Finite Volterra Series Research Articles

Criteria for the approximation of nonlinear systems

Conditions for the approximation of discrete-time time-invariant nonlinear systems that act between bounded real sequences are considered. It has been shown in different contexts that if the output of a nonlinear system at each moment is dependent on the remote past of the input only to an arbitrary small extent in a certain sense, then the system can be uniformly approximated arbitrarily well by the maps of certain simple structures such as lattice-map structures, finite Volterra-series structures, dynamic multilayered neural networks with sigmoidal hidden units, and dynamic radial-basis-function networks. Results are given to the effect that, for a certain broad class of system approximation problems, conditions introduced earlier and certain variations of other conditions are all equivalent, and are in fact necessary as well as sufficient.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Realization and generalized frequency response for non-linear input-output maps

A realization theory is developed and the ‘generalized frequency response’ is presented for an input-output map introduced in a previous work. Relations with the realization of finite Volterra series are outlined. A simple example is given to show that the new representations have improved convergence properties.

Finite Volterra-series realizations and input-output approximations of non-linear discrete-time systems

Abstract It is shown that a separability property of a given finite family of stationary kernels turns out to be necessary and sufficient for the realization of the associated functional by means of a polynomial affine system (i.e. a system that is polynomial in the input and affine in the state). Moreover, the discrete Volterra kernels associated with the input-output map of a non-linear analytic discrete-lime system, initialized at an equilibrium point, are shown to possess a separability property. On this basis, we state an approximation result for the given input-output map by considering the first kernels of the discrete Volterra series. Two explicit constructions of the approximating polynomial affine systems are proposed.

The Observation Space and Realizations of Finite Volterra Series

In this paper we extend the results on the theory of realizations of finite Volterra series by exploiting the structural properties of the observation space. In general, two distinct minimal realizations of a finite Volterra series may be constructed, one based on the properties of the observation space and the other based on the properties of the Lie algebra. Both these realizations display the canonical structure found earlier for such systems. The results given here also yield information on the observation algebra generated by functions in the observation space, just as previous work gave information on the Lie algebra. As an application, the structure found here is applied to the finite-dimensional filtering problem for these systems.

Dynamical Realizations of Homogeneous Hamiltonian Systems

In this work we consider homogeneous input–output systems defined by a single Volterra kernel, which have an internal state space realization in the form of a Hamiltonian system. Previous work by one of the authors considered the state space realization of input–output maps determined by finite Volterra series, the homogeneous case being characterized by an internal homogeneity. The present paper shows that minimal Hamiltonian realizations exist which exhibit the same polynomic structure whilst simultaneously displaying the canonical symplectic structure. The homogeneity is conveniently handled by extensive use of graded vector spaces, and their relation with nilpotent Lie algebras. The additional relation with symplectic structures is also utilized here. The work also provides a specialized version of the Darboux–Weinstein theorem, which is globally valid.

Nonlinear Input-Output Maps and Approximate Representations*

An approximation theorem is given for causal time-invariant nonlinear maps that take one set of functions defined on [0, ∞) into another. The theorem is used to show that, under some typically very reasonable conditions, an input-output map can be approximated arbitrarily well in a meaningful sense by a finite Volterra series, even though it may not have a Volterra series expansion. The set of inputs on which the approximation holds need not be compact, and the inputs need not be continuous.

Identification and Realization of Polynomial Systems

For stationary, homogeneous, continuous-time systems, we consider the discrete-time realization in the form of polynomial difference equations with a special internal structure of vectors and matrices. The system is described by a finite Volterra series. The different amplitude dependencies of the homogeneous subsystems are used to separate the response contributions of each subsystem. The identification and realization properties of degree 2 and 3 homogeneous subsystems are presented in terms of triangular kernels, and in terms of vectors and matrices. A procedure is given for constructing a minimal dimension, degree 2 and 3 homogeneous realization, with forms of observability and reachability, well known from the linear case.

On Finite Volterra Series and a Theorem of P. Crouch

In this paper, we give a new proof of Crouch’s theorem on the realisation of finite Volterra series and extend it to the $C^\infty $ case.

On finite volterra series which admit hamiltonian realizations

In this paper we give necessary and sufficient conditions for a stationary finite Volterra series to have a linear (in the controls) analytic realization, which at the same time has a Hamiltonian structure. The result generalizes that for linear systems, where the condition is that the impulse response should be an odd function, and is expressed as a particular symmetry condition on the Volterra kernals. The relation between this problem and that of the inverse problem in Newtonian mechanics is explored. The finiteness of the Volterra series implies a nilpotence condition on a certain Lie algebra defined by the realization. The additional requirement that the system be Hamiltonian adds further structure to the classical representation of a nilpotent Lie algebra by lower triangular matrices.

Structural Features and Discretization Of Polynomial Systems

For stationary. homogeneous. continuous-time systems permitting finite Volterra series representation. realizations are considered in the form of vector differential equations with a convenient structural form of matrices and vectors. A procedure is given for constructing the corresponding difference equation under the assumption of stepwise constant input. Bverything is done for the MIMO case.

Solvable Approximations to Control Systems

This paper is concerned with extending certain results of Rothschild and Stein [Acts. Math., 137 (1976), pp. 247–320] and Goodman [Lecture Notes in Mathematics 562, Springer-Verlag, New York, 1976], in which a finite set of vector fields $g_1 , \cdots ,g_m $ are lifted to vector fields approximated by generators of a free nilpotent Lie algebra. We wish to add a vector field f, to the set $g_1 , \cdots ,g_m $ and lift these vector fields to ones on a finite dimensional vector space V, approximated by generators F, $G_1 , \cdots ,G_m $ of a solvable Lie algebra, in which $ad^j F(G_i )$ generate a nilpotent, but not free, ideal. This procedure is accomplished in the context of a nonlinear control system, with outputs, in which f vanishes at the initial state, and in such a way that the output functions lift to the state space V, to define a system whose input output map is the same as the original system. The approximating system is obtained from a suitable realization of a truncation of the Volterra series expansion of the input output map. Such systems, with finite Volterra series, naturally exhibit the required Lie algebra structure.

Finite Dimensional Deterministic Nonlinear Filters via Riccati Transformation and Volterra Series

Filtering in two classes of nonlinear differential and difference systems is studied. The system structures admit the representation of the optimal recursive filter in the form of a finite dimensional differential or difference system. The filtering problem is posed as a set of fixed interval optimization problems. The deterministic least-squares problem statement results in the filter which is an ordinary nonstochastic differential or difference system driven by the observed signal. No stochastic concepts are used. The system classes considered are described by two linear subsystems with a polynomial link map between them. In one class the link map affects the input of the latter subsystem. In the other class the coefficient matrix of the latter subsystem is a polynomial in the state of the preceding subsystem satisfying a Lie algebraic nilpotency condition. The former class is a special case of the systems described by finite Volterra series for which finite dimensionality of the optimal filter is also proved. Some examples and comparisons on the basis of examples with the corresponding stochastic systems admitting finite dimensional optimal conditional mean filters studied recently are performed. The comparisons show that except in some special cases including a linear one the deterministic least-squares filter and the stochastic conditional mean filter are not equivalent.

Dynamical Realizations of Finite Volterra Series

In this paper, realizations of finite Volterra series are viewed as nonlinear analytic input–output systems, with state space described by an analytic manifold. For a minimal realization guaranteed by H. J. Sussmann, the state space, which is unique up to diffeomorphism, is shown to have the homogeneous space structure of a nilmanifold, the quotient of two nilpotent Lie groups. The structure of nilmanifolds as described by A. Malcev is used to show that for these systems, the state space has a vector space structure. As a consequence of this result, it is shown that a minimal realization of a finite Volterra series can be described as a cascade of linear subsystems with polynomial link maps, in which the dimension o f each linear subsystem is independent of the realization considered.

A note on the identification of discrete-time polynomial systems

Values of the kernels for a discrete-time system described by a finite Volterra series can be determined from the system responses to a set of inputs composed of unit pulses.

Optimal nonlinear estimation for a class of discrete-time stochastic systems

Recursive estimation for nonlinear discrete-time stochastic systems with additive white Gaussian observation noise is investigated. It is proved that for certain classes of systems, described either by finite Volterra series expansions or by state-linear realizations under certain algebraic conditions, the optimal conditional mean estimator is recursive and of fixed finite dimension. An example is presented to illustrate the structure of the estimators.

Optimal control of a class of nonlinear systems on Hilbert space

This paper presents necessary and sufficient conditions for the existence of an optimal control for a class of nonlinear systems described by an infinite series of Volterra type. Variational methods are applied to minimize quadratic cost functionals leading to a class of nonlinear integral equations. Sufficient conditions for the existence and uniqueness of a solution of this integral equation are presented using abstract analysis. To the best knowledge of the author these results are new. The technique presented here applies to linear and nonlinear stationary and time-varying systems described by input-output functional relations through finite or infinite Volterra series. An example of nonlinear systems described by an infinite series of Volterra type is presented for illustration.