robustness of nonlinear control systems with respect to unmodeled dynamics in theory it can be established that nonlinear control laws for linear or nonlinear plants can be used to meet strict performance requirements the success of these control designs in practical situations will very much depend on whether they can still meet the expected performance characteristics in the presence of inevitable modeling errors in this paper we develop necessary and sufficient conditions for a general class of nonlinear control laws in the presence of high frequency unmodeled dynamics under which global signal boundedness or asymptotic stability is guaranteed we show that a wide class of nonlinear control laws does not satisfy these conditions and therefore does not guarantee global stability in the presence of high frequency unmodeled dynamics

a generalized formula for integrable classes of surfaces in lie algebras we discuss relations between the approach of fokas and gelfand to immersions on lie algebras and the theory of soliton surfaces of sym we show that many results concerning immersions on lie algebras can be reduced to or interpreted within the soliton surfaces approach we present also some new results including a generalization of the fokas gelfand formula for integrable classes of surfaces in lie algebras and in particular in pseudo euclidean n dim spaces the generalized formula is used to formulate a method of constructing integrable classes of surfaces as an example we discuss the class of linear weingarten surfaces defined by the linear relationship between gaussian and mean curvatures we construct explicitly a one parameter family of linear weingarten surfaces parallel equidistant to a given pseudospherical surface

integrator backstepping for rounded controls and control rates we present a backstepping procedure for the design of globally stabilizing state feedback control laws such that the magnitudes of the control signals and their derivatives are bounded by constants which do not depend on the initial conditions we accomplish this by propagating such boundedness properties through each step of the recursive design

technical communique robust backstepping for the euler approximate model of sampled data strict feedback systems stabilization of uncertain sampled data strict feedback systems is addressed the stability study is carried out on the euler approximation of the exact discretized model of the plant firstly a class of state feedback controllers is developed that guarantees an input to state stability property for the closed loop system additionally assuming some hypotheses on the uncertain terms hold a practical asymptotic stability property is ensured by designing an appropriate class of controllers

backlund transformations of linear weingarten surfaces in minkowski three space the classical backlund transformation which transforms surfaces of constant negative curvature has been extended to surfaces in minkowski three space in this article the tangency requirement of the backlund transformation is relaxed to obtain a transformation of surfaces of minkowski three space with mean and gaussian curvatures satisfying a linear relationship

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

robustness of uncertain systems in the absence of matching assumptions abstract we consider a class of uncertain dynamical systems containing uncertain elements these may be due to model parameter uncertainty input disturbance and measurement noise we require no a priori information on the uncertain elements except that their restraint sets are assumed to be known and compact this paper describes a general approach for designing controls to assure practical stability in the absence of matching assumptions which most previous results have needed the result is applicable to non linear systems a specialization to linear systems is then given and a comparison is made with previous work

another integrable case in two dimensional plasticity in this paper we continue the investigation of the constant astigmatism equation zyy 1 z xx 2 0 we newly interpret its solutions as describing spherical orthogonal equiareal patterns which links them to principal stress lines under the tresca yield condition on the sphere by extending the classical bianchi superposition principle for the sine gordon equation we show how to generate an arbitrary number of solutions by algebraic manipulations finally we show that slip line fields on the sphere are determined by the sine gordon equation

bi hamiltonian structures of the coupled akns hierarchy and the coupled yajima oikawa hierarchy the hamiltonian theory for the two component akns hierarchy and yajima oikawa hierarchy is considered from the viewpoint of reduction we show that the second hamiltonian structures of the former is a dirac reduction of the sl 3 current algebra while the latter is related to the classical w 3 4 algebra

brief robust regulation with minimal parametrization for a class of nonlinear systems a class of nonlinear systems that are affected by unknown unmodelled time varying parameters which enter linearly in the system whose bounds are unknown is considered a systematic procedure is developed for the synthesis of robust regulation state feedback laws whose dynamics are minimal i e it consists of a single integrator and a smooth nonlinear mapping using as a key tool the so called small gain theorem the proposed scheme substantially improves previous solutions in which the regulator s dynamics order was equal to the space dimension

on the invariant theory of weingarten surfaces in euclidean space on any weingarten surface in euclidean space strongly regular or rotational we introduce locally geometric principal parameters and prove that such a surface is determined uniquely up to a motion by a special invariant function which satisfies a natural nonlinear partial differential equation this result can be interpreted as a solution to the lund regge reduction problem for weingarten surfaces in euclidean space we apply this theory to fractional linear weingarten surfaces and obtain the nonlinear partial differential equations describing them

multiple surface sliding control of a class of uncertain nonlinear systemsm we develop a multiple surface sliding control for a class of single input single output nonlinear systems whose uncertainties do not satisfy the standard matching condition a coordinate dependence condition on the uncertainty bound is assumed a computed normal form is defined to handle such uncertainties instead of the normal form used in conventional input output linearization a control algorithm which can be applied to global tracking problems has been developed and evaluated for a benchmark example

a novel adaptive fuzzy variable structure control for a class of nonlinear uncertain systems via backstepping abstract this paper presents a new design of adaptive fuzzy variable structure control to solve the traditional problem of model reference adaptive control mrac for a class of single input single output minimum phase uncertain nonlinear systems via backstepping instead of taking the tedious coordinate transformation and yielding a hard high gain controller we introduce smooth b spline type membership functions into the controller so as to compensate for the uncertainties much softer i e in a much smoother and locally weighted manner to be rigorous it is shown that the stability of the closed loop system can be assured and the tracking error can globally approach to an arbitrary preset dead zone range in order to demonstrate the effectiveness of the developed novel controller an example is extensively simulated to provide quite satisfactory performance

global robust stabilization of minimum phase nonlinear systems with uncertainty the problem of global stabilization via state feedback is studied for a class of minimum phase nonlinear systems with unknown unmodelled time varying parameters or disturbances belonging to a given compact set sufficient conditions are presented for the existence of a robust stabilizing state feedback controller the controller is constructed explicitly by using a lyapunov based recursive design procedure the proposed feedback control law is a smooth function of the state and requires information of bounding functions on the size of uncertainties in the case where the disturbance signals represent only an unknown constant vector parameter and appear linearly in the system an adaptive controller is developed based on similar arguments achieving global stability with state regulation the main results of this paper incorporate and generalize a number of robust and adaptive control schemes for global stabilization of uncertain feedback linearizable systems without zero dynamics

affine lie algebraic origin of constrained kp hierarchies an affine sl n 1 algebraic construction of the basic constrained kp hierarchy is presented this hierarchy is analyzed using two approaches namely linear matrix eigenvalue problem on hermitian symmetric space and constrained kp lax formulation and it is shown that these approaches are equivalent the model is recognized to be the generalized non linear schrodinger gnls hierarchy and it is used as a building block for a new class of constrained kp hierarchies these constrained kp hierarchies are connected via similarity backlund transformations and interpolate between gnls and multi boson kp toda hierarchies our construction uncovers the origin of the toda lattice structure behind the latter hierarchy

hierarchies of evolution equations associated with certain symmetric spaces hierarchies of evolution equations associated with certain symmetric spaces are presented main examples of such symmetric spaces are hermitian symmetric spaces and the obtained evolution equations include generalized nonlinear schrodinger equations those results are obtained by extending the lie algebra in the ablowitz kaup newell segur scheme from sl 2 c to more general one

on integrability of weingarten surfaces a forgotten class rediscovered by a systematic search a forgotten class of integrable surfaces is shown to disprove the finkel wu conjecture the associated integrable nonlinear partial differential equation possesses a zero curvature representation a third order symmetry and a nonlocal transformation to the sine gordon equation sin we leave open the problem of finding a backlund autotransformation and a recursion operator that would produce a local hierarchy

matrix nonlinear schrodinger equations and moment maps into loop algebras it is shown how darboux coordinates on a reduced symplectic vector space may be used to parametrize the phase space on which the finite gap solutions of matrix nonlinear schrodinger equations are realized as isospectral hamiltonian flows the parametrization follows from a moment map embedding of the symplectic vector space reduced by suitable group actions into the dual g of the algebra g of positive frequency loops in a lie algebra g the resulting phase space is identified with a poisson subspace of g consisting of elements that are rational in the loop parameter reduced coordinates associated to the various hermitian symmetric lie algebras g k corresponding to the classical lie algebras are obtained

stable neural controller design for unknown nonlinear systems using backstepping we propose from an adaptive control perspective a neural controller for a class of unknown minimum phase feedback linearizable nonlinear system with known relative degree the control scheme is based on the backstepping design technique in conjunction with a linearly parametrized neural network structure the resulting controller however moves the complex mechanics involved in a typical backstepping design from off line to online with appropriate choice of the network size and neural basis functions the same controller can be trained online to control different nonlinear plants with the same relative degree with semi global stability as shown by the simple lyapunov analysis meanwhile the controller also preserves some of the performance properties of the standard backstepping controllers simulation results are shown to demonstrate these properties and to compare the neural controller with a standard backstepping controller

design of softer robust nonlinear control laws abstract we show that the lyapunov function used in backstepping feedback designs for uncertain nonlinear systems leads to unnecessarily hard control laws having undesirable high gain properties we present a new lyapunov function and use it to design softer control laws which exhibit the high gain properties to a much lesser extent we show that the soft designs eliminate the chattering exhibited by the hard designs and achieve the same or better performance with less control effort

a new class of stabilizing controllers for uncertain dynamical systems this paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems the vector of uncertain parameters q cdot is time varying and its values q t lie within a prespecified bounding set q in r p furthermore no statistical description of q cdot is assumed and the controller is shown to render the closed loop system practically stable in a so called guaranteed sense that is the desired stability properties are assured no matter what admissible uncertainty q cdot is realized within the perspective of previous research in this area this paper contains one salient feature the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time invariant in contrast in much of the previous literature see for example 1 2 7 and 9 a linear system is stabilized via nonlinear control another feature of this paper is the fact that the

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

brief robust control of nonlinear systems with parametric uncertainty probabilistic robustness analysis and synthesis for nonlinear systems with uncertain parameters are presented monte carlo simulation is used to estimate the likelihood of system instability and violation of performance requirements subject to variations of the probabilistic system parameters stochastic robust control synthesis searches the controller design parameter space to minimize a cost that is a function of the probabilities that design criteria will not be satisfied the robust control design approach is illustrated by a simple nonlinear example a modified feedback linearization control is chosen as controller structure and the design parameters are searched by a genetic algorithm to achieve the tradeoff between stability and performance robustness

kac moody lie algebras and soliton equations ii lax equations associated with a1 1 the soliton equations associated with sl 2 eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment they are shown to be generated by a single family of commuting hamiltonians on a subalgebra of the loop algebra of sl 2 the conserved densities and fluxes of the usual anks hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems the hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment

robust tracking control of uncertain nonlinear systems with unknown control directions a global robust tracking control design procedure is proposed for a class of uncertain nonlinear systems the key point is that the signs of multiplicative uncertainties the so called control directions are not assumed to be known a priori the class of systems can be of arbitrary dynamic order and the unmatched additive uncertainties need not satisfy the global lipschitz condition it is proved that under the proposed control all closed loop states are bounded and the tracking error converges to any prescribed small neighborhood of the origin the results of this paper enlarge the class of uncertain nonlinear systems for which global robust tracking control can be designed

surfaces on lie groups on lie algebras and their integrability it is shown that the problem of the immersion of a 2 dimensional surface into a 3 dimensional euclidean space as well as then dimensional generalization of this problem is related to the problem of studying surfaces in lie groups and surfaces in lie algebras a particular case of the general formalism presented here implies that any surface can be characterized in terms of 2 2 matrices using an arbitrary parametrization it is also shown that this generality of parametrization is useful for studying integrable surfaces i e surfaces described by integrable equations in particular starting from a suitable lax pair i e a suitable integrable equation it is possible to construct explicitly large classes of integrable surfaces

decentralized global robust stabilization of a class of interconnected minimum phase nonlinear systems this paper focuses on a class of large scale interconnected minimum phase nonlinear systems with parameter uncertainty and nonlinear interconnections the uncertain parameters are allowed to be time varying and enter the systems nonlinearly the interconnections are bounded by nonlinear functions of states the problem we address is to design a decentralized robust controller such that the closed loop large scale interconnected nonlinear system is globally asymptotically stable for all admissible uncertain parameters and interconnections it is shown that decentralized global robust stabilization of the system can be achieved using a control law obtained by a recursive design method together with an appropriate lyapunov function

nonlinear schrodinger equations and simple lie algebras we associate a system of integrable generalised nonlinear schrodinger nls equations with each hermitian symmetric space these nls equations are considered as reductions of more general systems this time associated with a reductive homogeneous space the nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space the hamiltonian structure is investigated using r matrix techniques and shown to be canonical for all these equations throughout the reduction procedure this hamiltonian structure does not degenerate each of the above systems of equations is gauge equivalent to a generalised ferromagnet reductions of the latter are discussed in terms of the corresponding nls type equations

brief backstepping for nonsmooth systems the paper presents a constructive control design for integrator backstepping in nonsmooth systems the approach is based on non smooth analysis and lyapunov stability for nonsmooth systems and is similar in spirit with the robust control designs that have appeared in literature but is applicable to a larger class of systems the backstepping controller is first applied to the case of a unicycle driven by a new discontinuous kinematic controller yielding global asymptotic convergence with bounded inputs then it is used to implement a sliding mode controller in a hybrid system simulations results not only verify the convergence properties but also reveal the ability of the new backstepping controller to suppress chattering