ME190LLoop Shaping

Course IntroductionUC Berkeley

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Course Structure (Fall 2010)

Class detail– 1 hour of lecture/week, Friday, 10:00-11:00 AM.– 3113 Etcheverry Hall– BSpace

• Slides, Homeworks, Files

Workload– Weekly Homework assignments (1-2 hours/week workload)

• Hand in a clean notebook with all problems on Friday, December 10• Work steady, but go back and fix things as your understanding improves• Goal (serious): everybody eventually does every problem correctly• During semester, come to office hours, show me your work, and I’ll assess it

– Access to Matlab, Simulink, Control System Toolbox• ME students can get accounts in 2107, 2109; others see me if you need access

– 1 Final exam (take-home), available Dec 3, due (w/ homework) on Dec 10

Prereq– ME 132, or equivalent

Me:– Andrew Packard, 5116 Etcheverry Hall, apackard@berkeley.edu– Office hours: M 1:30-2:30, Tu 3:30-4:30, W 10:30-11:30

Supplementary Reading Material

SP: Skogestad and Postlewaite, Multivariable Feedback Control: Analysis and Design, 2nd edition, Wiley, 2005.

– Chapter 2 (pp. 15-66, especially pp. 42-54)– Section 9.4 (pp. 364-382)

DFT: Doyle, Francis, Tannenbaum, Feedback Control Theory, Macmillan, 1992, out-of-print

– available at http://www.control.utoronto.ca/people/profs/francis/dft.html– Chapter 7, Loopshaping– Chapter 6, Analytic Design constraints

FPE: Franklin & Powell & Emami-Naeini, Feedback Control of Dynamic Systems, Prentice Hall, 5th edition.

– popular, general undergrad control textbook, decent reference

MG: McFarlane and Glover, A Loop Shaping Design Procedure using H∞ Synthesis

– pp. 759-769, vol 37, June 1992, IEEE Trans. on Automatic Control.

Basic Feedback LoopStart with simplest feedback topology

Closed-Loop Transfer functions

Stability– Roots of characteristic equation in open-left-half plane. Alternatively…– Nyquist plot of P(jω)C(jω) must encircle -1 the correct number of times

Disturbance rejection– Goal is that d1 and/or d2 have little effect on y

Noise insensitivity– Goal is that n has little effect on u and y

Adequate robustness margins– Adequate gain/phase/time delay margins,

1

1 1 1 1

1 1 1 1

PC P PC

PC PC PC PC

C PC C C

PC PC PC PC

1

2

r

d

d

n

y

u

n

d2

u

d1

yPC-r

We’ll learn this starting in next lecture

What is Loopshaping?Control-Design technique: shaping, by choice of C, the magnitude/phase of PC, so that the closed-loop system has desired properties

– Stable– Disturbance rejection– Noise insensitivity– Adequate robustness margins

Advantages– PC depends linearly (simple) on C, moreover |PC| = |P| |C|– Some closed-loop properties are very simply related to |PC|

Requires:– Understanding how open-loop gain (|PC|) is related to closed-loop properties– Understanding what closed-loop properties are achievable for a given plant– Techniques (graphical, computer-based, etc) to shape PC

Easy for benign plants, with standard goals; more challenging for others– unstable poles, RHP zeros, flexible modes, etc, unusual objectives

The harder aspects can be partially automated– theory– computation

Disturbance rejection

Transfer functions–Open-Loop (C=0)

–Closed-Loop

–Improvement Ratio: Closed-Loop/Open-Loop

Usual goals of feedback–Make |S(jω)|<<1 in frequency ranges where d1 and d2 are large

–Keep |S(jω)|<2 in all frequency ranges

Question: Can S be made small at all frequencies?

-n

d2

u

d1

yPCr

1 2, 1d y d yG P G

1 2

1,

1 1d y d y

PG G

PC PC

1:

1i

i

CLd y

OLd y

GS

G PC

10-2

10-1

100

101

102

10-3

10-2

10-1

100

101

Ma

gn

itud

e (

ab

s)

Bode Diagram

Frequency (rad/sec)

Typical S for PI feedback around simple P

Noise Insensitivity

Transfer functions–Open-Loop

–Closed-Loop

–Feedback always introduces sensor noise into the loop

Note

Consequently, at frequencies where |S(jω)|<<1, there will be direct transmission, (Gn→y≈-1) of n to y

Basic Limitation:

0, 0n u n yG G

,1 1n u n y

C PCG G

PC PC

11 1

1 1n y

PCG S

PC PC

2

1d y n yG G

-n

d2

u

d1

yPCr

Disturbance/Noise Tradeoff

Basic Limitation: at all frequencies

So… for y to be unaffected by d2 and n, we need–at frequencies where n is large, it must be that

• d2 is small

• |T| is small

–at frequencies where d2 is large, it must be that• n is small• |S| is small

2

1d y n yG G S

Sensitivity function

TComplimentary Sensitivity function

-n

d2

u

d1

yPCr

Conditions on |PC|

Transfer functions–Sensitivity and Complementary Sensitivity

Simple (approximate) inequalities: for small β (relative to 1)

How do closed-loop stability and robustness margins enter?

1,

1 1

PCS T

PC PC

1 1

1PC

PC

1

PCPC

PC

Need large loop-gain where |S| is to be small relative to 1

Need small loop-gain where |T| is to be small relative to 1

-n

d2

u

d1

yPCr

Margins/Stability

Gain, Phase, Time-Delay margins–All measure how close P(jω)C(jω) approaches the point -1 from

different, special directions.• TImeDelay margin takes into account frequency, ω, too

–So, what is important for these margins to not be too small, is the phase of PC, when |PC|≈1

• previous bounds on |PC| were for very large and very small values• So, ensuring adequate margins is not addressed by the previous

constraints• Analytic function theory tells us (soon) that and are

related, so in loopshaping, margins are accounted for by properly adjusting the slope of |PC| in the frequency range where |PC|≈1

Stability–Nyquist theorem: closed-loop system is stable if and only if the plot of

P(jω)C(jω) encircles -1 the “correct number of times”

These are starting to both sound challenging, with regard to shaping PC by choosing C. But, we’ll address them.

( )L j( )d L j

d

-n

d2

u

d1

yPCr

Also in this course…

Theoretical Tool–P is given–User specifies a candidate (ie, proposed) controller Cprop

• This is chosen (typically) to satisfy the easy (|PC| large, |PC| small) constraints

• The issues of closed-loop stability and adequate margins are ignored–A “magic” process determines if there is a controller which

• preserves the large-loop gain of PCprop

• preserves the small-loop gain of PCprop

• achieves closed-loop stability, with modest gain/phase margins

We will learn/derive the theory behind this, as well as use it in a series of examples.

-n

d2

u

d1

yPCr