Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

M.G. Safonov University of Southern California

[msafonov@usc.edu]

Zames Falb Multipliers for

MIMO Nonlinearities

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

Background: Robustness Analysis Role of Zames-Falb Multiplier M(s)

x y

]1,0[ 0, , ,0 ,),(2

xTyxx

y

x

y

L

12

12 )()(0 xxxx

)(xi

x

Stable if topological separation Δxy

x

yΠ

x

y

L

, 0,),(2

0)(

00)

sMΠ(smonotone

• Theorem (Conic-Sector/IQC)

• Zames-Falb M(s)

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

Typical Applications:

• relay (& hysteresis) • deadzone• actuator/sensor saturation• anti-windup compensator analysis

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

Historical Highlights• Zames-Falb Discovery (1967-1968):

– includes small-gain, positivity, Popov, off-axis circle, RL/RC,…– ad hoc graphical criteria, no attempt to optimize

• No better multipliers exist (Willems 1969)• …long hiatus…• Multiple SISO (x)’s (Safonov 1984)• Optimal ZF multipliers SISO (x)’s :

– Safonov-Wyetzner, 1987; Gapski-Geromel 1994– Chen-Wen 1996; Kothare-Morari 1999

• Repeated SISO (x)’s: D’Amato-Rotea-Jonsson-Megretski 2001

• TODAY’s TALK: Kulkarni-Safonov – Generalizing Zames-Falb multipliers for MIMO nonlinearities

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

Overview

• Zames-falb multipliers (1968)

• Multiplier Theory, IQCs

• Zames-Falb Claim (1968)

• A Counter-Example

• Valid MIMO Extensions

• Repeated Nonlinearity Results

• Concluding Remarks

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

Monotone, TI, memoryless and norm-bounded

Stable and LTI

Investigated System

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

Multipliers: causal and stable with finite gain

Strongly positive MH & finite normed N Stability Zames and Falb (1968)

Transformed System

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

Zames-Falb Multipliers

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

m

|||| z

0

1

planes

Re

Im

Zames-Falb Multipliers (contd.)

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

<

Stable if graphs are topologically separated

T

1e 1y

2y 2e

Background: IQC Stability

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

.for IQC an :)(

where

] ... [

matrixIdentity :

where

)(

i)(

)()2()1(

j

X

NNI

IXj

i

Nijijijij

IQC for

block-diagonal

u y),...,,(diag 21 N

IQCs for diagonal

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

Some IQCs : Known Results

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

• Obtain component IQCs i(j• Stack them together

• Optimize multipliers Mi(j)

• Each i(j) depends linearly on a multiplier Mi(j) > 0

• Convex • LMI problem

Multipliers - A Broad Overview

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

Fact: It Does the Opposite!

The Zames-Falb Claim

MIMO Generalization

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

A Counter-example

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

The Source of Trouble

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

The Correct MIMO Extension

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

• Zames-Falb Claim (1968)

• A Counter-Example

• Valid MIMO Extensions

• Main Results

• Concluding Remarks

Overview

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

Three Problems for MIMO Nonlinearities

• Problem 1: Find the subclass of ZF multipliers that does preserve positivity

• Problem 2: Find the subclass of MIMO nonlinearities for which ZF multipliers do preserve positivity

• Problem 3: Find the greatest superclass of ZF multipliers the preserves positivity of repeated SISO nonlinearities.

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

Main Result 1

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

Main Result 2

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

and

Main Result 3Problem 3

Find the greatest superclass of ZF multipliers the preserves positivity of repeated SISO nonlinearities.

Solution Multiplier matrices with impulse response

That is, multipliers for repeated nonlinearitiesare L1-norm diagonally dominant matrices

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

SISO Artificial neural networks … Haykin (1997) Communication networks Electric circuits, e.g. Chua’s circuit … Gibbens & Kelly (1999)

MIMO Flexible structures: spacecraft, aircraft, drives Static electric fields: semiconductors, motors MEMS: microbots

Instances of Monotone Nonlinearities

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

xx

xN

0)(

skew . 0)( skew jM

• Zames-Falb multipliers M are positivity preserving for incrementally positive MIMO nonlinearities N if and only if

either or

•For repeated SISO nonlinearities, we now have L1-norm diagonally dominant matrices

–Ideal for repeated saturation & deadzone–Best anti-windup stability test

More reliable robustness analysis for aircraft, spacecraft, networks, missiles,…

Conclusions

Workshop in honor of J. Boyd Pearson JrRice University 9-10 March 2001

ftp://routh.usc.edu/pub/safonov/safo99f.pdf ftp://routh.usc.edu/pub/safonov/safo01e.pdfftp://routh.usc.edu/pub/safonov/safo01b.pdf

Q U E S T I O N S ?? routh.usc.edu