linearly degenerate hierarchies of quasiclassical sdym type we demonstrate that sdym self dual yang mills equations for the lie algebra of one dimensional vector fields represent a natural reduction in the framework of a general linearly degenerate dispersionless hierarchy we define the reduction in terms of wave functions and introduce a generating relation lax sato equations and the dressing scheme for the reduced hierarchy a multidimensional case is also discussed

solvable vector nonlinear riemann problems exact implicit solutions of dispersionless pdes and wave breaking we have recently solved the inverse spectral problem for integrable partial differential equations pdes in arbitrary dimensions arising as commutation of multidimensional vector fields depending on a spectral parameter the associated inverse problem in particular can be formulated as a nonlinear riemann hilbert nrh problem on a given contour of the complex plane the most distinguished examples of integrable pdes of this type like the dispersionless kadomtsev petviashivili dkp the heavenly and the two dimensional dispersionless toda equations are real pdes associated with hamiltonian vector fields the corresponding nrh data satisfy suitable reality and symplectic constraints in this paper generalizing the examples of solvable nrh problems illustrated in manakov and santini 2009 j phys a math theor 42 095203 2008 j phys a math theor 41 055204 2009 j phys a math theor 42 404013 we present a general procedure to construct solvable nrh problems for integrable real pdes associated with hamiltonian vector fields allowing one to construct exact implicit solutions of such pdes parametrized by an arbitrary number of real functions of a single variable then we illustrate this theory on few distinguished examples for the dkp and heavenly equations for the dkp case we characterize a class of similarity solutions of solutions constant on their parabolic wave front and breaking simultaneously on it of localized solutions whose breaking point travels with constant speed along the wave front and of localized solutions breaking in a point of the x y plane for the heavenly equation we characterize two classes of symmetry reductions

symmetries of hyper kahler or poisson gauge field hierarchy symmetry properties of the space of complex or formal hyper kahler metrics are studied in the language of hyper kahler hierarchies the construction of finite symmetries is analogous to the theory of riemann hilbert transformations loop group elements now taking values in a pseudo group of canonical transformations of a simplectic manifold in spite of their highly nonlinear and involved nature infinitesimal expressions of these symmetries are shown to have a rather simple form these infinitesimal transformations are extended to the plebanski key functions to give rise to a nonlinear realization of a poisson loop algebra the poisson algebra structure turns out to originate in a contact structure behind a set of symplectic structures inherent in the hyper kahler hierarchy possible relations to membrane theory are briefly discussed

adaptive observer for multiple input multiple output mimo linear time varying systems for joint state parameter estimation in linear time varying ltv multiple input multiple output mimo systems an approach to the design of adaptive observers is proposed it is conceptually simple and computationally efficient its global exponential convergence is established for noise free systems in the presence of noises it is proved that the estimation errors are bounded and converge in the mean to zero if the noises are bounded and have zero means potential applications are fault detection and isolation and adaptive control

on the heavenly equation hierarchy and its reductions second heavenly equation hierarchy is considered using the framework of hyper kahler hierarchy developed by takasaki 1989 j math phys 30 1515 21 1989 j math phys 31 1877 88 generating equations for the hierarchy are introduced they are used to construct generating equations for reduced hierarchies general n reductions logarithmic reduction and rational reduction for one of the lax sato functions are discussed it is demonstrated that rational reduction is equivalent to the symmetry constraint

anti self dual four manifolds with a parallel real spinor antiselfdual metrics in the signature that admit a covariantly constant real spinor are studied it is shown that finding such metrics reduces to solving a fourthorder integrable partial differ

adaptive observers with arbitrary exponential rate of convergence for nonlinear systems concerns the same class of linearly parameterized single output nonlinear systems that the authors previously identified in 1992 in terms of differential geometric conditions when persistency of excitation conditions are satisfied the adaptive observers presented in this note guarantee arbitrarily fast exponential convergence both of parameter and state estimates to actual parameters and states while previous adaptive observers guarantee only exponential not arbitrarily fast convergence this extends earlier results for linear systems

on the solutions of the second heavenly and pavlov equations we have recently solved the inverse scattering problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields such as the heavenly equation of plebanski the dispersionless kadomtsev petviashvili dkp equation and the two dimensional dispersionless toda 2ddt equation as well as with the commutation of one dimensional vector fields such as the pavlov equation we also showed that the associated riemann hilbert inverse problems are powerful tools to establish if the solutions of the cauchy problem break at finite time to construct their long time behaviour and characterize classes of implicit solutions in this paper using the above theory we concentrate on the heavenly and pavlov equations i establishing that their localized solutions evolve without breaking unlike the cases of dkp and 2ddt ii constructing the long time behaviour of the solutions of their cauchy problems iii characterizing a distinguished class of implicit solutions of the heavenly equation

the inverse spectral transform for the dunajski hierarchy and some of its reductions i cauchy problem and longtime behavior of solutions in this paper we apply the formal inverse spectral transform for integrable dispersionless partial differential equations pdes arising from the commutation condition of pairs of one parameter families of vector fields recently developed by s v manakov and one of the authors to one distinguished class of equations the so called dunajski hierarchy we concentrate for concreteness i on the system of pdes characterizing a general anti self dual conformal structure in neutral signature ii on its first commuting flow and iii on some of their basic and novel reductions we formally solve their cauchy problem and we use it to construct the longtime behavior of solutions showing in particular that unlike the case of soliton pdes different dispersionless pdes belonging to the same hierarchy of commuting flows evolve in time in very different ways exhibiting either a smooth dynamics or a gradient catastrophe at finite time

state and unknown input observers for nonlinear systems with bounded exogenous inputs a systematic design methodology for state observers for a large class of nonlinear systems with bounded exogenous inputs disturbance inputs and sensor noise is proposed the nonlinearities under consideration are characterized by an incremental quadratic constraint parameterized by a set of multiplier matrices linear matrix inequalities are developed to construct observer gains which ensure that a performance output based on the state estimation error satisfies a prescribed degree of accuracy furthermore conditions guaranteeing estimation of the unknown inputs to arbitrary degrees of accuracy are provided the proposed scheme is illustrated with a numerical example which does not satisfy the so called matching conditions

robust adaptive observers for nonlinear systems with bounded disturbances since existing adaptive observers for nonlinear systems may generate unbounded parameter estimates in the presence of bounded disturbances robust adaptive observers are presented which prevent parameter estimate drift in addition the input to state stability of the error dynamics with respect to disturbances and parameter time derivatives is guaranteed by generalizing a persistency of excitation result asymptotic convergence of state estimation errors is still achieved for systems in adaptive observer form when disturbances are not present by a suitable extension of barbalat s lemma

hyper kahler hierarchies and their twistor theory a twistor construction of the hierarchy associated with the hyper kahler equations on a metric the anti self dual einstein vacuum equations asdve in four dimensions is given the recursion operator r is constructed and used to build an infinite dimensional symmetry algebra and in particular higher flows for the hyper kahler equations it is shown that r acts on the twistor data by multiplication with a rational function the structures are illustrated by the example of the sparling tod eguchi hansen solution

stable adaptive observers for nonlinear time varying systems an adaptive observer identifier for single input single output observable nonlinear systems that can be transformed to a certain observable canonical form is described sufficient conditions for stability of this observer are provided these conditions are in terms of the structure of the system and canonical form the boundedness of the parameter variations and the sufficient richness of some signals the scope of the canonical form and the use of the observer identifier is motivated by the presentation of applications to time invariant bilinear systems nonlinear systems in phase variable form a biotechnological process and a robot manipulator in each case the specific stability conditions are presented

krein space based h adaptive smoother design for a class of lipschitz nonlinear discrete time systems in this paper the problem of h adaptive smoother design is addressed for a class of lipschitz nonlinear discrete time systems with l2 bounded disturbance input by comprehensively analyzing the h performance lipschitz conditions and unknown parameter s bounded condition a positive minimum problem for an indefinite quadratic form is introduced such that the h adaptive smoothing problem is achieved a krein space stochastic system with multiple fictitious outputs is constructed by associating with the minimum problem of the introduced indefinite quadratic form the minimum of indefinite quadratic form is derived in the form of innovations through utilizing krein space orthogonal projection and innovation analysis approach via choosing the suitable fictitious outputs to guarantee the minimum of indefinite quadratic form is positive the existence condition of the adaptive smoother and its analytical solutions are obtained in virtue of nonstandard riccati difference equations the quality of the estimator is checked on an example

ergodic properties of invariant measures for piecewise monotonic transformations during the last decade a lot of research has been done on onedimensional dynamics different kinds of invariant measures for certain classes of piecewise monotonic transformations have been considered in most of these cases the perron frobenius operator plays an important role in this paper we try to unify these different examples which are discussed in detail below first we give a discription of the results proved in this paper

on a class of reductions of the manakov santini hierarchy connected with the interpolating system using the lax sato formulation of the manakov santini hierarchy we introduce a class of reductions such that the zero order reduction of this class corresponds to the dkp hierarchy and the first order reduction gives the hierarchy associated with the interpolating system introduced by dunajski we present the lax sato form of a reduced hierarchy for the interpolating system and also for the reduction of arbitrary order similar to the dkp hierarchy the lax sato equations for l the lax function split from the lax sato equations for m the orlov function due to the reduction and the reduced hierarchy for an arbitrary order of reduction is defined by lax sato equations for l only a characterization of the class of reductions in terms of the dressing data is given we also consider a waterbag reduction of the interpolating system hierarchy which defines 1 1 dimensional systems of hydrodynamic type

dunajski generalization of the second heavenly equation dressing method and the hierarchy dunajski generalization of the second heavenly equation is studied a dressing scheme applicable to the dunajski equation is developed an example of constructing solutions in terms of implicit functions is considered a dunajski equation hierarchy is described and its lax sato form is presented the dunajski equation hierarchy is characterized by the conservation of a three dimensional volume form in which a spectral variable is taken into account

brief paper robust adaptive observer for nonlinear systems with unmodeled dynamics unmodeled dynamics exist in almost all applications of observers due to the impossibility of using exact and detailed models it is highly desired that the observers can dominate the effects of unmodeled dynamics independently to prevent the state estimations from diverging and to get the precise estimations based on adaptive nonlinear damping this paper presents a robust adaptive observer for multiple input multiple output nonlinear systems with unknown parameters uncertain nonlinearities disturbances and unmodeled dynamics the observer only has one adaptive parameter no matter how high the order of the system is and how many unknown parameters there are with the proposed observer neither estimating the unknown parameters or solving linear matrix inequalities is needed it is shown that the state estimation error is uniformly bounded and can be made arbitrarily small

hodograph solutions for the manakov santini equation we investigate the integrable 2 1 dimensional manakov santini equation from the lax sato form several particular two and three component reductions are considered so that the manakov santini equation can be reduced to systems of hydrodynamic type then one can construct infinitely many exact solutions of the equation by the hodograph method

non hamiltonian generalizations of the dispersionless 2dtl hierarchy we consider two component integrable generalizations of the dispersionless two dimensional toda lattice 2dtl hierarchy connected with non hamiltonian vector fields similar to the manakov santini hierarchy generalizing the dkp hierarchy they form a one parametric family connected by hodograph type transformations generating equations and lax sato equations are introduced and a dressing scheme based on the vector nonlinear riemann problem is formulated the simplest two component generalization of the dispersionless 2dtl equation is derived and its differential reduction analogous to the dunajski interpolating system is presented a symmetric two component generalization of the dispersionless elliptic 2dtl equation is also constructed

global adaptive observers for nonlinear systems via filtered transformations the problem of determining global adaptive observers for a class of single output nonlinear systems which are linear with respect to an unknown constant parameter vector is addressed sufficient conditions are given to observe asymptotically an equivalent state without persistency of excitation under additional geometric conditions the original state can be observed as well the results obtained are based on a nonlinear change of coordinates driven by auxiliary filters filtered transformations

the dispersionless 2d toda equation dressing cauchy problem longtime behaviour implicit solutions and wave breaking we have recently solved the inverse spectral problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions including the second heavenly equation of plebanski and the dispersionless kadomtsev petviashvili dkp equation arising as commutation of vector fields in this paper we make use of the above theory i to construct the nonlinear riemann hilbert dressing for the so called two dimensional dispersionless toda equation elucidating the spectral mechanism responsible for wave breaking ii we present the formal solution of the cauchy problem for the wave form of it exp tt xx yy iii we obtain the longtime behaviour of the solutions of such a cauchy problem showing that it is essentially described by the longtime breaking formulae of the dkp solutions confirming the expected universal character of the dkp equation as prototype model in the description of the gradient catastrophe of two dimensional waves iv we finally characterize a class of spectral data allowing one to linearize the rh problem corresponding to a class of implicit solutions of the pde

asymptotic periodicity of the iterates of positivity preserving operators soit x un espace de banach reel soit x un sous ensemble ferme de x tel que 1 si x x y x 0 alors x y x et x x 2 il existe m 0 0 tel que pour x x il existe x x et x x qui satisfont x x x x m 0 x m 0 x x et si x y y pour y x y x alors y x x 3 si x x y x alors x x y soit b un operateur lineaire borne sur x soit bx cx f0 est un sous ensemble compact non vide de x et lim n dist b n x f 0 0 quand x x et x 1 alors b n x est asymptotiquement periodique pour x x

adaptive observers for single output nonlinear systems an adaptive version of the nonlinear observer obtained by a j krener et al 1983 is presented this version involves the cancellation of nonlinear terms by output injection as an intermediate step necessary and sufficient conditions are given for transforming a nonlinear system by state space change of coordinates into the special adaptive observer form that was used by y bastin et al 1988 to design adaptive observers

an infinite number of hidden variables in hyper kahler metrics three types of hidden variables both independent and dependent are shown to underlie the hyper kahler geometry in a complexified setting the variables satisfy an infinite set of differential equations hierarchy just as in the case of most nonlinear integrable systems the notion of the plebanski key functions j math phys 16 2395 1975 is extended to this hierarchy to give an analog of the notion of the function two examples of special solutions which are reminiscent of several solution techniques in the theory of nonlinear integrable systems are presented for illustration

asymptotic periodicity of the iterates of markov operators on dit que p l 1 l 1 est un operateur de markov si 1 pf 0 pour f 0 et 2 pf f si f 0 on montre que tout operateur de markov p a une certaine decomposition spectrale si pout tout f l 1 avec f 0 et f 1 p n f 7b f quand n ou 7b f est un sous ensemble fortement compact de l 1

an improved estimation method for unmodeled dynamics based on anfis and its application to controller design by representing nonlinear systems as a combination of linear part and unmodeled dynamics in this paper an improved estimation algorithm using an adaptive neuro fuzzy inference system anfis for unmodeled dynamics is presented at first the unmodeled dynamics is divided into two parts using the differential expansion of the control input at the last time instant then the two parts are estimated by the anfis it has been shown that the proposed algorithm overcomes the problem that the unknown control input is embedded in unmodeled dynamics which makes the true value of unmodeled dynamics difficult obtain moreover the method improves the precision of the estimation of unmodeled dynamics second under the assumption that the growth rate of unmodeled dynamics does not exceed its input vector the one to one mapping and regularization technique are adopted to deal with the input and output data and the unmodeled dynamics respectively as a result the data vector can be guaranteed to lie inside a compact set which ensures the use of the universal approximation property of the anfis on the other hand it has been shown that datum of a system can be fully used to obtain the parameters centers widths in membership functions and the network connection weights in the anfis by offline training these parameters are tuned online to improve the estimation convergence rate of the unmodeled dynamics the effectiveness of the proposed estimation method is illustrated by comparing it with the simulation results that are obtained from the other existing methods finally the proposed estimation method is applied to the nonlinear switching control design both simulation and theoretical analysis have confirmed that the nonlinear switching control which adopts the proposed estimation method cannot only guarantee the stability and convergence of the system but can exhibit a desired dynamic performance for the closed loop system as well

robust control of nonlinear systems with input unmodeled dynamics in the rich repertoire of design methodologies for uncertain nonlinear systems the problem of unmodeled dynamics has thus far received little attention the authors initiate the investigation of nonlinear systems with input unmodeled dynamics first the authors show that even in their simplest forms they can result in dramatic shrinking of the region of attraction and in finite escape time in this paper the authors introduce a dynamic nonlinear damping design which guarantees global boundedness in the presence of input unmodeled dynamics with this design the authors achieve global asymptotic stability for strict feedback systems and output feedback systems

twistor theory at fifty from contour integrals to twistor strings we review aspects of twistor theory its aims and achievements spanning the last five decades in the twistor approach space time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold the twistor space after giving an elementary construction of this space we demonstrate how solutions to linear and nonlinear equations of mathematical physics anti self duality equations on yang mills or conformal curvature can be encoded into twistor cohomology these twistor correspondences yield explicit examples of yang mills and gravitational instantons which we review they also underlie the twistor approach to integrability the solitonic systems arise as symmetry reductions of anti self dual asd yang mills equations and einstein weyl dispersionless systems are reductions of asd conformal equations we then review the holomorphic string theories in twistor and ambitwistor spaces and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes finally we discuss the newtonian limit of twistor theory and its possible role in penrose s proposal for a role of gravity in quantum collapse of a wave function

observer design for uncertain nonlinear systems with unmodeled dynamics this paper is concerned with the observer design problem for a class of multi input multi output nonlinear systems with the unmodeled dynamics unknown parameters and external disturbance a dynamic signal which can dominate the unmodeled dynamics is firstly constructed then two types of observer schemes that is adaptive observer and robust observer are respectively proposed the observation error in the two schemes can be made arbitrarily small by choosing the appropriate design parameters an illustrative example is provided to demonstrate the validity of the proposed design methods

remarks on nonlinear adaptive observer design abstract a solution to the problem of state estimation for systems with unknown parameters is to use some on line adaptation of the observer parameters on the basis of various existing results for such adaptive observer designs a unifying adaptive observer form is proposed in this paper this form indeed emphasizes properties allowing some asymptotic state estimation in spite of unknown parameters as well as additional properties which further allow parameter estimation as an example it is shown how an adaptive observer can be designed for a class of state affine systems

nonlinear observers a circle criterion design and robustness analysis globally convergent observers are designed for a class of systems with monotonic nonlinearities the approach is to represent the observer error system as the feedback interconnection of a linear system and a time varying multivariable sector nonlinearity using lmi software observer gain matrices are computed to satisfy the circle criterion and hence to drive the observer error to zero in output feedback design the observer is combined with control laws that ensure input to state stability with respect to the observer error robustness to unmodeled dynamics is achieved with a small gain assignment design as illustrated on a jet engine compressor example

a systematic approach to adaptive observer synthesis for nonlinear systems geometric techniques of controller design for nonlinear systems have enjoyed great success a serious shortcoming however has been the need for access to full state feedback this paper addresses the issue of state estimation from limited sensor measurements in the presence of parameter uncertainty an adaptive nonlinear observer is suggested for lipschitz nonlinear systems and the stability of this observer is shown to be related to finding solutions to a quadratic inequality involving two variables a coordinate transformation is used to reformulate this inequality as a linear matrix inequality a systematic algorithm is presented which checks for feasibility of a solution to the quadratic inequality and yields an observer whenever the solution is feasible the state estimation errors then are guaranteed to converge to zero asymptotically the convergence of the parameters however is determined by a persistence of excitation type constraint