adaptive control of non linear continuous time systems using neural networks general relative degree and mimo cases multilayer neural networks are used in a non linear adaptive control problem the plant is an unknown feedback linearizable continuous time system with relative degree 1 the single input single output system is studied first and then the methodology is extended to control square multi input multi output systems the control objective is for the plant to track a reference trajectory and the control law is defined in terms of the outputs of the neural networks the parameters of the networks are updated on line according to an augmented tracking error and the network derivatives a local convergence theorem is given on the convergence of the tracking error this control algorithm is applied to control a two input two output relative degree two system

exponential spl epsiv regulation for multi input nonlinear systems using neural networks this paper considers the problem of robust exponential spl epsiv regulation for a class of multi input nonlinear systems with uncertainties the uncertainties appear not only in the feedback channel but also in the control channel under some mild assumptions an adaptive neural network control scheme is developed such that all the signals of the closed loop system are semiglobally uniformly ultimately bounded and under the control scheme with initial data starting in some compact set the states of the closed loop system is guaranteed to exponentially converge to an arbitrarily specified spl epsi neighborhood about the origin the important contributions of the present work are that a new exponential uniformly ultimately bounded performance is proposed and that the design parameters and initial condition set can be determined easily the development generalizes and improves earlier results for the single input case

dirac lie systems and schwarzian equations abstract a lie system is a system of differential equations admitting a superposition rule i e a function describing its general solution in terms of any generic set of particular solutions and some constants following ideas going back to dirac s description of constrained systems we introduce and analyze a particular class of lie systems on dirac manifolds called dirac lie systems which are associated with dirac lie hamiltonians our results enable us to investigate constants of the motion superposition rules and other general properties of such systems in a more effective way several concepts of the theory of lie systems are adapted to this dirac setting and new applications of dirac geometry in differential equations are presented as an application we analyze solutions of several types of schwarzian equations but our methods can be applied also to other classes of differential equations important for physics

non squareness properties of orlicz lorentz sequence spaces abstract in this paper criteria for non squareness properties non squareness local uniform non squareness and uniform non squareness of orlicz lorentz sequence spaces and of their n dimensional subspaces n n 2 as well as of the subspaces a of all order continuous elements in are given since degenerate orlicz functions and degenerate weight sequences are also admitted these investigations concern the most possible wide class of orlicz lorentz sequence spaces finally as immediate consequences criteria for all non squareness properties of orlicz sequence spaces which complete the results of sundaresan 1966 53 hudzik 1985 23 hudzik 1985 24 are deduced it is worth recalling that uniform non squareness is an important property because it implies super reflexivity as well as the fixed point property see james 1964 31 james 1972 33 and garcia falset et al 2006 19

newton s laws of motion in the form of a riccati equation we discuss two applications of a riccati equation to newton s laws of motion the first one is the motion of a particle under the influence of a power law central potential v r kr ensuremath epsilon for zero total energy we show that the equation of motion can be cast in the riccati form we briefly show here an analogy to barotropic friedmann robertson lemaitre cosmology where the expansion of the universe can be also shown to obey a riccati equation a second application in classical mechanics where again the riccati equation appears naturally are problems involving quadratic friction we use methods reminiscent to nonrelativistic supersymmetry to generalize and solve such problems

lie hamilton systems on the plane properties classification and applications we study lie hamilton systems on the plane i e systems of first order differential equations describing the integral curves of a t dependent vector field taking values in a finite dimensional real lie algebra of planar hamiltonian vector fields with respect to a poisson structure we start with the local classification of finite dimensional real lie algebras of vector fields on the plane obtained in a gonzalez lopez n kamran and p j olver proc london math soc 64 339 1992 and we interpret their results as a local classification of lie systems moreover by determining which of these real lie algebras consist of hamiltonian vector fields with respect to a poisson structure we provide the complete local classification of lie hamilton systems on the plane we present and study through our results new lie hamilton systems of interest which are used to investigate relevant non autonomous differential equations e g we get explicit local diffeomorphisms between such systems in particular the milne pinney second order kummer schwarz complex riccati and buchdahl equations as well as some lotka volterra and nonlinear biomathematical models are analysed from this lie hamilton approach

geometric hamilton jacobi theory on nambu poisson manifolds the hamilton jacobi theory is a formulation of classical mechanics equivalent to other formulations as newtonian lagrangian or hamiltonian mechanics the primordial observation of a geometric hamilton jacobi theory is that if a hamiltonian vector field xh can be projected into the configuration manifold by means of a 1 form dw then the integral curves of the projected vector field xhdwcan be transformed into integral curves of xh provided that w is a solution of the hamilton jacobi equation our aim is to derive a geometric hamilton jacobi theory for physical systems that are compatible with a nambu poisson structure for it we study lagrangian submanifolds of a nambu poisson manifold and obtain explicitly an expression for a hamilton jacobi equation on such a manifold we apply our results to two interesting examples in the physics literature the third order kummer schwarz equations and a system of n copies of a first order differential riccati equation from the first example we retrieve the origi

uniform nonsquareness and locally uniform nonsquareness in orlicz bochner function spaces and applications abstract in this paper criteria for uniform nonsquareness and locally uniform nonsquareness of orlicz bochner function spaces equipped with the orlicz norm are given although criteria for uniform nonsquareness and locally uniform nonsquareness in orlicz function spaces were known we can easily deduce them from our main results moreover we give a sufficient condition for an orlicz bochner function space to have the fixed point property

on lie systems and kummer schwarz equations a lie system is a system of first order differential equations admitting a superposition rule i e a map that expresses its general solution in terms of a generic family of particular solutions and certain constants in this work we use the geometric theory of lie systems to prove that the explicit integration of second and third order kummer schwarz equations is equivalent to obtaining a particular solution of a lie system on sl 2 r this same result can be extended to riccati milne pinney and to the here defined generalised kummer schwarz equations which include several types of kummer schwarz equations as particular cases we demonstrate that all the above mentioned equations related to the same lie system on sl 2 r can be integrated simultaneously which retrieves and generalizes in a unified and simpler manner previous results appearing in the literature as a byproduct we recover various properties of the schwarzian derivative

discrete time model reference adaptive control of nonlinear dynamical systems using neural networks a multilayer discrete time neural net nn controller is presented for the direct model reference adaptive control of a class of multi input multi output mim0 nonlinear dynamical systems the nonlinear dynamical system is assumed to be controllable and its state vector is available for measurement the nn controller exhibits learning while functioning features instead of learning then control so that control is immediate with no explicit learning phase needed the tracking error between the output of a nonlinear plant and an ideal linear model converges within a very short time persistence of excitation pe is not needed linearity in the parameters is not required and certainty equivalence is not used this overcomes several limitations of standard adaptive control the novel weight tuning paradigm for the nn controller is based on the well known delta rule but includes a modification to the learning rate parameter plus a correction term it guarantees tracking as well as bounded nn weights in non i

recursive hybrid algorithm for non linear system identification using radial basis function networks recursive identification of non linear systems is investigated using radial basis function networks a novel approach is adopted which employs a hybrid clustering and least squares algorithm the recursive clustering algorithm adjusts the centres of the radial basis function network while the recursive least squares algorithm estimates the connection weights of the network because these two recursive learning rules are both linear rapid convergence is guaranteed and this hybrid algorithm significantly enhances the real time or adaptive capability of radial basis function models the application to simulated real data are included to demonstrate the effectiveness of this hybrid approach

brief paper novel adaptive neural control design for nonlinear mimo time delay systems in this paper we address the problem of adaptive neural control for a class of multi input multi output mimo nonlinear time delay systems in block triangular form based on a neural network nn online approximation model a novel adaptive neural controller is obtained by constructing a novel quadratic type lyapunov krasovskii functional which not only efficiently avoids the controller singularity but also relaxes the restriction on unknown virtual control coefficients the merit of the suggested controller design scheme is that the number of online adapted parameters is independent of the number of nodes of the neural networks which reduces the number of the online adaptive learning laws considerably the proposed controller guarantees that all closed loop signals remain bounded while the output tracking error dynamics converges to a neighborhood of the origin a simulation example is given to illustrate the design procedure and performance of the proposed method

superposition principles for matrix riccati equations a superposition rule is obtained for the matrix riccati equation mre w a wb cw wdw where w t a t b t c t and d t are real n n matrices expressing the general solution in terms of five known solutions for all n 2 the symplectic mre w wt a at d dt c bt is treated separately and a superposition rule is derived involving only four known solutions for the unitary and gl n r subcases with d a and c bt or d a and c bt respectively superposition rules are obtained involving only two solutions the derivation of these results is based upon an interpretation of the mre in terms of the action of the groups sl 2n r sp 2n r u n and gl n r on the grassman manifold gn r2n

adaptive control of nonlinear systems using neural networks layered networks are used in a nonlinear adaptive control problem the plant is an unknown feedback linearizable discrete time system represented by an input output model a state space model of the plant is obtained to define the zero dynamics which are assumed to be stable a linearizing feedback control is derived in terms of some unknown nonlinear functions to identify these functions it is assumed that they can be modelled by layered neural networks the weights of the networks are updated and used to generate the control a local convergence result is given computer simulations verify the theoretical result

lie point symmetry analysis of a second order differential equation with singularity abstract by using lie symmetry methods we identify a class of second order nonlinear ordinary differential equations invariant under at least one dimensional subgroup of the symmetry group of the ermakov pinney equation in this context nonlinear superposition rule for second order kummer schwarz equation is rediscovered invariance under one dimensional symmetry group is also used to obtain first integrals ermakov lewis invariants our motivation is a type of equations with singular term that arises in many applications in particular in the study of general nls nonlinear schrodinger equations

robust neural network tracking controller using simultaneous perturbation stochastic approximation this paper considers the design of robust neural network tracking controllers for nonlinear systems the neural network is used in the closed loop system to estimate the nonlinear system function we introduce the conic sector theory to establish a robust neural control system with guaranteed boundedness for both the input output i o signals and the weights of the neural network the neural network is trained by the simultaneous perturbation stochastic approximation spsa method instead of the standard backpropagation bp algorithm the proposed neural control system guarantees closed loop stability of the estimation system and a good tracking performance the performance improvement of the proposed system over existing systems can be quantified in terms of preventing weight shifts fast convergence and robustness against system disturbance

invariance of second order ordinary differential equations under two dimensional affine subalgebras of ermakov pinney lie algebra abstract using the only admissible rank two realisations of the lie algebra of the affine group in one dimension in terms of the lie algebra of lie symmetries of the ermakov pinney ep equation some classes of second order nonlinear ordinary differential equations solvable by reduction method are constructed one class includes the standard ep equation as a special case a new ep equation with a perturbed potential but admitting the same solution formula as ep itself arises the solution of the dissipative ep equation is also discussed

erratum on lie systems and kummer schwarz equations j math phys 54 033505 2013 we show that several types of the so called generalised kummer schwarz equations cannot be treated through a lie system in the way described in our work on lie systems and kummer schwarz equations

practical identification of narmax models using radial basis functions a wide class of discrete time non linear systems can be represented by the nonlinear autoregressive moving average narmax model with exogenous inputs this paper develops a practical algorithm for identifying narmax models based on radial basis functions from noise corrupted data the algorithm consists of an iterative orthogonal forward regression routine coupled with model validity tests the orthogonal forward regression routine selects parsimonious radial basistunc tion models while the model validity tests measure the quality of fit the modelling of a liquid level system and an automotive diesel engine are included to demonstrate the effectiveness of the identification procedure

predicting the future advantages of semilocal units in investigating gaussian radial basis function rbf networks for their ability to model nonlinear time series we have found that while rbf networks are much faster than standard sigmoid unit backpropagation for low dimensional problems their advantages diminish in high dimensional input spaces this is particularly troublesome if the input space contains irrelevant variables we suggest that this limitation is due to the localized nature of rbfs to gain the advantages of the highly nonlocal sigmoids and the speed advantages of rbfs we propose a particular class of semilocal activation functions that is a natural interpolation between these two families we present evidence that networks using these gaussian bar units avoid the slow learning problem of sigmoid unit networks and very importantly are more accurate than rbf networks in the presence of irrelevant inputs on the mackey glass and coupled lattice map problems the speedup over sigmoid networks is so dramatic that the difference in training time between rbf and gaussian bar networks is minor gaussian bar architectures that superpose composed gaussians gaussians of gaussians to approximate the unknown function have the best performance we postulate that an interesing behavior displayed by gaussian bar functions under gradient descent dynamics which we call automatic connection pruning is an important factor in the success of this representation

stable neural network based adaptive control for sampled data nonlinear systems for a class of mimo sampled data nonlinear systems with unknown dynamic nonlinearities a stable neural network nn based adaptive control approach which is an integration of an nn approach and the adaptive implementation of the variable structure control with a sector is developed the sampled data nonlinear system is assumed to be controllable and its state vector is available for measurement the variable structure control with a sector serves two purposes one is to force the system state to be within the state region in which the nn s are used when the system goes out of neural control and the other is to provide an additional control until the system tracking error metric is controlled inside the sector within the network approximation region the proof of a complete stability and a tracking error convergence is given and the setting of the sector and the nn parameters is discussed it is demonstrated that the asymptotic error of the system can be made dependent only on inherent network approximation errors and the frequency range of unmodeled dynamics simulation studies of a two link manipulator show the effectiveness of the proposed control approach

symmetry and singularity properties of the generalised kummer schwarz and related equations abstract we examine the generalised kummer schwarz equation and some of its generalisations from the viewpoints of symmetry and singularity analyses we determine the complete symmetry group of the general equation and show that different forms of the fourth order representative illustrate the three possible classes of laurent series to be expected in the course of the singularity analysis

mixed superposition rules and the riccati hierarchy abstract mixed superposition rules i e functions describing the general solution of a system of first order differential equations in terms of a generic family of particular solutions of first order systems and some constants are studied the main achievement is a generalization of the celebrated lie scheffers theorem characterizing systems admitting a mixed superposition rule this somehow unexpected result says that such systems are exactly lie systems i e they admit a standard superposition rule this provides a new and powerful tool for finding lie systems which is applied here to studying the riccati hierarchy and to retrieving some known results in a more efficient and simpler way

b convexity the analytic radon nikodym property and individual stability ofc0 semigroups abstract lett t t t 0be ac0 semigroup on a banach spacex with generatoraand growth bound assume thatx0 xis such that the local resolvent r a x0admits a bounded holomorphic extension to the right half plane re 0 we prove the following results i ifhas fourier type 1 2 then lim 0 for all 1 and ii ifhas the analytic rnp then lim 0 for all 1 and iii ifis arbitrary then weak lim 0 for all 1 and as an application we prove a tauberian theorem for the laplace transform of functions with values in ab convex banach space

noncreasy and uniformly noncreasy orlicz bochner function spaces abstract we point out that uniform noncreasiness is a super property in banach spaces moreover we prove that orlicz bochner function spaces are noncreasy if and only if they are rotund or smooth finally we obtain that orlicz bochner function spaces are uniformly noncreasy if and only if they are uniformly rotund or uniformly smooth

superposition formulas for nonlinear superequations nonlinear superequations for which the general solution can be expressed algebraically in terms of a finite number of particular solutions are obtained they are based on the orthosymplectic supergroup osp m 2n and its action on a homogeneous superspace superposition formulas are discussed for the cases m 1 n arbitrary and m 2 n 1 for osp 2 2 the number of particular solutions needed to reconstruct the general solution depends on the dimension of the underlying grassmann algebra whereas for osp 1 2n it does not

the exp function approach to the schwarzian korteweg de vries equation by means of the exp function method and its generalization we report further exact traveling wave solutions in a concise form to the schwarzian korteweg de vries equation which admits physical significance in applications not only solitary and periodic waves but also rational solutions are observed

feedback linearization using neural networks abstract for a class of single input single output continuous time nonlinear systems a multilayer neural network based controller that feedback linearizes the system is presented control action is used to achieve tracking performance for a state feedback linearizable but unknown nonlinear system the control structure consists of a feedback linearization portion provided by two neural networks plus a robustifying portion that keeps the control magniture bounded a stability proof is given in the sense of lyapunov it is shown that all the signals in the closed loop system are uniformly ultimately bounded no off line learning phase is needed initialization of the network weights is straightforward

uniformly non ln 1 locally uniformly non ln 1 and non ln 1 properties in orlicz bochner function spaces endowed with the orlicz norm abstract some criteria are given in this paper for uniformly non l n 1 locally uniformly non l n 1 and non l n 1 properties of orlicz bochner function spaces endowed with the orlicz norm as the consequence we can immediately get the characterization of b convexity uniform nonsquareness local uniform nonsquareness and nonsquareness in orlicz bochner function spaces although some of them were obtained in recent years

on the generalizations of the kummer schwarz equation abstract we carry out complete group classifications of two non homogeneous generalizations of the kummer schwarz equation in particular we prove that the symmetry algebra for one of them is s l 2 r s l 2 r and that it can be mapped by an equivalence transformation to the homogeneous kummer schwarz equation for this purpose we calculate the continuous equivalence group for that equation we also study the linearization of the considered equations

gaussian networks for direct adaptive control a direct adaptive tracking control architecture is proposed and evaluated for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible the architecture uses a network of gaussian radial basis functions to adaptively compensate for the plant nonlinearities under mild assumptions about the degree of smoothness exhibit by the nonlinear functions the algorithm is proven to be globally stable with tracking errors converging to a neighborhood of zero a constructive procedure is detailed which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy a stable weight adjustment mechanism is determined using lyapunov theory the network construction and performance of the resulting controller are illustrated through simulations with example systems

uniformly nonsquare banach spaces have the fixed point property for nonexpansive mappings abstract it is shown that if the modulus x of nearly uniform smoothness of a reflexive banach space satisfies x 0 1 then every bounded closed convex subset of x has the fixed point property for nonexpansive mappings in particular uniformly nonsquare banach spaces have this property since they are properly included in this class of spaces this answers a long standing question in the theory

on the uniqueness of the schwarzian and linearisation by nonlocal contact transformation abstract we investigate the uniqueness of the schwarzian from a group theoretic point of view the kummer schwarz equation obtained from the schwarzian is also analysed for its group properties related equations arising in the study of geodesic curves on a surface of constant curvature are constructed these equations are generalised by reducing the symmetry imposed the unique generalisation is linearised under a nonlocal contact transformation

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

classroom note the lagrange charpit method we give a rigorous description of the lagrange charpit method used to find a complete integral of a nonlinear p d e adapted for a university course in differential equations

k symplectic lie systems theory and applications abstract a lie system is a system of first order ordinary differential equations describing the integral curves of a t dependent vector field taking values in a finite dimensional real lie algebra of vector fields a so called vessiot guldberg lie algebra we suggest the definition of a particular class of lie systems the k symplectic lie systems admitting a vessiot guldberg lie algebra of hamiltonian vector fields with respect to the presymplectic forms of a k symplectic structure we devise new k symplectic geometric methods to study their superposition rules t independent constants of motion and general properties our results are illustrated through examples of physical and mathematical interest as a byproduct we find a new interesting setting of application of the k symplectic geometry systems of first order ordinary differential equations

an adaptive variable structure model following control design for robot manipulators an adaptive variable structure model following control avsmfc design is proposed for trajectory tracking in a nonlinear robot system which ensures the stability of the intersection of the surfaces without necessarily stabilizing each individual one the approach avoids the difficulties linked to the strict positive realness requirement in traditional amfc by taking advantage of the inherent positive definiteness of the manipulator s inertia matrix and is easily extendible to a higher number of links the controller does not require any knowledge of a nonlinear robotic system and does not necessarily need the occurrence of a sliding mode at each individually stable discontinuity surface it thus greatly reduces the complexity of design in the closed loop the joint angles asymptotically converge to the reference trajectory with a prescribed transient response chattering is eliminated by restricting the state of the system to slide within a boundary layer rather than along the intersection of the surfaces

multi component ermakov systems structure and linearization abstract a symmetry reduction of a 2 1 dimensional nonlinearn layer fluid model is shown to lead to a prototype for ann component extension of the classical ermakov system the generaln component ermakov system introduced here has the attractive property that it may be iteratively reduced to a system ofn 2 linear equations augmented by a canonical 2 component ermakov system the recently established linearization procedure for the latter may then be used to solve then component systemn 2 in generality the procedure is illustrated in detail for a 3 component system sequences of classical 2 component ermakov systems are shown to be linked via darboux transformations

superposition formulas for pseudounitary matrix riccati equations the purpose of this article is to derive a superposition formula for the pseudounitary matrix riccati equation of dimension n 2 the superposition formula will be written in closed form in terms of five particular solutions satisfying certain well specified conditions defining a fundamental set examples will be studied in order to show how the superposition formula works and how it can be used in numerical calculations