/.s2C3sC9/.s2C0:48sC9/.s2C0:8sC1600/2 e

�0:075s:

Nyquist Diagram

Real Axis

Imagin

ary

Axis

−20 −15 −10 −5 0 5 10

−20

−15

−10

−5

0

5

10

15

20

Nyquist Diagram

Real Axis

Imagin

ary

Axis

−2 −1.5 −1 −0.5 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Pitfalls of the Nyquist plot:I becomes messy for systems with multiple crossover frequenciesI crossover region is imperceptible for systems with large resonant peaksI lacks system composition (superposition) properties of Bode diagrams

Remedy: Nichols chart

Nichols chart of transfer function L.s/ is plot of jL.j!/j (in dB) vs. †L.j!/(in degrees) as frequency ! changes from 0 to1.

Nichols Chart

Open−Loop Phase (deg)

Op

en

−L

oo

p G

ain

(d

B)

−720 −630 −540 −450 −360 −270 −180 −90 0−40

−30

−20

−10

0

10

20

30

40

6 dB

3 dB

1 dB

0.5 dB

0.25 dB

0 dB

−1 dB

−3 dB

−6 dB

−12 dB

−20 dB

−40 dB

Nichols chart: advantages

Since phase scale is linear rather than polar,I Nichols chart is typically cleaner than Nyquist plot

especially for systems with large phase lags, like time-delay systems.

As magnitude scale is in dB, regions with large magnitude don’t dominate,hence

I the crossover region is more visible.

Also the consequence of the logarithmic scale of jL.j!/j is thatI multiplication of systems results in superposition

on Nichols chart, almost as easy as on the Bode diagrams.

M-circles on Nichols charts

−360 −270 −180 −90 0 90 180 270 360−15

−10

−5

0

5

10

15

6 dB

3 dB

−3 dB

−6 dB

−12 dB

Open−Loop Phase (deg)

Op

en

−L

oo

p G

ain

(d

B)

M-circles on Nichols chart

N-circles on Nichols charts

−360 −270 −180 −90 0 90 180 270 360−15

−10

−5

0

5

10

15

−10o

−45o

−180o

−350o

−315o

Open−Loop Phase (deg)

Op

en

−L

oo

p G

ain

(d

B)

N-circles on Nichols chart

M-circles on Nichols charts: how to read

−250 −200 −150 −100 −50 0−25

−20

−15

−10

−5

0

5

10

−20dB

−12dB

−6dB

−3dB

0dB3dB

6dB

Open−Loop Phase (deg)

Op

en

−L

oo

p G

ain

(d

B)

L.j!/

!1!2!3

−20

−12

−6

−3

0

3

6

12

Frequency (rad/sec)

Ma

gn

itu

de

(d

B)

jT .j!/j

!1 !2 !3

Nichols charts of elementary systems

L.s/ Bode Nyquist Nichols

ks

−20

−15

−10

−5

0

5

10

15

20

Ma

gn

itu

de

(d

B)

10−1

100

101

−91

−90.5

−90

−89.5

−89

Ph

ase

(d

eg

)

Bode Diagram

Frequency (rad/sec)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

10 dB

−20 dB

−6 dB

−3 dB

−12 dB

20 dB

6 dB

3 dB

12 dB

Nyquist Diagram

Real Axis

Imagin

ary

Axis

−360 −315 −270 −225 −180 −135 −90 −45 0−20

−15

−10

−5

0

5

10

Nichols Chart

Open−Loop Phase (deg)

Open−

Loop G

ain

(dB

)

k�sC1

−15

−10

−5

0

5

10

Ma

gn

itu

de

(d

B)

10−1

100

101

−90

−45

0

Ph

ase

(d

eg

)

Bode Diagram

Frequency (rad/sec)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

10 dB

−20 dB

−6 dB

−3 dB

−12 dB

20 dB

6 dB

3 dB

12 dB

Nyquist Diagram

Real Axis

Imagin

ary

Axis

−360 −315 −270 −225 −180 −135 −90 −45 0−20

−15

−10

−5

0

5

10

Nichols Chart

Open−Loop Phase (deg)

Open−

Loop G

ain

(dB

)

k!2n

s2C2�!nsC!2n

−40

−30

−20

−10

0

10

Ma

gn

itu

de

(d

B)

10−1

100

101

−180

−135

−90

−45

0

Ph

ase

(d

eg

)

Bode Diagram

Frequency (rad/sec)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

10 dB

−20 dB

−6 dB

−3 dB

−12 dB

20 dB

6 dB

3 dB

12 dB

Nyquist Diagram

Real Axis

Imagin

ary

Axis

−360 −315 −270 −225 −180 −135 −90 −45 0−20

−15

−10

−5

0

5

10

Nichols Chart

Open−Loop Phase (deg)

Open−

Loop G

ain

(dB

)

e�sh −1

−0.5

0

0.5

1

Ma

gn

itu

de

(d

B)

10−1

100

101

−630

−540

−450

−360

−270

−180

−90

0

Ph

ase

(d

eg

)

Bode Diagram

Frequency (rad/sec)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

10 dB

−20 dB

−6 dB

−3 dB

−12 dB

20 dB

6 dB

3 dB

12 dB

Nyquist Diagram

Real Axis

Imagin

ary

Axis

−720 −630 −540 −450 −360 −270 −180 −90 0−20

−15

−10

−5

0

5

10

Nichols Chart

Open−Loop Phase (deg)

Open−

Loop G

ain

(dB

)

Gain and phase margins on Nichols chart

Gain margin (�g) and phase margin (�ph) are easily calculable from Nicholscharts:

I �g is the vertical distance from the critical point;I �ph is the horizontal distance from the critical point.

Nichols Chart

Open−Loop Phase (deg)

Open−

Loop G

ain

(dB

)

−360 −315 −270 −225 −180 −135 −90 −45 0−30

−25

−20

−15

−10

−5

0

5

10

15

20

�ph

�g

Nichols chart of lead controller

Lead controller: Clead.s/ Dp˛ s C !m

s Cp˛!m, ˛ > 1.

Frequency (rad/sec)

−15

−10

−5

0

5

10

15

10−2

10−1

100

101

102

0

30

60

90

Mag

nitude

(dB)

Phase(deg

)

Bode diagrams of Clead.s/ for !m D 1 and ˛ D f2; 4; 10; 20g

Open−Loop Phase (deg)

Op

en

−L

oo

p G

ain

(d

B)

−180 −135 −90 −45 0 45 90 135 180−20

−15

−10

−5

0

5

10

15

20

6 dB

3 dB

1 dB

Nichols chart of Clead.s/ for !m D 1 and ˛ D f2; 4; 10; 20g

Nichols chart of lag controller

Lag controller: Clag.s/ D10s C !m10s C !m=ˇ

, ˇ > 1.

0

10

20

30

40

50

10−3

10−2

10−1

100

−90

−45

0

Mag

nitude

(dB)

Phase(deg

)

Frequency (rad/sec)

Bode diagrams of Clag.s/ for !m D 1 and ˇ D f2; 4; 10; 30; 1g

Open−Loop Phase (deg)

Open−

Loop G

ain

(dB

)

−180 −135 −90 −45 0 45 90 135 180

−5

0

5

10

15

20

25

30

6 dB

3 dB

1 dB

0.5 dB

Nichols chart of Clag.s/ for !m D 1 and ˇ D f2; 4; 10; 30; 1g

Outline

Goals and means of loop shaping

M- and N-circles

Nichols chart

Bode’s gain-phase relation

Philosophical remark: Bode’s sensitivity integral

Dream loop shape

We’d prefer to have narrow crossover region, somethng like this:

low frequencies

high frequencies

0

Mag

nitude

(dB)

!c

Intuitively, it is hard to believe that this is possible (too good to be true:-). Itturns out that this “intuition” can be rigorously justified.

Bode’s gain-phase relation: minimum-phase loop

Let L.s/ be stable and minimum-phase and such that L.0/ > 0. Then 8!0

argL.j!0/ D 1

�

Z 1

�1d lnjL.j�/j

d�ln coth

j�j2d�; where �´ ln !

!0

(coth x´ exCe�x

ex�e�x ). Function ln coth j�j2D ln

ˇ̌!C!0

!�!0

ˇ̌:

−3 −2 −1 0 1 2 30

1

2

3

4

5

�

lnco

thj�j 2

Bode’s gain-phase relation: what does it mean

Since ln coth j�j2

decreases rapidly1 as ! deviates from !0,

I argL.j!0/ depends mostly on d lnjL. j�/jd� near frequency !0.

On the other hand,

Id lnjL.j�/j

d�is actually the slope of Bode plot of L.j�/.

It can be shown that

argL.j!0/ <

8<̂

:̂

�N � 65:3ı; if roll-off of jL.j!/j is N for 13� !

!0� 3

�N � 75:3ı; if roll-off of jL.j!/j is N for 15� !

!0� 5

�N � 82:7ı; if roll-off of jL.j!/j is N for 110� !

!0� 10

In other words,I high negative slope of jL.j!/j necessarily causes large phase lag.

1Think of it as � ı.! � !0/.

Bode’s gain-phase relation: implication

For systems with rigid loops, it is advisable toI keep loop roll-off 6� 1 in the crossover region2

to guarantee that L.j!/ is far enough from the critical point. This, in turn,means that

I low- and high-frequency regions should be well-separated.

This is the reason why our “dream shape” is not an option.

2I.e., not much smaller than �20dB/dec slope of jL. j!/j.

Gain-phase relation: one nonminimum-phase zero

Let L.s/ has one RHP zero at ´ > 0. Then

L.s/ D �s C ´s C ´ Lmp.s/

for a minimum-phase Lmp.s/. Sinceˇ̌� j!C´

j!C´ˇ̌ � 1, jL.j!/j D jLmp.j!/j and

argL.j!0/ D argLmp.j!0/C arg � j!0C´j!0C´

D 1

�

Z 1

�1d lnjL.j�/j

d�ln coth

j�j2d� � 2 arctan !0

´:

Thus,I nonminimum-phase zero adds phase lag (especially at ! > ´)

imposing additional constraints on the slope of jL.j!/j in crossover region.

Gain-phase relation: complex nonminimum-phase zeros

Now, let L.s/ has a pair of RHP zero at ´r ˙ j´i, ´r > 0. Then

L.s/ D �s C ´r C j´i

s C ´r C j´i

�s C ´r � j´i

s C ´r � j´iLmp.s/

and we have:

argL.j!0/ D 1

�

Z 1

�1d lnjL.j�/j

d�ln coth

j�j2d� C arg

�j!0 C ´r ˙ j´i

j!0 C ´r ˙ j´i

D 1

�

Z 1

�1d lnjL.j�/j

d�ln coth

j�j2d�

� 2�arctan !0 C ´i

´rC arctan

!0 � ´i

´r

�:

This harden constraints when !0 > ´i, though may soften when !0 � ´i.

Phase of all-pass systems

−180

−150

−120

−90

−60

−30

0

0:1´ 0:5´ ´ 10´

Frequency (rad/sec)

Phase(deg

)

−360

−300

−240

−180

−120

−60

0

� D 1

� D 0:01

0:1!n !n 10!n

Frequency (rad/sec)

Phase(deg

)

arg �sC´sC´ arg s2�2�!nsC!2

ns2C2�!nsC!2

n

Gain-phase relation: multiple nonminimum-phase zeros

In this caseL.s/ D �s C ´1

s C ´1�s C ´2s C ´2

� � � �s C ´ks C ´k

Lmp.s/

and we have:

argL.j!0/ D 1

�

Z 1

�1d lnjL.j�/j

d�ln coth

j�j2d� �

kX

iD1arg�j!0 C ´ij!0 C ´i

;

which further harden constraints.

Limitations due to nonminimum-phase zeros

For systems with a single crossover frequency3 RHP zeros near !c

I impose additional limitations on the roll-off in the crossover region.

Consequently, a well-known rule of thumb says that for RHP systemsI crossover frequency !c should be < the smallest RHP zero.

Also, it is safe to claim (regarding RHP zeros) thatI closer to the real axis H) more restrictive crossover limitations

3For systems with resonant frequencies it sometimes might be desirable to inject a phaselag by adding RHP zeros. Yet this must be done with maximal care (don’t try it at home!)

Outline

Goals and means of loop shaping

M- and N-circles

Nichols chart

Bode’s gain-phase relation

Philosophical remark: Bode’s sensitivity integral

Bode’s sensitivity integral

Let L.s/ be loop transfer function with pole excess � 2, then, provided S.s/is stable,

Z 1

0

lnjS.j!/jd! D(0 if L stable

�PmiD1Repi otherwise (pi—unstable poles of L)

0

+

−

jS.j!/j > 1

jS.j!/j < 1

Frequency, !

lnjS.

j!/j

What does it mean ?

Some conclusions:I Since �

PRepi � 0, jS.j!/j cannot4 be < 1 over all frequencies.

I Improvements in one region inevitable cause deterioration in other.(so-called waterbed effect)

On qualitative level,I controller can only redistribute jS.j!/j over frequencies

and the art of control may thus be seen as art of redistribution of jS.j!/j.

4If pole excess of L.s/ is � 2, of course. Yet this is typical in applications.