Stability of nonlinear systems with three time scales

We study the asymptotic stability of a singularly perturbed nonlinear time-invariant systemS _v , which has three vastly different time scales. The systemS _v is approximated by three simpler systems over different time intervals. We give a straightforward proof of the fact that the asymptotic stability ofS _v is guaranteed when the equilibrium points of the three simpler systems are exponentially stable and when the parameters and are sufficiently small.Research sponsored by the Joint Services Electronics Program, Contract Number F4962084-C-0057, and NASA, Grant NAG2-243.     

Alternative time scales for systems with random usage

In various reliability applications, there can be different time scales in which to measure time to failure, or assess the performance of objects. For instance, for automobiles the chronological age & the total number of driving hours, as measures of usage, are good candidates for the time scale. An impact of a random usage on some properties of time to failure distribution functions is studied. It is shown that a random usage can change the shape of the failure rate of an object compared with the shape of the failure rate for the deterministic usage. Specifically, the sharply increasing failure rate can turn into a decreasing one, which is, in fact, surprising. The random usage can also change the aging properties of distributions under consideration, and this should be taken into account in applications. Several simple examples illustrate the developed concept.     

T-S fuzzy controllers for nonlinear interconnected systems with multiple time delays

This paper investigates the effectiveness of a passive tuned mass damper (TMD) and fuzzy controller in reducing the structural responses subject to the external force. In general, TMD is good for linear systems. We proposed here an approach of Takagi-Sugeno (T-S) fuzzy controller to deal with the nonlinear system. To overcome the effect of modeling error between nonlinear multiple time-delay systems and T-S fuzzy models, a robustness design of fuzzy control via model-based approach is proposed in this paper. A stability criterion in terms of Lyapunov's direct method is derived to guarantee the stability of nonlinear multiple time-delay interconnected systems. Based on the decentralized control scheme and this criterion, a set of model-based fuzzy controllers is then synthesized via the technique of parallel distributed compensation (PDC) to stabilize the nonlinear multiple time-delay interconnected system and the H/sup /spl infin// control performance is achieved at the same time. Finally, the proposed methodology is illustrated by an example of a nonlinear TMD system.     

Genetic algorithm for optimizing the nonlinear time alignment ofautomatic speech recognition systems

Dynamic time warping (DTW) is a nonlinear time-alignment technique for automatic speech recognition (ASR) systems. It had been widely used in many commercial and industrial products, ranging from electronic dailies/dictionaries to wireless voice digit dialers. DTW has the advantages of fast training and searching times, which makes it more popular than other available ASR techniques. However, there exist some limitations to DTW, such as the stringent rule on slope weighting, the nontrivial computation of the K-best paths, and the significant increase in computational time when the endpoint constraint is relaxed or the variations of the length of pattern increased. In this paper, a stochastic method called the genetic algorithm (GA), which is used to solve the nonlinear time alignment problem, is presented. Experimental results show that the GA has a better performance than the DTW. In addition, two derivatives of GA: the hybrid GA and the parallel GA are also presented     

Absolute stability of nonlinear control systems with time lag

A nonlinear control system with time lag is transformed into a nonlinear control system with no time lag by means of certain types of transformations. If absolute stability for this transformed nonlinear control system is obtained by Popov's method, absolute stability in the nonlinear control system with time lag is realized.     

Instantaneous characterization of time varying nonlinear systems[neurophysiological responses]

A nonlinear system may be characterized by an orthogonal functional power series (FPS) computed from cross correlations between input and output variables. "Is the response changing over the course of the experiment?" is a fundamental question encountered in the analysis of both FPS and evoked potentials (EP's). Regression on closed-form functions of time produces a time-varying FPS or EP. Evaluation of these functions at a specified time point produces a system characterization for that instant.     

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     

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).     

Approximating linearizations for nonlinear systems

Given a nonlinear control system $$\dot x(t) = f(x(t)) + \sum\limits_{i = 1}^m {u_i (t)g_i (x(t))}$$ on ? ⁿ and a pointx ₀ in ? ⁿ , we want to approximate the system nearx ₀ by a linear system. Of course, one approach is to use the usual Taylor series linearization. However, the controllability properties of both the nonlinear and linear systems depend on certain Lie brackets of the vector field under consideration. This suggests that we should construct a linear approximation based on Lie bracket matching atx ₀. In general, the linearizations based on the Taylor method and the Lie bracket approach are different. However, under certain mild assumptions, we show that there is a coordinate system for ? ⁿ nearx ₀ in which these two types of linearizations agree. We indicate the importance of this agreement by examining the time responses of the nonlinear system and its linear approximation and comparing the lower-order kernels in Volterra expansions of each.     

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.     

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     

Some characteristics of commonly used time scales

Various examples of ideally defined time scales are given. Realizations of these scales occur with the construction and maintanance of various clocks, and in the broadcast dissemination of the scale information. Atomic and universal time scales disseminated via standard frequency and time-signal broadcasts are compared. There is a discussion of some studies of the associated problems suggested by the International Radio Consultative Committee (CCIR).     

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.     

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     

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.