Simplified neural networks for solving linear least squares andtotal least squares problems in real time

In this paper a new class of simplified low-cost analog artificial neural networks with on chip adaptive learning algorithms are proposed for solving linear systems of algebraic equations in real time. The proposed learning algorithms for linear least squares (LS), total least squares (TLS) and data least squares (DLS) problems can be considered as modifications and extensions of well known algorithms: the row-action projection-Kaczmarz algorithm and/or the LMS (Adaline) Widrow-Hoff algorithms. The algorithms can be applied to any problem which can be formulated as a linear regression problem. The correctness and high performance of the proposed neural networks are illustrated by extensive computer simulation results.     

Uniform stability of switched linear systems: extensions of LaSalle's Invariance Principle

This paper addresses the uniform stability of switched linear systems, where uniformity refers to the convergence rate of the multiple solutions that one obtains as the switching signal ranges over a given set. We provide a collection of results that can be viewed as extensions of LaSalle's Invariance Principle to certain classes of switched linear systems. Using these results one can deduce asymptotic stability using multiple Lyapunov functions whose Lie derivatives are only negative semidefinite. Depending on the regularity assumptions placed on the switching signals, one may be able to conclude just asymptotic stability or (uniform) exponential stability. We show by counter-example that the results obtained are tight.     

Exponential stability of general tracking algorithms

Tracking and adaptation algorithms are, from a formal point of view, nonlinear systems which depend on stochastic variables in a fairly complicated way. The analysis of such algorithms is thus quite complicated. A first step is to establish the exponential stability of these systems. This is of interest in its own right and a prerequisite for the practical use of the algorithm. It is also a necessary starting point to analyze the performance in terms of tracking and adaptation because that is how close the estimated parameters are to the time-varying true ones. In this paper we establish some general conditions for the exponential stability of a wide and common class of tracking algorithms. This includes least mean squares, recursive least squares, and Kalman filter based adaptation algorithms. We show how stability of an averaged (linear and deterministic) equation and stability of the actual algorithm are linked to each other under weak conditions on the involved stochastic processes. We also give explicit conditions for exponential stability of the most common algorithms. The tracking performance of the algorithms is studied in a companion paper     

Improving performance in model predictive control: Switching cost functionals under average dwell-time

In this paper, we propose a novel switched model predictive control (MPC) algorithm for nonlinear continuous-time systems, where we switch between different cost functionals in order to enhance performance. Thus, different performance criteria can be taken into account. In order to ensure stability of the resulting closed-loop system, we consider switching signals which exhibit a certain average dwell-time. When considering switching signals of this type, certain assumptions are common in the switched systems literature in order to ensure stability, like a matching condition for the different Lyapunov functions and the possibility to find Lyapunov functions with an exponential decay rate. In this paper, we show how these assumptions can be satisfied in the MPC context and thus stability of the proposed switched MPC algorithm can be established.     

Stability of nonlinear 2D systems described by the continuous-time Roesser model

This paper considers systems with two-dimensional dynamics (2D systems) described by the continuous-time nonlinear state-space Roesser model. The sufficient conditions of exponential stability in terms of vector Lyapunov functions are established. These conditions are then applied to analysis of the absolute stability of a certain class of systems comprising a linear continuous-time plant in the form of the Roesser model with a nonlinear characteristic in the feedback loop, which satisfies quadratic constraints. The absolute stability conditions are reduced to computable expressions in the form of linear matrix inequalities. The obtained results are extended to the class of continuous-time systems governed by the Roesser model with Markovian switching. The problems of absolute stability and stabilization via state- and output-feedback are solved for linear systems of the above class. The solution procedures for these problems are in the form of algorithms based on linear matrix inequalities.     

切换非线性正系统的有限时间稳定性

研究模型依赖平均驻留时间(MDADT)切换信号下一类齐次度为1的切换非线性正系统的有限时间稳定问题.首先,通过构造恰当的切换最大分离Lyapunov函数,借助于Dini导数,基于MDADT切换信号,给出切换非线性正系统有限时间稳定的充分条件.与已有的指数稳定性结果相比,进一步说明有限时间稳定与指数稳定的区别.其次,将所得结论应用于切换线性正系统,得到切换线性正系统在MDADT或平均驻留时间(ADT)切换信号下有限时间稳定的充分条件.最后,通过仿真算例验证所得结论的有效性.     

Estimating time-varying parameters by the Kalman filter basedalgorithm: stability and convergence

Convergence and stability properties of the Kalman filter-based parameter estimator are established for linear stochastic time-varying regression models. The main features are: both the variances and sample path averages of the parameter tracking error are shown to be bounded; the regression vector includes both stochastic and deterministic signals, and no assumptions of stationarity or independence are requires; and the unknown parameters are only assumed to have bounded variations in an average sense     

Vector Lyapunov functions for stability and stabilization of differential repetitive processes

Differential repetitive processes arise in the analysis and design of iterative learning control algorithms. They belong to a class of mathematical models whose dynamic properties are defined by two independent variables, such as a time and a spatial coordinate, also known as 2D systems in the literature. Moreover, standard stability analysis methods cannot be applied to such processes. This paper develops a vector Lyapunov function-based approach to the exponential stability analysis of differential repetitive processes and applies the resulting conditions to develop linear matrix inequality based iterative learning control law design algorithms in the presence of model uncertainty.     

Observer-based control of piecewise-affine systems

This paper presents a new synthesis method for both state and dynamic output feedback control of a class of hybrid systems called piecewise-affine (PWA) systems. The synthesis procedure delivers stabilizing controllers that can be proven to give either asymptotic or exponential convergence rates. The synthesis method builds on existing PWA stability analysis tools by transforming the design into a closed-loop analysis problem wherein the controller parameters are unknown. More specifically, the proposed technique formulates the search for a piecewise-quadratic control Lyapunov function and a piecewise-affine control law as an optimization problem subject to linear constraints and a bilinear matrix inequality. The linear constraints in the synthesis guarantee that sliding modes are not generated at the switching. The resulting optimization problem is known to be hard, but suboptimal solutions can be obtained using the three iterative algorithms presented in the paper. The new synthesis technique allows controllers to be designed with a specified structure, such as a combined regulator and observer. The observers in these controllers then enable switching based on state estimates rather than on measured outputs. The overall design approach, including a comparison of the synthesis algorithms and the performance of the resulting controllers, is clearly demonstrated in four simulation examples.     

Linear boundary control for a class of nonlinear PDE processes

This paper studies the problem of output regulation via boundary feedback control for a class of nonlinear PDE processes, including important industrial (bio)-chemical reactors. Under physically reasonable smoothness, stability and steady-state assumptions, it is proven that a linear stable controller with integral action yields global exponential regulation of the process output.     

Repetitive process based design and experimental verification of a dynamic iterative learning control law

This paper gives new results on iterative learning control (ILC) design and experimental verification using the stability theory of linear repetitive processes. Using this theory a control law can be designed in one step to force error convergence and produce acceptable transient dynamics. Previous research developed algorithms for the design of a static control law with supporting experimental verification. Should a static law not give the required levels of performance one option is to allow the control law to have internal dynamics. This paper develops a procedure for the design of such a control law with supporting experimental verification on a gantry robot, including a comparative performance against a static law applied to the same robot. The resulting ILC design is an efficient combination of linear matrix inequalities and optimization algorithms.     

基于NLMS算法的自适应滤波器的研究与应用

对改进的NLMS自适应滤波器算法与传统的LMS算法进行了较全面的性能比较,利用Matlab程序分别仿真分析了两种算法的误差、收敛速度和稳定性等特性。在改进的NLMS算的基础上给出了自适应滤波器的原理结构,并仿真了其在去除噪声信号中的应用。分析结果表明：改进的NLMS自适应滤波器算法相对于传统的LMS自适应滤波器算法在减小误差方面优势明显。以上NLMS自适应滤波器特性分析结果对于智能天线设计具有指导意义。     

Stability of linear discrete switched systems with delays based on average dwell time method

This paper deals with the problem of exponential stability for a class of linear discrete switched systems with constant delays.The switched systems consist of stable and unstable subsystems.Based on the average dwell time method, some switching signals will be found to guarantee exponential stability of these systems.The explicit state decay estimation is also given in the form of the solutions of linear matrix inequalities(LMIs).An example relating to networked control systems(NCSs) illustrates the effect...     

Stability analysis for switched positive linear systems under state-dependent switching

This paper addresses the state-dependent stability problem of switched positive linear systems. Some exponential stability criteria are established on the given partitions of the nonnegative state space. First, a exponential stability of systems without delays is established with the help of a single linear co-positive Lyapunov function. When this does not seem possible, we also prove the stability by using multiple linear co-positive Lyapunov functions. Moreover, we extend this result to the delayed systems in terms of the single and multiple linear co-positive Lyapunov functionals respectively. The proposed results can be applied to the general systems without any special restriction. Some numerical examples are given to illustrate the effectiveness of our results.     

Practical stabilization of a class of uncertain time-varying nonlinear delay systems

In this paper we deal with a class of uncertain time-varying nonlinear systems with a state delay. Under some assumptions, we construct some stabilizing continuous feedback, i.e. linear and nonlinear in the state, which can guarantee global uniform exponential stability and global uniform practical convergence of the considered system. The quadratic Lyapunov function for the nominal stable system is used as a Lyapunov candidate function for the global system. The results developed in this note are applicable to a class of dynamical systems with uncertain time-delay. Our result is illustrated by a numerical example.     

Adaptive algorithms with filtered regressor and filtered error

This paper presents a unified framework for the analysis of several discrete time adaptive parameter estimation algorithms, including RML with nonvanishing stepsize, several ARMAX identifiers, the Landau-style output error algorithms, and certain others for which no stability proof has yet appeared. A general algorithmic form is defined, incorporating a linear time-varying regressor filter and a linear time-varying error filter. Local convergence of the parameters in nonideal (or noisy) environments is shown via averaging theory under suitable assumptions of persistence of excitation, small stepsize, and passivity. The excitation conditions can often be transferred to conditions on external signals, and a small stepsize is appropriate in a wide range of applications. The required passivity is demonstrated for several special cases of the general algorithm. The first and third authors were supported by NSF Grants ECS-8506149, INT-8513400, and MIP-8608787. Research done while at the School of Electrical Engineering, Cornell University, Ithaca, New York 14853, U.S.A.     

A variable step-size selective partial update LMS algorithm

Selective partial update of the adaptive filter coefficients has been a popular method for reducing the computational complexity of least mean-square (LMS)-type adaptive algorithms. These algorithms use a fixed step-size that forces a performance compromise between fast convergence speed and small steady state misadjustment. This paper proposes a variable step-size (VSS) selective partial update LMS algorithm, where the VSS is an approximation of an optimal derived one. The VSS equations are controlled by only one parameter, and do not require any a priori information about the statistics of the system environment. Mean-square performance analysis will be provided for independent and identically distributed (i.i.d.) input signals, and an expression for the algorithm steady state excess mean-square error (MSE) will be presented. Simulation experiments are conducted to compare the proposed algorithm with existing full-update VSS LMS algorithms, which indicate that the proposed algorithm performs as well as these algorithms while requiring less computational complexity.     

Relaxed delay-dependent exponential stability condition for a class of neural networks with polytopic uncertainties and distributed delays

The global robust exponential stability of a class of neural networks with polytopic uncertainties and distributed delays is investigated in this paper.Parameter-dependent Lypaunov-Krasovskii functionals and free-weighting matrices are employed to obtain sufficient condition that guarantee the robust global exponential stability of the equilibrium point of the considered neural networks.The derived sufficient condition is proposed in terms of a set of relaxed linear matrix inequalities (LMIs),which can be checked easily by recently developed algorithms solving LMIs.A numerical example is given to demonstrate the effectiveness of the proposed criteria.     

An improved stability test and stabilisation of linear time-varying systems governed by second-order vector differential equations

In this article, the problems of exponential stability analysis and stabilisation of linear time-varying systems described by a class of second-order vector differential equations are considered. Using bounding techniques on the trajectories of a linear time-varying system, the stability problem of the time-varying system is transformed to that of a time-invariant system and a new sufficient condition for the exponential stability is obtained. Moreover, the new criterion is proven to be superior to a test presented in the recent literature. Finally, the proposed criterion is applied to the exponential stabilisation problem via state feedback. The results are illustrated by several numerical examples.     

WDOP‐based Summation Inequality and its Application to Exponential Stability of Linear Delay Difference Systems

This paper is concerned with the exponential stability analysis of linear delay difference systems. Firstly, a set of weighted discrete orthogonal polynomials (WDOPs) is established by using the Gram‐Schmidt orthogonalization process, and then two WDOPs‐based summation inequalities, including some existing summation inequalities as special cases, are developed. Secondly, these WDOPs‐based summation inequalities are applied to investigate the exponential stability criteria and explicit exponential estimates of solutions of linear delay difference systems. Finally, two numerical examples indicate that the proposed WDOPs‐based approach can derive the exponential stability condition with larger decay rate than the existing ones.