non autonomous he non heiles systems abstract scaling similarity solutions of three integrable pdes namely the sawada kotera fifth order kdv and kaup kupershmidt equations are considered it is shown that the resulting odes may be written as non autonomous hamiltonian equations which are time dependent generalizations of the well known integrable henon heiles systems the time dependent hamiltonians are given by logarithmic derivatives of the tau functions inherited from the original pdes the odes for the similarity solutions also have inherited backlund transformations which may be used to generate sequences of rational solutions as well as other special solutions related to the first painleve transcendent

solving quadratic distance problems an lmi based approach the computation of the minimum distance of a point to a surface in a finite dimensional space is a key issue in several system analysis and control problems the paper presents a general framework in which some classes of minimum distance problems are tackled via linear matrix inequality lmi techniques exploiting a suitable representation of homogeneous forms a lower bound to the solution of a canonical quadratic distance problem is obtained by solving a one parameter family of lmi optimization problems several properties of the proposed technique are discussed in particular tightness of the lower bound is investigated providing both a simple algorithmic procedure for a posteriori optimality testing and a structural condition on the related homogeneous form that ensures optimality a priori extensive numerical simulations are reported showing promising performances of the proposed method

a classification of sl n invariant valuations a classification of upper semicontinuous and sl n invariant valuations on the space of n dimensional convex bodies is established as a consequence complete characterizations of cen t ro affine and l p affine surface areas are obtained the proofs make use of a new sl n shaping process for convex bodies

some counterexamples in robust stability theory this note deals with the problem of determining if a linear system whose characteristics polynomial depends multilinearly on n independent uncertain real parameters i i 1 n is robustly stable it is shown by example that a polynomial in n variables may have a unique real root and that this observation disposes of several natural conjectures in robust stability theory in particular we show that in a certain sense there are no edge or m dimensional face kharitonov like theorems for the general multilinear case the result holds even when restricted to that subset of multilinear functions which can be written in the form f 1 n det i diag 1 n m for some complex matrix m

extremal robustness properties of multilinear interval systems this paper deals with robustness properties of a control system subject to parameter uncertainty when the parameters appear affine multilinearly in the coefficients of the characteristic polynomial the robust hurwitz stability of such a multilinear interval polynomial family reduces to that of the stability of a set of manifolds derived from the kharitonov polynomials several extremal properties of these stability manifolds are derived these properties are useful in determining the worst case parametric h and nonlinear sector bounded stability margins for control systems containing interval parametric uncertainty they generalize previous results obtained by the authors for the linear case

optimal estimation theory for dynamic systems with set membership uncertainty an overview in many problems such as linear and nonlinear regressions parameter and state estimation of dynamic systems state space and time series prediction interpolation smoothing and functions approximation one has to evaluate some unknown variable using available data the data are always associated with some uncertainty and it is necessary to evaluate how this uncertainty affects the estimated variables typically the problem is approached assuming a probabilistic description of uncertainty and applying statistical estimation theory an interesting alternative approach referred to as set membership or unknown but bounded ubb error description has been investigated since the late 1960s in this approach uncertainty is described by an additive noise which is known only to have given integral typically l2 of l1 or componentwise l bounds in this paper we review the main results of this theory with special attention to the most recent advances obtained in the case of componentwise bounds

improvement of parametric stability margin under pole assignment the improvement of the parametric stability margin of state space uncertain systems via a maximization formulation under the constraints of pole assignment is investigated the class of systems considered is where the uncertainty may be modeled as the possibly nonlinear variation of a parameter appearing in the entries of the system and input matrices the continuity and differentiability properties of the stability margin are discussed a gradient based approach is presented for the improvement of the stability margin and a compact formula to compute the gradient is provided numerical examples are used to demonstrate the effectiveness of the approach

robust control with structure perturbations the problem of making a given stabilizing controller robust so that the closed loop system remains stable for prescribed ranges of variations of a set of physical parameters in the plant the problem is treated in the state space and transfer function domains in the state space domain a stability hypersphere is determined in the parameter space using lyapunov theory the radius of this hypersphere is iteratively increased by adjusting the controller parameters until the prescribed perturbation ranges are contained in the stability hypersphere in the transfer function domain a corresponding stability margin is defined and optimized on the basis of the recently introduced concept of the largest stability hypersphere in the space of coefficients of the closed loop characteristic polynomial the design algorithms are illustrated by examples

grammian n soliton solutions of a coupled kdv system from the lax pair and the binary darboux transformation of a coupled korteweg de vries system we show that its nonlinear superposition formula is identical to that obtained for the kaup kupershmidt equation therefore the n soliton solution can be associated with a determinant of the gram type

on affine plane curve evolution abstract an affine invariant curve evolution process is presented in this work the evolution studied is the affine analogue of the euclidean curve shortening flow evolution equations for both affine and euclidean invariants are developed an affine version of the classical euclidean isoperimetric inequality is proved this inequality is used to show that in the case of affine evolution of convex plane curves the affine isoperimetric ratio is a non decreasing function of time convergence of this affine isoperimetric ratio to the ellipse s value 8 2 as well as convergence in the hausdorff metric of the evolving curve to an ellipse is also proved

analysis and applications of robust nonsingularity problem using the structured singular value this note uses the structured singular value technique to solve the robust nonsingularity analysis problem for matrices with unstructured and or quadratically coupled structured uncertainties with the ready spl mu analysis tools a new approach is proposed for finding a set of uncertainties within which the perturbed matrix keeps nonsingularity applications and examples in robust control theory are given to show the usefulness and the practicality of the proposed approach

backlund transformation of partial differential equations from the painleve gambier classification i kaup kupershmidt equation among the 50 nonlinear second order differential equations of painleve and gambier those which are linearizable provide a natural scheme for deriving the lax pair and the darboux transformation of a nonlinear partial differential equation when the order of the scattering problem is three this new method allows one to obtain from singularity analysis only the backlund transformation of the kaup kupershmidt equation a result which was missing until now

robustness analysis of polynomials with polynomial parameter dependency using bernstein expansion this paper considers the robust stability verification of polynomials with coefficients depending polynomially on parameters varying in given intervals two algorithms are presented both rely on the expansion of a multivariate polynomial into bernstein polynomials the first one is an improvement of the so called bernstein algorithm and checks the hurwitz determinant for positivity over the parameter set the second one is based on the analysis of the value set of the family of polynomials and profits from the convex hull property of the bernstein polynomials numerical results to real world control problems are presented showing the efficiency of both algorithms

a generalization of the routh hurwitz stability criteria and an application to a problem in robust controller design the classical results of routh and hurwitz on the stability of polynomials are generalized from the open left half plane to certain other subregions g of the complex plane by our method arbitrary conic sections can be handled the results are interpreted as criteria for g stability of linear systems a numerical algorithm to compute the criteria is given an application to a problem in robust controller design is outlined

on the affine heat equation for non convex curves 1998 american mathematical society first published in journal of the american mathematical society vol 11 no 3 july 1998 published by the american mathematical society

on soliton solutions of the kaup kupershmidt equation ii anomalous n soliton solutions abstract in the second of two articles designated i physica d 137 2000 25 and ii exact soliton solutions of the kaup kupershmidt equation are derived using a simplified approach based in part on hirota s bilinear transform these solutions have not previously been obtained explicitly by analytic methods it is found that at each order the n soliton solution is characterised by an additional parameter which accounts for the anomalous nature of these solutions the first four solitons are constructed explicitly and in a novel departure from hirota s method an iterative procedure is presented for obtaining the general n soliton solution some possible alternative approaches to solving the kaup kupershmidt equation are considered

nonlocal symmetries and the kaup kupershmidt equation computational and geometric aspects of nonlocal infinitesimal symmetries of nonlinear partial differential equations are considered in particular the relation of nonlocal symmetries with classical generalized and internal symmetries is briefly discussed a nonlocal symmetry for the kaup kupershmidt equation is introduced and studied in some detail some explicit particular solutions are found with its help and a darboux like transformation is also obtained

results on the stability robustness of systems with state space perturbations abstract some considerations on the stability robustness of a state space description for perturbed linear systems are presented on the basis of a determinantal criterion the two problems of the stability domain and of checking in the parameter space are considered and a number of computationally simple results are derived an extension to a general pole placement problem and some numerical examples are also proposed

perturbation bounds for robust stability in linear state space models abstract this paper addresses the aspect of stability robustness of linear systems in the time domain upper bounds on the linear perturbation of an asymptotically stable linear system are obtained to maintain stability both for structured as well as unstructured perturbations using the lyapunov approach for structured perturbation the resulting bound is such that it garners the structural information about the nominal as well as the perturbation matrix into a single unified expression in the case of unstructured perturbations special features of the nominally stable matrix are exploited resulting in simpler expressions for the bound without the need to solve the lyapunov equation improvement of the proposed measures is illustrated with the help of examples

an lmi approach to constrained optimization with homogeneous forms abstract this paper considers the problem of determining the minimum euclidean distance of a point from a polynomial surface in r n it is well known that this problem is in general non convex the main purpose of the paper is to investigate to what extent linear matrix inequality lmi techniques can be exploited for solving this problem the first result of the paper shows that a lower bound to the global minimum can be achieved via the solution of a one parameter family of linear matrix inequalities lmis it is also pointed out that for some classes of problems the solution of a single lmi problem provides the lower bound the second result concerns the tightness of the bound it is shown that optimality of the lower bound amounts to solving a system of linear equations an application example is finally presented to show the features of the approach

the he non heiles system revisited abstract the known integrable cases of the henon heiles system are shown to be closely related to the stationary flows of the known and only integrable fifth order single component and polynomial nonlinear evolution equations this is further evidence that these are the only integrable cases of the henon heiles system lax pairs are deduced for each of the integrable cases and used to construct the constants of motion a curious lax operator recently found by the painleve method is explained

an overview of extremal properties for robust control of interval plants abstract this paper provides an overview of recent results on the problem of robust stability and performance of feedback control systems in the presence of plant perturbations the widely studied class of interval plants is considered and an effort is made to cover different types of plant perturbations by incorporating in addition to the interval plant description unstructured norm bounded perturbations extremality results for these classes of uncertain systems are provided in a unifying framework robust performance of control systems is addressed in the paper and a number of extremal results are given for the computation of the structured singular value function for feedback systems with interval plants also classical frequency response analysis for interval plant controller families of open loop transfer functions is surveyed with the aim of providing extremal results for the computation of specification parameters such as phase or gain margins and sensitivity and complementary sensitivity peaks

computation of nonconservative stability perturbation bounds for systems with nonlinearly correlated uncertainties consideration is given to the problem of robust stability analysis of linear dynamic systems with uncertain physical parameters entering as polynomials in the state equation matrices a method is proposed giving necessary and sufficient conditions for computing the uncertain system stability margin in parameter space which provides a measure of maximal parameter perturbations preserving stability of the perturbed system around a known stable nominal system a globally convergent optimization algorithm that enables solutions to the problem to be obtained is presented using the polynomial structure of the problem the algorithm generates a convergent sequence of interval estimates of the global extremum these intervals provide a measure of the accuracy of the approximating solution achieved at each step of the iterative procedure some numerical examples are reported showing attractive features of the algorithm from the point of view of computational burden and convergence behavior

the centro affine invariant geometric heat flow in this paper a centro affine invariant curve evolution process is pre sented and the centro affine geometric heat flow is introduced it is shown that the original symmetric solution of the flow shrinks to a point in finite time and converges smoothly to an ellipse after suitable normalization

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

maximal polytopic stability domains in parameter space for uncertain systems abstract some results on the stability and pole location of families of uncertain systems with characteristic polynomial coefficients affected by structured perturbations are presented the case in which the coefficients are affine in a set of system physical parameters is considered assuming that the parameters are subject to polytopic perturbations of a given structure for these cases which often arise in system stability robustness analysis as for example in the robustification of controllers for multivariable closed loop systems a procedure is given to estimate maximal stability polytopic regions in the space of parameters this result is also used to derive simple necessary and sufficient conditions for the discrete time stability of convex combinations of two polynomials

rank preservation of matrices with structured uncertainties and its applications in robust control theory matrix rank is determined by the nonsingularity of its submatrices for matrices in which entries are quadratic functions of some uncertain parameters this paper derives sufficient conditions on parameters to that ensure the matrices preserve to some degrees the ranks they have when the parameters are all zero the rank preservation problem is converted to the nonsingularity analysis problem of the minors of the matrix in discussion and suitable tools such as the spl mu analysis method are used to solve the problem applications in robust control theory including tests for robust controllability observability minimum phaseness coprimeness and schur stability are given together with illustrative examples

frequency response of interval plant controller families abstract in this paper a general frequency domain result for families of interval plant with a fixed linear controller is given it is shown that the locus of the polar diagrams of frequency responses of the transfer functions of an interval plant controller family is bounded by the polar plots relative to the 32 segments of transfer functions of the interval plant family easy proofs of several important results such as the generalization of the theorem of kharitonov for feedback systems with interval plants or the robust version of the small gain theorem for the same class of systems are constructed by using the general result more importantly an immediate consequence of the main theorem is that extremal phase gain margins or sensitivity complementary sensitivity peaks for systems of the family can be deduced from those of the 32 segments of the interval plant family

nonlinear superposition formula for the kaup kupershmidt partial differential equation abstract the fifth order kaup kupershmidt kk equation is one of the solitonic equations related to the integrable cases of the henon heiles system as opposed to the sawada kotera equation which is its dual equation the construction of the n soliton solutions of kk is not an easy task in using a perturbation scheme or the hirota bilinear formalism from the backlund transformation obtained by singularity analysis considerations we here establish the permutability theorem for the kk equation this allows us to explain the anomalous structure of the n soliton solution empirically obtained by parker physica d 137 2000 25 33 34 48

stability robustness bounds for linear state space models with structured uncertainty in this note we consider the robust stability analysis problem in linear state space models we consider systems with structured uncertainty some lower bounds on allowable perturbations which maintain the stability of a nominally stable system are derived these bounds are shown to be less conservative than the existing ones

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

on soliton solutions of the kaup kupershmidt equation i direct bilinearisation and solitary wave abstract in this first of two papers designated i and ii physica d 137 2000 34 a direct procedure is given for bilinearising the kaup kupershmidt and sawada kotera equations it is found that whereas the sawada kotera equation can be written in bilinear form in two distinct ways which are shown to be equivalent the kaup kupershmidt equation has a single bilinear representation this distinguishing feature which has previously received no explanation has consequences for obtaining the soliton solutions of these two closely related fifth order integrable equations the anomalous solitary wave of the kaup kupershmidt equation is reconstructed and a simplified approach to obtaining its n soliton solutions is proposed this provides the basis for constructing these solutions explicitly in ii

a generalization of kharitonov s four polynomial concept for robust stability problems with linearly dependent coefficient perturbations kharitonov s four polynomial concept is generalized to the case of linearly dependent coefficient perturbations and more general zero location regions to this end a specially constructed scalar function of a scalar variable is instrumental to the robustness analysis the present work is motivated by two fundamental limitations of kharitonov s theorem namely 1 the theorem only applies to polynomials with independent coefficient perturbations and 2 it only applies to zeros in the left hand plane

robust analysis and design of control systems using interval arithmetic abstract several robustness problems such as stability and performance robustness analysis of feedback systems and robust design of control systems in the presence of mixed nonlinear parametric and nonparametric perturbations can be solved by means of algorithms based on interval arithmetic computation some of the main algorithms available in the literature are presented and their efficiency is tested on some examples of robustness analysis and design of control systems 1997 elsevier science ltd

robust hurwitz stability via sign definite decomposition abstract in this paper we first consider the problem of determining the robust positivity of a real function f x as the real vector x varies over a box x r l we show that it is sufficient to check a finite number of specially constructed points this is accomplished by using some results on sign definite decomposition we then apply this result to determine the robust hurwitz stability of a family of complex polynomials whose coefficients are polynomial functions of the parameters of interest we develop an eight polynomial vertex stability test that is a sufficient condition of hurwitz stability of the family this test reduces to kharitonov s well known result for the special case where the parameters are just the polynomial coefficients in this case the result is tight this test can be recursively and modularly used to construct an approximation of arbitrary accuracy to the actual stabilizing set the result is illustrated by examples

worst case design for robust compensation we consider the problem displaystyle rm max hspace 12pt raisebox 7pt scriptsize f hspace 20pt rm min hspace 26pt raisebox 7pt scriptsize c p f 0 hspace 4pt h p f noindent where f in mathcal r r p in mathcal r m and where c cdot and h cdot are c 1 let phi f rm min c p f 0 h p f we show by means of simple examples that phi cdot is in general discontinuous we develop in this paper necessary conditions for the case where phi cdot is continuous but not necessarily differentiable in an alternative approach which is computationally inferior we treat the discontinuous case as well we apply the results to robust control in linear systems where p stands for the structured real parameter uncertainty vector and f stands for the control parameters vector we demonstrate the results by means of examples

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

brief homogeneous lyapunov functions for systems with structured uncertainties the problem addressed in this paper is the construction of homogeneous polynomial lyapunov functions hplfs for linear systems with time varying structured uncertainties a sufficient condition for the existence of an hplf of given degree is formulated in terms of a linear matrix inequalities lmi feasibility problem this condition turns out to be also necessary in some cases depending on the dimension of the system and the degree of the lyapunov function the maximum norm of the parametric uncertainty for which there exists a homogeneous polynomial lyapunov function is computed by solving a generalized eigenvalue problem the construction of such lyapunov functions is efficiently performed by means of popular convex optimization tools for the solution of problems in lmi form comparisons with other classes of lyapunov functions through numerical examples taken from the literature show that hplfs are a powerful tool for robustness analysis

robust stability with structured real parameter perturbations this paper considers the problem of robust stabilization of a linear time invariant system subject to variations of a real parameter vector for a given controller the radius of the largest stability hypersphere in this parameter space is calculated this radius is a measure of the stability margin of the closed loop system the results developed are applicable to all systems where the closed loop characteristic polynomial coefficients are linear functions of the parameters of interest in particular this always occurs for single input multioutput or single output multiinput systems where the transfer function coefficients are linear or affine functions of the parameters many problems with transfer function coefficients which are nonlinear functions of physical parameters can be cast into this mathematical framework by suitable weighting and redefinition of functions of physical parameters as new parameters the largest stability hyperellipsoid for the case of weighted perturbations and a stability polytope in parameter space are also determined based on these calculations a design procedure is proposed to robustify a given stabilizing controller this algorithm iteratively enlarges the stability hypersphere or hyperellipsoid in parameter space and can be used to design a controller io stabilize a plant subject to given ranges of parameter excursions these results are illustrated by an example

exact calculation of the multiloop stability margin a mapping theorem by l a zadeh and c a desoer 1963 serves as the basis for an algorithm that computes the stability margin k sub m of diagonally perturbed multivariable feedback systems without conservatism the stability margin determination not only verifies system stability but also quantifies how much further the plan uncertainties can be extended before instability occurs the essence of the computational approach is the realization that the true image of a given domain can be approximated with arbitrary accuracy by first subdividing the domain into the requisite number of subdomains and then forming the union of the convex hulls of their images the technique is illustrated in an example with a plant model having three uncertainties in general the method is applicable to either siso or mimo lti systems whose uncertainties can be modeled with noninteracting coefficients in the open loop transfer function representations

on sufficient conditions for the stability of interval matrices using gershgorin s diagonal dominance theorem new sufficient conditions for the stability of interval matrices are obtained the new conditions remove a restriction in heinen s 1983 analysis and are applicable to a wider class of problems than previously possible

extended soliton solutions for the kaup kupershmidt equation we extend the n soliton solutions of the kaup kupershmidt equation on a nonzero background decreasing as x 1 a 2 these new solutions describe the interaction of n solitary waves with a static bell shaped wave we give the conditions so that the backlund transformation relating those solutions and the n solitons of the sawada kotera equation will be satisfied

on robust stability of polynomials with polynomial parameter dependency two three parameter cases abstract we consider real polynomials whose coefficients depend polynomially on the elements of an uncertain parameter vector the size of perturbation is characterized by the weighted norm of the parameter vector the smallest destabilizing perturbation defines the stability radius of the set of uncertain polynomials it is shown that determining this radius is equivalent to solving a finite set of systems of algebraic equations and picking out the real solution with the smallest norm the number of systems of equations depends crucially on the dimension of the parameter vector whereas the complexity of systems of equations increases mainly with the kind of polynomial dependency and the degree of the polynomial this method also yields the smallest destabilizing parameter combination and the corresponding critical frequency for two or three parameters this transformed problem can be solved using symbolic and numeric computations

brief proving set inclusion via intervals application to parametric robust stability proving that an uncertain parametric model is stable amounts to prove the inclusion of two sets the set a of all feasible parameters and the set b of all parameters for which the model is stable in this paper a new algorithm able to decide whether or not a is included in b is presented the method is based on interval analysis which is a numerical tool able to deal with inequalities in a global and guaranteed way convergence properties of the algorithm are provided the algorithm is then applied to the robust stability of a discrete time model where the information on the parameters is given through bounded error data the behavior of the algorithm with respect to the number of parameters is illustrated on a continuous time model