Linearization of discrete-time systems

We characterize the equivalence of single-input single-output discrete-time nonlinear systems to linear ones, via a state-coordinate change and with or without feedback. Four cases are distinguished by allowing or disallowing feedback as well as by including the output map or not; the interdependence of these problems is analyzed. An important feature that distinguishes these discrete-time problems from the corresponding problem in continuous-time is that the state-coordinate transformation is here directly computable as a higher composition of the system and output maps. Finally, certain connections are made with the continuous-time case.     

Linearization of discrete-time systems by exogenous dynamic feedback

It is shown that a discrete-time system may be linearizable by exogenous dynamic feedback, even if it cannot be linearized by endogenous feedback. This property is completely unexpected and constitutes a fundamental difference with respect to the continuous-time case. The notion of exogenous linearizing output is introduced. It is shown that existence of an exogenous linearizing output is a sufficient condition for dynamic linearizability. Necessary and sufficient conditions for the existence of an exogenous linearizing output are provided. The results of the paper are obtained using transformal operator matrices. The properties of such operators are studied. The theory is applied to the exact discrete-time model of a mobile robot, showing that the above-mentioned property concerns not only academic examples, but also physical systems.     

Linearization of discrete-time systems via restricted dynamic feedback

We extend the results from a previous paper of ours to multiple-input systems, and utilize these to obtain necessary and sufficient conditions for the linearization of discrete-time nonlinear systems via restricted dynamic feedback. We observe that for discrete-time nonlinear systems, the bound on the number of delays (or integrators) needed to synthesize the linearizing dynamic feedback differs from the continuous-time analogue.     

Observers for discrete-time nonlinear systems

The problem of observers for discrete-time nonlinear systems has been considered and a simple, easy-to-implement algorithm is given whose convergence properties are guaranteed for autonomous and forced systems. Some numerical examples show the effectiveness of the proposed observer.     

Fuzzy controllers for a class of discrete-time nonlinear systems

In consideration of system stability, two fuzzy controllers for a class of discrete-time nonlinear systems are proposed. When the knowledge on the control process is available, a fuzzy controller will maintain it as much as possible. In the other case where such knowledge is unavailable, another fuzzy controller can be used where the fuzzy rules are viewed as approximators to deal with some unknown parts in the system to be controlled. Computer simulations are provided to illustrate the effectiveness of the fuzzy controllers proposed in this paper.     

Functional expansions for nonlinear discrete-time systems

Given an input-output map associated with a nonlinear discrete-time state equationx(t + 1) =f(x(t);u(t)) and a nonlinear outputy(t) =h(x(t)), we present a method for obtaining a “discrete Volterra series” representation of the outputy(t) in terms of the controlsu(0), ...,u(t − 1). The proof is based on Taylor-type expansions of the iterated composition of analytic functions. It allows us to make an explicit construction of each kernel, that is, each coefficient of the series expansion ofy(t) in powers of the controls. This is achieved by making use of successive directional derivatives associated with a family of vector fields which are deduced from the discrete state equations. We discuss the use of these vector fields for the analysis and control of nonlinear discrete-time systems. This work was carried out while D. Normand-Cyrot was working at the I.A.S.I. (from March to October 1984) and with the financial support of the Italian C.N.R. (Consiglio Nazionale delle Ricerche).     

Minimax filters for nonlinear discrete-time systems

We consider a dynamic system consisting of two controlled plants whose motions are modeled by nonlinear recurrences. Minimax state estimates for one plant are obtained from observations contaminated with errors that are attributable to the nonlinearity and depend on the motions of both plants.Translated from Kibernetika, No. 4, pp. 97–103, 129, July–August, 1990.     

Generating series for discrete-time nonlinear systems

A simple definition of generating series for discrete-time nonlinear systems is given and a connection is pointed out with Ritt's differential groups introduced 30 years ago.     

Stable inversion for nonlinear discrete-time systems

We solve the exact output tracking problem for square nonlinear discrete-time systems through stable inversion. This stable inversion problem involves finding a bounded solution of a dynamical system driven by a bounded signal. Such a solution is given by a difference representation formula, whose existence, under the proper assumptions, can be proven using a Picard process     

Linearization and input-output decoupling for general nonlinear systems

Necessary and sufficient conditions are obtained for the linearization and input decoupling (by state feedback) of general nonlinear systems. It is shown how these conditions can be derived from the already known conditions for affine nonlinear systems, thereby also elucidating the existing theory for affine systems.     

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.     

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.     

Stabilization of nonlinear discrete-time systems using statedetection

Some recent results concerning stabilization of continuous-time systems using state detection are extended to the case of discrete-time nonlinear systems. It is shown that if a discrete-time locally detectable system can be stabilized by a state feedback law, then it can also be stabilized by a feedback law that depends on the output of a weak detector     

Linearization of nonlinear systems about constant operating points

For bilinear-realizable nonlinear systems described by a Volterra series, an input-output definition of the linearized system about a constant operating point is given. This definition is then compared to the usual notion of linearization in terms of a bilinear realization of the nonlinear system. Properties of the linearized system with respect to the original nonlinear system are developed, and implications for analysis and design of nonlinear systems are discussed.     

Adaptive control of a class of discrete-time affine nonlinear systems

The adaptive control problem is addressed in the paper for a class of discrete-time affine nonlinear input/output stochastic models with linear unknown parameters. The controller is a certainty equivalence weighted one-step-ahead control and is constructed by using the weighted-least-squares and random regularization methods. Global stability of the closed-loop systems is established, which shows that arbitrarily large growth rate is allowed for the multiplicative nonlinear part of the systems.     

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.     

Stable adaptive neurocontrol for nonlinear discrete-time systems

This paper presents a novel approach in designing neural network based adaptive controllers for a class of nonlinear discrete-time systems. This type of controllers has its simplicity in parallelism to linear generalized minimum variance (GMV) controller design and efficiency to deal with complex nonlinear dynamics. A recurrent neural network is introduced as a bridge to compensation simplify controller design procedure and efficiently to deal with nonlinearity. The network weight adaptation law is derived from Lyapunov stability analysis and the connection between convergence of the network weight and the reconstruction error of the network is established. A theorem is presented for the conditions of the stability of the closed-loop systems. Two simulation examples are provided to demonstrate the efficiency of the approach.     

Observer design for autonomous discrete-time nonlinear systems

We obtain a necessary and sufficient condition for a discrete-time nonlinear system to be locally state equivalent to the nonlinear observer form. The result looks similar to the continuous counterpart except for the fact that Ad-operation is utilized instead of ad-operation.     

Nonsmooth stabilizability and feedback linearization of discrete-time nonlinear systems

We consider the problem of stabilizing a discrete-time nonlinear system using a feedback which is not necessarily smooth. A sufficient condition for global dynamical stabilizability of single-input triangular systems is given. We obtain conditions expressed in terms of distributions for the nonsmooth feedback triangularization and linearization of discrete-time systems. Relations between stabilization and linearization of discrete-time systems are given.