Helton Talk

7/23/2019 Helton Talk

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Various Developments in Control

Bill Helton UCSD

Department of Mathematics


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Numerical Optimization 2

Unconstrained: Gradient descent, Newton - nds a localoptimum efficiently.

Constrained: is harderOld idea is interior point method .Example subject to g(x ) 0

minX

f (x 1 , . . . , x g ) subject to g (x ) 0

Solution use a barrier function. Example

b (x ) := ln g (x )


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Numerical Optimization 3

Solution use a barrier function.Example b (x ) := ln y y = g(x ).

y

Set f (x ) := f (x ) ln g (x )

Solve minx

f (x ) with a unconstrained algorithm. Take

short steps. Send 0


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INTERIOR POINT METHOD 4

Set f (x ) := f (x ) ln g (x ). Solve minx

f (x ) withan unconstrained algorithm.

Rough History:

1. Solution 197? SOL paper said is VERY unreliable,

since f (x

) is badly conditioned if solution x

is onboundary, g(x ) = 0.

2. Kamarkar, Bell labs 1985ish - showed works on

many examples.

3. Nestrov-Nimerovski 1990ish introduced LMIs, andnumerical solution


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LINEAR MATRIX INEQUALITIES 5

GIVEN a linear pencil L (x ) := A 0 + A 1 x 1 + + A g x gsymmetric matricesFIND numbers (or matrices) X := {X 1 , X 2 , , X g }

making L (X ) Pos SemiDef .SOLN: Interior Pt using barrier b (X ) := n det L (X )

APPLICATIONS:Combinatorics 1993ish I Do not know this area,Control - S. Boyd 1993ish. A Large crowd convertedcontrol problems to LMI (about 2 dozen are clean).Elimination Method, Change of Variables ( CarstenScherer ). Also lots of tricks with Schur Complements.Computational RAG appl to Lin and Nonlin Sys. 2000 Lasserre and Parrilo


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LINEAR MATRIX INEQUALITIES 6

1. Many many uses of LMIs in compromises.

2. Vast vast vast literature.

3. Bilinear Matrix Inequalities BMI

(contains most systems problems)

XAY + Y A T X + BX + XB + Q P osDef.

People here who are experienced in converting:Linear Sys Probs LMIs


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LIN MATRIX INEQS vs CONVEX 7

Stabilize two linear systems A, B j with a state feedbackcontroller, as in gain scheduling.

SOLN:Lyapunov inequality: Find X which simultaneouslysolves

XA + AT

X XB 1 BT 1 X + Q P osDef

XA + A T X XB 2 B T 2 X + Q P osDef

Can do with an Matrix Ineq package for 150 150matrices. Not possible with a Riccati equation solver.

Solution identify if model 1 or 2 applies; then use the

controller u = B T j Xx with B 1 or B 2 .


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Sums of Squares 8

POSITIVE POLYNOMIALS vs SUMS of SQUARESOF POLYNOMIALS

CONSTRUCTING A LYAPUNOV FUNCTION

FROM LALL- PARRILO SLIDESIn Appendices


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LIN MATRIX INEQS vs CONVEX 9

LINEAR MATRIX INEQUALITIES LMIsGIVEN a linear pencil L (x ) := A 0 + A 1 x 1 + + A g x gsymmetric matricesFIND matrices X := {X 1 , X 2 , , X g } making L (X )Pos SemiDef.

CONVEX MATRIX INEQUALITIES CMIs

QUESTION: How much more general are ConvexMatrix Inequalities than Linear Matrix Inequalities?

Bill believes there are two different situations with verydifferent behavior

Dimension Free , Dimension Dependent


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Systems problems Matrix Ineq 10

Given

Find

Many such problems Eg. H control

Example: Get Riccati expressions like

A X + X A T X BB T X + CC T 0

OR Linear Matrix Inequalities (LMI) like

A X + X A T + C T C X BB T X I

0

which is equivalent to the Riccati inequality.ncDimlessPart


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DIMENSIONLESS FORMULAS 11

1. DIMENSIONLESS FORMULAS THIS TALKTopology is xed; but many systems . E.g.

+ S 1 S 2

S 3WANT FORMULAS: which hold regardless of thedimension of system S 1 , S 2 , S 3 . Then unknowns arematrices and formulas respect matrix multiplication.Eg. Most classical control text problems:

Control pre 1990: Zhou,Doyle,Glover.LMIs in Control: Skelton, Iwasaki,. Grigoriadis.

Get noncommutative formulas


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Keeping Matrices Whole 12

Matrices Whole

A X + X A T + C T C X B

BT

X I

0 (1)

Looks the same regardless of system size.


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Keeping Matrices Whole 12

Matrices Whole

A X + X A T + C T C X B

BT

X I

0 (1)

Looks the same regardless of system size.

Matrices Entry by Entry DisaggregatedIf dimensions of the matrices A, B, C, X are specied, wecan write formula (1) with matrices L 0 , . . . , L m as

m

j =0

L j X j 0

with the unknown numbers X j taken as entries of X .


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Disaggregated Matrices 13

Example: If A R 2 2 , B R 2 1 , C R 1 2 , then X S 2and we would take

X =X 1 X 2

X 2 X 3 and the LMI becomes

3

j =0L j X j 0

where the 4 4 symmetric matrices L 0 , L 1 , L 2 , L 3 are:

L 0 :=C T C 0

0 I L 1 :=

2 a 11 a 21 b 11 b 12a 21 0 0 0b 11 0 0 0

b 12 0 0 0

L 2 :=

2 a 12 a 11 + a 22 b 21 b 22a 22 + a 11 2 a 21 b 11 b 12

b 21 b 11 0 0b 22 b 12 0 0

L 3 :=

0 0 0 a 120 0 0 a 220 0 0 b 21

a 12 a 22 b 21 2 b 22

Down with vec


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+ and of Keeping Matrices Whole 14

+ not many variables

+ short formulas

Trouble is formulas are noncommutative.

+ NCAlgebra package does symbolic noncommutativealgebra.


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1. INTRO NC RAG - SEMIRAG 15

R eal Algebraic G eometry = RAGStudy of Polynomials on R g

RAG Semialgebraic Geometry = poly inequalities.

What about NonCommutative Polys?Part I. A FEW PHENOMENA

NC Positive Polynomial

NC Convex Polynomials

introNCPolyPart


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1. POSITIVE NC POLYS 16

p 0 Means What ?

p is a symmetric polynomial in non-commutativevariables

x = {x 1 , x 2 } with real coefficients, eg.

p(x ) = x 1 2 + ( x 1 2 ) T + x 2 T x 2

Dene MATRIX POSITIVE polynomial

Plug in n n matrices X j for x j in p

always get

p(X 1 , X 2 ) is a PosSD n n matrix.

MATRIX POSITIVE POLYNOMIALS


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MATRIX POSITIVE POLYNOMIALS-Examples 17

Example

x = {x 1 , x 2 }

p ( x ) = (3 x 21 + x 2 ) T (3 x 21 + x 2 ) +( x 1 5 ) T ( x 1 5 )

p (X ) = PosSD + PosSD

This is a sum of squares (SoS). More generally

p(

x ) =

c

j =1L j (

x ) T L j (

x )

is clearly always Matrix Positive.

NON COM POLYS ARE BETTER


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NON-COM POLYS ARE BETTERBEHAVED THAN COMMUTATIVE

POLYS 18

THM:Matrix positive non-commutative

polynomials are sums of squares

Can compute SoS, thus CAN TEST Matrix Positivity.

Proof: Operator theory techniques

See 2001 H preprint.

S. McCullough 2000 preprint - when all variables x jare unitary.


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19

NC CONVEXPOLYNOMIALS


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20

Function p of noncommutative variables x := ( x 1 , x 2 ) is

MATRIX CONVEX (geometric def.) 0 1

p(X + (1 )

Y ) p (

X ) + (1 ) p(

Y )

12

p(X ) + 1

2 p(

Y ) p( 1

2X + 1

2Y ) is Pos Def ?

Question: Consider the noncommutative polynomial

p(x ) := x 4 + ( x 4 ) T .

Is it matrix convex?

CONVEX POLYNOMIALS ARE


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CONVEX POLYNOMIALS AREWIMPS 21

The riddle revealed: p(x ) := x 4 + ( x 4) T is not matrix convex.

THM: ( McC + H)Every symmetric noncommutative polynomialwhich is matrix convex has degree 2 or less.

COR A Convex NC Polys is the Schur complement of somelinear pencil. Proof:

1. NC Positive Polynomials

2. NC Second Derivatives

3. Put the two together


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Convexity Algorithm 22

In[1] :=


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SUMMARY 23

1. THM EVERY POSITIVE NC POLYNOMIAL ISA SUM OF SQUARES. COMPUTABLE.

2. THM EVERY CONVEX NC POLYNOMIAL HASDEGREE 2. Equiv. to an LMI.

3. ALGORTHIM CHECKS CONVEXITY OF NC

RATIONAL FUNCTION. TRY NCAlgebra

ALGORITHMS and


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ALGORITHMS andIMPLEMENTATIONS 24

1. Convexity Checker - Camino, Skelton, H Public

2. Realization Builder: Convex Rational to LMI -Slinglend, Shopple in progress

3. Numerical matrix unknowns - Camino, Skelton, H

in house

4. LMI Producer (uses existing methods on specialproblems) de Oliveira, H in house (out soon)

Try NCAlgebra

SUMMARY CONVEXITY LMI


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SUMMARY: CONVEXITY vs LMI 25

CONJ: FOR DIM INDEP LINEAR SYS PROBS

CONVEX MIs are equivalent to LINEAR MI.

FOR DIM DEPENDENT LINEAR SYS PROBSNOT EQUIVALENT: There is an extra constraint,the line test

WEAK CONJ: the line test is necessary and sufficient

Proved in two dimensions Vinnikov + H

THERE IS NEW FIELDNC REAL ALGEBRAIC GEOMETRY.

THERE IS NC computer algebra software.

END 2006


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END 2006 26

END 2006

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