A new symbolic calculus for the response of nonlinear systems

We present a new operational calculus for computing the response of nonlinear systems to various deterministic excitations. The use of a new tool: noncommutative generating power series, allows us to derive, by simple algebraic manipulations, the Volterra functional series of the solution of a large class of nonlinear forced differential equations. The symbolic calculus introduced appears as a natural generalization to the nonlinear domain, of the well known Heaviside operational calculus. Moreover, this method has the advantage of allowing the use of a computer.     

Series expansions for analytic systems linear in control

This paper presents a series expansion for the evolution of a class of nonlinear systems characterized by constant input vector fields. We present a series expansion that can be computed via explicit recursive expressions, and we derive sufficient conditions for uniform convergence over the finite- and infinite-time horizon. Furthermore, we present a simplified series and convergence analysis for the setting of second-order polynomial vector fields. The treatment only relies on elementary notions on analytic functions, number theory, and operator norms.     

On input-output linearization of discrete-time nonlinear systems

We consider a nonlinear discrete-time system of the form Σ: x(t+1)=f(x(t), u(t)), y(t) =h(x(t)), where x ε R^N, u ε R^m, y ε R^q and f and h are analytic. Necessary and sufficient conditions for local input-output linearizability are given. We show that these conditions are also sufficient for a formal solution to the global input-output linearization problem. Finally, we show that zeros at infinity of ε can be obtained by the structure algorithm for locally input-output linearizable systems.     

A cumulant based algorithm for the identification of input-output quadratic systems

The identification of a special class of polynomial models is pursued in this paper. In particular a parameter estimation algorithm is developed for the identification of an input-output quadratic model excited by a zero mean white Gaussian input and with the output corrupted by additive measurement noise. Input-output crosscumulants up to the fifth order are employed and the identification problem of the unknown model parameters is reduced to the solution of successive triangular linear systems of equations that are solved at each step of the algorithm. Simulation studies are carried out and the proposed methodology is compared with two least squares type identification algorithms, the output error method and a combination of the instrumental variables and the output error approach. The proposed cumulant based algorithm and the output error method are tested with real data produced by a robotic manipulator.     

Feedback passivity of nonlinear discrete-time systems with direct input-output link

This paper is devoted to the study of the feedback passivity property in nonlinear discrete-time systems. The relative degree and zero dynamics of the non-passive system are related to the feedback passivity of the system. Two main results are presented. First, some relative degree-related properties of passive systems in general form are stated. Second, sufficient conditions in order to render a multiple-input multiple-output (MIMO) system passive by means of a static state feedback control law are obtained.     

On the realization of nonlinear discrete-time systems

It is shown, on the basis of a compact expression of the kernels of the discrete Volterra series associated to a linear analytic discrete-time system, that the kernels satisfy a suitable property which enables their inductive characterization. We also give a first result on the realization of a given family of functions by a linear analytic discrete-time system with linear invertible drift term.     

Dynamic non-minimum phase compensation for SISO nonlinear, affine in the input systems

Non-minimum phase difficulties prevent the use of input-output feedback linearization. To avoid this problem earlier work proposed the use of synthetic outputs in order to place zeros in prescribed locations under a condition of static equivalence. The objective here is also to introduce a synthetic output but this is defined through the use of perturbations aimed at yielding maximum relative degree. Resetting of the initial conditions allows design freedom for closing the gap between the behaviour of the synthetic and actual output.     

Global state and feedback equivalence of nonlinear systems

The problem of finding global state space transformations and global feedback of the form u(t)= α(x) + ν(t) to transform a given nonlinear system to a controllable linear system on Rⁿ or on an open subset of Rⁿ, is considered here. We give a complete set of differential geometric conditions which are equivalent to the existence of a solution to the above problem.     

Input identification to a class of nonlinear input-output causal systems

In this paper, we adapt Lavrent'ev method so to obtain a reconstruction procedure for the input u to a nonlinear input-output system described by a Volterra integral equation. The proof is based on a monotonicity assumption which does not imply the monotonicity of the operators appearing in the Volterra equation.     

Automatic differentiation and nonlinear controller design by exact linearization

Various modern algorithms for controller design are based on differential-geometric concepts. A method of particular importance is called exact linearization via feedback. In this case, the implementation of the controller requires the computation of Lie derivatives, which have been computed symbolically. This can be very time consuming. We present a new computation method relying on automatic differentiation.     

On linear models for nonlinear systems

Best linear time-invariant (LTI) approximations are analysed for several interesting classes of discrete nonlinear time-invariant systems. These include nonlinear finite impulse response systems and a class of nonsmooth systems called bi-gain systems. The Fréchet derivative of a smooth nonlinear system is studied as a potential good LTI model candidate. The Fréchet derivative is determined for nonlinear finite memory systems and for a class of Wiener systems. Most of the concrete results are derived in an ?_∞ signal setting. Applications to linear controller design, to identification of linear models and to estimation of the size of the unmodelled dynamics are discussed.     

Minimality of dynamic input-output decoupling for nonlinear systems

In this note we study the strong dynamic input-output decoupling problem for nonlinear systems. Using an algebraic theory for nonlinear control systems, we obtain for a dynamic input-output decouplable nonlinear system a compensator of minimal dimension that solves the decoupling problem.     

Output frequency response function of nonlinear Volterra systems

An expression for the output frequency response function (OFRF), which defines the explicit analytical relationship between the output spectrum and the system parameters, is derived for nonlinear systems which can be described by a polynomial form differential equation model. An effective algorithm is developed to determine the OFRF directly from system simulation or experimental data. Simulation studies demonstrate the significance of the OFRF concept, and verify the effectiveness of the algorithm which evaluates the OFRF numerically. These new results provide an important basis for the analytical study and design of a wide class of nonlinear systems in the frequency domain.     

On approximate linearization of nonlinear control systems

A method for the approximate linearization of nonlinear control systems based on the ‘state-space exact linearization’ method is presented. An explicit procedure, both for the single-input and for the multiple-input case, is given, which is straightforward to implement.     

Local stabilization of minimum-phase nonlinear systems

In this note we briefly review the notion of a nonlinear minimum-phase system, i.e. a system which when constrained in such a way as to produce zero output, evolves with an asymptotically stable dynamics. The main purpose of the paper is to show that any minimum-phase nonlinear system can always be locally stabilized by smooth state-feedback     

On the problem of feedback linearization

The paper studies and solves in a geometric framework the problem of partial feedback linearization for discrete-time dynamics. An algorithm for computing the largest linearizable subsystem is proposed. This approach can be considered as dual to the one already proposed in literature in an algebraic context.     

Feedback linearization and driftless systems

The problem of dynamic feedback linearization is recast using the notion of dynamic immersion. We investigate here a generic property which holds at every point of a dense open subset, but may fail at some points of interest, such as equilibrium points. Linearizable systems are then systems that can be immersed into linear controllable ones. This setting is used to study the linearization of driftless systems: a geometric sufficient condition in terms of Lie brackets is given; this condition is also shown to be necessary when the number of inputs equals two. Though noninvertible feedbacks are nota priori excluded, it turns out that linearizable driftless systems with two inputs can be linearized using only invertible feedbacks, and can also be put into a chained form by (invertible) static feedback. Most of the developments are done within the framework of differential forms and Pfaffian systems.This work was partially supported by INRIA, NSF Grant ECS-9203491, GR Automatique (CNRS), and DRED (Ministère de l'Éducation Nationale). Part of it was done while the first author was visiting the Center for Control Engineering and Computation, University of California at Santa Barbara.     

Approximate observability of infinite dimensional bilinear systems using a Volterra series expansion

In this paper, the approximate observability of a class of infinite dimensional bilinear systems is investigated. The observability conditions are discussed based on a Volterra series representation of this class of systems. A testable observability criterion is derived for the case where the infinitesimal generator is self-adjoint or Riesz-spectral operator. The theoretical results are illustrated by examples.