CDS 101: Lecture 8.2Tools for PID & Loop Shaping

Richard M. Murray17 November 2004

Goals:Show how to use “loop shaping” to achieve a performance specificationIntroduce new tools for loop shaping design: Ziegler-Nichols, root locus, lead compensationWork through some example control design problems

Reading: Åström and Murray, Analysis and Design of Feedback Systems, Ch 8

17 Nov 04 R. M. Murray, Caltech CDS 2

Tools for Designing PID controllers

Zeigler-Nichols tuningDesign PID gains based on step responseWorks OK for many plants (but underdamped)Good way to get a first cut controllerFrequency domain version also exists

Caution: PID amplifies high frequency noiseSol’n: pole at high frequency

Caution: Integrator windupProlonged error causes large integrated errorEffect: large undershoot (to reset integrator)Sol’n: move pole at zero to very small valueFancier sol’n: anti-windup compensation

C(s) ++-

d

r ye uP(s)

1( ) (1 )DI

C s K T sT s

= + +

1.2 /K a= 2*IT L= / 2DT L=

Point of maximumslope

L

a

Step response

Bode Diagrams

Frequency (rad/sec)

Mag

nitu

de (d

B)

0

20

40

60

80

10-4 10-2 100 102 104

17 Nov 04 R. M. Murray, Caltech CDS 3

Example: PID cruise control

Ziegler-Nichols design for cruise controllerPlot step response, extract L and a, compute gains

Result: sluggish ⇒ increase loop gain Frequency (rad/sec)

Phas

e (d

eg);

Mag

nitu

de (d

B)

Bode Diagrams

-100

-50

0

50

100

10-3 10-2 10-1 100 101-200

-100

0

100

Time (sec.)

Ampl

itude

Step Response

0 10 20 30 40 50 600

0.5

1

1.5

1/( )/

m rP ss b m s a

= ⋅+ +

0 10 20 30 40 50-0.1

0

0.1

0.2

0.3

0.4

0.5

2.490.039

La== step

slope

1.2 /K a= 2*IT L= / 2DT L=

1( ) (1 )DI

C s K T sT s

= + +

( )L s( )P s

( )C s

( )L s( )C s( )P s

17 Nov 04 R. M. Murray, Caltech CDS 4

Pole Zero Diagrams and Root Locus Plots

Pole zero diagram verifies stability

Roots of 1 + PC give closed loop polesCan trace the poles as a parameter is changed:

Root locus = locus of roots as parameter value is changedCan plot pole location versus any parameter; just repeatedly solve for rootsCommon choice in control is to vary the loop gain (K)

C(s)+-

r ye uP(s)

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

-6

-4

-2

0

2

4

6

8

Real AxisIm

agAx

is1( ) (1 )DI

C s K T sT s

= + +

α

α

Original polelocation (α = 0)

Pole goesto ∞

Pole goesunstable for some α

Pole goesto terminalvalue

Poles mergeand split

17 Nov 04 R. M. Murray, Caltech CDS 5

One Parameter Root LocusBasic idea: convert to “standard problem”:

Look at location of roots as α is varied over positive real numbersIf “phase” of a(s)/b(s) = 180°, we can always choose a real α to solve eqnCan compute the phase from the pole/zero diagram

Trace out positions in plane where phase = 180°

At each of these points, there exists gain α to satisfy a(s) + αb(s) = 0All such points are on root locus

( ) ( ) 0a s b sα+ =

1 2

1 2

( ) ( )( ) ( )( )( ) ( )( ) ( )

m

n

a s s z s z s zG s kb s s p s p s p

+ + += =

+ + +L

L

0 0 1 0

0 1 0

( ) ( ) ( )( ) ( )

m

n

G s s z s zs p s p

∠ = ∠ + + +∠ + −∠ + − −∠ +

L

L

φi = phase contribution from s0 to -pi

ψI = phase contribution from s0 to -zi

17 Nov 04 R. M. Murray, Caltech CDS 6

Root Locus for Loop Gain

Loop gain as root locus parameterCommon choice for control designSpecial properties for loop gain

Roots go from poles of PC to zeros of PCExcess poles go to infinityCan compute asymptotes, break points, etc

Very useful tool for control designMATLAB: rlocus

Additional commentsAlthough loop gain is the most common parameter, don’t forget that you can plot roots versus any parameterNeed to link root location to performance…

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

-6

-4

-2

0

2

4

6

8

Real AxisIm

agAx

is

C(s)+-

r ye uP(s)α

Open loop polelocation (α = 0)

Closed pole goes to openloop zeros

Asymptotes for excess poles at

(360˚/(P-Z))

Real axis to theleft of odd # of real poles & zeros is on root locus( )1 ( ) ( ) 0

( )n s d s n sd s

α α+ → + =

17 Nov 04 R. M. Murray, Caltech CDS 7

Second Order System Response

Second order system responseSpring mass dynamics, written in canonical form

Performance specifications

Guidelines for pole placementDamping ratio gives Re/Im ratioSetting time determined by –Re(λ)

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0-10

-8

-6

-4

-2

0

2

4

6

8

10

Imag

Axis

Real Axis

Root Locus Editor (C)

2 2

2 2( )2 ( )( )

n n

n n n d n d

H ss s s j s j

ω ωςω ω ςω ω ςω ω

= =+ + + + + −

1.8 /3.9 /

r n

s n

TT

ωςω

≈≈

21d nω ω ς= −

2/ 1

SS 0pM e

e

πς ς− −≈

=

44%16%4%Mp

-3.9-1.7-1

Slope

0.250.5

0.707ζ

Ts < x

Mp < yDesired regionfor closed loop poles

17 Nov 04 R. M. Murray, Caltech CDS 8

Effect of pole location on performance

Idea: look at “dominant poles”Poles nearest the imaginary axis (nearest to instability)Analyze using analogy to second order system

PZmap complements informa-tion on Bode/Nyquist plots

Similar to gain and phase calculationsShows performance in terms of the closed loop polesParticularly useful for choosing system gainAlso useful for deciding where to put controller poles and zeros (with practice [and SISOtool])

Time (sec.)

Am

plitu

de

Step Response

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4From: U(1)

To: Y

(1)

Real Axis

Imag

Axi

s

Pole-zero map

-5 -4 -3 -2 -1 0 1-1.5

-1

-0.5

0

0.5

1

1.5

Time (sec.)

Am

plitu

de

Step Response

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4From: U(1)

To: Y

(1)

Real Axis

Imag

Axi

s

Pole-zero map

-5 -4 -3 -2 -1 0 1-1.5

-1

-0.5

0

0.5

1

1.5

Time (sec.)

Am

plitu

de

Step Response

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5From: U(1)

To: Y

(1)

Real Axis

Imag

Axi

s

Pole-zero map

-5 -4 -3 -2 -1 0 1-1.5

-1

-0.5

0

0.5

1

1.5

x

x

x

x

x

x

x

x

x

x

x

x

x

17 Nov 04 R. M. Murray, Caltech CDS 9

Example: PID cruise controlStart with PID control design: Modify gain to improve performance

Use MATLAB sisotoolAdjust loop gain (K) to reduce overshoot and decrease settling time

ζ ≈ 1 ⇒ less than 5% overshootRe(p) < -0.5 ⇒ Ts less than 2 sec

1/( )/

m rP ss b m s a

= ⋅+ +

1( ) (1 )DI

C s K T sT s

= + +

17 Nov 04 R. M. Murray, Caltech CDS 10

Example: Pitch Control for Caltech Ducted Fan

System descriptionVector thrust engine attached to wingInputs: fan thrust, thrust angle (vectored)Outputs: position and orientationStates: x, y, θ + derivativesDynamics: flight aerodynamics

Control approachDesign “inner loop” control law to regulate pitch (θ ) using thrust vectoringSecond “outer loop” controller regulates the position and altitude by commanding the pitch and thrustBasically the same approach as aircraft control laws

17 Nov 04 R. M. Murray, Caltech CDS 11

Performance Specification and Design Approach

Design approachOpen loop plant has poor phase marginAdd phase lead in 5-50 rad/sec rangeIncrease the gain to achieve steady state and tracking performance specsAvoid integrator to minimize phase

Performance Specification≤ 1% steady state error

Zero frequency gain > 100≤ 10% tracking error up to 10 rad/sec

Gain > 10 from 0-10 rad/sec≥ 45˚ phase margin

Gives good relative stabilityProvides robustness to uncertaintyFrequency (rad/sec)

Phas

e (d

eg);

Mag

nitu

de (d

B)

-50

0

50

100 101-200

-150

-100

-50

0

2( ) rP sJs ds mgl

=+ +

( ) s aC s Ks b+

=+

2530015 300

ab

K

=== ⋅

17 Nov 04 R. M. Murray, Caltech CDS 12

Summary: PID and Root Locus

PID control designVery common (and classical) control techniqueGood tools for choosing gains

Root locusShow closed loop poles as function of a free parameter

Performance limitsRHP poles and zeros place limits on achievable performanceWaterbed effect

p I Du K e K e K e= + +∫ &

Frequency (rad/sec)

Phas

e (d

eg);

Mag

nitu

de (d

B)

Bode Diagrams

0

10

20

30

40

50

10-3 10-2 10-1 100 101 102 103-100

-50

0

50

100

11

ITω = 2

1

DTω =

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

-6

-4

-2

0

2

4

6

8

Real Axis

Imag

Axis