model structure simplification of nonlinear systems via immersion this paper concerns the simplification of model structures of nonlinear systems while preserving their input output maps the basic technique is an immersion of a system which is a mapping of the initial state from the original system to another system of an identical input output map the necessary and sufficient conditions are presented for the immersibility of a given system into a state space representation of such particular structures as rational functions or polynomials with respect to the state the conditions are so mild that many types of nonlinear systems can be represented by rational or polynomial model structures moreover it is also shown that a polynomial state space representation can always be constructed to be at most quadratic with respect to the state

a recursive elimination method for finite horizon optimal control problems of discrete time rational systems in this technical note a method of solving finite horizon optimal control problems involving discrete time rational systems is proposed sequences of algebraic equations for the control input and costate at each time are constructed backward starting from the terminal condition by the recursive elimination of variables in the optimality conditions this recursive elimination can be viewed as a generalization of the classical backward sweep method to obtain the discrete time riccati equation for finite horizon linear quadratic control sufficient conditions are given for the existence and uniqueness of locally optimal state feedback laws in the form of algebraic functions of the state

system impulse response identification using a multiresolution neural network abstract this paper proposes a new identification method for the discrete time impulse response model of a linear system from sampled input output data our attention is especially focused on identification of the impulse response which includes high frequency components locally the continuous time impulse response of the system is approximated by a multiresolution neural network composed of the scaling and wavelet functions hence the system under study can be viewed as the weighted sum of a group of subsystems in which the scaling functions and wavelet functions are interpreted as their impulse responses respectively then the genetic algorithm and the aic are introduced to select significant subsystems at each resolution level such that some redundant subsystems that are sensitive to the noise effects are discarded it is shown through a simulation that the proposed method yields accurate estimate of the impulse response even in the ill conditioned cases 1997 elsevier science ltd

a continuation gmres method for fast computation of nonlinear receding horizon control in this paper a fast numerical algorithm for nonlinear receding horizon control is proposed the control input is updated by a differential equation to trace the solution of an associated state dependent two point boundary value problem a linear equation involved in the differential equation is solved by the generalized minimum residual method one of the krylov subspace methods with jacobians approximated by forward differences the error in the entire algorithm is analyzed and is shown to be bounded under some conditions the proposed algorithm is applied to a two link arm whose dynamics is highly nonlinear simulation results show that the proposed algorithm is faster than the conventional algorithms

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

nonlinear optimal control as quantum mechanical eigenvalue problems a novel approach for approximating the nonlinear optimal feedback control of a system with a terminal cost is proposed to lessen the difficulty due to nonlinearity we try to treat the system in a framework of linear theories for this we assume a quantum mechanical linear wave associated with the system since the control system is constrained by state equations we handle the system according to quantum mechanics of constrained dynamics a hamiltonian is represented as a linear operator acting on a function that describes behavior of waves subsequently nonlinear feedback is calculated without any time integration in the backward direction using eigenvalues and eigenfunctions of the linear hamiltonian operator an optimal feedback law is given as a combination of analytic functions of time and state variables we take as an example a system described by two scalar variables for state and control input simulation studies on the system by the eigenvalue analysis show that the proposed method reduces calculation time to nearly a tenth that of a numerical calculation of a hamilton jacobi equation by a finite difference method

nonlinear input normal realizations based on the differential eigenstructure of hankel operators this paper investigates the differential eigenstructure of hankel operators for nonlinear systems first it is proven that the variational system and the hamiltonian extension with extended input and output spaces can be interpreted as the ga spl circ teaux differential and its adjoint of a dynamical input output system respectively second the ga spl circ teaux differential is utilized to clarify the main result the differential eigenstructure of the nonlinear hankel operator which is closely related to the hankel norm of the original system third a new characterization of the nonlinear extension of hankel singular values are given based on the differential eigenstructure finally a balancing procedure to obtain a new input normal output diagonal realization is derived the results in this paper thus provide new insights to the realization and balancing theory for nonlinear systems

solutions to the hamilton jacobi equation with algebraic gradients in this paper the hamilton jacobi equation hje with coefficients consisting of rational functions is considered and its solutions with algebraic gradients are characterized in terms of commutative algebra it is shown that there exists a solution with an algebraic gradient if and only if an involutive maximal ideal containing the hamiltonian exists in a polynomial ring over the rational function field if such an ideal is found the gradient of the solution is defined implicitly by a set of algebraic equations then the gradient is determined by solving the set of algebraic equations pointwise without storing the solution over a domain in the state space thus the so called curse of dimensionality can be removed when a solution to the hje with an algebraic gradient exists new classes of explicit solutions for a nonlinear optimal regulator problem are given as applications of the present approach

frobenius algebras of corepresentations and group graded vector spaces abstract we consider frobenius algebras in the monoidal category of right comodules over a hopf algebra h if h is a group hopf algebra we study a more general frobenius type property uncover the structure of graded frobenius algebras and investigate graded symmetric algebras the graded frobenius concept is related to frobenius functors

graded semisimple algebras are symmetric abstract we study graded symmetric algebras which are the symmetric monoids in the monoidal category of vector spaces graded by a group we show that a finite dimensional graded semisimple algebra is graded symmetric the center of a symmetric algebra is not necessarily symmetric but we prove that the center of a finite dimensional graded division algebra is symmetric provided that the order of the grading group is not divisible by the characteristic of the base field

on a state space approach to nonlinear h control we study the standard h optimal control problem using state feedback for smooth nonlinear control systems the main theorem obtained roughly states that the l2 induced norm from disturbances to inputs and outputs can be made smaller than a constant 0 if the corresponding h norm for the system linearized at the equilibrium can be made smaller than by linear state feedback necessary and sufficient conditions for the latter problem are by now well known e g from the state space approach to linear h optimal control our approach to the nonlinear h optimal control problem generalizes the state space approach to the linear h problem by replacing the hamiltonian matrix and corresponding riccati equation as used in the linear context by a hamiltonian vector field together with a hamiltonian jacobi equation corresponding to its stable invariant manifold

frobenius monads and pseudomonoids six equivalent definitions of frobenius algebra in a monoidal category are provided in a monoidal bicategory a pseudoalgebra is frobenius if and only if it is star autonomous autonomous pseudoalgebras are also frobenius what it means for a morphism of a bicategory to be a projective equivalence is defined this concept is related to strongly separable frobenius algebras and weak monoidal morita equivalence wreath products of frobenius algebras are discussed

analytical approximation methods for the stabilizing solution of the hamilton jacobi equation in this paper two methods for approximating the stabilizing solution of the hamilton jacobi equation are proposed using symplectic geometry and a hamiltonian perturbation technique as well as stable manifold theory the first method uses the fact that the hamiltonian lifted system of an integrable system is also integrable and regards the corresponding hamiltonian system of the hamilton jacobi equation as an integrable hamiltonian system with a perturbation caused by control the second method directly approximates the stable flow of the hamiltonian systems using a modification of stable manifold theory both methods provide analytical approximations of the stable lagrangian submanifold from which the stabilizing solution is derived two examples illustrate the effectiveness of the methods

analysis of the hamilton jacobi equation in nonlinear control theory by symplectic geometry in this paper the geometric property and structure of the hamilton jacobi equation arising from nonlinear control theory are investigated using symplectic geometry the generating function of symplectic transforms plays an important role in revealing the structure of the hamilton jacobi equation it is seen that many fundamental properties of the riccati equation can be generalized in the hamilton jacobi equation and therefore the theory of the hamilton jacobi equation naturally contains that of the riccati equation

brief a transformation approach for solving the hamilton jacobi bellman equation in h2 deterministic and stochastic optimal control of affine nonlinear systems in this paper we present a transformation approach for solving the hamilton jacobi bellman equations hjbes arising in h 2 or quadratic cost control of nonlinear deterministic and stochastic systems we show that the hjbe can be solved analogously to a scalar quadratic equation and we give a parameterization of solutions to the hjbe characterizing the solution of the optimal control problem the procedure is also generalized to include nonsmooth or viscosity solutions of the s hjbe

optimal control using an algebraic method for control affine non linear systems a deterministic optimal control problem is solved for a control affine non linear system with a non quadratic cost function we algebraically solve the hamilton jacobi equation for the gradient of the value function this eliminates the need to explicitly solve the solution of a hamilton jacobi partial differential equation we interpret the value function in terms of the control lyapunov function then we provide the stabilizing controller and the stability margins furthermore we derive an optimal controller for a control affine non linear system using the state dependent riccati equation sdre method this method gives a similar optimal controller as the controller from the algebraic method we also find the optimal controller when the cost function is the exponential of integral case which is known as risk sensitive rs control finally we show that sdre and rs methods give equivalent optimal controllers for non linear deterministic systems examples demonstrate the proposed methods

modules comodules and cotensor products over frobenius algebras abstract we characterize noncommutative frobenius algebras a in terms of the existence of a coproduct which is a map of left ae modules we show that the category of right left comodules over a relative to this coproduct is isomorphic to the category of right left modules this isomorphism enables a reformulation of the cotensor product of eilenberg and moore as a functor of modules rather than comodules we prove that the cotensor product m n of a right a module m and a left a module n is isomorphic to the vector space of homomorphisms from a particular left ae module d to n m viewed as a left ae module some properties of d are described finally we show that when a is a symmetric algebra the cotensor product m n and its derived functors are given by the hochschild cohomology over a of n m

algebraic structures in nonlinear systems over rings obtained by immersion an immersion of a system is a mapping of the initial state exactly preserving the input output map as has already been shown a system is immersible into a rational in the state representation rsr and a polynomial in the state representation psr if and only if the field generated by the observation space is finitely generated which is true in most practical systems in this paper some algebraic structures and their geometric counterparts associated with an rsr and a psr obtained via an immersion are discussed first rsrs and psrs are viewed as systems over rings in a unified framework and the notions of an invariant ideal and an invariant variety which are related to a differential algebraic equation are introduced then it is shown that an rsr and a psr have invariant ideals and invariant varieties associated with an immersion in particular an invariant variety of an rsr or a psr is the zariski closure of the image of the immersion i e the smallest variety containing the image of the immersion the degrees of freedom in rsrs and psrs obtained via immersion are also investigated and are characterized in terms of invariant ideals

galerkin approximations of the generalized hamilton jacobi bellman equation in this paper we study the convergence of the galerkin approximation method applied to the generalized hamilton jacobi bellman ghjb equation over a compact set containing the origin the ghjb equation gives the cost of an arbitrary control law and can be used to improve the performance of this control the ghjb equation can also be used to successively approximate the hamilton jacobi bellman equation we state sufficient conditions that guarantee that the galerkin approximation converges to the solution of the ghjb equation and that the resulting approximate control is stabilizing on the same region as the initial control the method is demonstrated on a simple nonlinear system and is compared to a result obtained by using exact feedback linearization in conjunction with the lqr design method