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                     =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


    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


    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


    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

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    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.

    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

    Survey paper on LMI Representations and NC Inequalities

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

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    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

    Bi H-infinity Control

    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
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    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

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    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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    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

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    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.

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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.

    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


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

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