CDS 101/110: Lecture 10.1Limits on Performance

November 28, 2016

Goals: Introduce concept of limits on performance of feedback systems Introduce Bodes integral formula and the waterbed effect Show some of the limitations of feedback due to RHP poles and zeros

Reading: strm and Murray, Feedback Systems, Section 12.6

2

Loop Shaping: Design Loop Transfer Function

BWGM

PM

C(s) P(s)++

d

e u

-1

r +

n

y Translate specs to loop shape

= ()

Design C(s) to obey constraints

= =1 ( )

=1 ( )

Poles/Zeros from PID Poles/Zeros from

- Lead - Lag

Check the Gang of Four

=1

1 + (); =

()1 + ()

=()

1 + (); =

()1 + ()

3

Algebraic Constraints on Performance

Goal: keep S & T small S small low tracking error T small good noise rejection (and

robustness)

Problem: S + T = 1 Cant make both S & T small at the

same frequency Solution: keep S small at low frequency

and T small at high frequency Loop gain interpretation: keep L large

at low frequency, and small at high frequency

Transition between large gain and small gain complicated by stability (phase margin)

Sensitivityfunction

Complementarysensitivityfunction

Mag

nitu

de (d

B)

C(s) P(s)++

d

e u

-1

r +

n

y

4

Bodes Integral Formula and the Waterbed Effect

Bodes integral formula for = 11+()

= = = =

Let be the unstable poles of () and assume relative degree of () 2

Theorem: the area under the sensitivity function is a conserved quantity:

Waterbed effect:Making sensitivity smaller over some frequency range requires increase in sensitivity someplace elsePresence of RHP poles makes this effect worseActuator bandwidth further limits what you can doNote: area formula is linear in ; Bode plots are logarithmic

Frequency (rad/sec)

Mag

nitu

de (d

B)

Sensitivity Function

-40

-30

-20

-10

0

10

100

101

102

103

104

Area below 0 dB + area above 0 dB = Re pk = constant

5

Example: Magnetic Levitation

System description Ball levitated by electromagnet Inputs: current thru electromagnet Outputs: position of ball, , (from IR sensor) States: , Dynamics: = , = magnetic force

generated by wire coilIR

receivierIR

transmitter

Electro-magnet

Ball

System Dynamics

= 2/2

= + 0where:

= current to electromagnet = voltage from IR sensor

Linearization:

=

2 2

Poles at = open loop unstable

6

Bode Plot of Open Loop System

IRreceivier

IRtransmitter

Electro-magnet

Ball

Real Axis

Imag

inar

y Ax

is

Nyquist Diagram

-4 -2 0 2 4 6 8-1.5

-1

-0.5

0

0.5

1

1.5

Frequency (rad/sec)

Phas

e (d

eg);

Mag

nitu

de (d

B)

Bode Diagram

-100

-50

0

50

100

101

102

103

104-200

-150-100-50

050

Note: RHP pole in L need one net encirclement (CCW)

Linearization:

=

2 2

Poles at = open loop unstable

7

Control Design

Need to create encirclement To offset RHP pole Loop shaping is not useful here Flip gain to bring Nyquist plot over -1

point Insert phase to create CCW

encirclement

Can accomplish using a lead compensator Produce phase lead at crossover Generates loop in Nyquist plot

Real Axis

Nyquist Diagram

-4 -2 0 2 4 6 8-1.5

-1

-0.5

0

0.5

1

1.5

Frequency (rad/sec)

Phas

e (d

eg);

Mag

nitu

de (d

B)

Bode Diagram

-100

-50

0

50

100 101 102 103 104-200-150

-100-50

050

=0=1=0

8

Control Design

Need to create encirclement

Loop shaping is not useful here Flip gain to bring Nyquist plot over -1

point Insert phase to create CCW

encirclement

Can accomplish via lead compensator Produce phase lead at crossover Generates loop in Nyquist plot

Real Axis

Imag

inar

y Ax

is

Nyquist Diagram

-4 -2 0 2 4 6 8-1.5

-1

-0.5

0

0.5

1

1.5

Frequency (rad/sec)

Phas

e (d

eg);

Mag

nitu

de (d

B)

Bode Diagram

-100

-50

0

50

100 101 102 103 104-200-150

-100-50

050

=0=1=0

9

Performance LimitsNominal design gives low perf

Not enough gain at low frequency Try to adjust overall gain to improve low

frequency response Works well at moderate gain, but notice

waterbed effect

Bode integral limits improvement

Must increase sensitivity at some point

Frequency (rad/sec)

Mag

nitu

de (d

B)

Sensitivity Function

-40

-30

-20

-10

0

10

100

101

102

103

104

Time (sec.)

Ampl

itude

Step Response

0 0.04 0.08 0.12 0.160

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

10

Right Half Plane ZerosRight half plane zeros produce non-minimum phase behavior

Phase vs. frequency has additional lag (not minimum) for a given magnitude Can cause output to move opposite from input for a short period of time

Example: vs

Frequency (rad/sec)

Phas

e (d

eg);

Mag

nitu

de (d

B)

Bode Diagrams

-30

-20

-10

0

10

100 101 102-300

-200

-100

0

term

H1

H1

H2

, H2

Time (sec.)

Ampl

itude

Step Response

0 0.2 0.4 0.6 0.8 1 1.2-0.2

0

0.2

0.4

0.6

0.8

1

1.2

H1

H2

11

Example: Lateral Control of the Ducted Fan

Source of non-minimum phase behavior To move left, need to make > 0 To generate positive , need 1 > 0 Positive 1 causes fan to move right

initially Fan starts to move left after short time

(as fan rotates)

Poles: 0, 0, Zeros:

Time (sec.)

Ampl

itude

Step Response

0 0.2 0.4 0.6 0.8 1-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Fan moves right andthen moves to the left

12

Stability in the Presence of (RHP) ZerosLoop gain limitations Poles of closed loop = poles of 1 + L. Suppose () = ()/(), where k is the

controller gain

For large k, closed loop poles approach open loop zeros RHP zeros limit maximum gain serious design constraint!

Root locus interpretation Plot location of eigenvalues as a

function of the loop gain k Can show that closed loop poles go

from open loop poles (k = 0) to openloop zeros ( = )

-7 -6 -5 -4 -3 -2 -1 0 1 2 3-8

-6

-4

-2

0

2

4

6

8

Real Axis

Imag

Axi

s

Original polelocation (k = 0)

Closed loopzeros

13

-150

-100

-50

0

50

Frequency (rad/sec)

Mag

nitu

de (d

B)

Additional performance limits due to RHP zerosAnother waterbed-like effect: look at maximum of over frequency range:

Theorem: Suppose that () has a RHP zero at . Then there exist constants 1 and 2(depending on 1, 2, ) such that 1 log1 + 22 0.

M1 typically

14

Summary: Limits of PerformanceMany limits to performance Algebraic: S + T = 1 RHP poles: Bode integral formula RHP zeros: Waterbed effect on peak of S

Main message: try to avoid RHP poles and zeros when-ever possible (eg, re-design)

Frequency (rad/sec)

Mag

nitu

de (d

B)

Sensitivity Function

-40

-30

-20

-10

0

10

100

101

102

103

104

0

log | |2

= 1

RHP poles

Announcements

Homework #8 is due on Friday, 2 pm In class or HW slot (102 STL)

Final exam Out on 5 Dec (Mon.) Due on Fri. December 9, by 5 pm:

- turn in to Sonya Lincoln - 250 Gates-Thomas

Final exam review: December 2 from 2-3 pm, 105 Annenberg

Office hours during study period- 5 Dec (Mon), 3-5 pm- 6 Dec (Tue), 3-5 pm

15

YouTube: Chicken Head Tracking

CDS 101/110: Lecture 10.1Limits on PerformanceLoop Shaping: Design Loop Transfer FunctionAlgebraic Constraints on PerformanceBodes Integral Formula and the Waterbed EffectExample: Magnetic LevitationBode Plot of Open Loop SystemControl DesignControl DesignPerformance LimitsRight Half Plane ZerosExample: Lateral Control of the Ducted FanStability in the Presence of (RHP) ZerosAdditional performance limits due to RHP zerosSummary: Limits of PerformanceAnnouncements