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
To download a General Audience talk talkSiam01.pdf
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.
To download file rigidconvexity.pdf
     
rigidconvexity.ps
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
mtnsMI.ps        
mtnsMI.pdf
-- MTNS 2002 Plenary Talk
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
Bad News about BMI's (Bi Convex Matrix Inequlities)
This paper gives strong evidence that co-ordinate descent
use of LMI's on bi-convex LMI's almost never hits a local optimum.
To download ps file of prepint.
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.
To download file dymHmer.ps
oooooooo
dymHmer.pdf
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
A frequency domain type of gain scheduling which relies
on results in several complex variables.
To appear TAC tech notes section + added on comments.
To download scvTAC.pdf file .
Older Analytic Function Optimization Papers, Engineering, etc
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
Skelton, Helton, Adhikari, Pinaud, Chan
To appear a a chapter
in Handbook of Mechanical Systems Design, CRC Press.
Order from crcpress.com
To download (big file =4.7 Meg) tensegrity.pdf
To download the CDC Proceedings Really short version of tensegrity.pdf
- 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
.
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
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
Control Systems production of LMI symbolical Oliveira and Helton CDC 2003
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.
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.
-
Download Ups and Downs
- By Ed Bender, J. William Helton and Bruce Richmond
-
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