Approximating nonlinear fading-memory operators using neural network models

A fading-memory system is defined as a system whose response map has unique asymptotic properties over some set of inputs. It is shown that any discrete-time fading-memory system can be uniformly approximated arbitrarily closely over a compact set of input sequences by uniformly approximating either its external or internal representation sufficiently closely. In other words, the problem of uniformly approximating a fading-memory system reduces to the problem of uniformly approximating continuous real-valued functions on compact sets. The perceptron is shown to realize a set of continuous real-valued functions that is uniformly dense on compacta in the set of all continuous functions. Using the perceptron to uniformly approximate the external and internal representations of a fading-memory system results, respectively, in simple nonlinear finite-memory and infinite-memory system models.     

Expansions for discrete-time nonlinear systems

Results are given concerning the existence and other properties of Volterralike input-output expansions for discrete-time systems whose inputs and outputs are defined on the time set {0,1,2,}. The results, which establish the existence of (and show how to obtain) a locally convergent input-output expansion for a very large class of systems, are along the lines of those reported recently by this writer for continuous-time systems. With regard to one of the theorems given here, new ground is broken in that the theorem directly addresses the case where the nonlinear part of the system need not be assumed to be memoryless. Systems of the kind considered arise, for example, in the area of communication channel modeling for digital data transmission.     

Structure theorems for nonlinear systems

Structure theorems for nonlinear systems     

Structure theorems for nonlinear systems

One of the main results is a proposition to the effect that under some typically mild conditions finite sums of the form $$\sum\limits_\ell {K_\ell \sigma } \left[ {\sum\limits_m {\eta _{\ell m} Q_m (\cdot) + \rho _\ell } } \right]$$ are dense in an important sense in the set of shift-invariant approximately-finite-memory mapsG(·) that take a certain type of subsetU ofR intoR, whereR is the set of real-valued functions defined onR ⁿ orZ ⁿ . Here theQ _m (·) are linear, σ is any element of a certain set of nonlinear maps fromR toR, and the κ_?, ρ_?, and η_?m are real constants. Approximate representations comprising only affine elements and lattice nonlinearities are also presented.     

Series expansions for nonlinear systems

In this paper we generalize recent theorems concerning the existence (and algorithms for obtaining) a power-series-type expansion for a nonlinear system's outputs in terms of its inputs. The results given here are directly applicable to a much larger class of systems.This paper was presented at the 1982 International Symposium on Circuits and Systems (Rome, Italy, May 1982).     

Multiple time scales for nonlinear systems

In this paper we extend the results on the multiple time-scale structure for linear autonomous systems of the form $$\dot x = A( \in )x$$ (cf. Coderchet al. [1]) to nonlinear autonomous systems. Our main result is in obtaining conditions under which the linearized system and the nonlinear system around an equilibrium point have the same time-scale structure.     

Synchronous sampling theorem for nonlinear systems

It is shown that the sampled output of a nonlinear system with sampled-data input is given by the discrete equivalent of a Volterra series in which the kernels are the sampled Volterra kernels of the system. This theorem is then extended to the common case in which the sampled-data input is applied to the nonlinear system through a zero-order hold.     

Bandpass sampling criteria for nonlinear systems

Sampling criteria for nonlinear systems with a band-pass input are developed in this paper. It is well known that nonlinear systems may produce an output signal with a larger bandwidth than that of their input signal. According to the Nyquist sampling theorem, the sampling rate needs to be at least twice the maximum frequency of the output signal; otherwise, the sampled output would be aliased. However, if the input is a bandpass signal, the spectrum of the output signal often occupies multiple frequency bands. In this case, it is possible, by using the bandpass sampling concept, to sample the output signal at a rate much lower than the Nyquist sampling frequency. In this paper, all conditions in which bandpass sampling can be achieved are derived for nonlinear systems up to the third order. Furthermore, for nonlinear systems higher than the third order, some conditions in which bandpass sampling can be guaranteed are derived. The result can be used to choose an appropriate sampling frequency for nonlinear systems of an arbitrary order     

Output feedback nonlinear control for electro-hydraulic systems

In this paper we present an output feedback nonlinear control for position tracking of electro-hydraulic systems (EHSs). Although previous nonlinear control methods improved the position tracking performance of EHS, all of the methods require full state feedback. However, due to cost and space limitations, it is not always possible to measure the full state of the EHS. The proposed method consists of a high gain observer and a passivity-based controller. The high gain observer is designed to estimate the full state, and the passivity-based control is implemented for position tracking. In order to design the passivity-based controller with the high gain observer, a defined Lyapunov condition guarantee that the origin of the tacking error dynamics is exponentially stable by selecting the controller gain. The stability of the closed-loop is studied using the singular perturbation theorem. The performance of the proposed method is validated through simulations and experiments.     

Disturbance observer based control for nonlinear systems

This work presents a general framework for nonlinear systems subject to disturbances using disturbance observer based control (DOBC) techniques. A two-stage design procedure to improve disturbance attenuation ability of current linear/nonlinear controllers is proposed where the disturbance observer design is separated from the controller design. To facilitate this concept, a nonlinear disturbance observer is developed for disturbances generated by an exogenous system, and global exponential stability is established under certain condition. Furthermore, semiglobal stability condition of the composite controller consisting of a nonlinear controller and the nonlinear disturbance observer is established. The developed method is illustrated by the application to control of a two-link robotic manipulator.     

A fuzzy sliding controller for nonlinear systems

It is well known that sliding-mode control can give good transient performance and system robustness. However, the presence of chattering may introduce problems to the actuators. Many chattering elimination methods use a finite DC gain controller which leads to a finite steady-state error. One method to ensure zero steady-state error is using a proportional plus integral (PI) controller. This paper proposes a fuzzy logic controller which combines a sliding-mode controller (SMC) and a PI controller. The advantages of the SMC and the PI controller can be combined and their disadvantages can be removed. The system stability is proved, although there is one more state variable to be considered in the PI subsystem. An illustrative example shows that good transient and steady-state responses can be obtained by applying the proposed controller     

Adaptive nonlinear compensation for CDMA communication systems

This paper deals with nonlinear compensation in code-division multiple-access communications systems. We analyze the performance of the generalized cerebellar model articulation controller neural network in two problems: predistortion in nonlinear transmitters and adaptive interference cancellation in multiuser and multipath scenarios. Our examples demonstrate the performance of the compensators in typical mobile channels     

Adaptive nonlinear cancellation for high-speed fiber-optic systems

The authors propose techniques for adaptive nonlinear cancellation of intersymbol interference (ISI) in the electrical signal at the receiver in Gb/s lightwave systems and describe several demonstrations of these techniques. Techniques for adjustable nonlinear cancellations are discussed and demonstrations of these techniques using commercially available integrated circuits (ICs) at data rates as high as 1.7 Gb/s are described. Techniques for automatic adjustment are discussed, and a demonstration of adaptive nonlinear cancellation at 450 Mb/s is described. The authors discuss how these techniques can be integrated onto the detector IC for operations at 2.5 Gb/s and higher data rates. These techniques allow a single IC detector with adaptive nonlinear cancellation to be used in long-haul and undersea lightwave systems to optimize the detector threshold and compensate for the ISI     

Actively adaptive control for nonlinear stochastic systems

An adaptive control method can be classified into two categories, actively adaptive or passively adaptive, according to how the available information is being utilized in the on-line calculation of the control. An actively adaptive controller utilizes, in addition to the , available real-time information, the knowledge that future observations will be made, and regulates its adaptation (learning). This is done by anticipating how future estimation will be beneficial to the control objective. On the other hand, a passively adaptive controller, while utilizing the available real-time measurements, does not account for the fact that future observations will be made. Thus any learning in such a case will occur in an accidental manner. This paper summarizes a recent research effort in the development of an actively adaptive control method for nonlinear stochastic systems. A new and simpler set of equations for the original algorithm is given that provides further insight into the concepts of probing and caution in adaptive control. Several examples are chosen to illustrate the actively adaptive control method.     

Identification of nonlinear systems

A new formula for the autocovariances of the output from a nonlinear system in terms of the input autocorrelations and the nonlinearity has been derived. From this, a new method for identifying a nonlinear system has been developed and applied to extreme clipping, for which an analytical solution is available.     

Linearisation of nonlinear systems

It is shown that the application of a technique used in numerical analysis to the problem of linearising nonlinear systems yields linearised responses. The systems considered contain single-valued nonlinearities which can be described by low-order power series.