• ## NONCOMMUTATIVE INEQUALITIES= SYMBOLIC MATRIX INEQUALITIES                  =NC REAL ALGEBRAIC GEOMETRY

SURVEY Article.pdf plus some new results- in preperation (may change some).   J. W. Helton, M. Putinar, Positive polynomials, the spectral theorem and optimization, pp 106

### NC POSITIVITY

We call a noncommutative polynomial matrix positive provided that when we plug matrices of any size in for its variables, the matrix value which the polynomial takes is positive semi-definite.

The paper     posPoly.ps   posPoly.pdf     shows that every matrix positive polynomial is a sum of squares.

--- to appear Annals of Mathematics   Sept 2002

### NC POSITIVSTELLENSATZ

Helton and Scott McCullough       PosSS.ps     PosSS.pdf       gives a Noncommutative generalization of the classical commutative strict Positivstellensatz. It then turns to the extreme nonstrict case, namely, the NC Real Nullstellensatz, it gives a counterexample and an affirmative result.

Helton, Scott McCullough and Mihai Putinar       onNCsphere.ps     onNCsphere.pdf       gives a class of Noncommutative situations where one has a nonstrict Positivstellensatz. This result is false for commutative polynomials.

Helton, Scott McCullough and Mihai Putinar       NCheredNSS.ps     NCheredNSS.pdf       shows a NC Real Nullstellensatz holds for hereditary polynomials.

Helton, Scott McCullough and Mihai Putinar       xandh.ps     xandh.pdf       gives a type NC Real Positivestellensatz representation in terms of positive semidefinite matrices of polynomials rather than sums of squares.

Helton, Scott McCullough and Mihai Putinar       HMPnullSS.ps     HMPnullSS.pdf       gives a NC Real Nullstellensatz and NC Nichtnegativstellensatz

### NC CONVEX FUNCTIONS

Helton, Scott McCullough and Victor Vinnikov       convRat.pdf       convRat.ps       Proves that any matrix convex Noncommutative rational function R (in many variables) is the Schur complement of a monic linear pencil. Proves the matrix inequalities based on R are equivalent to Linear matrix Inequlities!
Proves that every polynomial p (in g commutative variables) has a determinantal representation. That is p is the determinant of a linear pencil.
Algorithms by N. Slinglend mke these results constructive. They have been implemented by J. Shopple.

Helton and Scott McCullough       Published article SIAM 2004     Old Version convPoly.pdf       Proves that any matrix convex polynomial (in many variables) has degree 2 or less.

Camino, Helton, Skelton, Ye       convCheck.ps     convCheck.pdf
Gives a computer algebra algorithm for computing the domain on which a noncommutative function is "convex". The key mathematical theorem expresses a symbolic function Q in noncommuting variables z and h which is quadratic in h as a weighted sum of squares. This is a noncommutative positivstellensatz for a special class of functions. The surprising thing is that the weights in this decomposition determine precisely the domain on which Q is "matrix positive". - To appear:
J.~F Camino, J.~W. Helton, and R.~E. Skelton and J. Ye, Matrix inequalities: A Symbolic Procedure to Determine Convexity Automatically, Integral Eq and Operator Thy Vol 46, issue 4, August 2003 on pp. 399-454

# LMI Representations

### WHICH SETS C in R^m have a Linear Matrix Inequality REPRESENTATION? that is, C = { x : L0 + L1 x1 + ... + Lm xm }

Little is currently known about such problems. In this article we give a necessary condition, we call "rigid convexity", which must hold for a set C in R^m in order for C to have an LMI representation. Rigid convexity is proved to be necessary and sufficient when m=2.

Helton and Victor Vinnikov.

Surprisingly a theorem in this paper was used by Adrian S. Lewis, Pablo A. Parrilo, Motakuri V. Ramana to solve a 1958 conjecture of Peter Lax when (n=2).       To download LPR Lax conjecture paper.pdf     Lax conjecture paper.ps

# Survey paper on LMI Representations and NC Inequalities

mtnsMI.ps         mtnsMI.pdf -- MTNS 2002 Plenary Talk

## NUMERICAL MATRIX OPTIMIZATION

To download pdf file of prepint. Gives an approach to numerical algorithms which exploits the matrix structure of unknowns provided the unknowns are matrices. It begins with NC symbolic computation and carries this as far as possible.
J. F. Camino, J. W. Helton and R.E. Skelton, Solving Matrix Inequalities whose Unknowns are Matrices
To appear SIAM Jour of Optimization, 17 (2006), issue 1, p1-36

This paper gives strong evidence that co-ordinate descent use of LMI's on bi-convex LMI's almost never hits a local optimum.

## OPTIMIZING OVER SPACES of ANALYTIC FUNCTIONS

Multidisk Problems in $H^\infty$ Optimization: a Method for Analysing Numerical Algorithms. H. Dym, J. W. Helton, O. Merino, Preprint 2000 to Appear Indiana Jour of Math.

Analyticity (ie. stability if you are an engineer) makes uniqueness (even for nonconvex problems) much more common that one would think. Global Uniqueness Tests for $H^\infty$ Optima. J. W. Helton and M. Whittlesey, Preprint 2000. To download file cdcWh00.pdf

## Bi H-infinity Control

A frequency domain type of gain scheduling which relies on results in several complex variables.

## TENSEGRITY STRUCTURES

Some basic properties of tensegrity structures are surmised (no proofs are known) from numereous examples:

The role of pretension

A structure with high strength vs. mass. under compression

A structure with high strength vs. mass. under tension

To appear a a chapter in Handbook of Mechanical Systems Design, CRC Press. Order from crcpress.com

## NONLINEAR CONTROL AND NON LINEAR OPERATOR THEORY

THE BEST SOURCE FOR MY RECENT WORK IS:
See The Monograph by Helton and James SIAM Dec 1999, Samples are on the Helton Homepage.

To actually compute nonlinear measurement feedback controllers one must solve the information state PDE online. If there are many measurements (just a little less than full state feedback) this seems possible using the theory and algorthims described in the paper with Matt James and Bill MCEneaney, preprint 2000. We call this cheap sensor control. In control terms this corresponds to a type of singular control (D's very noninvertible). In mathematical terms this corresponds to a natural type of J inner-outer factoring of nonlinear operators where some state info flows back.
Many Measurements.ps

If you want to solve Belman equations, see papers with Mike Hardt and Ken Kreutz Delgado, which take a shot at this question.
Control Systems Technology 2000. Numerical implementation of the nonlinear theory.pdf

Power gain optimization, with Peter Dower CDC 1999. This applies to systems which can only be controled to a region not a point; as would be the case with deadzone nonlinearities.
From the mathpoint of view it probably is the nonlinear generalization of some type of boundary interpolation, like Loevner interpolation;
this has not been explored.
Control to achieve prescribed power (rather than energy) gain.ps

Path planning using the same methods- leads to the question why do people take such long steps? OR more realistically what in the math model disposes it to such short steps?

Minimum Energy Walking.ps Minimum Energy Walking.pdf
Helton James McEneaney-- Cheap Sensor Control.pdf
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OLDER NONLINEAR PAPERS

Control with Command but not Disturbance Signal Access (Posted Aug 21, 1996)
By J. W. Helton and Wei Zhan

Some Preliminary Results on Information State System Stability (Posted Aug 9, 1996)
By J. W. Helton and M. R. James

An Information State Approach to Nonlinear j-Inner/Outer Factorization (Posted Dec 1, 1995)
By J. W. Helton and M. R. James

Reduction of Controller Complexity in Nonlinear H-inf Control(Posted Dec 1, 1995)
By J. W. Helton and M. R. James

Dissipative Control Systems Synthesis with Full State Feedback(Posted Dec 1, 1995)
By J. W. Helton, S. Yuliar and M. R. James

A New Type of HJBI Inequality Governing Situations Where Certainty equivalence Fails(Posted Dec 1, 1995)
By J. William Helton and Andrei Vityaev

Piecewise Riccati Equations and the Bounded Real Lemma(Posted Dec 1, 1995)
By J. William Helton and Wei Zhan

Viscosity Solutions of Hamilton-Jacobi Equations Arising in Nonlinear H-Inf Control(Posted Dec 1, 1995)
By Joseph A. Ball and J. William Helton

## NONCOMMUTATIVE COMPUTER ALGEBRA

More detail on many topics is on the NCAlgebra Homepage.
To download pdf file of prepint. de Oliveria and J. W. Helton, "Computer Algebra Tailored to Matrix Inequalities in Control",
To appear Special Issue of the International Journal of Control, on the Use of Computer Algebra Systems for Computer Aided Control System Design

This gives symbolic implementation to change of variables like methods by Scherer et al for producing LMIs from lucky control problems.
By Oliveira and Helton

We have a noncommutative function and want to determine automatically the region on which it is "convex". This type of problem when engineers are manipulating a set of matrix inequalities. Our symbolic algorithm computes the "region of convexity of F". Here is an announcement; proving that the domian is the best possible requires an enjoyable operator theoretic proof and is currently being written up.
By Juan Camino, J. William Helton and Robert E. Skelton. A Symbolic Algorithm For Determining Convexity of A Matrix Function: How To Get Schur Complements Out of Your Life

International Jour. of Nonlinear and Robust Control, 10: p983-1003, 2000 J.W. Helton F. Dell Kronewitter W.M. McEneaney and Mark Stankus Singularly perturbed control systems using noncommutative computer algebra.

Computer Assistance in Discovering Formulas and Theorems in System Engineering with Mark Stankus, Journal of Functional Analysis 1999. It is available Dvi or PostScript formats.

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Older Papers and mostly announcements

Computer Assistance in Discovering Formulas and Theorems in System Engineering ANNOUNCEMENT of partial results(Posted Jul 1, 1996)
By J. William Helton and Mark Stankus

Rules for Computer Simplification of the Formulas in Operator model Theory and Linear Systems(Posted Dec 1, 1995)
By J. William Helton and John Wavrik

Computer Simplification of Engineering Systems Formulas(Posted Dec 1, 1995)
By J. William Helton, Mark Stankus and John Wavrik

## MISC: Combinatorics, Monotone Maps

A paper with Lev Sahnovic on applications of fixed points of monotone maps. HSakhnovic.ps HSakhnovic.pdf

A combinatics paper which fortunately for Bill needs a Perron -Frobeneous argument.

By Ed Bender, J. William Helton and Bruce Richmond

## Older OPTIMIZATION OVER ANALYTIC FUNCTIONS, ENGINEERING, SEVERAL Complex Variables

Optimization with Plant Uncertainty and Semidefinite Programming (Posted Aug 8, 1996)
By J. William Helton, Orlando Merino and Trent E. Walker

Algorithms for Optimizing Over Analytic Functions (Posted Dec 1, 1995)
By J. William Helton, Orlando Merino and Trent E. Walker

Optimization Over Analytic Functions Whose Fourier Coefficients are Constrained (Posted Dec 1, 1995)
By J. William Helton, Orlando Merino and Trent E. Walker

An Optimization with Competing Performance Criteria (Posted Dec 1, 1995)
By J. William Helton and Andrei Vityaev

H-infty Optimization With Uncertainty in the Plant (Posted Dec 1, 1995)
By J. William Helton, Orlando Merino and Trent E. Walker

A Fibered Polynomial Hull Without an Analytic Selection (Posted Dec 1, 1995)
By J. William Helton and Orlando Merino

H-infty Optimization and Semidefinite Programming (Posted Dec 1, 1995)
By J. William Helton, Orlando Merino and Trent E. Walker

### Miscelleneous

Some Systems Theorems Arising From The Bieberbach Conjecture (Posted Dec 1, 1995)
By J. William Helton and Frederick Weening