Observability and Forward–Backward Observability of Discrete-Time Nonlinear Systems

 In this paper we study the observability properties of nonlinear discrete-time systems. Two types of contributions are given. First we present observability criteria in terms of appropriate codistributions. For particular, but significant, classes of systems we provide criteria that require only a finite number of computations. Then we consider invertible systems (which includes discrete-time models obtained by sampling continuous-time systems) and prove that the weaker notion of forward–backward observability is equivalent to the stronger notion of (forward) observability. Date received: January 19, 2001. Date revised: May 14, 2002.     

Comments on “Discrete-Time Adaptive Backstepping Nonlinear Control via High-Order Neural Networks”

The purpose of this comment is to point out some mistakes in the above paper. It is shown that the main results of the paper cannot stand in general. Also, it is pointed out that after some corrections, the proposed control algorithm is still applicable to a more simple system. For simplicity, all the symbols in this comment are the same as those in the above paper.     

A “Globally” Convergent Controller for Multi-Machine Power Systems Using Structure-Preserving Models

The design of excitation controllers to improve transient stabilization of power systems is a topic of renewed interest in the control community. Existence of a state-feedback stabilizing law for multi-machine aggregated reduced network models has recently been established. In this paper we extend this result in two directions: first, in contrast with aggregated models, we consider the more natural and widely popular structure-preserving models that preserve the identity of the network components (generators, loads and lines) and allow for a more realistic treatment of the loads. Second, we explicitly compute a control law that, under a detectability assumption, ensures that all trajectories converge to the desired equilibrium point, provided that they start and remain in the region where the model makes physical sense.     

非线性系统的能观性和状态观测器*

本文首先介绍了非线性系统能观性的各种定义,探讨了这些定义的背景,叙述了在各种定义下的研究成果,指出了尚未解决的问题,并介绍了Morse-Smale系统的能观性。其次,叙述了非线性系统设计状态观测器的五种方法,详细介绍了各种方法的假设条件和设计步骤。     

非线性离散系统的反馈线性化

本文研究一类多输入非线性离散系统的线性化问题，给出了非线性离散系统状态反馈线性化的充要条件。     

“Vague-to-Crisp” Neural Mechanism of Perception

This brief describes neural modeling fields (NMFs) for object perception, a bio-inspired paradigm. We discuss previous difficulties in object perception algorithms encountered since the 1950s, and describe how NMF overcomes these difficulties. NMF mechanisms are compared to recent experimental neuroimaging observations, which have demonstrated that initial top-down signals are vague and during perception they evolve into crisp representations matching the bottom-up signals from observed objects. Neural and mathematical mechanisms are described and future research directions outlined.      

Comments on “Controller Synthesis of Fuzzy-Dynamic Systems Based on Piecewise Lyapunov Functions”

This comment tries to describe a theoretical mistake made in the aforementioned paper [G. Feng, IEEE Trans. Fuzzy Syst., vol. 11, no. 5, pp. 605–612, Oct. 2003] to formulate the inverse of matrices used to construct the piecewise-quadratic Lyapunov functions. Derivation of these inverse matrices is the most critical step toward transforming the design constraints into linear matrix inequalities (LMIs). Therefore, the erroneous formulation essentially affects the validity of the approach and final results. Unfortunately, it seems that there is no simple correction for this problem. However, some close alternative approaches are suggested by the same authors in their other works.      

Observability Under Sampling for Nonlinear Systems

A previous article has investigated the problem of the preservation of the observability under sampling in the framework of compact manifolds. A natural generalization of the cited result would be to extend it to the systems globally Lipschitzian defined on non compact manifolds. A simple representative of this class of systems is constituted by the set of bilinear systems, for this particular class of systems the preservation of the property of observability has been shown in a previous work. In this paper, we deal with the preservation of the observability under sampling for globally Lipschitzian systems defined on R ⁿ.     

On Uniform Global Asymptotic Stability of Nonlinear Discrete-Time Systems With Applications

This paper presents new characterizations of uniform global asymptotic stability for nonlinear and time-varying discrete-time systems. Under mild assumptions, it is shown that weak zero-state detectability is equivalent to uniform global asymptotic stability for globally uniformly stable systems. By employing the notion of reduced limiting systems, another characterization of uniform global asymptotic stability is proposed on the basis of the detectability for the reduced limiting systems associated with the original system. As a by-product, we derive a generalized, discrete-time version of the well-known Krasovskii-LaSalle theorem for general time-varying, not necessarily periodic, systems. Furthermore, we apply the obtained stability results to analyze uniform asymptotic stability of cascaded time-varying systems, and show that some technical assumptions in recent papers can be relaxed. Through a practical application, it is shown that our results play a similar role to the classic LaSalle invariance principle in guaranteeing attractivity, noting that reduced limiting systems are used instead of the original system. To validate the conceptual characterizations, we study the problem of sampled-data stabilization for the benchmark example of nonholonomic mobile robots via the exact discrete-time model rather than approximate models. This case study also reveals that in general, sampled-data systems may become non-periodic even though their original continuous-time system is periodic. A novel sampled-data stabilizer design is proposed using the new stability results and is supported via simulation results.     

Corrections to “Basins of Attraction in Fully Asynchronous Discrete-Time Discrete-State Dynamic Networks” [Mar 06 397-408]

This paper brings a correction to the formulation of the basins of fixed-point states of fully asynchronous discrete-time discrete-state dynamic networks presented in the above titled paper (ibid., vol. 17, no. 2, pp. 397-408, Mar 06). In our subsequent works on totally asynchronous systems, we have discovered that the formulation given in that previous paper lacks an additional condition. We present in this paper why the previous formulation is incomplete and give the correct formulation.     

Comments on “The Extreme Learning Machine”

This comment letter points out that the essence of the “extreme learning machine (ELM)” recently appeared has been proposed earlier by Broomhead and Lowe and Pao , and discussed by other authors. Hence, it is not necessary to introduce a new name “ELM.”      

Remarks on “Measuring Ambiguity in the Evidence Theory”

In a recent paper, a functional AM (ambiguity measure) is introduced and an attempt is made to show that this functional qualifies as a measure of total aggregated uncertainty in the Dempster-Shafer theory. We show that this attempt fails due to a particular error in the proof of one of the principal theorems in the paper. Some additional remarks are made regarding recent research pertaining to the subject of the discussed paper.     

Fuzzy Set-Membership Filtering for Discrete-Time Nonlinear Systems

In this article, the problem of state estimation is addressed for discrete-time nonlinear systems subject to additive unknown-but-bounded noises by using fuzzy set-membership filtering. First, an improved T-S fuzzy model is introduced to achieve highly accurate approximation via an affine model under each fuzzy rule. Then, compared to traditional prediction-based ones, two types of fuzzy set-membership filters are proposed to effectively improve filtering performance, where the structure of both filters consists of two parts: prediction and filtering. Under the locally Lipschitz continuous condition of membership functions, unknown membership values in the estimation error system can be treated as multiplicative noises with respect to the estimation error. Real-time recursive algorithms are given to find the minimal ellipsoid containing the true state. Finally, the proposed optimization approaches are validated via numerical simulations of a one-dimensional and a three-dimensional discrete-time nonlinear systems.      

Evolutionary-Programming-Based Kalman Filter for Discrete-Time Nonlinear Uncertain Systems

Some observations and improvements on the conventional Kalman filtering scheme to function properly are presented. The improvements can be achieved using the minimal principle evolutionary programming (EP) technique. A new linearization methodology is presented to obtain the exact linear models of a class of discrete-time nonlinear time-invariant systems at operating states of interest, so that the conventional Kalman filter can work for the nonlinear stochastic systems. Furthermore, a Kalman innovation filtering algorithm and such an algorithm based on the evolutionary programming optimal-search technique are proposed in this paper for discrete-time time-invariant nonlinear stochastic systems with unknown-but-bounded plant uncertainties and noise uncertainties to find a practically implementable “best” Kalman filter. The worst-case realization of the discrete-time nonlinear stochastic uncertain systems represented by the interval form with respect to the implemented “best” nominal filter is also found in this paper for demonstrating the effectiveness of the proposed filtering scheme.     

Erratum to “On the Synthesis of Haptic Textures” [Jun 08 527-536]

In the above titled paper (ibid., vol. 24, no. 3, pp. 527-536, Jun 08), it is stated in Sec. IV-D3 that the characteristic number of the first variant of the "god object" method applied to texture synthesis is 1. This statement is not always true, and clarification of this issue is presented here.     

Comments on “Scalable Services via Egress Admission Control”

In the recent paper by Cetinkaya et al , admission control tests are derived by approximating the sum of two Gumbel distributed random variables by a Gumbel distributed random variable. In fact, the sum of two Gumbel distributed random variables does not follow a Gumbel distribution. Here, explicit expressions are derived for the probability density function (pdf) and the cumulative distribution function (cdf) of the exact distribution of the sum. The discrepancy between the exact and the approximate distributions is studied numerically.     

Comment on “On Siphon Computation for Deadlock Control in a Class of Petri Nets”

It was claimed recently in the paper by Li and Zhou that ldquoa polynomial time algorithm for finding the set of elementary siphons in S³PRs is proposed, which avoids complete siphon enumerationrdquo. However, this is incorrect because Proposition 1 and Corollary 6 of the aforementioned paper, which lead to the claim, are both incorrect. In this correspondence paper, Proposition 1 and Corollary 6 are disproved. As a consequence, some claims of the aforesaid paper are denied. Two examples are presented to disprove some claims of the previously mentioned paper.     

非线性离散系统基于观测器的反馈控制

针对一类非线性离散系统,首先提出了一种新的容易实现的状态观测器设计了方案,并证明了观测器的收敛性,其次设计了系统基于观测器的输出反馈稳定化控制器,最后给出了数值算例,仿真结果表明,本文设计方法的有效性。